Embed Size (px)
Citation preview
6lql1s
ANHARIVIONICITY IN ALKALI METALS : ATI X-RAY APPROACH
WITH PARTICULAR REFERENCE TO
POTASSII.JM AND LITHIUM.
B. Bednarz , B.Sc. (Hons . )
A ïhesis submitted for the Degree of
Do,ctor of Philosophy
in the Departnent of PhYsics at the
University of Adelaide.
by
August
(Irrir'.l.rc\ 1!*'
L977
t "(\
,+
CONTENTS
SUMMARY
DECLARATION
ACIG{OIVLEDGEVIENTS
CFIAP.IER 1 : INTRODUCTION
1.1 The Ain of the Project
I.2 Properties of Potassium and Lithiun
1.3 The Generalized Structure FactorFormalisn of Dawson
L.4 The One Particle Potential l,fodel andthe Corresponding Tenperature Factor
PAGE
11
I7
2L
26
28
31
33
37
1
2
1.5 The Quasi-harmonic Approxirnation and theSignificance of the Isotropic AnharmonicParameter y
Previous X-ray Studies of Potassiun andLithiun
1.6
CTIAPTER 2 : INTENSITY MEASUREMENT PROCEDURES
2.L Crystal Growth
2.2 Apparatus
3 Data Collection Procedures2
2 4 Low Temperature MeasurementsData Set I :
for Potassiun
2 Data Set 2 : Hig}. Ternperature lvfeasurementsfor Potassiun
55B
CONTENTS Continued
2.6 Data Set 3 : Low Temperature N{easurementsfor Lithium
2.7 Data Set 4 : High Ternperature Measurenentsfor Lithiun
2.8 Sumrnary of Experimental Work
CHAPTER 3 CORRECTION FACTORS FOR MEASURED INTENSITIES
3.1 The Polarízation Correction
3.2 The Anomalous Dispersion Comection
3.3 The Absorption Correction
3.4 The Lorentz Factor and the Correction forTher¡nal Diffuse Scattering
5.5 The Corrected Intensity of a BraggReflection
CHAPTER 4 : T}IE ABSOLI.ITE SCALE FACTOR
4.7 Definition of the Sca1e Factor for aNon-unifont Incident Beam Distribution
4.2 Measurement of the Scale Factor
CTIAPTER 5 DETERMINATION OF ANHAR}'IONIC PARAMETERS OF
POTASSIU[,I AND LITHII.M
5.1 Data Analysis
Anharnonic Thermal Parameters of Potassium:Results of Analysis of Data Sets 1 and 2
PAGE
40
47
42
43
46
46
47
56
58
62
54
s.27T
CONTENTS Continued
5.3
5.4
Anharmonic Thermal Parameters of Lithiun:Results of Analysis of Data Sets 3 and 4
Surunary of Results and Review of Previouslr{easurements of Vibration Anplitudes inPotassiun and Lithium
PAGE
77
83
95
r02
110
119
122
CHAPTER 6 : THE RELATION OF THE ONE PARTICLE POTENTIALTO THE INTERIONIC INTERACTION POTENTIAL
6.1 Einstein Models of the Harmonic Parameter cr 86
Anisotropy of the Tine Averaged EinsteinPotential
6.3 The Effect of Correlation
6.4 Relation of the One Particle Potentialto the Specific Heat
6.s The Monte-Carlo Method Applied to theLattice Dynanics of Alkali Metals
CFIAPTER 7 : CONCLUSIONS AND DISCUSSION
APPENDIX 1 : RECORD OF EXPERIMENTAL DATA
BIBLIOGRAPHY
6.2
SUMMARY
This thesis describes an accurate x-ray study of anharmoni_c
lattice dynamics of crystalline potassium and lithiun. The variation
with tenperature of the Debye-lValler factor iras been measured'for these
tlo metals. Classical statistical mechanics were used to express the
observed momentum space representation of the probability distribution
function of the atornic displacements in terms of an effective one particle
potential model of thermal vibrations. The values of model pararneters
describing this potential are presented for the first tine for these two
rnetals and their relation to other crystal properties is discussed.
Single crystals of potassium of purity 99.97% ulere prepared.
Intensities hrere recorded from two crystals, one spherical, the other
cylindrical. The ternperature range covered by these measurements extends
fron 207K to the nelting point of potassium at 337K.
Single crystals of lithium of purity 99.95% h¡ere prepared.
Data were collected at 248K and 296K from a spherical crystal. In
addition, experiments h¡ere carried out on a cylindrical lithiun crystal
from 293K to 423K which is within 3lK of the nelting point of lithiun.
Absolute scale factors were deternined for the spherical crystals
of lithium anC potassiun. The results confirrn the existence of an
isotropic anharrnonic contribution to the crystal field in potassiun but
inply that the isotropic part of the one particle potential in lithium
is quasi-harmonic.
The anharmonic properties of the three a1kali netals lithium,
sodiun and potassium are reviewed. It is shown that the harmonic component
of the one particle potential represents the mean inverse square frequency
of the quasi-harnonic phonons and that the sign of the fourth order
isotropic anharmonic parameter may be deduced from the tenperature
dependence of the specific heat at constant volume. The atomic
vibration arnplitudes for sodium and potassium are anisotropic and are
such that there is a greater probability of vibration in the nearest
neighbour directions than in the next nearest neighbour directions.
Perturbation of the one particle potential by thernally generated lattice
vacancies is postulated in order to e¡plain this phenomenon.
A ce11-cluster expansion of the total crystal eneïgy has been
used to show analytícally that atomic displacements are correlated at high
temperatures. A fundamental distinction between the effective one particle
potential and the Einstein potential is pointed out. Arguments are
advanced to support the proposition that an analytical real space analysis
of anhannonic lattice dynamics can be made.
DECLARATION
The work described in this thesis was
carried out in the Department of Physics between
February 1973 and August L977. No naterial contained
in this thesis has been subnitted for the award of any
othêr degree or diplona in this or any other University.
To the best of the author's knowledge and belief, the
thesis contains no material previously published or
written by another person except where due reference is
made in the text.
ACKNO1VLEDGEMENTS
I would like to thank my supervisor Dr. E.H. l4edlin
for suggesting the project and for his guidance and insight
throughout the project. I would also like to thank Dr. S.G. Tonlin
who acted as an interin supervisor during the absence on study
leave of Dr. E.H. Medlin.
The facilities of the Physics Departnent were provided
during the Chairnanship of Professors J.H. Carver and J.R. Prescott.
I am grateful to Dr. D.W. Field for helpful discussion
and advice throughout the project.
I would like to thank Dr. M.R. Snow of the Physical and
Inorganic Chenistry Department of the University of Adelaide with
whom the autornatic two-circle Wiessenberg diffractometer is shared.
I an grateful to Mr.A.G. Ewart for technical assistance
during the course of this work.
This work was made possible by the tenure of a
Cornmonwealth Postgraduate Award (1973 - 7976).
I
CFI,APTER ONE
INTRODUCTION
1.1 The Ain of the Pro ect
In recent years there has been considerable interest in the
anharmonic lattice dynarnics of sirnple structures and in particular
crystals of elenents. The advent of inelastic neutron scattering has
made possible the direct measurement of phonon frequencies in the solid
state (Dolling and Woods, 1965). The dispersion curves so derived have
been related to the interatonic potential, a knowledge of which is
essential to the understanding of anharnonic interactions (Wi11is and
Pryor, 1975). A conplernentary approach to this subject of anharmonicity
is the study of the Debye-Wa1ler factor by elastic X-ray or neutron
scattering. The mean inverse square phonon frequency, for example,
may be determined frorn the harnonic part of the Debye-Waller factor
(Blacknan, 1955). Furthermore, harmonic and anharrnonic conponents of
the probability distribution function of atonic displacenents rnay be
deduced fron the temperature dependence of, the Debye-Wa1ler factor
(Willis and Pryor, 1975).
Perhaps the nost fundanental of metal crystals are alkali
metals characterized by an inert gas core and a nearly free conduction
electron. However, there have been very few measurements of amplitudes
of vibrati.on in these metals. The aim of this project was to extend
the X-ray study of anharrnonicity already carried out on sodium (Field,
L97L; Field and Medlin, L974) to potassium and lithium and to relate
the results fron these three elements to each other and to the inter-
atornic potential. As the nelting points of lithium and potassium are
readily accessible, this is also an opportunity to study atomic
vibrations near a phase transition.
2
I.2 Properties of Potassium and Lithium
The average peri-od of vibration in a solid is of the order
of 10-13 seconds. The tine scale of an X-ray diffraction experirnent is
such that the electron density, averaged over all instantaneous
configurations of the atoms in a crystal, is derived by Fourier
transforrnation of the measured structure factors. In general it is
not possible from a consideration of X-ray data alone to distinguish
unequivocally between crystal field effects on the charge distribution
and atomic displacements. A judgement must be made to decide which of
these possibilities is predoninant. In the case of potassiun and
lithium, the decision is based on the following discussion of their
crystalline properties.
Potassium is a silvery, polycrystalline solid at roorn temperatures
Its specific gravity is 0.858 and it nelts at 336.8 K The crystal
structure is body-centred cubic and the space group is In3m. There
aïe two atoms per unit ce11. The roon temperature lattice parameter
has been measured by Posnjak (1923) and by Bohm and Klemn (1959) who
obtained 5.344 Å and 5.32I ^
respectively. However, the value that
was adopted for the present work is 5.329 ^
calculated from the
¡neasured density of potassiun (Stokes, 1966; Schouten and Stvenson, I974).
At roorn ternperature, lithium is body-centred cubic. Its specific
gravity is 0.554 and its nelting point is 453.7 K The roon
tenperature lattice parameter. was taken to be 3.5095 Â, (Owen and
Itlill.ians, 1954). Unlike potassiun, but in common with sodium, lithiun
undergoes a martensitic phase transformation at liq.uid air temperatures
and two additional crystal forms have been observed, nanely the hexagonal
close-packed and the face-centred cubic phases (Barrett, 1956).
0.0
-2.39
- 6.29
- 5L-75
Fig. l.lEn erg yscate
Vocuum
Conduct i onbond
1s
-2.15
- lr'1.5
-17.B
- 33.9
Vocuum
Conductionbond
3p
3s
2p2s
Lithium
l5
t evets f or lithiumtogarithmic and
-3608 1s
Pot ossi u m
potass ium metats. The
is in etectron votts.,and
ener9y
-)
An energy leve1 diagram for rnetallic potassiunl and lithium is
gi-ven in Figure 1.1. The energies of the core states, lSt in lithium,
and 1s2 , 2t' , 2pu , 3s' , 3p' in potassium, were obtained from the
X-ray enission wavelengths (Bearden and Burr, L967). The rvi.dth of the
conduction band is given by the Ferni level and was taken to be 2,3 eY
for potassiun and 3.9 eV for lithiun (Kennard and l{aber, 1976). The
energy for the top of the conduction band is the work function taken
from Michaelson (1950) for both rneta,Is.
Yegorov, Kuzlretsov, Shirokovskiy and Ganin (1975) have
calculated the solid state electron distribution for the alkali netals
at O K. Sone physical parameters of their charge density, rvhich is
spherically symnetrical, are listed in Table 1.I for lithiun and
potassium. Here a o is the relevant lattice constant and "rr' the
corresponding nearest neighbour separation; -, is the radius of the
Slater sphere inscribed in the l{i.gner-Seitz cell (t, = rnn/Z)i Zn is
the atomic number; a is the electron charge within the Slater sphere;
Qat is the contribution to a from the atom at the centre of the
Wigner-Seitz ce11. The radial density of atonic tithiun and potassium
calculated from a Hartree-Fock wavefunction by Yegorov e't aL' (1975) is
shown in Figure 1.2. Assuning that the core distributions in the free
atom and the crystal lattice are identical, it follows that the core
charge ín the crystal is contained almost entirely within the Slater
sphere. The number q of valence electrons within this sphere is
approximately
q = Q - ( zrr:11
Of this the anount Irt given bY
9at = Qat - (zTr-1)
is derived frorn the atorn at the origin of the lt/igler-Seitz cell. As
4
TABLE 1.1 Sone physical pararneters of potassium andlithiun (After Yegorov et aL. , 1975) .
Potassium
s.225
4.525
2.263
19
18.6s4
0. 654
18. 339
0. 339
L.T2
Lithiun
3.49r
3.023
t.5L2
2.669
0.669
2.293
0.293
3
1.0s
Unit ce1l parameter at OK (Â)
Radius of Slater sphere (A)
Charge enclosed by Slater sphere
Contribution to a fromthe atom at the centre ofthe Wigner-Seitz cel1
Ratio of charge density at theradius of the Slater sphereto the uniforn charge density
ao
(Â)
Atonic number
a- (z-7)n
p(rr) / (2ao-3 )
ïS
zn
a
q
(n
'7
Qat
1)Qat Qat
rnn Nearest neighbourseparation
t-
a_(\
¡_
0.25 0
0.r25
2.0
¡-
o- l'0C.¡
0
Fi9. 1.2
Radiat charge densitY in
end potassium. ( After
0-2 0.4 0.6 0.8 1.0
r/rs
0.2 0.t. 0.6 0.8 1.0
r /rt
ato m Ic
Yegorovunits of tithiumet at., 1975 )
0
t-
t-our[
qìCt-c,
õu
U>
0.8
0.¿
0
- 0.2
0 0 2
0 t-t 0.8 1.2 t.6 2.0
Fig. L3S catt eri ng
A-fromThe firstpo in ts 01
L¡sin0/À=0 (a.u)-l
factors for vatence etectrons in tithium'pseUdo - atom density : B- f rom 2s atornic wâvef unction
and second reciprocaI tattice vectors occur at
and Q 2 respectivety. ( Af ter Perrin et at., 1975 )
5
approximately two-thirds of the charge of a valence electron is rvithin
the Slater sphere whose volurne is approxirnately two-thirds of the
electrically neutral ltrigner-Seitz cell (the exact value is , S'/' /g), the
conduction electron density is alnost uniforn. In fact the total charge
density p(rs) at the radius of the Slater sphere is alnost equal to
2%-t which would be obtained if the conduction electrons were unifornly
distributed throughout the lattice.
An outstanding feature of the a1kali netals is this unarnbiguous
distinction between tightly bound core and delocalized valence electrons.
It is reflected by the success of the free electron model in describing
numerous ploperties of these elements (Seitz, 1940). In this respect
potassium is alnost ideal. The number of free charge carriers per
aton, related to the Hall coefficient B" by n-r = eR" , is 0.99 for
potassiun (Goodman, 1968). Furthermole, the Ferni surface is almost
perfectly spherical. The distortion D(x, Y, z) of the Ferni sphere in
nomentum space may be expressed as'
D(x, Y, z) lk(x, y, z) - k(F)l/k(F)
where k(x, Y, z) is the wave-vector length at the Fermi sphere in the
(x, y, z) direction and k(F) is the rnean radius. For all the alkali
metals the rnean radius is, within experinental error' the same as for a
free electron Ferni sphere (Shoenberg, 1969). For potassiun lO(*, y, ùl
is less than 0.2eo in any direction (Dagens and Perrot, L973) .
unlike potassium, and indeed sodium, lithiun shows some
significant departures frorn ideal free electron behaviour. The number of
free charge carriers deduced fron the Hall coefficient is 0.78 (Kittel,
Ig74). The Fermi surface anisotropy parameter D(x, y, z) attains a
maximum value of 4.Teo in the < 111 > directions (Dagens and Perrot,
fgTg). In addition the Compton profile, which measures the nomentum
6
space wavefunction, is anisotropic (Lundqvist ancl Lyden, L97L;
Eisenberger, Lam, Platzrnan and Schmidt, 1972), To date no Comptqn
profile rneasurements have been reported for potassium. A recent
calculation of the charge density in crystalline lithium by Perrin,
Taylor and March (1975) indicates that the real space anisotropy is
small. Their scattering factors f, for the conduction electron,
described by a pseudo-atom model , are shown in Figure 1.3 together
\^/ith the scattering factors fb of the valence electron in atomic
lithium. The positions of the first two reciprocal lattice vectors,
which repre-sent the (110) and (200) lattice planes, are also indicated.
It can be seen that for the 110 reflection fa - 0.04 but
the dorninant contribution to the total scattering factor fa is fron the
two core electrons and fufr-r * 2eo . Thus any difference between f^
and f. is difficult to detect via X-ray diffraction. The saneb
argument applies to metallic potassium. In each case, the unifornity of
the conduction electron distribution in real space renders the Fourier
coefficients virtually unobservable at non-zero reciprocal lattice
points. As solid state effects have a major influence on valence
electrons, this may be regarded as a somewhat frustrating aspect of
scattering factor neasurements for these metals. However, this fact
introduces an important sinplification in the interpretation of such
experinents (Section 1.3) .
An invaluable tool in deconvoluting the observed time-averaged
electron density in terms of a solid state atom and a probability
distribution function for the nuclear displacement is the adiabatic
approxination of Born and Oppenheimer (Born and Huang, 1954). In this
treatment it is assumed that the electronic wavefunctions adjust
instantaneously to the nuclear displacenent. The configurational part
0 of the total crystal energy is then a function only of thè nuclear
7
co-ordinates. For a nonatonic sol-id with one atom per cell, 0 nay be
written in the form
0 t+0
Iz
1
3!
+
+
0,Q, U
.c p .crv
L V 9.rv [rrE
T (r,)uu (L)
tuu(u, ¡' )uu (1,) uu ([' )
ouu6([, L' ,L" )uu([)uu(.cr)ut(.c" )
+
0 +
0 ( r.)u
ouv (.0, l,' )
ðuu (f,)
(1.2.1)
(r.2.2)
(r.2.3)
0+0+0+20 3
where u.,(,Q,) is the U-component of the displacement of the atom ín theu
Î,th unit cel1 (see for exanple Maradudin, 1974) .
The coeffr.ì"nt, of the displacement terms are referred to
as force constants. For the first three orders they are given by
a0
a2o
t,
t,
t,
ðuu (1,) âuu ([' )
(r.2.4)âuu (.0) âuu (l' ) Aug ([' ' )
where the subscript 'r 0 rt indicates that the derivatives are evaluated
at the equilibriun configuration of the lattice. The first order force
constants 0 f,e,l are therefore zero. In what is called the harnonicu'-approximation ternr 0r, , for n larger than two, aîe onitted. The
motion of any nucleus is a superposition of non interacting normal modes
and the distribution of any nucleus is Gaussian (Maradudin, Montroll and
Weiss, 1963). The mean square amplitude of vibration ( u2.. ) in theu
p-direction is given by
ouug ([, L' ,L' ')
r,"(u2 >
a3o
u(h/6Nrú) g (trl)trr- I coth (|$hu,) dur (1.2.s)
B
where M is the atomic mass; ß-1 = O"t, ," is the rnaximum frequency
of the cïystal; g(o) is the normalized frequency distribution of the
phonons, that is,OJ
L
g (trr) dur = 3N0
for a rnonatomic crystal of N atorns (lrlaradudin et aL. , 1963) .
These results fol1ow fron the existence of a function 0
with properties described by equation 1.2.1. It is assumed that the
adiabatic approximation is vatid for the ion in lithiun and potassiun
netals. It is unlikely that an energy typically in the vicinity of
1/40 eV (k"T at room temperature) could significantly perturb even the
3p electrons of potassiun at -17.8 eV (see Figure 1.1) [ but the
enigrnatic behaviour of aluminium should be noted : see page I7l.
Since the ion core is confined to the Slater sphere, core overlap is
negligible. Vosko (1964) has shown that short range overlap repulsive
forces have a negligible effect on the normal modes in alkali metals.
Thus the ion, with core electrons moving together with the nucleus, fiâY
be described as rigid.
It has been realized for some time that the adiabatic
approximation need not apply to the valence electrons. This is evident
in the free electron model of an alkali netal which has been investigated
by Chester (1961) who has shown that the adiabatic approxination is in
fact valid for the conduction electrons except thoserrvery close" to the
Ferni level. This view has been confirned by the calculations of Brorrman
and Kagan (1967). . The conduction electrons form a screening charge about
each ion sufficient to neutralize it (Cohen, L962). The screening charge
noves together with the ion in the crystal rvhich nay be thought of as an
assembly of screened ions, the neutral pseudo-atoms (Ziman, 1964).
Individual electrons, however, are not localized in the screening cloud
9
and the overall conduction electron density is almost uniforn. For
lithiurn, for exanple, f ^
i.s sma1l even for the 110 Bragg reff ect:-oi
(see Figure 1.2) .
The interionic interaction potential 0 is anharmonic and
may be expressed in the fornt
0 (r)Z'e2
r G (Q) (r .2.6)æ
0
dQ
(see for example Shyu, Singwi and Tosi, 1971). The scalar
function G , introduced by Cochran (1963), is related to the bare
electron-ion pseudopotential w(Q) and the dielectric screening function
Ê(Q) of the electron gas by
G (Q)w(o) (1 e-l (Q) ) (r.2.7)
2
_4rZe2 /ç¿Qr
where Ze is the charge of a bare-ion and A is the volume per ion.
The potential Q assumes its simplest forn if the ions are ?epresented
as point charges and the dielectric screening function of Thornas and Fermi
is invoked. Under these conditions, ô is given by
0(r) = t Ê t*, (-Àr) (1 '2's)
where À is a screening parameter (Kittel, 1971). The results of a
more accurate calculation of 0 in potassium and lithium by Dagens,
Rasolt and Taylor (1975) are illustrated in Figure 1.4. It can be seen
that the nearest neighbour separation ( r/as=O.866) coincides
approximately with the ninimum of the interionic potential rve11. The
oscillatory nature of 0 for large ionic separations is known as the
Friedel oscillation and is due to a logarithmic singularity in e(Q) at
the Fermi surface (Harrison, 1966).
The validity of the description of Q given by equation 1.2.6
has been extensively tested in studies of the lattice dynanics of alkali
netals including, in particular, lithiurn and potassium. The dispersion
0.08
0
- 0-04
LiFLl¿
je
0'04
-0-08
0.04
0.02
1-3
r /eo0.7 1.0 t'ô 1.9
't'6 1-9
K
0
L9
:e
-0-02
0.7 1.0 l-3
rleo
Fig.1.4Interionic potent iat in tithium en
at 0 K . The nearest neigh bouris indicated by an arrow.( Af ter Dagens et at l, 1975 )
d potassium metalsseparat ion ( r/ao= 0'866)
10
cutves have been measured by inelastic neutron scattering for potassiurn
(Cowley, Woods ancl Dol1ing, 1966; Buyers and Cowley, I969) and lithium
(Smith, Dolling, Nicklow, Yíjayaraghavan and þJilkinson, 1968; Beg and
NieIsen, 1976). The experimental results have been successfully
accounted for by various groups (for a review, see for example Joshi
and Rajagopal, 1968). Differences in treatment arise in choice of
pseudopotential (for exanple Ashcroft, l96B; Rasolt and Taylor, 1975)
and dielectric screening function (for example Singwi, Sjolander, Tosi
and Land, 1970; Geldart and Taylor, I970a, 1970b). In all of these
exanples ì^/, e and therefore Q are isotropic which means that the
interionic forces are taken to be central. However, the total crystal
energy consists not only of pair potential terms but also a volume
contribution fron the nearly free electron gas. Thus the Cauchy relations,
given by Cr, = C4+ , are violated for the a1kalí metals in spite of
the fact that each atom is at a centre of symnetry and the forces central
(Cochran; I973; Martin, 1975). This interpletation is in agreenìent
with the work of Bertoni, Bortolani, Calandra and Nizzoli (1974) who
considered the lattice dynanics of sinple netals to third order in
perturbation theory and showed that unpaired three body non central
forces may be neglected in alka1i netals.
There is another interesting aspect of the elastic constants.
A body-centred cubic lattice is unstable unless Cr, - Cr, ) 0 (Born
and Huang, 1954). Using the elastic constants tabulated by Kittel (197I),
the ratio CLL/C* ir 1.18 at room temperature for lithium, sodium and
potassiun. These metals are soft, have comparatively low melting points
and, of elemental crystals, only the inert gases have comparable
vibration anplitudes. It is sufficient, at this stage, to point out
that the vibration anplitude of sodium at room temperature is greater
than that of any of the inert gas crystals at their rnelting points
11.
(Kanney, 1975) with the exception of heliun (Sears and Khanna, L972).
In view of this discussion of cry-stalline properties, an
alkali metal may be regarded as an assernbly of rigid ions interacting
with each other through a well defined anharmonic potential 0 . This
potential is derived from the screening of the Coulomb potential by a.
nearly free electron gas that is in effect unobservable at non zero
reciprocal lattice vectors. As the atomic displacements are very large
it is expected that anharmonic effects rather than solid state effects
on electron wavefunctions wil.1 be manifest in the experimental structure
factor:s. In this respect the situation in potassium and lithiun is in
marked contrast with that in diamond. Even at room temperature, the
vibration amplitudes in dj anond are sufficiently snal1 to be described by
a Gaussian distribution and the extent of covalent bonding has been
deduced from the stTucture factors (Dawson, 1967b).
1.3 The Generalized Structure Factor Formalisn of Dawson
The Bragg condition for a reflection
2d(h k 1) sin0 = tr
means that the intensity in momentum space is linited to a sphere of
radius 4n À-r and centred at the origin. For MoKo radiation this
radius is I7.7 ^-r
In alkali metals there is a more severe
restriction on the accessible infornation. The decrease in intensity
associated with atomic vibrations is such that structure factors beyond
a radius -10Â-t are undetectable. In sodium, for exanple, only
fifteen independent reflections have been observed at room tenperature
(Field and Medlin, 1974).
It nay also be pointed out that the structure factors of
sodiurn are anisotropic. Anisotropy in structure factors may be
attributed to anharnonicity or to distortion of the electron density of
L2.
solid state atoms. In general a combination of these two factors
may be required to describe the momentum space data. It is therefore
necessary to adopt a method of data analysis which recognises both
possibilíties. Dawso¡r (f964), for example, has shown that the constraint
of spherical symmetry inposed on the scattering factors of bonded atoms
may lead to considerable apparent therinal anisotropy and to spurious
atornic shifts in a structure determination.
There is an additional consideration. At the present time
the highest accuracy attainable in measurements of structure factors is
approxinately I% (Sirota, 1969; lVeiss, 1969; Miyake, 1969; Mathieson,
f969) but this is of the order of magnitude expected of contributions
of crystal field effects (see page 6 ) . Kurki-Suonio (1968) has argued
that the analysis of a linited set of data of finite accuracy nay be
better carried out in the same space as the data is collected, that is,
momentum -space. A generaLized structure factor formalism which
incorporates anharmonicity and solid state effects on electronic wave-
functions has been given by Dawson (I967a). His fornulation of the
structure factor may be described as fo1lows.
Suppose that O(1) is an atonic density and that t(u)du is
the probability of finding the nucleus in the volume element dg at u
Assuning that the electrons follow the nuclear motion exactly, the time
averaged density P'(r) is given bY
p' (r) p(r - g) t(g)dg
(p * t) (r)
The functions p and t nray be expressed as
p 9a* pa
+ta
t-t c
(1.3.1)
where p" and a" are centrosymmetric and 9u and t^ are
antisyrrunetric. Thus the scattering factor f and tenperature factor
r , which are the transforms of p and t respectively, assume the
form
f _ f +ifcca
1Ta
a' and i ,^ are the transforms of g" '
c
^. ^) jcos (Q. r . )
13.
(r.3.2)
(1.3.3)
considered as
r.; theJ
(1.3.4)
(1.5. s)
where fa ,
tandtca
where
A(Q)
if T
T=T+
(0.
respectively. Thus
þa'c
c)
*
has a transform fr given bY
fr
paor)+
aaaccf t -f r
(t +tc
)*i(f
)
( r +f ra
n'= f exp(iQ. r.-)
A(q) + i B(Q)
c
If the tine averaged electron density in the unit cell is
a superposition of distributions such as p' located at
structure factor F becomes
s
j)
sr
j
j
J
f
^r^). sin(9.f¡)
f t -f( fft'ca * f"..) .sin(Q.rr) ,c c
B(g) ( fccT + (f"." + f"t.) j cos (a.f¡ ) (1 . 3.6)
and the scattering vector a has the property that
a = 4r sin 0/À
It is pointed out that for elastic neutron scattering the
scattering factors tj in the expressions for A and B are replaced
by point-atom scattering factors b', which are independent of q andJ-
information concerning the temperature factort .j only is derived in
such experiments.
14
The two atoms in each unit cell of a body-centred cubic
lattice are at centres of syrnmetry and are identical and
a
Hence
F (Q) 2fc
(Q) r (a) (r .3.7)
and
F, (g) 412rc
(a) (a) (1.3.8)
at the reciprocal lattice points. The observed intensity is proportional
to F2 and the reduction in intensity as a result of thermal vibration
is given by T' which is the Debye-l\raller factor. In equation I.3.7
there aïe no cross-combinations of centrosymmetric components of f with
antisyrnmetric components of 'r and vice versa. Thus there is no
possibility of I'forbidden" reflections in lithiun and potassium. On the
other hand, although the diamond structure has a centre.of synmetry (hence
B = 0), the local site symnetry is 43m which is not centrosymmetric and
the existence of an antisyrunetric component f, of f accounts for the
trforbidden" 222 reflection (Dawson, 1967b). In fluorite structures the
local site symnetry of the anion is 43m and an antisyrunetric component
T of 'r has been observed in neutron scattering experirnents (Dawson,a
Hurley and Maslen, 1967).
Dawson (1975) has suggested the following extension of the rigid
atom nodel:
a-B=0Tf
c
T2c
Pt = P*t +p *t
core core valence valence
where t and t , _ describe the vibration of the core andcore valence
valence charge densities 9.or" and gvalence respectively' If
taot" ' Tvalence ' faot" and fu"l"rra" are the co*esponding transforms
in the same ordet, this relation becomes
Tfr f core core + fvalence Tvalence
in nomentum space. For alkali metals fvalence is srnall and its
contribution to f is further reduced by the factorr characterized
by large vj.bration anrplitudes
In effect
f Tcole core
+c
t=T+ôtccc
where the centrosymmetric corrections 6p
Hence
I = 2 f.r.
and ôt nay be anisotropic.c
I
15.
(1.3.s)
(1 . 3. 10)
The distortion by the crystal field of the spherically
syrnnetrical ion core % (see Figu're I.2) and the Gaussian distribution
t (described by equation 1,2.5) nay be expressed in the form
o=o'c 'c ôp
c
c
= ) (T r +F ôt *T- -c c c c c
6flc'
c)ôtT
c
ôf)c
where ôf and 6'rccThus the effects of
sodium, Field (1971)
structure factors via
are transforms of ô9" and ôt. respectively.
ô9" and &. are not separable. In the case of
has shown that the descriptions of the experinental
., (f.F
F 2(f rtc c
T+Tc c
and + (1.5.11)
are mathernatically equivalent. 0n the basis of argunents presented in
Section 1.2, it is assumed that the latter interpretation is the correct
one for alkali netals. However it is not claimed that ôf. = 0 is a
good approximation for all netals. In particular there has been no
satisfactory explanation of the experinental structure factors of
16.
TABLE 1.2 Free atom forn factors of potassium and
lithium derived fron the nine-parameter-fittables of Doyle and Turner (1968) for assuned
unit cell paraneters of 5.329Å and 3.5095Â
respectively.
hk1
110
200
2TT
220
310
'2 2 2
32L
400
330
4LL
420
3s2
422
43r
510
52L
440
4 3.3
Lithium
L.738
I .546
I .395
.J,.264
I .1s0
I .049
0.961
0.885
0.815
0. 815
a.754
0.700
o.6s2
0.609
0.609
0 .535
0.505
0.47s
Potassium
L5.763
14. 100
12.B4I
11.851
11 .059
10 .416
9.888
9 .451
9 .086
9 .086
8.778
8.516
8.290
8 .093
8.093
7.766
7.627
7.501
530 0.475 7.501
t7.
alurni¡riurn (Dawson, 1975). A brief discussion of the actual scattering
factors chosen for data analysis fo1lows. In all future references to
f and T the subscript rr c rr will be omitted.
In view of the screening of the ion in the metal (see page B ),
free atorn form factors were thought to be more appropriate than those of
the free ion. In any case the difference is less than L%. The free
atom form factors derived from the relativistic Hartree-Fock wavefunctions
of Coulthard (1967) were taken frorn Doyle and Turner (1968). The fornt
factors are listed in Table I.2 for potassiurn and lithiun. The
scattering factors of lithium agree to 0.2eo wittr those of Benesch and
Snith (1970) calculated from a 100-term Hylleraas-type wavefunction rvhich
takes electron correlation into account explicitly.
7.4 The One Partice Pot ential lvlodel and the Corresponding
Temperature Factor.
The physics of anharmonicity is under constant review. In
one dirnension, exact, that is, non-perturbative solutions of the equations
of notion have only recently been obtained for model anharmonic potentials
(Varna, 1976). In three dinensions the solution is knorun exactly only
for a harmonic lattice. However, such a lattice is not physically
realízable and, even if the interatonic potential rþ were exactly
harmonic, the total crystal energy would contain third order terms. This
is the so-ca11ed induced anharrnonicity (Leibfried, 1965). The harmonic
approximation is the starting point for perturbative treatments of
lattice dynamics in which anharrnonic terms are responsible for interactions
between phonons leading to frequency shifts and finite lifetimes (Cowley,
1968). Experimentally, the high temperature region, where these effects
are observable, is of interest. If the temperature is greater than the
Debye temperature, classical statistical mechanics rnay be adopted'
Nfaradudin and Flinn (1963) have evaluated the tenperature factor in this
18
classical limit by treating third and fourth order potential terms as
a perturbation of the harrnonic tlaniltonian. Their calculations for a
monatonic face-centred cubic lattice involve various approxínations and,
in particular, only nearest neighbour interactions are considered.
Their results rnay be reproduced and extended to any crystal by assuming
a model in r^¡hich each atom moves in an average effective potential of
the rest of the lattice. This one particle potential (OPP) is defined
by
t(u) exp (-Vop" (u)/k"r)¡ exp (-Vop "
(u) /k"r) du (1.4. 1)
and for a cubic crystal may be expressed as a linear combination of
Kubic Harmonics of von der Lage and Bethe (1947). To fourth order the
forn for V consistent r{rith the m3m site synnetry appropriate to aOPP
body-centred cubic solid is
V lu) =\u,ÍoP P --'^Yu' ô lua u4
vtf, - 3/tuo) (r .4 .2)+ + ++
Here u = (u--, u--, u-) and Y is a fourth order isotropic anharmonic'x' y' z'
parameter and ô represents a fourth order anisotropic anharmonic
contribution to V^-- Thus
t(g) exp(i Q.g)auT
(exp(iQ.u))
At the reciprocal lattice points (h, k, 1), 'r becornes
N, [ - 2n2 (h2 +k2 +r2 Ik.T / u a2 f
+ t0 (k,T)' (2n / a)2 (y/ct' )
- (k,T)' (2r/a)a Q/ú I
where *y=[1-
at tenperature T
(k"T)t (2r/a)a (ô/cro ) (ha +ka *Lo -3/, (h2 +k2 +12 )2) ] (t .4-4)
ISQ/& )k"T]-r and a is the unit cell parameter
(l{i1l is, 1969) .
(1 .4. 3)
L- {r-tst"t 0/ú )
(h2 +k2 +12 )
(h2 +k2 *I')'
19.
This exprossion for t is in agreernent with the work of Mair and
Wilkins (1976) who did not restrict their cälculations to high
tenìperatures and is the basic result that rvill be used in the analysis
of experi¡nental date (Chapter 5).
It is useful to consider the expectation values of some
additional quantities. If n is any unit vector then u.n is the
component of u in the n direction. Since Vo". (g) is centrosymrnetric,
for any odd integer n
< (rr.n-)t ) = o (1'4'5)
wherea-s
< (*.!-), >
= Ny { (kBT/o) - 35(y/cr3 ) (k"T)' }
-- <ú>¡3 (1.4.6)
(Fie1d, lg74) and is isotropic, consistent with the demands of synnetry.
Thus in the harnonic approximation ( y=o ) t may be expressed in the
form
r = exp [-8n' < ui > (sin'?o/À'?)]
exP [-B(sin2 o/x")] (r-4.7)
where B = 8t2 < u'n > is the Debye-Waller B-factor. In the general
case ( y + 0 ) an harmonic Debye-Waller B-factor Bh rnay be defined as
Br, = 8ïr2 a "T
tr, (1 .4. 8)
where k T/s,
(see for exanple Cooper and Rouse, 1973)
It can also be shown that
(u2 >.nn
(u4 ) t, [15 (k"T/o)'? - s45(Y/ao ) (k"T)' ] (1 .4. s)
20.
However ( (u.n)o >
V is given byOPP
vo"" ('' o'
is anisotropic. In the three principal directions
0)=L0,u2+yu4 *7t ôua
(u,0,0)-Lôua (1.4.10)Vo"" (9, u, o)
'Ìf \EVo"" (t, u, g)
,/T ,/ts ,/f
V
V (u, 0 0) '/, ô ua
OPP
OPP t
and the monents by
< (r.n)o )[,roo] Ny[3(kBT/a12 -t89 (y/ao ) (k"T)' -o"/, (6/oo ) (k"T)']
< (rr.n_)o t[rro] = < (rr.n_)o t[roo, + 12Nr(ô/cro)(k.T)' ( 1.4.11)
< (n.n)o )[ rrr] = < (rr.n)o t[ roo, + 16Nr(ô/oo) (krT)t
where the subscripts refer to the direction in the crystal. Thus if the
sign of 6 is positive there is a greater probability of vibration in
the nearest neighbour directions, <111> , than in the next nearest
neighbour directions, <100> At the same tirne, for any given lql
in rnomentum space r(q) is greater in the <111> directions than in
the <100> directions. The interpretation of the anisotropy of the
structure factors (Chapter 5) will be based on this result which is
consistent vüith the invariance of the Kubic Harnonics under Fourier
transforrnation (Kurki-Suonio and Meisalo, 1967).
An inportant distinction is now nade. For the tirne being the
anisotropic ô term will be ignored and Vor" will be taken to be
vo"r(*) = \uú +Yu4 (1.4.12)
In the expansion of the crystal energy 0 given by equation 1.2.I it
is understood that the derivatives defined by equations 1.2.2, I.2.3
and 1.2.4 are evaluated at O K. Thus the force constants are temperatule
2r.
independent by definition ancl Õ, determines the thermal expansion of
the solid. On the other hand equations I.4.2 and 1.4.L2 represent an
expansion of Vo"" about the nean position of the atoln at the relevant
temperature. This is essentially an extension of the quasi-harmonic
theory of Gruneisen (see for example Leibfried and Ludwig, 1961) in which,
in effect, the anharmonic parameter Y is taken to be zeto. Thus
V (u) Lcu2OPP
but cr, depends on the crystal volume. This variation of 0 with
crystal volurne is derived in the next section.
1.5 The Quasi-harrnonic Appro xination and the Sienificance
of the Isotropic Anharmonic Parameter
In the quasi-harmonic approxination it is assumed that the
change in volune arising frorn thermal expansion gives rise to a
proportional change in the inverse square frequency of the normal modes
which rnay be referred to as quasi-harmonic phonons. The frequency shift
of these phonons. is expressed in terms of the Gruneisen parameter YG
defined by
cl(lnur) - -Ycd(lnV) (1.5.1)
where v is the crystal volume (Donovan and Angress, I97I). The
vibration of any atom of a monatonic cubic solid is isotropic and
Gaussian and a "'n
t is given by equation 1.2.5, that is,
f'(h/6Nì,t) g (r¡)t l- I coth (åßhur) dt¡
depends on V At high
L (1.s.2)(u2n
but the frequency distribution g(t¡)
temperatures it is possible to express
(Bul4!) (h/k"T)aP, + . . .l
(u'n
'"i (kBT/M) [u_, * (82/2!) (h/kBT)'? +
(1.s.3)
22.
where the Un are the moments of g defined by
Un = (1/3N
1,"tì(¡ g(ut)do (n > -3) (1 . s.4)
and the But are Bernoulli nunbers ' Sr= å ,
(Barron, Leadbetter, Morrison and Salter, I963). The conponents of
T arise from the(u2n
existence of the zero-point energy.
It is conventional to describe thernodynamic properties such
as the specific heat C' at constant volume and < ui > by Debye
frequencies or alternatively Debye temperatures. They are defined as
follows. The Debye frequency, denoted by t.ro(n) , is related to the nth
nonent of g by
Itür'(n) = lt/r(n+s)prrl/n (n*0,n>-3) (1.5.5)
and the corresponding Debye temperature O (n) is defined by
k"O(n) = ht¡o(n) (1.5.6)
The shift in frequency of the Debye frequencies tto(n) is described by
mean Gruniesen parameters denoted by y(n) and defined b;'
d(lnoo(n)) = -Y(n)d(lnV) (1.5.7)
(Barron, Leadbetter and Morrison, 1964).
For a Debye frequency spectrurn, that is,
g (t¡) * t¡' 'it can be shown that for all n
t¡ fn) = û)D'' L
For a real crystal this is not the case and tlo(n) or equivalently
O(n) depends on n . In particular it can be shown from equation 1.5.3
that
1Bq= 30'"'
tt2n
(sfir7n**a' (-2)) lL * '/ru (o(-z)/"t)' + ... l
)7
(1.s.8)
(1 . s. 10)
(1.s.11)
(1. s. 12)
(1.s.13)
whereas the specific heat C,, at high temperatures is given by
c,, = 3Rlr-'/ro@Q)/T)2 +... 1 (1.5.9)
(Blacknan, 1955). Thus Debye tenperatures derived from measurements of
any two crystalline properti.es need not be equa1. Blacknan has proposed
that the Debye temperatule , denoted by 0o , should be referred to
C He has also suggested tl'rat the Debye temperature derived from X-ray
measurements of
temperatures 0o
o,=Mu-
(u'n
0(2) but 0(X-raY) = O(-2)
-2
I,fklO' (X-tay) / SIf
In the one particle potential model of lattice dynamics the
mean square amplitude of vibration given by equation 1.4.6. reduces to
u')n
k T/crEl
In the linit as T + - , equation 1.5.3. yields
t2\un
u kT/M'-2 8'
Thus cl is related to the mean inverse square frequency U by
I2
From equations I.5.7 and I.5-I2 it follows that
d(ln cx) - -2Y(-2)d(1nv) .
If X,, is the volume coefficient of expansion
d(lnv) xrdr
Thus
and provided that 2Y(-2)X.,rT"1
0o exp Ç2y (-z)YuT)cl=
0= cio (1-2Y(-Z)&r) (r. s. 14)
24.
wlìere cr' is the val,ue of o at OK. It is pointcd out that \(-2)
and & depend on V and in any given temperature range it is necessary
to adopt appropriate values of y(-2) and 4 in applying equation
1.5.14. If YG is independent of frequency then y(-2) Y
cr = 0o(1-2YoX.'T) (1.5.15)
whj-ch is the expression adopted in the literature (see for example l{illis
and Pryor, L975). Thus in the quasi-harmonic a-pproximati-on
Vo., (g) -- \ cro(l-2y.x.rT)u2 {1.5. 16)
and the distribution function t given by
(t(g) - exp(-vo"" (u)/k,r)/
J exp (-vo"" (g)/k,r)¿g
is isotropic and Gaussian with the property that
( u2 ) = 3krT/a (1- 5.17)
and (ua ) = 15(k"T/cr)2 (1.5.18)
For a normal distribution these two moments are related by
(ua ) / <ú t = S/3
The significance of the isotropic anharmonic contribution
yu4 to V^_^ (u) is now considered. Physically the Y term dependingPP'_
on its sign describes tire softening or hardening of the one particle
potential at la:rge displacements. Fornally it clescribes the deviation
of the ratio ( ua ) / I u2 * fron the ideal value of 5/3. The cumulant
expansion for the centrosymmetric temperature factor r is given by
r - ( exp(i Q.u) >
"*p {-} <(Q.g)'}+ht . (Q.g)o >-3 < (Q.g)'l I + ...}
- exp{ $<u'>*{ql}.rt-å(u2 ll+"'}
(1. s.1s)
andG
25.
l{ith the aid of equations I .4.6 and 1.4.9 and the substitution :
f = (2r/a)2 (h2 +k2 +12 ) at the reciprocal lattice points, 't beco¡nes
r = exp¡-q'?t"r7zcrl [1 + 10(y/o3¡¡t"r¡'q'? - 0/a! ) ß,T)t d ]
(1. s.20)
which agrees with equation I.4.4 In fact a general expression for
the ternperature factor has been formulated entirely in terms of the
rnoments of the function t (Johnson, 1969).
It can be seen that Y contributes thro terns to r , one
or order two, the other of order four in a . In many cases the second
order term is rnuch larger than the fourth order term and
r È exp [-({k"T/2u)(r - 20(v/a2 )k,T) ] (1.s.2r)
(Cooper and Rouse, 1973) which is equivalent in real space to the
approxirnation
1t2 >n
Thus o and y are highly correlated and it is possible to define an
effective harmonic parameter o" by
o" = o0(1-2y.X.,rT) (1 * 20(y/a2 ) (k"T) ) (I.5.23)
with the property that
r N exp(-Q'?k;/2a.) Q.s.24)
It is for this reason that it is difficult to extract unequivocally o0
and \ from a single tenpelature data set. On the other hand the
correlation between isotropic and anisotropic paraneters is negligible.
Experinental, data were therefore collected at several ternperatures.
26.
1.6 Previous X-ray Studies of Potassiun and Lithiun
Arakatzu and Scherrer (1950) have measured the scatteÌing
factors of lithiun for six low angle reflections at Toom temperature
using the porvcler method. Their data analysis was based on an assumed
X-ray Debye temperature of 510K. Pankow (1936), also using the powder
nethod, has neasured the intensities of eight low angle reflections at
three temperatures 90K, 190K and 293K to obtain an X-ray Debye
tenperature of 352 ! IzK which means that at 2g3K ( u2 ;" is
0.39 .A and < u' >\/ , is LI.2eo. However no corrections for thermal'nnciiffuse scattering (to be described in Section 3.4) were applied to his
data. To the best of this authorts knowledge no reliable experirnental
structure factors are available for lithiun and no single crystal studies
have been reported in the literature-
In the case of potassium, Krishna Kumar and Viswanitra (1971)
have determined the mean square anplitude of vibration at roorn temperature
by measuring the Bragg intensities of the 110, 200,220,310, 400 and 530
reflections from a single cïystal. They obtained <,r' >t = 0.60 Å
( ( u' >t/rrrr, = LI.s%) which is equivalent to an X-ray Debye temperature
of 96.1K. However the intensities of the reflections 110, 200, 220 and
310 were affected by extinction and it will be seen that there is a
significant discrepancy between their result and the present work.
It was pointed out in Section 1.5 that there is a variation in
Debye temperatures deternined from different crystal properties. The
extent of this variation is summarized in Table 1.3.
In a preceding section it was anticipated that the vibration
amplitudes in lithiun and potassium are large (see page 11). The results
of pankow (1936) and Krishna Kumar and Viswanitra (1971) confirm this
supposition. The limited momentum space data available requires that
27.
TABLE 1.3 Debye temperatures of potassium and f,ithiun
Reference
see text
Martin (1965b)
Martin (1965a)
Kel1y andNfacDonald (1953)
Derivation
(u2 >
C
entlopy
electricalresistance
Debye Te;np.
0(X-ray) (K)
0 (K)D
os(K)
o (K)R
Lithium
352
407
373
330-3s0
Potassiurn
96
103
87
T2B-145
both its precision and its accuracy be of the highest order. Accordingly
the greatest possible care has been taken first with the collecting of
the data and secondly with its correcting and thirdly uiith its
interpreting. At all stages attenpts have been made to present physical
justifications for the arithmetic and algebraic operations perforrned.
In particular, as anharmonic contributions to the crystal enelrgy are
expected to be significant, the enphasis in the interpretation of data
has been placed on anharmonicity.
28.
CHAPTER TWO
INTENSITY },ÍEASUREMENT PROCEDURES
The arnbiguity in the interpretation of X-ray structure factors
may be resolved by taking measurements at several tenperatures. The
anisotropy at room temperature in the structure factors of the body-
centred cubic netal vanadiun persists down to tenperatures as low as 4K
(Korhonen, Rantavuori and Linkoaho, 1971) where the probability
distribution function for atonic displacenents may be considered to be
isotropic and Gaussian. Under these circunstances the anisotropy has
been attributed to asphericity of the wavefunction of the valence electrons
in the solid state rather than to anharmonic effects (Linkoaho, 1972).
This chapter describes the measurernents of the structure factors of
potassium and lithium at high and low tenperatures. It will be seen
that experiments below 200K were not possible and the aim of measuring
the temperature dependence of the Debye-llaller factor ü/as the separation
of the highly correlated isotropic harmonic and anharrnonic components
of the one particle potential (see page 25).
2.I Crystal Growth
Potassium and lithium rnetals of purity 99.97eo and 99.95q"
respectively vlere supplied by Koch Light Laboratories. Single crystals
r4¡ere grown in cornmercial glass capillaries designed for X-ray work.
Their wal1 thickness is -0.01 mm and absorption and background
scattering r^/ere negligible for MdO radiation. At this wavelength the
optinum diameter of a crystal, 2 V-' , where U is the linear absorption
coefficient, is -17 cn for lithium and -0.1 cm for potassium.
However all crystals prepared were less than 0.5 nm in dianeter to
conform to the requirement that the entire crystal be bathed in the beam.'
?o
The preparation of four single crystals wilI be described.
Crystal I was a cylinder of potassium. A snall sphere of
potassium unde:r rrVaselinerr petroleum je1ly which had pre','iously been
outgassed by noderate heating under vacuum was sucked into a capillary.
The tube was cut to an appropriate length ( - 1.0 c¡r' ) and both ends
were sealed in a gas flarne. The sanple was heated fron room temperature
to 350K in a bath of petroleum jelly on a hotplate. When the
tenperature reached the desired va1ue, the hotplate u¡as switched off.
ll¡ith few exceptions the result was a single crystal.
Crystal 2 was a sphere of potassium and uras prepared in a
transparent plastic glove bag filled with argon gas. As the glass
capillaries are slightly tapered, a suitable sphere of potassium was
gently pushed down an argon filled tube, rvhich had previously been sealed
at one end, until it just touched the sides. The open end u¡as closed
with a lump of potassiun and then sealed in a gas flane. Only the
potassium protecting the sphere was contaninated in the process. The
sample was crystallízed by the method used for Crystal I
Crystal 3 was a sphere of lithium. The lithiun supplied was
in the form of pellets coated with a very thin, shel1-like, oxide laye-r:.
TLis oxide layer was pierced with a sharp need.le point and the pellet
placed under a glass slide under petroleu.m jeI1y and heated to 460K
The lithiun within the rigid oxide layer was under pressurc because of
its large relative expansion. At the nelting point a stream of lithium
spheres energed through the hole in the oxide layer. The¡r were collected
in a glass tube before they reached the surface of the protective nedium
and transferred to a bath of petroleun je1ly at 550K The spheres were
sucked into suitable glass capillaries which were then cut and sealed.
30
No crystallization was required as these spheres were single crystals.
Crystal 4 was a cylinder of lithiun. A 0.5 nn length of
lithium, cut from 0.5 mn dianeter lithiun wire in an inert atnosphere,
was carefully pushed down an argon filled capillary, that had been closed
at one end, until it just touched the sides. The open end was sealed in
a gas flane. It was not possible to seal the tube in the same way as
for potassium. By comparison with potassiun, lithiun is very hard and
any atternpt to block the open end with a piece cf lithiun cracked the
capillary. Thus a thin oxide layer at one end of the crystal was
unavoidable. The sample u¡as crystallized by heating it gradually to
460K in a paraffj-n oj-l bath, lowering the tenperature to 440K ovea
three hours, and cooling to room temperature. Of thirty such specimens,
only one hras successfully crystallized as expansion of lithiun at high
temperatures was sufficient to shatter the capillaries.
The data sets derived fron these crystal sarnples, whose
properties are summarized in Table 2.I, were labelled according to crystal
nurnber.
TABLE 2.I The nature of the four crystal. samples
Crystal number
Element
Crystal shape
Protectivemediun
petroleunj elly
petroleumj e1ly
I
K
2
K
43
Li Li
cylinder sphere sphere cYlinder
aI'gon argon
31.
2.2 Appa.ratus
The following equipnent was used in data collection:
(i) a Philips PW 1130 X-ray generator with tube current and
EHT stability rated at O.Iga. It was operated at 20mA
and 50 KV. The 002 reflection frorn a planar graphite
crystal was used to rnonochrornatize the prinary bean
from a nolybdenum tube. The radiation detector was a
Na I scintillation counter coupled to a Philips PW 4620
single chanltel analyser incorporating an H.T. supply for
the detector, linear anrplifier, discriminator and rateneter.
(ii) a Stadi 2 which is a Stoe on-line automatic two-circle
diffractometer comprising the Stoe lViessenberg counter
diffractometer enploying equi-inclination geometry on
line, via the Stoe interface, to a PDP 8/E Digital
Equipment mini-conpr.rter with teletype. lnformation was
either printed out or punched on paper tape.
Data from crystals 1, 2 and 3 were collected on this system.
Data fron crystal 4 were coltected on a manually operated four-circle
diffractonìeter consisting of a Philips PI\l 1164 Eulerian Cradle rnounted
on a Philips PW 1380 Horizontal Gonioneter. Output from a Philips
Pl{ 4630 counter-timer-printer control, tinked to a system identical to
that described in (i) , was print-ed on a Victor Pri-nter.
Low temperature measurements v,rere carried cut on the Stadi 2
using a standard Stoe low temperature attachment capable of producing
temperatures a-s low as liquid nitrogen temperature. The crystal, mounted
on a brass screw fixed to an insulating teflon plug set in the goniometer
head, was cooled by an air stream which had been passed through heat
5¿.
exchanging coils in a stainless-stee1 dewar vessel of liquid nitrogen.
The source of air was an external compressor and the boil off from the
cooling agent. The rate of flow deternined the temperature rvhich was
nonitored b)' a chronel-alumel thermocouple tip placed about 5 nm frorn
the crystal. The air stream was diffused by a sma1l disc (see Figure 2.I)
to provide a uniform temperature distribution in the vicinity of the
crystal. Gas flow was controlled by an on-off valve switched fron the
tlrerrnocouple output, a large pressurized container to act as a nechanical
buffer against surges in gas f1ow, and by needle valves. The precisiou
of the temperatule contTol was 1K The crys-'al was enclosed in a
chanber formed by three concentric cylinders of plastic foi1. The
function of the warn air stream between the outer foils was to prevent
formation of ice on the crystal. The original mylar foils were replaced
with a light flexible plastic to avoid crystal displacement by frictional
drag on the frame of the goniometer head. The attenuation of X-rays
passing through six 1-ayers of foil was shown to be negligible.
This same chamber carrier with concentric foils was used for
high temperature measurements on potassium. The source of heat was a 150
lVatt lamp located above the foils : the higher the 1anp, the lower the
temperature, and vice-versa. The ternperature stability was I 0.5K and
could be naintained indefinitely with little supervision. In fact, if
roon temperature were constant, no adjustment of the height of the lamp
would be required.
A niniature attachrnent for a standard ACA gonioneter head was
built for high temperature measurements on lithium on the Eulerian Cradle.
The construction of the attachment is shown in Fígure 2.2 where all
linear dinensions have been doubled. The dimensions of the device are
such that the reduction of the accessible volume of reciprocal space is
Tip of thermocouPte
For[ 3
Foit 2
Fort I
<_ Heat ed arr
- Cotd N2 gas
Goniometer headwith frame
.<_
___->
rystaI Oiffusing disc
Fig.2.tStoe tow t em Perature attachment (not to scate)
Heated ai r
Gtass rod
Atuminium top
Ptastic f oit
Drffusrng dtsc
Brass pin
Thermocoupte ttPC rystat
Iuminium base
Fig.2.2High temperature attachment ( tw¡ce actuat size )
33..
sma11. Heat exchanging coils, coupled to the device by a flexible
silicone hose, provided a warm stream of nitrogen over the c::ysta1. A
smal1 disc (see Figure 2.2,) diffused the florv to produce a Lrniform
temperature throughout the volume containi-ng the crystal and the tip of
a chromel-alumel thermocouple. The gas escaped through holes in the top
an<l the attachrnent \{as fixed to a standard ACA goniometer head by the
same screw on which the ciystal rvas mounted. The heating of the goniometer
head, even at 420K was no more than 10K above room tenperature.
The relation between thermocouple EMF and temperature was
taken from tables in the Handbook of Chenistry and Physics (I97I=2). The
reference tempe-rature ','Ias 273.16K (the ice point) . The calibration of
the thernocouple was checked at 234.3K (the freezing-point of nercur¡')
and at S7S.16K (the steam point). At both tenperatures the agreenent
of the tables rvith the ternperature indicated by the therrtocouple l{as
within 0.2K.
2.3 Data Collection Procedures
Crystal orientation v¡as determined by stereographic projection
fron flat plate transnissíon Laue photographs according to the nethod
described by Nufficld (1966). Crystals 1, 2 and 3 (see Table 2.1)
hreTe mounted to rota.te about a < 110 > axis for neasurements on the
Stadi 2. Prelininary adjustment ltlas carried out using the double
oscillation method of Davies (1950). Final adjustments weïe nade on the
diffractometer. The crystal was set to the reflecting position for a
1lo-type reflection and the detector set accordingly. The arcs of the
goniometer head tvere systematically adjusted until no variation in count
rate was observed as the crystal was rotated through 360 degrees.
Crystal 4 (see Table 2.1) was mounted on the Eulerian Cradle with a
< 111 > zone paTallel to the Ô axis. The notation adopted in
34.
connection with the four circle diffractometer is that of Busing and
Levy (1967). Alignnent r^ras based on the sarne principle as for the other
crystals but in this case the relevant refLection was a 222-type
reflection.
The scanning mode used fol a1l, measurements was the u/2u
method defined by Kheiker, Gorbatyi and Lube (1969). 0n the zero layer
this coincides with the a/20 method. It was observed experinentally
that background was significantly reduced for this scan node as compared
with an o:sc¿uì for the same volume swept out in reciprocal space. Since a
major component of this background was thermal diffuse scattering (TDS),
the TDS correction (to be described in Section 3.4) was also reduced.
This was confirned by calculation of the TDS for bcth scan modes and
applied to both litliium and potassium. The background was measured by
taking stationary counts at r,he ends of the scan lange. Optinum counting
tines have been discussed by Young (1965). Since potassium and lithiun
have large vibration arnplitudes, intensities of nost reflections are weak.
Thus equal times were spent on measuring integrated intensity and back-
ground.
In the case of neasuremcnts on the Stadi 2, the choice of axis
was governed by certain advantages of using a < 110 > zone. For
exanple, the reflections ilO and IT4 occur at the same scattering angle
on the zero Tayer of the [110] zonei the reflections 303 and 411
occur at the sanre scattering angle on the third layer; 43L and 105
occur at the same scattering angle on the first 1ayer. These are ideal
conditions for assessing anisotropy in intensity data (ltleiss and De Marco,
re6s) .
It has been pointed out by various authors (e.g. Yakel and
Fankuchen, 1962; Zachariasen; 1965; Young, 1969) that intensity data
35
shoulcl be checked for effects of rnultiple diffraction. This occurs if
two or more reciprocal lattice points are sinultaneously in contact wittr
the Ervald sphere. Such events may be intrinsic to the method of data
collecti.on or acciclental. If H, is the reflection being measured and
H2 .is on oï near the Ewald spher'e, then the intensity effect AE, on Ht
may be rvritten as
aE, = -kR(Hr)R(H2) - kr R[Hr)R(trr2) + krr R(H2)R(Hr2)
(Azaroff, Kaplow, Kato, Weiss, I{ilson and Young, 1974) 'whete R(Hr) , for
example, is the integrated reflectivity per unit volurne for reflection Ht t
Hrz is the coupli-ng reflection clefined by Hrz = Hr - H, and the
proportionality factors k, kr and kil depend on Lorentz an'i pol-arization
factors and path lengths in the crystal (Moon and Shu1l, 1964). Although
it is difficult to aoply this expression quantitatively for a clystal
smaller than the closs-section of the incident beam, it may be used to
predict the occurTence of significant intensity perturbations. As
intensity varies approxirnately as exp(-16n2 ( u2 ) sin2O/3X2 ) (see
equation 1.4.7) and < 'Í >
in alkali metals nay be significa¡rt if Hr is weak and at the samc tine
H2 and H, , are strong.
For equi-inclination geometry and a < 110 > rotation axis,
a17 data collected on any non zero even layer is collected under conditions
of mrltiple diffraction. However, with the exception of the 420-type
reflections, the intensities of all reflections from 110 to 440 may
be measured at least twice on only the zero and first layers allorving
possible nultiple diffraction effects to be assessed. No such effects
were observed as data collected on different layers v/ere self-collsistent.
Of the observable reflections for lithiun and potassiun there are fout
additional cases. Th-ey are characteristic not necessarily of equi-
inclination geometry but of a cubj.c structure. These are 220, 422 and
36
440-type reflections on the zero layer and 32I-type reflections on
the first layer. However, no anomalies were observed eitirer for lithium
or potassium. For exantple, if the crystal is set to the reflecting
position for the TZS reflection on the first layer of the [110] zone,
the 2OZ and I2L reflections sinultaneously satisfy the Bragg condition
irrespective of the wavelength of the radiation. On the other hand 23I,
also on the first layer, does not occuï simultaneously witl-r any other
reflection but there were no significant tiifferences between the
intensities of Tzs and Zst reflections.
Accidelttal rnultiple diffraction was also considered. For each
reflection neasured the position of every other reciprocal lattice point
with respect to the Ewald sphere h¡ith the given reflection in the
reflecting condition was calculated. The effective size of a reciprocal
lattice point was estimated fron the scan range. With the exception of
crystal 3 (see Section 2.6) the half-width Wn at half-height of all
reflections recorded was less than 0.08 degrees. If a reciprocal lattice
point. was within Wn lUl , where H2 is the vector extending from the
origin of the reciprocal lattice to the point H, , of the Ewald sphere
the possibility of rnultiple dj-ffraction was considered. The distance
Wt lÐl was - 0.005 Â-1 which is in the same range as the criterion
adopted by Coppens (1968) although his experinental arrangement was quite
different. As the lattice constants of lithiun and potassium are small,
these extrinsic coincidences were rare and no intensity changes were
observed.
5/
2.4 Data Set 1 : Lotv Tenperature Measurements for Potassiunt
Data set 1 was clerived from the cylindrical crystal of
pcrtassium under petroleuln je1ly (see Table 2.I) The temperature
dependence of the intensity of one rcflection for each of the types 222 ,
400, 330 and 4IL was observed in the tenperatule lange 207K to
glSK The upper limit in +-enperature was determined by the stability
of the column of petroleurn jelly supporting the crystal whose position
was constantly checked. At low tenperatures the crystal shorved signs of
thermal shock precluding experiments below 200K A total of 80
integrated intensities was obtained. Each of these intensities is the
meart of at least two, and on the average three, measurements made under
the sane conditions. In addition, the intensity of all syrnmetry related
reflections on the zero and first layers of a < 110 > zone were
collected at 308K, 296K, 260K and 233K Excluding extinction
affected reflections, this con-Eributed 47 data points which are the neans
of the intensities of equivalent reflections at the same temperature.
The crystal disintegrated into a polycrystalline sanple during
a rlrn at 233K . An oscillation in temperature of amplitude - 10K about
the set temperature and the large thernal expansion of potassium were
responsible for generating a thernal strain field within the crystal
sufficient in the first instance to increase the mosaic spread and
ultimately to destroy the crystal. Tenperature instabilities of this
magnitude hreïe caused by formation of ice within, and subsequent ejection
from, the cooling coils in the dewar vessel of liqi-rid rritrogen. In fact
in an earlier experiment at 2I3K the nosaic spread, taken to be the
half-width at half-height of the Bragg peak, increased from 0.16 clegrees
to about 0.40 degrees. The crystal was slowly returned to 1.oom
tenperature and within ten hours had annealed to its original state. This
38.
recovery of potassium is consistent with a low defect nigration energy
and has been studied by Gugan (1975).
The decrease in intensity associated with extit'tction depends on
the size and orientation of the mosaic domains that constitute the crystal
(see for. exarnple Azaroff et aL., I974). In potassiun the reflections
affecte<l by extinction are of the type 110, 200, zlL, 220 and 310
The combination of a large thermal expansion coefficient and room
ternpe::ature annealing nteans that the size of the nosaic dornains is
variable. Under these conditions tlìe intensities of exti-nction affected
reflections are not ïeploducible. For exanple, a variation - 20% was
observed in room temperature measurements of 220-type reflections. Under
these circunstances extinction corrections to data are not possible. It
is pointed out that room temperatu::e annealing has been obser'.'ed by
Field (1971) in an X-ray study of sodium.
2.5 Data Set 2 : High Tenp erature l''leasurements for Potassium
High tenperature measurements were carried out on the spherical
crystal of potassium under argon (see TabIe 2.I) The tenperatule
dependence of the intensity of all sytnmetry related reflections on the zero
and first layers and the 420-type reflections on the second layer of a
< 110 > zone was observed from room temperature to the nelting point.
Excluding extinction affected reflections 108 data points, each of which
was the mean of equivalent reflections collected, ü/ere retained for
analysis. Table 2.2 Lists the corrected intensities (Chapter 3) of the
110, 220, 330 and 440 reflections at 297K and at 324K It can
be seen that the 1atio r (h k 1) of the intensity at 297K to the
intensity at 324K is such that
r(440) > r(110) r(330) > r(220)
39
but the ordering expected from an analysis of equations I.4.4 and
I.4.7 is
r(1r0)<r(220)<r(530)<r(440)
This anonalous decrease in observed intensity of the 110 and 220
reflectíons is consistent with an increase with ternperature in the size
of the mosaic dornains. For this reason extinction corections for 1I0,
200, 2II, 220 and 310-type reflections were not attempted.
TABLE 2.2 Corrected intensities ofand 440 reflections at
110,297K
220,and
330324K
Ref lect-ion
Corrected intensityat 297K
110 220 330
367033 87398 4858
183029 57s90 2923
440
154
Corrected intensityat 324K 57
Ratio (r) 2.Or t.52 1 .66 2.70
In this experirnent a temperature difference between the thermo-
couple and crystal rvas detected. Potassium melts at 336.8K However,
the ternperature indicated by the thermocouple when the crystal did in
fact melt was 34IK This difference could not be attributed to any
de..'iation of the therrnocouple temperature fron the calibration tables
(see page 53). Although the crystal was isolated from the goniometer head
b¡* the glass capillaty and teflon insulation (see page 31) , it appears
that the thermal conductivity of the glass wa1l, or argon gas within, h¡as
sufficient to lower the crystal temnerature to the extent observed. In
any case the discrepancy A at the meltíng point T, was small. If it
40.
is assumed that the crystal tenperature T. was equal to the thermo-
couple temperature Tt at room temperature Tr , the relation betleen
T and T. may be written asCE
T" N Tt - ¡(Tt - T")/(Tn - Tï) (2.s.r)
This correction to the thernocouple temperature was applied to data set 2
It is enphasized that in all other temperature measurements
the experimental arrangement was quite different. The crystals were
cooled or heated by a stream of gas whose temperature u/as monitored by
the thermocouple. If in fact there were any deviation between T_ andc
Ta , then the relation would be of the forn
T (2.s .2)c
In the case of low tenperature measurements it h/as not possible for the
crystal to have been cooler than the air flow, and in the high temperature
experinent on lithiurn (to be described in Section 2.7) it was not possible
for the crystal to have been warrner than the nitrogen gas. In both
examples this neant that d > 0 The effect of such a systematic error
was investigated in the data analyses.
2.6 Data Set 3 : Low Temp erature Measurements for Lithium
Data set 5 is derived frorn the spherical crystal of lithium
in petroleum jelly (see Table 2.L) AII symmetry related reflections
on the zero, first and second layers of a < 110 > zone li¡ere collected
at 296K and 248K During the initial low ternperature run the crystal
showed signs of thernal shock. On returning to room temperature it was
found that the mosaic spread was anisotropic and for any given hk1-type
varied fron the original 0.16 degrees up to 1.0 degrees for synnetry
T.N d(Tt - Tr)
4L.
related reflections. Further, this change was permanent and no room
temperature annealing was observed. Thus two independent Toom temperature
da-ua sets were collected - one characterized by a sntal 1 isotropic inosaic
spread (data set 38) and the other by a large anisotropic mosaic spread
(data set 3A) In the first run at 200K a temperature instability
of the kind described (see page 37) permanently destroyed the single
crystal and no further data were obtained.
¿. t Data Set 4 : High TemP erature lvleasurenents for Lithium
The cylindrical lithium crystal under argon (see Table 2.1)
was suitable for high tenpelature measurements. Initially a room
temperature data set was collected on the four-circle diffractometer.
Thezonesusedwere <111>, <011>, <001>, <2I0>, <135>
and < ll3 > Data collected on different zones wer:e self-consistent
and no multiple diffraction effects (see Section 2.3) were observed.
The temperature dependence of one 220-type reflection was
rneasured in the range 293K to 423K At each temperature the intensity
was recorded at least four tines and the statistical counting erlcor was
less than leo During the cooling cycle the reaclings at 293K,'363K,
S23K and 313K were repeated. The intensities at 393K , 363K and 323K
were within 0.Seo of their previous values. However, the reâding at 313K
was 3.5% lower than the earlier measurement, as was the room temperature
value. It was found tjìat ihe crystal had moved slightiy within the
capillary Curing the experinent, apparently a'v about 313K In view of
this instability fur:thcr high temperature measurements were inpracticable.
As thj s crystal was unique (see page 50) the high ternperature data were
retained though any conclusions 'jra'*n frorn an analysis were qualified in
vier,¡ of the inconsistency in the measurements.
42.
2.8 Summary of Experimental Vrrork
The experiments carried out on potassium and lithium are
summarized in Table 2.3. For potassiun the measurements of intensity
extend up to the nelting point at 337K In the case of lithium no
data beyond a tenperature of 423K , which is within 31K of the
nelting point, have been recorded. The lotver limits in temperature
were not deternined by lirnitations of the apparatus but by thernal
shock which is consistent with the softness and large thermal expansion
of these rnetals.
TABLE 2.3 Sumrnary of data sets for potassiun and lithiun
Crystal number I
Element
Crystal shape cylinder sphere sphere sphere cylinder
4
Li
3
Li
3
ti
2
KK
Protectivenediun
Apparatus
Data set
Mosaic spread
Tenperaturerange
petroleunjeIly
argon
snal I ,isotropic
petroleunj e1ly
large,anisotropic
petïoleun argonjelly
Stadi 2Stadí 2 Stadi 2 Stadi 2
I
srnal l ,isotropic
3A2
snal1,isotropic
4-circlediffract-ometer
4
smal1,isotropic
5B
207K-308K 297K-337K 248K,296K 248K,296K 293K-423K
43.
CFIAPTER TIIREE
CORRECTION I'-ACTORS FOR MEASURED INTENSITIES
This chapter presents an account of the procedures for
correcting and processing the intensity measurenìents described in the
previous chapter to a stage where the experimental data nay be related
to the païameters of the one particle potential '
3.1 The Polarization Correction
The absolute integrated intensity for an ideally irnperfect
single crystal snal1 enough to be bathed in the incident X-ray beam is
given by
Eu/r = NI Àt (e2 /nc2 )'vlFlt llp Ao(l+crr(ro=1)lTu (3.1.1)
(see for exanple James, 1948) where
E is the reflected energY;
I is the incident energy per unit area per unit tirne;
F is the structure factor;
N., is the number of unit cells per unit volume;
À is the wavelength;
e, rn and c are fundanental constants;
V is the crYstal volune;
û) is the angular velocity of the crystal;
P is the Polarization factor;
Ao is the correction for anomalous dispersion;
t is the Lorentz factor;
T is the transmission factor;u
01 (ror ) it the correction for, thennal diffuse scattering (TDS) '
It is customary to adopt the symbol r' 0, " to Tepresent the TDS correction'
44.
The subscript rrTDSil has been included to avoid confusion with the
harrnonic parameter cL of the one particle potential. P, Ao, L, TU and
or qros) are standard corrections (see for example Buerger 1960;
I{eiss, 19661' Azatoff et aL., 1974) but are appropriatety rnodified for
the physical and geometrical properties of the present work. This chapter
begins with a discussion of the correction for polarization.
The polarization factor for unpolarízed radiation is given by
P - (1+cos2 2g) /2 (3.r.2)
If a crystal nonochromator is used the incident bearn is partially
polarized and P rnay be expressed in the form
p = (cos2X + M(sin2X sin2To * cos' T0))/(1+M) (s.1.3)
where M is a parameter which depends on the nature of the monochromator.
Ttre angles X and T0 are defined by l{hittaker (1953); X is the angle
between the tr\rice reflected ray and the equatorial plane of the
monochromator; To is the angle between the projection of the twice
reflected ray on to the sarne plane, and the once reflected ray. The
experimental arrangement of the Stadi 2 is such that for the zero layer
the equatorial planes of the monochromator and Ciffractometer are
perpendicular. It can be shown that
cos'x = cos2 20(I-tat vtan-2 o) + tan2 Vtan-2 0 (3,r.4)
and that
sin2 x sin2 To * cos2 Ts = 1-tan2 vtan-2 0sin2 20 (3. 1.5)
where v is the ínclination angle of the detector.
For the zero Layer v = 0 and P reduces to
p = (1 + M-t cos'20)/ (L + M-l ) (3.1.6)
in agreement with Azaroff et aL., (1974).
given by
For an ideally imperfect monochromator crysta-l, lt{ is
M cos'20¡?l
45.
(3.r.7)
where Onn is the Bragg attgle cf the monochromator.
For a perfect crystal, M is given bY
M = lcos2onol (3.1.8)
For the 002 reflection from graphìte at the MoKd wavelength
cos'20, and cos20, take the values 0.9556 and 0.9776 respectively.
Since the nature of the monochromator crystal was not determined, the
mean value was adopted in equation 3.1.3. In making this choice it is
assumed that the actual value of M lies between the trvo ideal values.
This is consistent with an experiment of Hope (1971) on a graphite
monochromator but it is not clained that this is the case in general.
(see for example Jennings, 1968). For potassiurn the difference between
the polarization factors for the two types is less tlian 0.4% for aII
observable reflections. For Iithium the rnaximurn difference is 0.9% and
the diffeïence between the polarization factors given by equations 3.1.2
and S.1.3 is at most l.4eo. Thus any errors introduced into the data
analyses through the use of an estimated value of I\'l were considered to
be negligible.
In the case of the 4-circle diffractometer the equatorial
planes of the rnonochromator and diffractometer are para11e1. Hence
X=0, To = 20 and PisgivenbY
P = (1 + M cos2 2O) / (I + M) (3.1.9)
The same estinated value of I'l was adopted to calculate P.
46.
5.1 The Anomalous Dispersi.on Correction
Anomalous dispersion descrj-bes the deviation from the Born
approxination of the interaction betleen X-rays and atonic electrons. If
the frequency of the radiation is much higher than any absorption edge. of
the atom, the total atonic scattering factor fa is frequency independent
and nay be represented by a real number. In general fa is complex and
may be expressed in the fonn
f +^f'
+ i. af ,' (s .2.r)
where f^ is independent of frequency and Aft and ^fil
are the realo
and inaginary frequency dependent components of the anomalous dispersion
(Jarnes, I 948) . The observed integrated intensity is proportional to
! t.l' given by
lt.l' - (fo + af ')2 + (^f ' ')2 G.2.2)
It is convenient to define a factor A by
of t
D
AD I r. l' / lrol" (3.2.3)
Thus the correction to the data is applied by nultiplying the observed
integ::ated intens.ity by Ao-t The values of Af I and ^f
t ' were taken
fron International Tables (1974). For lithiun A -r = 1.00 and for
potassiurn Ao-t is in the range (0.95, 0.98).
3.3 The Absorption Correction
The transmission factor is defined as the ratio of the intensity
which is diffracted to the intensity which would be diffracted if there
lrrêTê'rro absorption. Thus
v-1 exp (-Ut) CV (5.3.1)u
T
where U is the linear absorption coefficient and the integration is over
47
all possible total path lengths t within a crystal of volurne V
(Buerger, 1960). 1'he values of U were derived fron the mass absorptron
coefficients tabulated in International Tables (1962). For cylinders
and spheres and zero'Layer geometry, tU is a function of UR and 0
where R is the radius and 0 the Bragg angle.
For both líthium crystals (crystals 3 and 4: see Table 2.3)
T = 1.00.u
For crystals 1 and 2 (see Table 2.3) UR took the values 0.271
and 0.296 respectively. Values of tU were evaluated numerically and,
as UR is small, convergence was rapid. In the case of the cylindrical
potassiun crystal (crystal 1 : see Table 2.3) and upper level equi-
inclination geoÍnetry, the procedure given by Buerger (1960) was fo1loled.
Correctiorls to tU for the ends of the cylinder were shcwn to be
negligible.
3.4 The Lorentz Factor and the Correction for Thernal
Diffuse Scattering
The correction for first order
If Eo is the integrated Bragg intensity,
E for an inperfect crystal is given by
E - Eo(t + ülror¡)
TDS may be expressed as follows.
then the total observed inten-sity
where o¡ro=¡Eo is the one-phonon differential scattering cross-section
integrated over the volune of reciprocal space that is sh'ept out in the
neasurement of the intensity of a Bragg reflection. The procedure that is
usually adopted to calculate o¡"o=) it based on the nethod of Rouse and
Cooper (1969) which takes into account the anisotropy in reciprocal space
of the TDS. However their treatment, though applicable to a clystal of
48
arbitrary synnetry, is valid oniy for ze'ro layer geonetry. The
extension to upper 1eve1 equi-inclination geometry rvill be described.
Two scan modes will be considered, the o scan and the u/2u scan.
The first step in evaluating o _ - is to describe the[ros )
volume of integration. The geonetry of the equi-inclination method is
set out in figures 3.1 and 3.2. Here k0 and L are the incident and
diffracted wave vectors respectively; P0 is the reciprocal lattice point
and I is the reci,procal lattice vector; 10,, 1 and L are the
projections of k0, k and G respectively on to the layer plane O"It
containing 1o and 1 ; T is the angle between 10 and 1 ; the area
Pr P2 P3 Pr in momentum space is the projection of the detector slits,
which are at a distanc. *g (in real space) from the crystal, on to the
Ewald sphere. The position of any point P in the neighbourhood of the
reciprocal lattice point Po rnay be described by three co-ordinates u,
v and w defined thus : u is the angular di-splacement about the rotation
axis ur required to bring P into contact with the Ewald sphere; v and
r¡, are the vertical and horizontal divergence angles respectively of the
scattered bearn and are mea.su:red with respect to the instantaneous position
of the centre of the detector slits. The origin in ( u, v, w ) space is
P0 , that is, u(Po) = v(Ps) = w(Po) = Q and v is measured upwards
and hr in the direction of increasing T
A right handed Cartesian co-ordinate' systen ( x, y, z ) where
the y-axis is parallel to ûJ and the x-axis is parallel to ! is
illustrated in figure 3.1. On the zero layer this triad coincides with
the reference frame of Cooper and Rouse (1968). The relation betleen
( u, v, w ) and (x, y, z ) for both scan nodes is shown in figute 3.2.
There is a distinction between w( x, )r, z ) for the ûJ and ø/2u scans,
hence the notation w(t¡) and w(u/2u). The transformation from
( tr, v, w ) to (x,y,z ) isgivenby
-kcosvsinTtan(T/2) 0
0 kcosv
kcosvsinT
for an o-scan and
kcosvsinTcolc(T /2) 0
v kcosv
kcosvsinT
ux kcosv
kcosv
49.
(3.4.1)
(3 .4 .2)
V
w
0
00
v
ux
V0
00
0
141
for an u/2u scan. Here k = ltl=2tt\- I and for v= 0 these equations
reduce to those of Cooper and Rouse (1968).
The Jacobian J of the transformations, in both cases, is
given by
k3 cos3 vsinT (3.4.3)J
Thus if the width and height of the detector slits are b and H
respectively (see figure 3.1), then
Aw = b/ (R-cosv)
and
Av
c
H/Rc'g
and the volume AV*
shrept out in a scan of range Au is
t 0
b
/ I'I
II
vl{c//ì..- - -
S
0
Fig. 3.I Equi- inctination geometry'
o"S
z
Fig. 3.2
The Iayer Ptanet att ic e point Po .
Io
P (x,zl
containingln this
w(r¡)
,x
!o and the reciProcatptane v = 0.
50.
*AV JAuAvÀw
k3 (cos2 vsinT) [Hcb/R;]Au (3.4.4)
for both scan modes
(1e69) .
This is consistent r4rith the result of Kheiker et aL.
The volu¡ne scanned per unit tine is the produce of the
projected area of the detector slits on to the Eweld sphere and the*
velocity r' of the reciprocal lattice point P0 normal to that area.
Since
Au t¡At
*Av /At kt¡ (cos2 vsinT) lk'? (Hcb/R;) ]
and
*vn
V
ko(cos2 vsinT)
The linear velocity ": of a reciprocal lattice point at a distance k
fron the origin of the reciprocal lattice and in a plane normal to the
rotation axis t¡ is
*L
kt¡
**The ratio of v to v- is the Lorentz factor LLn which corrects the
observed integrated intensity for the different rates at which reciprocal
lattice points sweep through the Ewald sphere. Hence
L-l = cos'vsinT (3.4. s)
in agreement with Arndt and Willis (1966). Thus the Jacobian of the
transformations is of the correct nagnitude.
The steps involved in the actual integration of the TDS are
given in detail by Cochran (1969) and by Rouse and Cooper (1969) for the
zero Layer and ct(ros )
nay be expressed as
o¿
(ros )G2 k T/ (2r)e (s (o.) /t )d!
where g ÞÞo it the phonon wave-vector and
51.
(3.4 .7)
(3.4. B)
s (g)3
Ðcoj=1
s
irj
s'Y, (g) ¡ pti'r(g,)
where Y., (q) is the angle between G and the polarization directionJ_
of the normal mode ( j, g ) with associated velocity Vj (g) The
only change in the present work is the use of the appropriate forn for
the volume dq in rnonentum space.
It can be shown that in a Cartesian co-ordinate system S(q)
takes the form (Wooster, 1962)
s (q) sisj (A-')ij
where the 91 are the direction cosines of G. The inverse matrix A-1
is derived fron the natrix A by contracting the four-rank elastic
constants tensor Cif*i accordirig to
Aij (3.4. s)
where the ft are the direction cosines of q Rouse and Cooper (1969)
chose to work in the ( x,y,z ) frame defined previously in which
S (g) beco.nes
s(g) = cos'0(A-t)rr + sin20(A-t)r, * sin20(A-t)r, (3.4.10)
In general it is preferable to work in a co-ordinate system defined by
the crystal axes. For a cubic crystal, for the reflection hl h2 h3 ,
91 = h1/ ft'?, * h', * h'3)
and
S(g) = E"r(A-t)rr* E"r{A-')rr* e;(A-t)33 (3.4.11)
+ 2gtyz(A-t)r, * ZErEr(A-t)r, * 2ErEr(A-t)rs
c.- f-f].lml I m
Although three additional terms are involved (unavoidable, in any case,
for non zero layers) the transforrnation of the cit^i to the ( x, y, z)
frane is avoided. In fact for a cubic crystal the Orj are given by
A fc + (f + f2)C2 3 t+t+
52.
(3.4.12)
It I rt
A f2c +22 2 t7
É + f2)C3 4l+
A C33
G + f)c2 t+4I
P3
fI
tl+
A=L2
Al3
A
f(c +c)2 L2 4lr
)
C
c
16TÉ (sin2 o/x')(kBr/À3 )cos3 vsinr I I I s(g)/q'dudvdw (3.4.1s)
úvw
)l+4
= f f (CI 3 t2
+
+ff2 4l{
(c3 L223
(Wooster, L962). Thus the intensity of the TDS at any point P may be
found by transforming its co-ordinates ( u, v, w ) to the crystal axes via
the ( x, y, z ) frame of Rouse and Cooper and using the direction cosines
so obtaíned to calculate S (g) . From equations 3 .4 .3 and 3 .4 .7 0 (ro= 1
is given by
d¡to"¡
It remains to choose a suitable three dimensional grid to perform the'
above integration.
The procedure described above over corrects for TDS as some
TDS is included in the background reading. Cochran (1969) has shown that
if the range of integration is replaced by a sphere of equal volume
: Pryor, 1966) then the ovqr correction oftro=¡
h,r"=, and the actual correction o, ¡"o.) it(Pryort s approxination
is given by c["o"1 -
thus
20¡(ror) = 0ltot; CI (ros )
N3*(ros )
53
Florvever the argurnent presented to this stage has overlooked
any resolution effects. In order to evaluate dr¡ros, exactly the
integral over the scanned volume of the actual TDS profile is required.
If cr, - and c[r are thc values of cx - - and cr I-n (ros ) *"- * n (tos )
*- * (ros ) *"- (ros )
respectivel¡' after taking resolution into account then the true TDS
correction is
- clr0r ,n (ros )
0,R [TDs J n (ros )
The effect of resolution is to smear out the TDS such that
cL<*n(tos) - *(ros)
and c[r >n [ros ) (rÞs J
In oractice ü and cr I*n(ros) *"* * n(ros) are difficult to evaluate as the
required integrations involve the resolution function of the diffractometer
and are over six and five dinensions respectively. Model calculations
have been perforned by Scheringer (1973). He concludes that the difference
between or 1"or, and o,,*¡"ot) i.' large if the range of integration is
snall and approaches the Bragg peak closely. It rnay then, attain 20% or rnore
of or ¡ros) However, if the background measurement is made far from
the Bragg peak and the detector slits are large, this difference is snal1.
It is this principle that was adopted in collecting data. For examole,
the half-width of the Bragg peak for the cylindrical potassium crystal
(crystal I : see Table 2.3) was < 0.08 degrees for: all reflections but
the scan range was set at 1.2 degrees. The slit height and r,¡idth H"
and b ü/ere -1.5 degrees, approxinately four times the area required.
It is pointed out that to achieve convergence of ctr to 0.5%
a three dinensional grid within ^Vn
of at least 12000 points is required
although the integration need onl¡'be carried out for half of this volume
54.
as S(g) = S(-g) Thus conrputing tine and storage requirements to
take resolution into account is excessive. The correction applied to all
experinental data rras or (ros ) = o¡r.,u ) - d, (ros ) calculated by the
procedure described. The elastic constants were taken frorn li{arquardt and
Trivisonno (1965) and Snith and Snith (1965) for potassiullt, and fron Nash
and Snith (1959) and Trivisonno and Smith (1961) for lithium. At each
temperatuïe, or¡rou) was derived fron the elastic constants interpolated
to tirat temperature from the published values.
Since or,n(ros) < or¡rosl the effect of neglecting resolution
is to underestinate the harrnonic potential parameter cll . Nevertheless
the values of cx, obtained from the three lithium data sets 54, 3B and 4
(see TabIe 2.3), collected under different conditions, agree to better
tharn 1%. It may be thought that the possible over correction rvould be
more serious for potassium. However, for data collected at room temperature,
for example, the effect of naking no TDS, :correction at aIL is to change
the refined value of s by no rnore than 7eo bu:u in view of the high
degree of correlation between cL and Y (see Section 1.5) it was
imperative that the TDS cortection be made as accurately as possible.
3.5 The Corrected IntensitY of a Bragg Reflection
The expression for the absolute integrated Bragg intensity given
by equation 3.1.1 na¡' be expre-ssed as
EIADLI(1 + crr(ros))Tu] t - *,, Àt(e, /^.,)' lrl'tv7o (s.s.t)
If the corrected intensity Ec is defined by
E E IADLP (1 + o, (ros ) )I ]-
t (3.s.2)c
then since F 2fr (see p age 14)
E c 4NXÀ3 (e2 /mc2 )2 (tv/u)f 'f (3.s.3)
55.
Thus E has the property thet
k f2'c2S
c
Ec
(3.s.4)
and the observed intensity is in a form suitable fot data analysis
(Chapter 5).
The scale factor k- given byS
k, = 4NXÀ3 (e2 /nc2 )2 rv /u (3. 5 .5)
is the subject of the following chapter.
5ó.
CHAPTER 4
THE ABSOLUTE SCALE FACTOR
For a large set of data the scale factor k, , defined by
E^ = k^f 't' , ilâI be deduced by statistical rnethods, for example, bycsthe nethod of Wilson (1942). However, a scale factor so derived nay
depend on the model of scattering factor f or thermal vibration factor
T asstrmed in the refinement of experimental data (see for example
Coppens, 1972). Under these conditions an absolute measurement of the
scale factor nay allow the selection of the appropriate nodel of two or
rnore mathematically equivalent nodels. One of the ains of this study of
potassium and lithium ütas to distinguish the description of 'r by an
effective harnonic parameter (see page 25) and the fully anharnonic
noclel. As spherical crystals were available the absolute scale factors
were measured for both netals.
4.r Definition of the Scale Factor for a Non-uniform
Incident Beam Distribution
The derivation of the scale factor k, in Section 3.5 is based
on the assunption that the incident bean distribution is uniform over the
cross-section of the crystal. In general this condition is not satisfied
if a crystal rnonochromator is used. Thus it is necessary to redefine the
scale factor.
For a non-uniform distribution of intensi-ty I, equation 3.1.1
nay be nodified as follows. If dE is the reflected enelgy fron the
volume element dV , then
uúE/r(x,y) = NIÀt (e2 fmczl'lel' [lp Ao(l+cr,¡ror¡)] exp(-u(t, * .r) )
(4.1.1)
57.
where the structure of the incident bearn is explicitly taken into
account by I (x,y) and the co-ordinates x and y are shown in
Figure 4.1. Here tl is the path length from the bounclary of the crystal
to the volume elernent dV along the primary beam direction and t, is
the path length of the diffractecl bean rvithin the crystal. Thus the
total diffracted ene::gy E is given by
E dEI
(I (x,y) /o) exp (-pt)dV
(4.r.2)
where t , the total path length, is tr * tz If as in Section 3.3
a transmission factor tU is defined as the ratio of the intensity which
is diffracted to the intensity which would be diffracted if there were no
absorption then
Tu
(< r >v)-' I (x, y) exp (-itt) dV (4. 1. 3)
N.r'Àt (e2 f mc2)' lp l' IADLP (1 + cr, ("r= ) ] f
I
where
<r> v-r I (x,y) dV (4 .r.4)
Thus the integrated intensity becones
Eu/1 I t = N.r'Àt (e' /mc')'lrl'¡noln(1 + o,r(ros))lT-V (4.1.5)
and the forrn of equation 3.1.1 has been retained.
Thus k, is given by equation 5.5.5, that is,
k, - 4N*,' Àt (et /^J )" ( I )V/t¡ (4. i .6)
The scale factor is well defined only if < I > is the same for all
reflections. This restricts the shape of the crystals to spherical or
cylindrical and has been pointed out by Burbank (1965).
(
I ( x.y )
vFig.4.1Coordinate system to describe the intensity I (x,y) of
X-rays that have been reftected by a planar graphitemonochromator and are incident on a sphericaI crystatof diameter A B. The normaI to the ptanar graphitemonochromator is in the x-z Ptane.
t0
0)
90
80
z
I
(¡:C2
t-rút-
-_-ot-fú
o
I
70
A
Fig. t.2Observed incident beam d istribution Io ( x, y)
to data set 2 ( high temperature potassium )
A B represents the d iameter of the .sphericatol potassium.
B
retevant
cr ys taI
X
58.
This condition has been satisfied by all crystals used in collecting data
by tlris author (see Table 2.3). Also the transmission factor is no longer
given by the conventional expression, equation 3.5.1, but by ecluation
4.r .3.
4.2 l.{easurenent of the Scale Factor
The absolute scale factor rvas deternined for data sets 2 and 3
(see Tabl e 2.3) as fo11orvs.
The incident beam distributions were measured with a pinhole in
gold foil of thickness 125 nicrons (nominal). The author is indebted to
Mr.B.L. Green for making the pinhole. It was punched through the gold
foil by the pivot of abalance staff nounted in a watchnakerrs staking tool.
The hole was punched rather than drilled to avert distortion of the foil
in the vicinity of the hoLe. Any burrs were removed by spinning the foil
about a taut hair threaded through the hole. This rvas followed by gentle
sanding with very fine enery paper. The diameter of the pinhole was
measured on a travelling microscope to be 72.7 ! 0.5 rnicrons.
The distribution of intensity over the cross-section of the
crystal rnay be described with the aid of a Cartesian co-ordínate system
(x, y, z) such that the z-axis is parallel to the monochromated beant
and the x and y - axes are as in figure 4.1. The nornal to the planar
graphite monochromator (see page 3i) is in the x-z plane and, for V=0,
the x-axis is paralle1 to the rotation axis of the l¡liessenberg goniometer.
The pinhole in gold foi1, mounted on a standard ACA goniorneter head, was
centred optícal1y with the foil normal to the incident bean direction.
The observed incident beam distrjbution Io(x,I) was measurerl over the
cross-section that would be taken up by the crystal. To naint-ain counting
rates within the linear range of the detector, a seties of attenuators
59,
was used. The overall attenuation factor was measured to be f34.5 ! 2.2.
Each mea-surenent of Io(x,y) was imnediately repeated and counting
statistical errors were negligible in view of the large count rates. It
was found that lo(x,y) rvas independent of y over the area of
neasurement to 2% but as a function of x was not constant. Its shape
is shown in Figure 4.2. Although the beam distribution was not uniform
it was shown that the change in integrated intensity of a Bragg reflection
was less than 0.5eo for displacements Ax of the crystal from the centre
of the Wiessenberg goniometer less than 0.03nm. The crystal centre was
naintained within these linits in all experiments carried out on this
apparatus (see Table 2.3).
There is a distinction between the observed distribution Io(x,y)
and the actual distríbution I(x,y). They are related by the integral
equation
(ro (x,r) = çt/rf ) ) r ds (4.2 -r)
pinhole
where r is the radius of the pinhole.
If it is assumed that the circular pinhole nay be replaced by a
square aperture of equal area without significantly affecting Io then,
in view of the ínvariance of Io with y , the problem of reconstructing
I from Io
nay be treated approximately in one dimension as follows.
to (x,I) Ql 2Ð2 I
Ix
I
IdSsquare aperturex+ß U+g
I t (v,w) dvdw)-ß Y-g
x+ßI (v,w)dv
0/zÐ'z
(r/29)x-ß
where (29)2 'ÍÍî" and v and w are duruny variables.
(4.2.2)
60.
If it is further assumed that
I (v,w) al + azy * arf * a4vt (4.2.3)
then
(x,y) Ia, * (ar/3)ß2 J + fa, * au}'')x + agx2 + âr.x3o
(4.2.4)
Thus by fitting Io(x,I) to a polynomial of order three the coefficients
4t , ?z , ãs and 3u of I (x,y) rnay be deduced.
This procedure was carried out for the sets of rneasurements
carried out in conjunction with data sets 2 and 3. The fit of the observed
incident intensity Io(x,I) to the polynonial of equation 4.2.4 was
excellent. The average incident intensity < I > was calculated
according to equation 4.1.4 and it was shown that the difference between
< I > and < I- > is less than Ieo. Furthermore, the discrepancyo
between the transnission factors given by equation 4.1.3 and the
conventional expression, equation 3.3.I, is negligible not only for the
lithium crystal (TU = t.OO : see page 47) but also for the potassium
crysta.l.
In the case of potassiun there is an additional cornplication.
The values of the mass absorption coefficients (U/p) of elements have
recently been revised (International Tabies, 1974). The new value for
potassiun is 16.20 cm2g-r compared with 15.8 cn2g-1 which was the
value used in calculations of corrected intensities and scale factor ks.
As UR is snall it can be shown that the effect of this change in U is
tc increase the calculatecl value of k, by leo. However this cha.nge in
the value of l/ p has been overlooked as the possible error in the
revised value is - S9o and the two values agree to 29o.
I
6i.
The results of the calculations of k. are presented in
Table 4.1. Taking into account errors in crystal volume, attenuation
factor, linear absorption factor ancl < I > , the estinated errors in
are -Beo and - Seo for potassiun and lithium lespectively. Thus
ks
5960 I 470
for the potassiun data set and
k 2.II t 0.10 x 10sS
for the lithium data set.
TABLE 4.1 Va1ues of paraneters for absolute determinationof the scale factor. Quantities in brackets
represent estimated errors.
kS
À
a
wavelength
unit cel1 parameterat 293K
number of unit cellsper unit volume
x 10 "(.r-t ) 23.L35
x 106 (ctn t )
x 102 (cm)
x los (.rt )
x loa (s-t )
x 108 (cn)
x 108 (cm)
Lithium
0.7r07
3.5095
61 .02s (o .2%)
2.40 (0 .7e")
s .7s (2 .L%)
L.74s
2rL(s%)
Potassiun
o.7ro7
5.329
6 .608
4 .s7s (o . s%)
2.I7 (P;)
4.28(3e")
3.49r
s . e6 (8%)
Nv
1
V
(¡
4N2 ¡.3 çe' /ncz 7'
crystal radius
crystal volume
angular velocity ofcrystal
nean incidentintensity
scale factor
<r>
S
x 10-e (cm 2 s-l ) ro.4(2.5%) 9.76(2.5%)
x 10-3k
62
CHAPTER FIVE
DETERMINATTON OIì ANHAT{UON IC PARAI4E'IERS
OF POTASS]UI.I AND LI'I'FIIUT,I
Thi-s Chaptel describes the refinenent from the experimental
data of the thermal parameters c[ , Y and ô of the one particle
potential
V (u) 30u2 Yu4 ôlua * uo * uo'x y z 5
u4)1
z + +OPP
of potassiuln and lithiun. A lot of time and computing is of course
involved in such an exercise; judgenent is also required. This account
is reduced to its essentials. The experintental data are recorded for
the sake of completeness in Appendix 1
5.1 Data Arrall'sis
The data were analysed by ninirnizing the quantity
x' Ec (ci) )
2 (s.1. D
where E-,^,\ represents the ith corrected,c [o1J
Section 3.5) with weighting factor Wi and
conputed value given bY
observed intensity (see
E ^ , ^r., is the correspondingc 1c1.,
(s. 1 .2)E
? lvi (8. (oi)
k f .12sclc].c (ci)
The scattering factors fci weïe generated fron the nine-pararneter-fit
tables of Doyle and Turner (1968), The temperature factort rci htere
derived from equation 1.4.4, that is,
r^: = ¡^. exp[-2tr2 (h2 +k2 +r2l.k T/ua2 ] {l-lskBT(l/a')-Cl- -'Y 1 B
+ 10 (krT)' çzr ¡ a7' (y/ot ) (h2 +k2 + 12 ) i
- (k"T)' çzr¡ a¡a (y/ao ) (h2 +k2 *1')'í
- (k"T)t (2n/a)a (6/c¿o ) [(ha +ka+14 )i - å,n' +k2 +r2)2rf ] (s.1.3)
63.
-1where [1- 1skBT (y /a' )]N
Y
and (h k 1).
ith reflection.
parameter a was
The scale factor
ate the lt'liller indices of the latti-ce planes of the
The variation with tenìperature of the '.init cel1 lattice
explicitly accounted for in evaluating fci and rci
k is siven byJ
Wst
iT2cf
cf.-f.cI c]'
W
i
EI c (oi) f2. 12cl c1kS (s. r .4)
1 a 1
which satisfies the condition AX2 /ðk, = Q
The tenperature dependence of s is given by quasi-harmonic theory.
Hence
0 = %(1-2yoX,,T) (5.1.5)
(see Secticn 1.5) and the values of the volume coefficient of expansion
X., and Gruneisen constant Yc were taken from the literature. It has
been assumed by various authors (e.g. Wi1lis, 1969; Mair, Barnea,
Cooper and Rouse, 1974) that the variation with temperature of the
anharnonic parameters y and ô is also described by the quasi-harmonic
approxination, that is,
\/\^ - ô/ô_ = I-2ynyuT (5.1.6)'00
where yÒ and ôo are the values of Y and ô respectively at OK.
To the best of this authorrs knowledge there has been no formal
justification to support equation 5.1.6 which predicts a decreasa - 7eo
in the retative value of Y and ô in lithium and potassium for an
incre¿rse in temperature - 100K (values of Yc and & are given in
Section 5.2) It wilt be seen that the errors in refined values of
y and ô are nuch larger than any change predicted by equation 5.1.6
64.
and as an increase r\rith'temperature of tìre paraneter ô has been
observed in sodium (Field and Bednarz, 1976) no correction for the
temperature dependence of Y and ô was made in the computation of
the temperature factors rci by equation 5.1.3.
Four models were considered in the data analysis. They are
summarized in Table 5.1. In nodel I X" was minimized subject to the
constraint that the isotropic anhannonic y terms in the expression for
Tci aîe zeto. The parameters that were refj-ned were oo , ô and the
corresponding scale factor k, These optimum values of oo and ô
were the starting point for nodel 2 in which the constraint on y was
lifted. There was little correlaticn between the isotropic and anisotropic
quantities and the search for an optinum "t' was carried out essentially
in a two dimensional ( oo , y ) space. Dawson (1975) has pointecl out
that the y-term nay be simulated by choosing values of yc or Xv
larger than the actual values. The extent to which this is possible was
investigated in nodel 3 in which Yc was considered as a variable and
f was minimized with "( set to zero. The resultant value of y., uras
denoted bv y and compared to the actual value of the Gnrneisen' 'G(3_l
constant. In nodel 4, which is an extension of model 3, the Gruneisen
constant in equation 5. 15 v¡as set to Yo (, ) and the refinement oí thernal
parameters ca¡ried out as in nodel 2.
The weighting factors Wi were given by Wi = I/ (oi)' where
01. was the estirnated error in E The observed intensities for- 1 c [or_J
potassium and lithium were sensitive to sma1l variations in temperature.
At room ternperature, for example, the relative change in intensity per
degree K was - 0.6eo for the 220 tefLection of lithium and - 2.0eo
for the 222 refiection of potassium. To take sone accor¡nt of the instabilitv
65.
in temperature, the follorving scheme was adopted. The rnean relative errol
e for a data set of N points was
ã N-1' > ( oi i/Ec loi I )
(s. 1.7)
where o, is the statistical error in E^,_,. . The o. were taken to1 c(o1j -1
be the statisticai counting errors where the ternperature dependence of one
reflection of a given hkl-type was deterrnineC (sce for exarnple Section 2.i)
and were taken to be the standard deviation of the mean wl"rere all
accessible sylTunetry related reflcctions of a gi.ren hkl-type hrere measured.
TABLE 5.1 Sumrnary of nodels used in data analysis
Fixed Parameters
Y (y=o)
(literature value)(literature value)
YG
yc (literature value)
)Ç (literature value)
Y (Y=o)
Xv (literature value)
xv
\ G(Y.= Yo ,r, )
Derived Parameterslufodel
10
CX
ô
c[o
Y
ô
2
3 00
ö
YG
do
Y
6
4
Xv (literature value)
66.
The estinate<l errors 01 , for tl're purpose of weightíng the data,
taken to be
weTe
{ot1
o.(eIe
)eE if
ifE
c (oi)c (oi)
+
l rN-lR ,o .2 5
(s.1. B)a o E
1
Thus no data point was assi-gned a relative error less than e
The measure of agreement between observed data and theoretical
rnodels was expressed in terms of an R-factor R defined by
c (oi)
1
>tvi i(Ec(oi) - Ec(ci) 2
)
R (s. 1. e)> l\r. E2 .i 1 ctolj
The tables of Hamilton (1965) were used as a guide to assess the
significance of the anharmonic terrns. In addition these tables were
used to estimate the errors in the optinum thernal parameters as follows.
For a data set of N points described by m refined parameters
pI... Pi... p* with a corresponding R-factor R(Pr. P1... Pr) the
error Api in p1 was taken to be that value which altered the R-factor
ratio ¡(pr... p1 + Api... pr)/R(pr... p'... pr) to the extent that
P(Pr. . . P1 + Api. .. Pn,)
& (s.1.10)R(p Pi Pn')I
where & is a distri.bution of R-factor ratios (Hamilton, 1965). In
this case the probability of an R-factor ratio greater than *r,*_,,o.2s
is 0.25 and Api corresponds approximately to a standard deviation. The
scale factor was obtained fron equation 1.5.4 for a given { qo, Y, ô }
parameter set. The error ôk, in k, was deternined vi.a
ôk, = ao.o (ãk=/ðc*o) * Ày(ðks/ðY) + aô (ðkr/ðô) (5.1- 11)
where Acl , Ây and Aô were obtained by the nethod described above.0
To avoid confusion with any previous notation and for convenience, the
symboLs to be used in this chapter are summarized in Table 5.2.
67.
TABi,H 5. 2 Definition of symbols.
a Unit ce11 pa:rameter
Nearest neighbour separation
Displacement vector
Mean square displacenent in n-direction.
Root mean square displacement (t*n,r= < u' )L )
Ratio of (ua ) to <ri,
One particle potential (OPP)
Harmonic part of the OPP
Value of 0, at 293K
Fourth order isotropic anharmonic parameter of the OPP
Fourth order anisotropic anhar¡nonic parameter of the OPP
Gruneisen constant
Volume coefficient of expansion
Harmonic Debye-ltra1ler B-factor (=Brr2 krT/cl)
X-ray Debye temperature
Mean inverse square phonon frequency
Atomic mass
Corected observed integrated intensity
Corresponding conputed value of integrated intensity
R-factor (=R(0,Y,5))
R(a,Y,o)
Probability distribution of R-factor ratios
Scale factor
rnn
u
,În
uFII\'IS
r
VOPP
OI
0,293
Y
6
yG
x,
Bt
0(X-ray)
u-2
M
Ec (oi)
Ec (ci)
R
RI
&
ks
68.
TABLE 5.'5 Parameters refined from data set 1 (1orv
temperature potassium) .
Model
o¿ (eV.A-'? I0
y x 103 (eVÅ-a I
6 x 103 (evÄ-a I
k x 1O-3s
Rx102
Rr x 102
."? at 2%K(fJ )
1.15 1.22
0.181r0.001 0.r85!0.002
0.0 0.0
3.7lI .0 3.7!I.0
9.6710. 30 9.5910.30
6.84
7.44
6.51
7.r3
0.16810.001 0.16810.001
r.667 7.667
I 1 I
r.34
0.18710.001
0.0
3.7tI.0
9.5410 . 30
5 .96
6.62
0.15010.001
0. 168t0. 001
L.667
3
I .80
0. 20610 .001
0.0
3.8r0.8
B. 9BtC. 20
4.76
5.52
0. 15210. 001
0. 166!0 .001
r.667
YG
ct (eV.A-'?¡ 0. 15010.001 0. 150r0.001293
1
Model
vG
o (eVÅ-2 )0
y x 103 (evÄ:a ¡
ô x 103 (ev,{-a I
k x 10-3s
Rx102
il x1O2
cl (eV,4-2 I293
2
1.15
0 . 18610. 001
-5.610.6
5.511 .0
10. 9310.65
5.7r
6.26
0. 15510. 001
0. 18110.003
I . 70910.003
2
r.22
0. 188Ì0. 001
-5.010.6
3.5r1 .0
10.70r0.60
5.s4
6.12
0. 15510 . 001
0.179r0.003
L.707!0.004
2
r.34
0.19210.001
-3.910.6
3.611.0
10.2910.60
s.29
5.92
0. 15410.001
0.17710.003
1 . 704r0. 003
4
r.B0
0.20610.001
0. 0r0.5
3. 810. B
8.9810.50
4.76
5 .52
0.15210.001
0. 16610. 001
1. 66710.003
<uf>at 2%K(13 )
r
69.
TABLE 5.4 Parameters refinecl frorn a subset of data
set I (low temperature potassiurn)
Model
YG
o (eVA-2 )0
y x 1û3 (eV,{-a I
ô x 103 (evÄ,-a;
k x 10-3s
Rx102
É x102
(¡2) at 29sK(Å2 )
r
0.0
5 .0r0. I
9.I4!0.25
s.74
7.62
0.0
5 .0r0. B
9. 0610.25
s.27
7 .29
31I1
1 .1s r.22 L.34 r.70
0. 18210. 001 0. 18510.001 0 .18910 .001 0. 20310. 001
0.0
5.1r0.7
0.0
CI (eVÂ-'z1 0. 15110.001 0. 15210. 001 0. 15210. 001 0. 152t0 .001293
5 . 4r0.6
9. 0110. 20 8.67t0 .20
4.50 3.20
6.77 6. 0B
0. 16710. 001 0. 16610.001 0. 166t0.001 0. 16610.001
t.667 r.667 L.667 L.667
Model
YG
q fevÂ-2 )o'
.¡x 103 (evr\-a ¡
6 x 103 (eVÂ-a 1
k x 10-3s
Rx102
Rr x 102
1.15
0. 185r0. 001
2
t.22
0.18710.001
-4.2!L.0
5.010.8
10.4110. 70
4.32
6.55
0. 153t0.001
0. 17910. 004
I . 70510.004
42 2
r.34 L.70
0. 191r0 .001 0. 20310. 001
-5.0r1.0 0. 010.8
s.2!0.6 5.410.6
9.8310.65 8.6710.50
-4.611 .0
5 .010. 8
10.5510. 70
4.63
6.75
3.83
6.3r
3.20
6 .08
cl (eVÄ:'zI 0 . 15410.001293
0.15310.001 0.15210.001
0.175f0.005 0.16610.001.";t at zwKQ?) 0.17e10.004
r 1 . 706t0.004 1 .69810.006 1 .66710. 005
70.
TABI,E 5.5 Pararneters refined from data set 2 (high
temperature potassium) .
Model
ye
cr (eVÂ-'?10
"y x 103 (evÅ-a ¡
ô x 103 (evÄ.-a I
k x l0-3s
Rx102
Rt x 102
4.8r0.5 4 . 8r0.4
4.07!0.r2 4.08r0.12
s.69 s.42
9.76 9.60
J1II
1. 15 I.24 r, -<4 1.93
0.17910.001 0.18210.001 0.186j0.001 0.210!0.001
0.0 0.0 0.0
4.9r0.5
4.04!0.I2
s.12
9.47
0.0
4.910.4
4 .0510. 10
4.08
8.90
ct (eVÂ:'?I 0. 14910. 001 0 .14910.001 0. 14910 .001 0. 15i10.001293
4Í)at2%K(N) 0.169i0.001 0.16910.001 0.16910.001 0.16710.001n-r.667 r.667 1.667 I.667T
Model
vG
2
1.15 r.24
0. 17910. 001 0 .181r0 .001
42 2
r.34
0.18410.001
-5 .1r0.8
4.510.4
5 . 6810.50
4.04
I .95
0. 14810. 001
0. 189t0. 004
1 .710j0.004
1 .93
0.21010.001cr. (eVa 2 ¡
0
y x 103 (evÅ-+ I
ô x 103 (evÂ-a ¡
k x 10-3s
Rx102
Rrxld
-5. 611 . C
4.510.5
-5 .4t0. 8
4.510.4
5 .7610.50
4.24
9.05
-2.2!0 .8
4.7!0.4
5.81r0.55
4.44
9. 16
4.66!0.40
3.86
8. 82
0. 15110 .001
0. 17510.006
d, (eVA:2 I 0. 148r0. 001 0 . 148r0. 00129?
*? at 2%K(F? ) 0.1e1t0.00s 0.19010.004
T t.712!0.00r 1.711r0.003 I .691r0. 001
7I,
5.2 furharmonic Ther'mal Paranleters of Potassiurn : Results
of Analysis of Data Sets I and 2
The results of the analysis of data set 1 , which describes
the tenperature dependence of the Debye-lValler factor of potassium in
the tenperature r¿nge 207K to 50BK , aTe presented in Table 5.3.
Trial values of yc used to describe the variation with temperature
of the harrnonic palametel G were I.15 (Gerlich, 1975), I.22 (Schouten
and Slenson, L974) and I.34 (Slaier, 1939) and cover the ganut of
published values. The volume coeffj-cient of expansion X' was taken
as 2.4g x I¡-a per degree K (Kittel , IgTl). Rl is the R-factor for
the paraneter set in rvhich the anisotropic ô conponent has been onítted
fron the refinement (as recorded in Table 5.2) . For each parameter set
the value of < ui > was determined from the relation
< u1 > = (r - r5(y/a2 )k.T)-' ¡¡t"r7o¡-3s(y/d ) (k,T)'ln
(s .2 . 1)
(see Section 1.4) and the root mean square displacement üIas denoted by
u The ratio r of ( uo >RMS
of equation 1.4.9.
For any assumed trial value of Yc in the range 1.15 to 1.34
it can be seen that the agreement of the nodel calculations with experinent
is better for nodel 2 than for nodel 1 This indicates that there is a
genuine isotropic fourth order "( component of the one particle
potential (see Table 5.1) It is evident that the nagnitude of Y
decreases as Yc increases. Nevertheless the value for Y of
-3.9 x 10-3eVÅ-a , corresponding to a YG of 1.34 in model 2, is
significant at a 1evel of 0.005. The optirnum YG deduced via nodel 3
is 1.80 with a colresponding refined Y in nodel 4 of zero. Thus the
72
parameters of models 3 and 4 ale identical. The R-factor R of models
3 and 4 is less than that of nodel 2. This confinns the proposition of
Dawson (1975) nentioned earlier (see page 64) . The dependence of YG
on phonon frequency has been studied by Slivastava and Singh (1970);
Kushwaha and Rajput (1975); Nfeyer, Dolling, Ka1us, Vettier and Paureau
(1976) and Taylor and Glyde (1976). Their results inply that the nean
value of yc in the Briltouin zone is - I.3 and the most probable value
- I.4. Thus the value of 1.80 is unrealistic and on this basis it is
concluded that Y is real.
It was furiher shown that Y is not an artifact of the
weighting scheme. Tabte 5.4 shows tl're results of calculations in which
only data collected at 233K, 260K, 296K and 30BK (see Section 2.4)
were used. The mean relative error for this subset of data set 1 is 2.08%
conpared with 1 .65% for the entire data set. There is close agreenent
between corresponding sets of paraneters and the best Yc deduced via
nodel 5 for the subset of data is 1.70. The possibility of systenatic
errors between the thernocouple ternperature and crystal temperature ï/as
considered. In particular it was shown expl,icitly that the effect of a
relationship of the forn of equation 2.5.2 is to increase the magnitude
of y (which is negative).
The R-factor R of model 2 is srnallest for a YG of I.34
and the corresponding thermal parameters at 293K are
cr, = 0. 154 t 0.002 eV,4-2
Y = -3.9 t 0.6 x lO-3eV'{-a
ô = 3.6 I 1.0 x 16-r "Yfi-a
At this stage no claim is nade as to the actual value at 0K of o which
0is equal to q in Table 5.3 only if YG and Xv are indepenrlent of
73"
tempeïature from OK to 293K As this is not the case the valüe of
o has been onitted fron the list above. The quantities derived from0
ct , y and ô are as follows:293
(u2n
0 .L77 t 0.003¿V
5J
uRMS
0.729 r 0.0064
ll L .704 1 0. 003
The value of r is only slightly larger than the quasi-harmonic value of
Thus the deviation from a Gaussian function of the distribution of
atonic displacements is not large.
A notable difference between model 2 and models I and , (it
both of which y is fixed at ze'ro as recorded in Table 5.1) is the
refined scale factor k, which is approxinately L}e" latger for a value
for y - -4 x l0-3eV,{-a than for Y=0. If equations 1.5.10 and 1.5.11
were valid, tl're three nodels would have a commcn value of k, . However
these equations are based on the assumption that the ratio of the fourth
order to second order terrns involving \ in equation 5.1.3 are negligible
(see Section J..5). For the 420 reflection at room ternperature, for
example, this ratio --0.5 and it is no longer the case that the effect
of y is equivalent only to a contribution to the harrnonic parameter o .
The two y terrns are of opposi-te sign and if the sign of Y is negative
the second order anharmonic term is compensated by the fourth order term
and a larger scale factor.
The refined paraneters for data set 2 (where the neasurements of
intensity span the temperature range 297K to the rnelting point of
potassium at 337K) are given in Table 5.4. The analysis of this data
set was carried out for trial values for yG of 1.15, I.34 (see
page7l) arìd I.24 Schouten and Swenson (L974) The value for YG of
L.24 ís appropriate to the temperature range (sce Ta-ble 2.3).
74.
As in the case of data set 1, rnodel 2 provides a better fit
to the experimental data than does nodel 1 (in which y is zero)
The value of the y parameter in rnodel 2 is in the range -5.6x10-a"y[-a
to -5.1 x 10-3eVÂ-a and is significant at a level of 0.005 The
optinun value of Yc determined via nodel 3 is 1.93 but, unlike data
set 1, there is a residual y of -2x10-3eVÅ-a in the pararneter set of
nodel 4. It is clear from an inspection of R-faciors that nodels 5 and 4
fit the data better than nodel 2 but, as has already been pointed out
(see page/l), an assumed value of yc - 1.8 or greater is not physically
reasonable. Thus y is real. The R-factor R of nodel 2 is least for
a yc of I.34 and the corresponding parameters arîe
0 = Q. 148 t 0. 001eV,{- 2
293
Y = -5.1 1 0.8 x 10-3 e\A-a
ô - 4.5 !0.4 x10-3eVÅ-a
Hence
0. 189 t 0.004 Fe
uRÀ4iS
0.753 r 0.008 Å
r 1.7I0 r 0.004
and in particular the scale factor is
5680 t 500
On the other hand the scale factors of all models in which Y=0 are
- 4050 t 100 but the absolute scale factor is
k, = 5960 !470
(see Sectíon 4.2). The agreement of the measured value of
refined value of nodel 2 confirms the existence of Y
ks
2tIn
ks
with the
75.
In view of the magnitude of the errors in refined .¿alues of
"( and 6 ( - 20"," ) it is not possible to assess the validity of the
qua,si-harnonic approximation of equation 5.1.6 which predicts a change
in y and ô -0.07vo per degree K. Although there is a difference
between values of or* of nodel 2 in ð,ata sets 1 anð,2 of the order of
4eo this nay be due in part to the fact that the data were taken in different
temperature ranges. In particular,data set 2 extends up to the nelting
point and as no account was taken of the yariation with tenrperature of "'(
(there is negligible correlation of cx and 6 ) the discrepancy is not
necessarily due to experimental error. In fact values of oru. for
nodel 1 of data sets 1 and 2, agree to 2%.
The Wilson pluts for the four ternperature data subs.et of data'
set 1, and the high temperature data set 2 are shown in figures 5.I ano 5.2
respectively. The straight lines are drawn for a y of zero and the
values of o derived from nodel 1 for a yc of I.34. The difference
between the scale factors of nodels 1 and 2 in figure 5.2 is apparent but
there is no obvious curvature in the ltlilson plot. It can be shown fron
equation I.4.4 that
- +.,î t * i. "îr cfis lq' 5.2.2
As r - 1.71 and sin2 0/À2 is proportional to f , it can be shown
that the change in åi*#àÐ , from 222 to 440 is only - 3eo.
An alternative representation of data set 2 is given in Figure
5.3. This is a plot of ln(E.¡oi)/krf'?"r) against tenperature. The
straight lines are drawn for values of o and k, for model 1 for an
assuned yG of I.34. The deviation of the experinental points fron the
straight lines is determined essentially by the anisotropy factor
76
h4 + k4
and in particular the splitting of the intensities of the 330, 4Il
and 431, 510 pairs of reflections is clearly shown. There is no
evidence of pre-melting phenomena characterizeri by a decrease in intensity
significantly greater than that predicted by anharmonic theory. At the
nelting point
+ f å,n' + k2
= 0.143 I
= .5.1 !
= 0.230 !
= 10.3 t
+ f)'
0.001 eVÂ-2
o . 8x t0- ÈvÄ- 4
0.00s f3
0.L%
ot,
Y
Hence
with
(u2n
< ,Í )\/rn nn-
where r-- is the appropriate nearest neighbour seperation.nn
The anharnonic ô-parameter is significant at a level of 0.005
in both data sets and is 3.6 t 1.0 x 10-3eVÂ-a in data set 1 and
4.5 t 0.4 x 10-3 eV,4-a in data set 2. ]'he sign of ô is consistent rvith
the intensity ratios of the 330 , 4II and 43L , 510 pairs of
reflections. Using equation L.4,4 and ignoring the isotropic anharmonic
"( terms of T , it can be shown that
!-ç:ett'ç+rr1
!-E:ttt'(sto)
1+ r92(2r/a)o (6/oo ) (k.T)'
r' C_sso-l - 2
t2 ¡+tr;3and
Figure 5.4 shows the values of ô deduced fron the intensity ratios of
the two pairs of reflections for the,four temperature data subset of data
set 1 and for data set 2. The mean r4alue of ô is 6 I 1 x 10-3eVÂ-a
a.nd as the sign of ô is positive there is a greater probability of
vibration in the nearest neighbour directions than in the next nearest
l-'8
1..2
3.6
3.0
1.8
1.2
0.6
0
\=
222 32' 400811Ì 20 332 rrrÉi| 52t aar[í3å]
E
T
rT.
EÃ
Y
\
Þ.
T
Ã
\ å\E
-'3 2'1.-\ rJ
L'
UJ
c
ïT
ï.
I
TIt
-ð_á
T
q
I\
rr.
E
ï
ï
T
r
0.21 0.30
ïrT
T
ï
0.0 6 0.12
sin0.18
o/^22
Fig. 5. I
Witson ptot f or data set t ( tow. temperature potassium ).
Measurements were et 233K (o),260K(a); 296K(o)and 30BK (e) Verticat bars rLpresent estimated errors.
1.,5
3.9
3.3
2.7
222 3214,0081fl420 332 .rr[ái¡ sz' aao[3f
3
r$to
(, a
olxÂ+c,o
_'!IJJ
C
2
5
0.9
0'3
-0.30 0.06 0.12
sin
0.2t,
Fig. 5.2
Witson ptot for data set 2 ( high temperature potassium).
Measurements were åt 2g7K (').302K("), 307K(^),3ttK(x)315K(a), 32OK (+), 32t,K (v), 328 K (r) and 333K(o).
The asterisk represents the absotute scate factor.
1
a
alxÁ+vIo
aoIoAIv
¡oIao
a
o¡><ô+.
v¡N;
\)ìî
¡ao
¡
0.18
o¡À2
0'3 0
2
-2
-3
-l-
v
ÅA
A
A A
¡
Â=...-a
v-- v
LI
^
Y
V
N
v
--..-x
<.¡ 'õrF
t¡.Y
^
v
^
Vou
A
o
+
x
+
¡
++
o
UJ
C
-5
+
-6 ¡
-7
290 300
Fig. 5.3
The variation of
222 (^), 32t (v),1.22 G), 431 (r.) and 510 (o )
from data set 2 ( highT¡¡ is the metting Point
310 320
Temperoture ( K )
tr+
o
o o
o
330 3
Tm
0l-
tn ( Ectoi)/krf.z¡ ) with temperature f or the
400 (^), 330 (.) , 4l I (o) , 420 (v), 332 (x)reftections. Data are taken
temperature potassium ).
of potassium.
\tI
o
10
6
t,
I
(u
frlox
tro
2
0210
Fig. 5.4 .
0 bser ved variat ion
260 280Temperoture
320 310300(K)
of 6 with temperature for Potassium
77.
neighbour directions (see page 20)
5.3 Anharmonic Thermal Paraneters of Lithiurn : Results
of Analysis of Data Sets 3 and 4
Data set 3, derived fron the spherical crystal of lithium
(see Table 2.3), was analysed via modeis 1 and 2. The value of the
Gruneisen constant was taken to be 0.86 (Martin, 1965a) and the volune
coefficient of expansion Xv is 1.35 x 10-a per degree K (Kittel,
1971). 0nly 110-type reflections of lithium were affected by extinctj.on
and have been excluded fron the refinemerìt of thennal parameters. An
extension of model 1 to the anharmonic theory of model 2 did not produce
a value of y significantly diffel'ent from zero. Thus the isotropic
part of the one particle potential is quasi-harmonic.
The results for nodel I for data sets 3Á. and 3Ð (see Section
2.6) are given in Table 5.6. The mean values of cx, and k- listed299 s
in that table are
cl C.422 I 0.002 eV,{-2293
2.04 ! 0.02 x 10skS
The value for 2.04 x I}s is in good agreement with the value
i.e.,obtained in Chapter 4,
kS
2.ll I 0.1C x 10s
This agreernent is evident in the Wilson plot for data set 3A in Figure
5.5 a-nd supports the conclusion that there is no isotropic fourth order
contribution to the one particle potential for tenperatures up to room
tempcrature. However the argument is different for experimental data for
elevated temperatures for lithiun.
k, of
78.
TABLE 5.6 Parameters refined via model I fron data
set 3 of lithiun.
y x 102 (evÄ-a ¡
ô x 102 (ev,{-a ¡
k x 10-5s
Rx102
Rr x 102
3A
24BK
0 .454!0 .002
0.0
3. Btl .6
2.O5!0.02
r.44
1.96
0.423!0.0c2
L.667
3B
248K
0.447!0.002
0.0
1.9r1.1
2.05t0.02
I .00
I .15
0 .41 710 . 002
0. 060510. 0003
t.667
0293
r
(eV,{-'z1
( u2_ ) at 293K (¡¡ ) o. o597to . ooo3n
Data set
Tenperature
ct (eV Â-2 I0
5A
296K
0.455
0.0
4.0r1 .4
2 . 05t0.04
2.86
3 .98
0.424!0.002
0. 059510. 0003
t.667
3B
296K
0.45st0 .002
0.0
4.610 . B
1 .9910. 03
L.43
3.7s
0.424!0.002
0.0s9510. 0003
L.667
y x ld (evÅ-a ¡
ô x 102 (ev.&-a ¡
k x 10-ss
Rxl02
Rt x 102
cr, (eVÂ-2¡293
< u'?O > at 293K (Â')
r
79
TABLE 5.7 Parameters refined via nodel 1 from the
roon temperature data of data set 4 of lithiun.
0.453 ! O.OO2
0.0
ct (eV,{-'?¡0
Y (eVÂ-a ¡
6 x 102 (ev,4. a ¡ 2.0 r 1.0
k x 10-5S
Rx102
Rt x 102
1.07 ! 0.02
2.55
3.29
0.422 r 0.002c[ (eV,{-'?¡293
< u'?- > at 2%K (I? ) 0.0598 t o. ooo3n
T r.667
The refined pararneters of rnodel 1 for the room temperature
data of data set 4 are presented in Table 5.7. The 110 and 200-type
reflections r^/ere affected by extinction and have been onitted from the
data analysis. It is clear that the value for or* of 0.422!0.002eVi\-2
is in excellent agreement with the results derived fron data set 3 at
293K
(u2 >n
u*r" = 0.424 I 0.001 .A'
r = L.667
The high temperature data of data set 4 describcs the
tenperature dependence of the intensity of the 220 reflections of
lithium. The variation with ternperature of the quantity
(-x2 /zsi* o ) rn(E"(oi) (Ti)/ksf'?ci),
6.0
5.5
3.5
\f
r0 200 z, zz0 310 222 rr' ,ooftfl]rrn trr,rr[fl]
Þ
521 .,0 []i]
0-56 0.70
E
E
T
1
u
5.0
l-,5
l-'0
Þ
r E
EI
E
É
T
ïo(\¡ v
-- r..rl
UJ
C
ï
I
ï
3.0
0 0.14 0.28 0'1.2
e/ ^2
2stn
Fig.5.5Witson ptot for tithium.Data ere f rom data set 3A at 21,8 K (" ) and 296 K (')'Verticat bars represent estimated errors.The asterisk rePresents the absotute scale factor.
8.0
7.5
6.5
6.0
T
T
(\'õ¡+-
llr
T:_ 7,0t--
'õ(,
u-J
C
(Dßt
C'tn
Nñl<
I
II
IT
T
5'5
5.0
{
/
/
/I/I
T4.5
290 320
Fig. s.6Variation w¡th temiereture of
350 380
Temperoture (K )
lr10 l-l-0
for the 220 reftection of Lithium compared
erpected on the basis of room temperature data.represent errors of 1% in E.1o¡¡
r-^2t zsin20 ) tn(E.roil (Ti ) t Rrr!¡)witn the variationVerticat bars
80.
rvhere Ec¡oi¡ (Tr) is the observed conected intensity of the 220
reflection at temperature Ti , is illustrated in Figure 5.6. In the
quasi-hannonic approxintation ( y=0) this quantity is the Deby'e-tVa1ler
B-factor at the relevant temperature. The heavy line in Figure 5.6
represents the temperature dependence of B given by
c[
B 8d k T/crB'
(s.3.1)
where
c[ (1 2\.Xu(T-293) ) (s .3.2)293
and 0 was taken to be O.422eVÃ-2293
If it is assumed that the dífference between the experimental and
theoretical values is real, then there is an isotropic fourth order
component of the one particle potentíal. The combination of roorn
tenperature and high temperature data fron crystal 4 was analyzed vía
rnodel 2 and yielded a value for y of - 4.6x10-z "y[-a which is
equivalent to a value for r of L.72 at 423K , the highest temperature
reached in the experiment (see Table 2.3). It was pointed out in Section
2.7 that there was an inconsistency in the high temperature measurements
on account of the instability of the crystal within the capillary. As
the effect of any movement of the crystal was to decrease the observed
intensity, the value for Y of - 4.6x10-2eVÄ-a can only be regarded as
a lower limit for y up to 423K Nevertheless the experiment has
established that r < I.72 rrithin 30K of the nelting point of 454K.
At the nelting point
0 = 0.406 1 0. 002 eVÂ-2
and if it is assumed that Y=0 then
81.
u2n
k T/crEI
0.0963 t 0.0005 ff
\and ( u2n
/'r'nn 10. 14 I 0.03 %
There is inconsistency in the values of ô deduced from data
sets 3 and 4. Although both data sets indicate a positive value for the
sign of delta it is only in data set 5ì] that ô is significant at a level
of 0.005. In Table 5.8 the values of ô deduced from the intensity
ratios of the paired reflections 330 , 4Il and 43I , 510 are denoted
by 6 and 6 respectively. It can be seen that there is a wide'1 2
variation in ô and no real ô term can be claimed to exist for lithium.
TABLE 5. B Values of ô deduced from the intensity ratiosof the 530 , 411. (ôr) and 43I , 510 (6r)
pairs of reflections of lithiun.
Paraneter
6 x 102 (ev,4.-a ¡I
ô x IO2 (eV,{-a ¡2
6 x 102 (eV'4;a I
ô x 102 ( ev,&- a
¡2
Temp.
248K
248K
296K
296K
Data set 38
-1!1
5t4
7!2
Data set 4
2!2
1t1
Data set 3A
I!2
5!2
7!2
1t1
82
TABLE 5.9 Sumnary of paramet.ers of the one particlepotential of lithiun, sodium and potassium.
The paraneters for sodiun have been taken fronField and Bednarz (1976).
a at 293K (,q)
c[ (e\',{-'? ¡293
y x 103 (evi\-a I
ô x 102 (eV.Â-a;
Temperature rangecovered by experiment
-2
Lithium Sodium Potassium
5. 5095 4.2872 5.329
0 .422!0 .002 0 .264!0 .024 0. 15110. 005
0.0 0.0 -4.510. 6
unresolved 3.3!1.2 0.4010. 05
248K-423K 148K-371 K 207K-337K
0.487Ì0.003 0.4510.04 0. 40r0. 01
454 371 337
Bh at 2g3K (.43 ) 4.72!0.02 7. 5510.63 13 .20!0 .26
< u1 > x 10 ar 2%K(f( ) 0. s98t0. 003 0. 9610.08n
I .8310. 06
u^.._ at 293K (Å)RM.s
0 .424!0.002 0.5410. 03 0.7410.01
r.667 r.667 I . 70610. 004
O(X-ray) at 293K (K) 326LT 140r6 80. 810. 8
MU-t at 293K (eV.{-'? ) O.436tO .OO2 O .266!0 .024 0. 15110. 005
T
I:0¿ 293(rnn/2)2 (ev)
Melting point (K)
(u2n
\ x 102 atTnn
the nelting point 10. 1410.03 9 .610.6 10.1r0.2
83.
5.4 Surunary of Results and Review of Previous lvleasurements
of Vibration Amplitudes in Potassium and Lithiun
The experinental results for potassium and lithium are
summarized in Table 5.9. The values of orr, , \ and ô for
potassiun are the means of values taken from tables 5.2 and 5.4 for
nodel 2 with Yc set to L.34. In the case of lithium the value for
ct of 0.422eYK ' is conmon to data sets 3 ancl 4. For each elenent293
an harmonic Debye-lVal1er B-factor Bn
Bf, = 8T2 a "'n
tn
where
. 4 tn= k, T/o (s.4.2)
(see Section 1 .4) and the corresponding X-ray Debye temperature is given
by
a "'n
tr, = (sh2 T/MkBo2 (x-ray))[r + t/ra (O(x-ray) /T)" ] (5.4.3)
was calculated fron the relation
(s.4.1)
-l-2
(see equation 1.5.8). The quantity MU was derived via
Mu- t = rk1 @ (x-ray) / 3f.2 (5 .4 . 4)-2 E¡
It is pointed out that o and MU:; are equal only in the linit of
high tenperatures. where the effects of the zero point energ)/ are negligible
(see Section 1.5) . This condition is satisfied at 293K in potassium
as T >> o(X-ray) In lithiun T - o(X-ray) and there is a difference
oî. 4% between 0, and MU-r-2
For the sake of conpleteness the thernal parameters of sodiun
at 293K (Field and Bednarz, 1976) have also been included in Table 5.9.
These parameters of sodium require some qualification. The Debye-Waller
factor of sodiun was measured from 148K to the nelting point at 37LK
However it was not possible to describe the tenperature factorr through-
B4
out the entile temperature range by the one particle potential model
described in Section 1.4 Beyond a temperature - 310K an anomalous
decrease in intensity was observed and
0.264 eV,{-2 describes 'r from 148K
a ,r; t* of rnn at the rnelting point
by extrapolating or* to 371K In
T the value of 9.6eo maf underestirnate
the value for or* of
to 310K only. The ratio ofwas derived via equation 5.4.1..
view of the anomalous decrease in^t< un >'/Tnn
In the case of potassium it is possible to relate the thermal
parameters o and Y to the results of a recent experiment of Meyer,
Dolling, Scherm and Glyde (L976) who measured the dynamic form factors
S(Q,¿,1) of potassiun at 4.5K , 99K and I50K by inelastic neutron
scattering at fivc points a in the [ 110 ] direction in reciprocal
space. From the integrated intensity under the one phonon peaks they
deduced the effective (X-ray) Debye temperatures. which are 96K, 9lK
and 88K at 4.5K , 99K and l-50K respectively. In order to cornpare
their results with the present work equations I.5.23, 5.4.1 and 5.4.2
were combined as follows:
8n2 k"T Bn-I = o"
= cr^(1-2y.4T)(1+20¡ylo'z1t"r) (5.4.5)0
In this way the values of o" at 99K and 150K are found to be
0.1917eV.{-2 and 0.1795eV4-' respectively. If it is assumed that.(o = I.L7 and Xr, = 1.87 x 10-a per degree K in this temperature range
(Schouten and Swenson, I974), the twt¡ simultaneous equations derived fron
the above expression may be solved with the result that at 150K
ct = O .200 eVÂt 2
Y = -15.8 x10-3 eVÅ-a
It is pointed out that the approxination for T implied by equation
B5
5.4.5 is sufficiently accurate in the tenperature ïange 99K to 150K
for the comparison to be valid. If the value of oruo is extrapolated
to 293K, equation 5.1.5 with "(o = L.20 a.nd Xu = 2.16xlO-a per
degree K (Schouten and Swenson, I974) yields
cr. = Q.185 eV,{-2293
Meyer, Dolling, Scherrn and Glyde (1976) give no estinate of the errors in
their values of 0(X-ray) and it is not possible to assess the reliability
of the calculated values of o and y (which is large). Nevertheless
their resul"ts and those of this author are in disagreenent with those of
Krishna Kumar and Viswamitra (197I) whose value of or* ir 0.20910.004eV4-2
Only two of the six observed intensities used in deriving their result
were unaffected by extinction. It is not unreasonable to suggest that
extinction has been underestimated in their calculations.
There is only one previous measurement of vibration amplitudes
in lithium (see Section 1.6). The value of O(X-ray) given by Pankorù
(1936) is 352K and is equivalent to a value for or* of 0.490eV,&-2
This is - 16% larger than the present result. The effect of applying a
TDS comection to the Pankow data would be to reduce the difference.
However lack of infornation concerning that experirnental situation prevents
further comparison with the present work.
86.
CFTAPTER 6
THE RELATION OF THE ONE PARTICLE POTENTIAL TO T}IE
INTERIONIC INTERACTION POTENTIAL
There are thro interpretations of the one particle potential
V___ The parameters o , Y and 6 may be viewed as a representationOPP
of the moments of the probability distribution function t (g) of the
atomic displacernents. Alternatively Vo"" may be regarded as a real
potential deterrnined by the properties of the anharmonic interaction
potential 0 This Chapter describes the extent to which it is possible
to relate Vo"" to 0 which is well knoln for the three alkali metals
lithiun, sodiun and potassium. The starting point of this discussion is
the Einstein nodel of lattice dynamics.
6.1 Einstein l"lodels of the Harrnonic Parameter o
Perhaps the sirnplest approach to anharnonic lattice dynamics
is via the Einstein model which has been successful in describing such
crystalline properties as elastic constants, thernal erpansion, Gruneisen
parameter and specific heat (see for example Holt, Hoover, Gray and Shortle
1970). Recently Cowley and Shukla (L974) have shown that the Einstein
model accurately accounts for these same properties even at the melting
point. The OPP model described in Section 1.4 is closely related to
the Einstein nodel. Indeed, it is not uncommon for the OPP to be
referred to as an Einstein potential in the literature (for example I''Iillis,
1969). In both models a physical system of N (say) interacting atoms is
reduced to one atom with only three degrees of freedon. It will be seen
that the EinStein potential nay be related analytically to the interionic
87
potential Q and in view of the success of the Einstein model a
detailed analysis of the Einstein potential was undertaken.
For a monatornic solid the configurational part 0 of the energy
of a crystal may be expressed in the forn.
0 O(l&j.sj-Bi-eil) (6.1.1)i<j
N0 * > or(gi) + fu.z'-1 u.
-Ji<j)
0
where
and
and
0 (u=)l-I j
0 i I o( oi-Bi )J-
(6.r.2)
R._J -u,_J l)f (6.1.3)
0
R.-R.-u.-J -1 -1
) - 0( I
0 (u, ,u= )2_L_J o(l !¡ * ej - Bi - ei l)* orl Å¡ - Bi l)
0(l R. 4 u: - R= l)- O(l n. - R. - u. ll-J -l -1 -J -r -l t' (6. 1 .4)
The notation is that of Westera and Cowley (1975). Here &i is the
position vector of the origin of the ith l\rigner-Seitz cell and s1 is
the instantaneous displacement of the atom in that ce11.. The tern i = j
is excluded from all double sums and equation 6.1.1, unlike its counterpart,
equation I.2.I, is an identity. The potential field 0rlfil ir determined
only by the interaction potential S and the configuration of neighbouring
lattice sites of atom i displaced fron 81 by si In the harrnonic
approximation 0r is given by
q,t u2 (6.1.s)0Iz
crr =ål(0,J
I
* zþìjwith (6. I .6)
(Lloyd, 1964) where ôrj and Oaj are the radial and tangential force
constants respectively of the jth neighbour of any atom, that is,
88.
(6.1.7)
(6.1.8)
(6.1.e)
(6.1.10)
0ïJ
0tj
a
d0 (R)
RdR
R
R=R.)
t_
R.J
and
The classical partition function takes the form
(2nmk"T/hz,3N/2 Ozc
where the configuration integral a given by
Iñ exp(-30(9r... s1... r*) )
N
]T du.-1a 1
is an integral over all N position vectors u1 over the entire crystal
volume and ß-1 = k"T For a nonatomic crystal this volume can be
divided into N ltrigner¡Seitz cells and, as there are N! ways of
arranging N distinguishable atoms in N ce1ls with one atom per cell,
Q becomes
rNa = | exp(-ß0(9,... s1... \) )iIr d\ (6.1.11)
with the integration over the co-ordinate s1 of each atom restricted to
one particular Wigner-Seitz cel1.
The probability of finding the aton in cell í , for i=I,2...N,
in the volume element at 1S
Pfu ... u.. .. r ,)'-t -l ---N
N
]I du
u.-1
du.-1
1^
where the density function P is related to 0 by
89.
o(9r .. s1... \) = Q-l exp(-go(gr... s1...\) ) (6.1 . i2)
The one particle probability distribution functii;n t may be derived fron
P thus :
r(u )-l
j oto, ,uN
IT du.. ^-1r=¿
(6.1. 13)
(6.1.14)
(6.1.1s)
-2
The experimentally determined one particle potential Vo"e is related
to t by equation 1.4.1 repeated here for convenience :
t (u) exp (-ßVo"" (g) ) / exn l-ßV lu) lduoPP '-' - -
Although t rnay be expressed as a function of one variable the atomic
displacements u., are not independent, that is',_1
Pfu ...u....u )'-l -]- --+'l'
In an Einstein nodel, however, it is assumed that P is separable and
of the forrn
Pfu ...u....u )'-l -1 --+t -
N
+ ]I tfu.)\-1 'a=l
Ii=1
where p(u.) is the probability of finding the aton in the volume elernent
du. at u: irrespective of the location of any neighbouring atoms.-1 -L
Kirkwood (1950) obtained an integral equation for p by
ninimizing the free energy F given by
F -kTlnZ c(6.1.16)
subject to the constraint that each atom is confined to its own Wigner-
Seitz cel1. Using his result and invoking the harnonic approximation
Barker (1963) has shown that the Gaussian f.unction that best describes p
is
with
p (e) (2rk.T/a)-3/2 exp (-o=r' /2kBT)
90.
c[E
2þ (6.1.r7)
This is also the quantun rnechanical solution at 0K of the tine independent
Schrodinger equation in the self-consistent Hartree approxirnation
(Nosanow and Shaw, 1962). Thus or and o, are equel and in a self-
consistent Einstein nodel the vibration of any atom is influenced only
by its neighbours at their equilibrium positions. Th.e experimental values
of o, ,clenoted by oexpt ,and calculated values of o¿ are compared in
Table 6.1 for lithiur,",, sodiun and potassium at 293K The force constarìts
of sodium, and potassiun were taken from Sil-rkla and Taylor (1976) and for
lithiurn fron Beg and Nielsen (1976). 0n1y nearest and next nearest
neighbour interactions were included in equation 6.1.6. AIso listeci in
Table 6.1 are the values of o deduced fron the theoretical vibration
amplitudes of Dobrzynski and Masri (I972) who used an Einstein approximation
in evaluating the dynamical matrix.
TABLE 6.1
ål ,t,.)
r)j
Conparison of experinental withtheoretical values of cr forlithium, sodiun and potassium at 293K
E lenent Lithiurn Sodiun Potassiun
c[r (ev,q-2 ) T.T67 0.s73 0.364
cr(evÅ-21 0 .656 0.s46 0.503
c[expt (eV,{-2 I 0.422 0.264 0.ls1
Reference
from forceconstants
Dobrzynski Ç
Masri (I972)
experirnent
91.
The difference between oexpt and values of o¿ based on
Einstein models is considerable though not unexpected. No allowance has
been made for correlation in equation 6.1.15. Furthernore anharnonic
cornponents of 0 may be significant in view of the large vibration
amplitudes in the alkalis but have not been taken into account. The
relative inportance of these thro factors will be assessed.
It is well known that the usual Taylor series for 0 of
equation I.2.1 is inapplicable to the lattice dynanics of inert gas
crystals (Guyer, 1969). Hooton (1955a, 1955b, 1958) and rnore recently
Gi11is, Wertharner and Koehler (1968) have extended the conventional
harmonic theory to describe such solids. In this revised approach, the
self-consistent harmonic approximation, the classical force constant is
replaced by the thernal average of the second derivative of 0 or
alternatively by the second derivative of the average potential with
respect to the equilibrium position (Schnepp and Jacobi, 1972). For
alkali metals it is possible to express S as
ô (r) (2r)-t 0(g) exp(-iQ.r)dQ (6.1.18)
Thus
ô lu. +R. -u. -R.l' --r -1 -J -J'(2r)-t
(I 0 (a) exp [-iQ. (u. -u., ) ] exp [-iQ. (R; -R.: ) ] dQ) - -L-J -r-J
(6. 1 . re)
Ih the Einstein approxination, the displacements u,' and u¡
independent (see equation 6.1.15) and the thernal average of
are
0 denoted
where 'r is the ternperature factor given by
is given by
(2r)-tJ OCO .'(q) exp[-iQ. (!i-B¡)]dQ rc.r.20)
T (Q) ( exp(iq.g) >
92.
(see Section 1.4). Thus the tine averaged potential < 0(R) > of
two atons separated on average by R is
< 0(R) > (2r)-t 0(E) r' (q)exp(-i a.R)dq (6.L.2r)
If it is assumed that r(Q) is isotropic then
< 0(R) > < 0(R) >
and tjme averaged force constants 0 and 0 rnay be defined asTJ rj
0& < o(R) > (6.r.22)TJdR2
R=R.J
and
where
T (Q)
d<0 (R) >0tj RdR
R=R.)
In a self consistent theory otj and 0 may be related to r byrj
exp(-f kBT/2a)
(6.L.23)
(6.r.24)
The procedure outlined above is very closely related to the self-consistent
Einstein nodel of Nakanishi and lr{atsubara (1975) and Matsubara (L976) .
The tine averaged force constants 0rj and 6tj were evaluated
for lithiurn, sodiurn and potassium at 293K for the nearest and next
nearest neighbours using the quasi-hannonic expression for T , that is,
CÌ, å ì,õ, . zþì i
exp (-Q' k.T/2o"*pt)T=
In reciprocal space 0(Q) assumes the forn
(6. 1 .2s)
0 (Q) (4122 e" /t ) tl-c(Q)l
93
(6.r.26)
(6.r.27)
rvith
G (Q) w (Q)
-+¡rze2 /l¿o3
(1-e-'(Q))
2
where Ze is the charge of a bare-ion and ç¿ is the volume per ion
(see equation 1.2,7). It has been pointed out in Section L.2 that there
are several independent treatments of the pseudopotential w and
dielectric screening function e For the present purposes the
pseudopotential of Ashcroft (1968) and dielectric screening function of
Singwi et aL.(1970) are the rnost convenient versions as they have
straightforward analytical representations in momentum space. Thus the
model parameters of Price, Singwi and Tosi(1970), who combined the
Ashcroft pseudopotential with the dielectric screening function of Singrr'i
et aL. (1970) to describe the observed dispersion curves in lithiun, sodium
and potassium, were used to generate values of G(Q).
The classical and time averaged force constants are given in
TabIe 6.2. It can be seen that o is in fact "harder:rr than or
It is pointed out that in the case of inert gas crystals this rthardening"
is in certain cases sufficient to invert the sign of the classical force
constants from negative to positive. It is recalled that icx,ru2 is the
potential field in a $/igner-Seitz cel1 given that the neighbouring atorns
are fixed at their lattice sites. The result
cr>d>cl expt
means that the arrangement of atoms in a crystal is not rigid. Furthermore
displacements s1 and rj of any pair of atoms are not independent.
In terms of moments of the distribution functions of any pair of atons this
conclusion may be expressed as
94
TABLE 6.2 Classical and time averaged force constants
and corresponding values for o for lithiun,sodiun and potassium at 293K
Paraneter Lithiun
0 .495 7
-0.0409
0.1128
0 .001 3
0.s625
-0.0s82
0.1607
-0.00s8
I .488
1.335
o.422
Sodiun
o.2232
-0 .0093
0.0264
0.0064
0.2683
-0 .0198
0 .0570
o.0024
0.733
0.624
0.264
Potassium
0. 1346
-0.0046
0 .0153
0.0047
0. 1605
-0.0115
0.0555
0.0019
0.44s
0.384
0.151
O"(nn) (evÅ-2 ¡
0. (nn) (evff2 ¡
0"(nnn) (evÄ-2 ¡
0r(nnn) (eVÂ-'z¡
õ"(nn) (evÄ-'?¡
õ. (nn) (evtr2 ¡
õ"(nnn) (ev.{-2 ¡
õ.(nnn) (evÄ-2 ¡
o (eVff2 ¡
ç¡r (eVff2 ¡
0,expr(eVff2 ¡
(u2)ifi=j{
9s
aTe
(u.-t
11.-J 0 otherwise
In the following section the anharmonic cornponents of 0
cons idered.
If the vector gi * R.-t-
R, is denoted by R_J
6.2 Anisotropy of the Tine Averaged Einstein Potential
The observed one particle potential is anisotropic. It is of
interest to calculate the anharmonic components of the Einstein potential
0 (u) and conpare tham with the experirnentally determined values of Y1-
and ô However, the process of evaluating the sums in equation 6.1.3 is
tedious as the integral for 0 , equation 6.1.18, converges slolly. In
view of the preceding discussion of tine averaged force constants, a
quantity rnore realistic than 0l is the time averaged Einstein potential
which may be described as fol1ows.
The contribution of atom j to the time averaged potential seen
by aton i in the volume element du. at u: is of the form-1
(I O(,r, ,. R, - u. - R,) t(u.,)du.,j ' '-1 -1 -J -J' --J' -)
itoR. joining atom_J
the above expressiolt becomes
(I 0(R - u.) t(u.)du,j ''- -J' '-J' -)
which is the convolution of 0 and t
A body-centred cubic lattice nay be regarded as two interlocking sinple
cubic lattices separated by a translation of Ttr, l, 1) Each atom of
the body-centred crystal is at the centre of a simple cubic lattice which
includes, in particular,, the nearest neighbours. The contribution Vr.
of this simple cubic lattice to the total time averaged Einstein potenti"al
within the lVigner-Seitz ce11 of the body-centred lattice may be written as
96.
a Fourier series thus:
Y (x, y, z) > O (Q) t (Q) cos [2n (hx+ky +Iz)] (6 .2 .I)I
sl
hSC -d
3
co
sk
-@
where q = (2n/a) (h, k, 1) and x, y, z àTê the fractional co-ordinates
of the point in the unit cell at which Vr. is being deternined. The
product 0(q)r(q) is derived fron the convolution in real space of 0
and t Since the vibration amplitudes of the alkalis are large, the
factor T ensures rapid convergence of Vr. . There is no difficulty in
the tern 0(q) for Q = 0 as the screening of the Coulonb potential by
the ele.ctron gas ensures that the linit 1in0(Q) exists. This nay beQ'o
deduced fron the properties of e for snall a (see for example Harrison,
1966). Alternatively it can be seen that the Fourier transform of
R-lexpt(-ÀR) (see page 9 ) is an(f * À2)-r which is well defined for
aLI a The potential Vr. h¡as expressed as a function of displacernent
u fron the rest position + , , , Þ in the simple cubic latice and
calculated for lithiun, sodiun and potassium in the three principal
directions. The values of Q and T were derived by methods already
described.
The results for the three metals are listed in Table 6.3 where theco
sk
_æh
a
v,.( å' :' þ\'ó \riJ t/5
3 > 0(Q)t(Q) hasI
contribution of the constant term given by
been subtracted fron f.[x, f, z).
It is evident that the rates of Vr. to the harmonic potential
is an increasing function of u and thatÞ"*n. t'
Y*("' o'Jå,å'
Thus the sign of the effective values of the parametels Y and ô of
Vr" are opposite to those observed experirnentally.
97.
TABLE 6.3 The time aver:aged Einstein potential Vr.the three principal directions for lithiun,sodi-un and potassium at 293K. R the ratio
1n
E
of or",Ë,Ë, o, Þ"*p.t'to
G/ a)
Lithiun
V ( u, o, o) x 102 (ev) 0 .00 0 .29SC
0.00 0 .02 0.04 0.06 0.08 0.10
V u-5
%"Ë'.,ä'"ä',1,
x
expt ,Í x t02 (eV)
Sodiun
c( u, o, o) X
RE
Potassium
V (u,o,o) x
t,o) x,1,
( 102 (ev) 0. oo 0.29 1 .15 2.59 4.62 7 .26
102 (ev) 0.00 0.29 1.1s 2.60 4.66 7 .37
I.I4 2.ss 4.49 6.94
2.75 2 .76 2.77 2.78 2.79
SC
1fR
E
vs
0.00 0.10 0.42 0.94 1.66 2.60
102 (ev) 0.00 0.22 0.87 1.94 3.42 5.28
%
".,Ë'.Ë'
o' x
v_-( g, t, 9) xt" .,6 ,fs \E
Þ"*n." x to2 (ev)
102 (ev) 0.00 0 .22 o .87 L .97 s .s2 s . ss
102 (ev) 0 .00 0 .22 0.88 1 .98 3. s6 s .63
0.00 0.10 0.39 0.87 1.55 2.43
2'.24 2.25 2.26 2.27 2.28
102 (ev) 0.00 o .20 0. B1 I .82 3.22 4 .97sc
V^^( 9, 9, o) x 102 (ev)tt xE tEV_-( g, ', 9) x to2 (ev)t" tB \E .,8
Þ"*n." x 102 (ev¡
0.00 0 .2r 0.82 I .85 3.3I 5.19
0.00 0.2r 0.82 1.86 3.34 s.27
0.00 0.09 0.34 0.77 I.37 2.t4
RE
2.39 2.39 2.40 2.4r 2.42
98.
Nearest neighbour interactions nay be taken into account by a
related procedure. Each atom in a bocly-centred crystal is at the centre
of a face-centred lattice with a unit cel1 of side 2a containing the
set of its next nearest neighbours. This face-centred lattice and the
simple cubic lattice already considered are mutually exclusive. The time
averaged Einstein potential Vf.. of this face-centred subset of atoms
is given by a Fourier series of the forn
V¡.. (x, I, z) (2a)-tÌ
Ar""(hkl)Q (Q) r (Q) cos [2n (hx+kv+Lz)l
(6.2 .2)
@
k-æ
h
where
Afcc (hkl) {1 if h,k and I are of the same parity0 otlierwise
It was sho',vn tha.t Vf.. is an order of nagnitude smaller than Vr. and
although the anisotropy of Vf.. is consistent with a positive sign for
ô the previous conclusions regarding the erffectir¡e y and ô conponents
of the Ej-nstein potential are unchanged.
Intuitively the negative sign of the 6 term in Vr. is not
unexpected. The superposition of spherical pseudo-atoms in a body-cen-'re,j
cubic crystal may be expected to produce a conduction electron density
p(x,y,z) that is higher in the < 111 > directions than in the < 100 >
directions. Under these conditions for any given displacenent the potential,
arising fron the overlap of the screening charge of a displaced atom with
the conduction electron density of the rest of the crystal, would have the
same asphericity as g(x,y,z) . However Perrin et aL. (1975), using the
Korringa-Kohn-Rostoker (KKR) method applied to lithitxî metal, have shown
that p is in fact higher ín the next nearest neighbour directions than
in the nearest neighbour directions. Now p(x,y,z) may
99.
be written as a Fourier series as
g(x,Y,z) a-3
where
h k 1 bcc'_oo
Abcc (hk1) {1o
if h+k+l is an even number
otherwise
and f(hkl) is the form factor of the conduction electrons. By direct
substitution into equation 6.2.3 it can be shown that p(x,y,z) is
higher in the < 100 > directions than in the < 111 > directions if
f(1I0) is negative and f(200) is negligible in comparison with f(110)
This is in fact the case for the'scattering factors of the pseudo-atom of
Perrin et aL.(see Figure 1.3). Contrary to the conclusions of those
authors it is therefore possible to:reproduce at least qualitatively the
KKR angularity with a spherical pseudo-atom. The possibility that the
anisotropy of \tr. is determined by the overlap of charge distributions
was therefore considered.
The form factors of the screening charge density were derived
fron the Ashcroft pseudopotentia.l via Poissonrs equation and are given by
f (Q) = cos ¡Qr.a¡) (1-e-' (Q) ) 6 .2 .4)
where u^ is the Bohr radius and r"a" is an effective radius of the metal
ion (Price et aL.1970) Values of f calculated from the nodel
parameters of Price et aL.(1970) are preserrted in Table 6.4 for lithiun,
sodium and potassium. For reciprocal lattice vectors beyold (2r/a)(2,0,0)
f(Q) is negligible. There is, horvever, a significant contribution to
p(x,y,z) in equation 6,2.3. from f (200) with the result that p(x,y,z)
has the same asphericity as Vr" .A comparison of this charge density,
based on the Ashcroft pseudopotential, with the pseudo-atom of Perrin et aL.
(1975) inplies that the asphericity in the conputed density p is model
r00.
dependent. However Vr. is i.nsensitive to the ratio of f(110) to
f(200) This was verified by calculating Vr. for a nodified dielectric
screening function defined by
e (Q)e (Q)1
{of Singrvi et aL. (f970) forotherwise
lql<l çzra't) (1, r, o) |
whiclr ensures that f (200) given by equation 6.2.4. is zero. Thus the
observed anisotropy in Vor" cannot be reconciled with the time averaged
Einstein potential. This result suggests that there may be an entirely
different interpretation of the parameter ô in V--- This point will
be taken up in Section 6.4.
TABLE 6.4 Pseudo-atom forin factors for lithium, sodiun
and potassium.
Elenent f (110) f (200)
Lithiun
Sodium
-0. 036
-0.041
-0.069
-0.023
-0.029
-0.038Potassiu:r,
It may also be noted that if the interionic potential Q were
replaced by its harmonic approximation 0h given by
I0¡(r) = þ-r¡r-rnn)2
rvhere r__ is the nearest neighbour separation Ëhen to second order in uTln
0(u) = Q(u)t- t
= ltþ;"" (6.2 ¿s)
with only nearest neighbour interactions being included in equation 6.1.5.
Although 0h is exactly harmonic, 0r is anisotropic to fourth order in u
101.
and is given in Table 6.5 for potassiun. As for Ur.
0 (u, o, o) < o ( 9-, 9,0) < o ¡ s, e, s)I ItD,/î ''ßrß.ß
This is the induced anisotropy referred to in Section 1.4.
The most significant outcome of Sections 6.1 and 6.2 is the
failure of the Einstein nodels to preclict the vibration anplitudes. The
calculated potential parameters have been consistently too rrhardr'. In a
real crystal for any displacernent s1 of aton i (say) there is a
relaxation of the neighbouring atoms in such a way as to ninimize the
increase in crystal energy associated with that displacement thereby
sof+.ening the Einstein potential.
TA.BLE 6 .5 Induced anisotropy of 0 in potassiunI
(u/a)
Ö, (u, o, o ) x 102 (ev)
O ( g, g, o) x 102 (ev)I t/T'/'
\/3
u x to2 (ev)
x to2 (ev)
0.00 0 .02 0.04
0.00 0.20 0.81
0.00 0.20 0.82
0.00 0.20 0.82
0.00 0.20 0.82
0 ( uu--t -)* \/f
0 .06
1 .83
1 .84
1 .84
1.84
0 .08
3.25
3.26
3.27
3.26
0.10
5 .07
5. t0
5.11
5.10Ir,n
r02.
6.3 The Effect of Correlation
It is now clear that a nost irnportant consideration is
correlation of atomic displacements. Nelson, Thompson and lrfontgomery
(L962) and Jackson, Powell and Dolling (I975) have shown that correlation
increases with temperature. The treatment to be presented is based on a
ce11-cluster expansion of the configuration integral a given by
equation 6. 1 . 1 1, that is ,
a ))i
(6 .3. 1)N
]T du.-t-I
A useful starting point is the Einstein approximation for 0
>0
WS
0 N0 +0
l¡vs
lu.I '-1 )
1
(6.3 .2)
(6. s. srr
(6.3.4)
The corresponding configuration integral a- is given by
Q" exp(-ßNoo)exp(-u T
*,(91))d9, . d5
As the integrand is separable a- reduces to
a=N
exp(-ßN0 )G0
where G is the free volume defined by
exp (-ß0 (u., ) ) du.l-r-r
o-rE exp (-ßNQo) exp (-ß >0, (ur) )du1
du
G
llvs
The one particle probability distribution function
Einstein approxination becomes
t_ (u ) in thisE-I
t (u
exp (-ßO (u., ) ) G- t- l-r
) -+ll_E -l
which is consistent with a previous result, that is, CI, 0r
(6.3. s)
f
103
1l
Correlation may be taken into account by introducing factors
defined by
exp(-ßôr(u'ui)) - 1 + fr, (6.3.6)
This approach is reminiscent of that adopted in the dynamics of quantum
crystals using Jastrow functions (Jastrow, 1955; Nosanow, 1966; Sears
and Khanna, 1972). The configuration integral becomes
aws
The product of the factors ( 1 + f ij ) may be written as
II(1 +fi<j I)
and a assumes the form
a Q"[1*
where
< f.....f.. >LJ KI
ft, ) * ...] (6.s.9)
) ttj ftr +
1J
(6.3.7)
(6 .3. B)
du--1
(6. 5. 10)
i<j
i<j ij i<j k<I
t)
:, I I r'(exp(-ß0, (sr))]ri.j...rrr dui
and the product of y factors, [' , has one factor for each atom whose
nunber appears as a subscript in the product fii...ftf
In particular a fij t is given by
( f,, ) = I I t"*pC-gO (u..)-80 (u,))lf..du,du, (6.3.11)ij
G2 ) r- ¡ t I '-1' ' t'-J"' lJ -1 -J
Taylor (1956) has applied equation 6.3.9 to evaluate a for a lattice
of N atoms each of which has z nearest neighbours. If only nearest
neighbour interactions are considered then all < fij > are identical and
nay be denoted by q Further
"tjfu>
I
rc4.
(6.3.r2)
(6.3. 16)
ws
<f..> <fk1
provided that (i, j) and (k, 1) have no index in cornmon. Taylor made
the approximation
(6. 3. 13)
indices. l\¡ith this approximation
where
a(u)-l -rexp (-ßN0o) exP (-ß (gi f.-
LJ )du-2 du-N
<f..... f >r_J mn L)
irrespective of the number of repeated
the sum in equation 6.3.9 becomes
a = Q,[1*(þ"1q,*(þ")è2N,-r)qi,* 1
a=( r * or)*"/' (6.s.14)
(Taylor, 1956). The one particle probability distribution function is
given by
t(u) = Q_(u)Q-' (6.g.t5)-t I -l
= Qrr
))II (1 *i<j
>0ir
In particular
a (0) exp (-ßNQo) exp (- ß0r (q) )
x I exp(-ß ,ïr0,
(ur) ) [1 *i<j 1J i<j k<t L"! Kr -2
ws
(q) becomesIt9.u
-lf
I jNow as =0fora1fj and a
a (o) - exp(-ßr'rq )exp(-ß0 (0)) c*-1t- 0 l-
x { r . c }CN-z)z)er . lf}C*-rl z1 ç l6-z)r-r)e",
- exp(-ßi,lp )exp(-ß4 (o)) cN-r [ 1* q 1à(N-2)z0 I - 'I
Thus
+Ì
105.
(6.3.17)
(6.3.18)
't(N-2) z(t * qr)
(1+e)-"
\ltzt (0) / (r * qr)
and
If it is assumed that t(u) is Gaussian then
7
t(o)/r_ (o)
t (u) (2rk.T / a) t "*p ç-aú ¡zurtl
(see equation 6 .I.I7) and
-3tt(q) = (znk T/a) t2
Hence
t/"t (o) /tE (q) = ¡crlo,=)
= (1 * qr)-' (6.3. 19)
This argument may be extended to include next nearest neighbours and more
distant interactions. If next nearest neighbour forces are considered
t(0) t=(q) = (1 * er)-"(L * er')-"' (6.3.20)
where zt is the number of next nearest neighbours and Qr' is the
corresponding correlation factor. Thus
o o.-' - (1 + qr r-22/3 (1 + qr ,;22''/3 G.3.zt)
The only nodel calculation of which the author is aware to test
this result is the calculation of the Debye-l{aller B-factor for a
face-centred cubic crystal with nearest neighbour interactions only by
Flinn and Maradudin (1962). In the harnonic approxination it can be
shown that
106
(6.3.22)q [ (r- (ôt/cr '!)' )' (1-Or/0' )' )]-2- II
(Westera and Cowley, 1975). In this particular example z = L2,0t = 0
and 0r - 40r
Thus q = Q.03280I
and ¡x = (1 .05280)-8 cl"
3.0898 0r
The result of Flinn and Maradudin is o, = 2.3861 0t Thus the calculated
value of 0 is too hard.
The author has applied equation 6,3.2I which includes next
nearest neighbour interactions to some well known structures which are
listed in Table 6.6 together with values of force constants, 0t and
ct__-..- The force constants of potassium chloride were calculated fromexpt
the room tenperature elastic constants tabulated by Kittel (1971) for a
central force nodel described by Feyrunan, Leighton and Sands (1964). Here
the force constants for the interaction between like ions at the next
nearest neighbour separation are equal. This is not the case in reality
(Copley, Macpherson and Timrsk, 1969) and o'(K+) <
However the difference is snal1 and was ignored. The force constants at
room temperature of calciun fluoride and strontium fluoride were taken
fro¡n Elconbe and Pryor (1970) and Elconbe (L972) respectively and are
based on a rigid-ion nodel. For aluminium the force constants were taken
from Gilat and Nicklow (1966) and are derived fron the experinental
dispersion curves at 293K
707 .
TABLE 6.6 Force constants for nearest neighbour (nn) and
next nearest neighbour separation (nnn) and
values of cr deduced from the cell-cluster nodel
for potassiun chloride, calciu¡n fluoride and
stlontium fluoride at 293K.
Halide Potassium Chloride Calciurn Fluoride Strontium Fluoride
K*
IIz
Ion
0"(nn) (evÅ-'z;
0. (nn) (ev'&-'z ¡
zt
0r(nnn) (evÂ:'?1
0.(nnn) (eVÅ-'z¡
0r (eV.{-2I
cr, (eV.A-'?)
o"*pt (eV,{- 2 ¡
I
6
0.538 0.538
0.000 0 .0
L2 L2
0.L27 0.I27
0.0 0.0
1 .583 1 .583
t.2c7 r.207
0.916 0.921
c
6
c*+ sr'*
4
F-
2.686 2.686 3.200 3.200
0.r77 0.117 -0.148 -0.148
L2 6 T2
0.519 r .049 0 .592 0.355
-0 .2r9 -0.078 -0.148 0.021
8. 108 5 .679 8.924 4 .66s
5 .813 3.774 6.054 r.97r
3.62 2.49 3.50 2.32
F
4
6
108
TABLE 6.7 Force constants for nearest neighbour, (nn)
and next nearest neighbour separations (nnn) and
values of o deduced from the cell-clusternodel for aluninium, lithiun, sodiun and
potassium at 293K' .
z
Elenent Aluminium
L2
)..325
-0. 101
0.156
-0.032
4.677
3.332
2.36
Lithium
o .407
-0 .006
6
0 .0s4
0.001
I.L67
0.822
o.422
888
Sodiun Potassium
0.207 0.129
-0 .006 -0. 005
0. 014 0.014
0.007 0. 004
0.573 0.364
0.394 0.252
0.264 0.1s1
0r(nn) (evÄ-'?1
0.(nn) (evÂ-'?¡
zl
0r(nnn) (eVÂ-2 1
0. (nnn) (eV,4:'? I
0r (eV,{-'¡
CI (eVÄ-'?¡
oexpt (eV'4:'?1
666
6.1. From a comparison of
general
.,t>c,>ocxpt
109 .
For the alkalis the force constants are those used to evaluate o¿r in Table
cr, with cr it can be seen that inexpt
Westera and Cowley (1975) have evaluated the configuration
integral without invoking Taylorrs approxination for a face-centred
cubic crystal vrith nearest neighbour interactions on1y. a tvas written
in the forn
-Nz/2a = Q"!t * ", * t, + "')
where ï is linear in correlation factors, r is quadratic etc.y2
With the result they were able to derive an analytical approximation, which
is within 0.5% of the lattice dynanical value of Hoover (1968), for the
geometric mean frequency of the crystal. In tcrrns of phonon frequencies
o = MU-l and equation 6.5.18, based on Taylor's approximation, in effect-2
underestinates l_, . llowever the author has been unable to extend the
form for t(0)tE-1 (0) along the lines of Westera and Cowley (1975). The
difficulty appears to be to find a polynomial forn compatible with a and
0 , that is, such that the quotient a- /a is independent of N It-t I
nay be pointed out that the expression for a given by Westera and Cowley
is not a unique representation of a
' Assuming that such a cell-cluster approach to lattice dynamics
is viable it would be possible ultimately to obtain an analytical expression
for V^-- (u) using the relationoP P --'
t (g) = QrQ- t
(- exp(-ßVo," (g) )/
,J "*n(-ßVop" (u))du
110.
where the parameters of Vo"" are determined by the properties of t
Anharmonic interactions would then be taken into account as corrections
to correlation factors t, and t, . The correlation factor Q, , for
exanple, mây be expressed as
q = e, : 0.1108(k,T) fG) ¡IO(')]' + 0.00688s(kBr) ¡p(r) l' /lþ(' )1''l 'r ,h
where er,h is the harntonic part of Q, given by equation 6.3.22; and
*(') , O(t) und *(o) are derivatives of the interatomic potential o
(Westela and Cowley, 1975). In this wa-y phonon-phonçn interactions, which
are difficult to assess in conventional theories, would be taken into
account without explicit reference to then.
6.4 Relation of the One Particle Potential to the Specific Heat
It has been shown in Chapter 1 that the tnean square anplitude of
vibration and specific heat C., are related to the moments of the density
ft¡rction g(0r) (see Section 1.5). Martin (1965a) has analysed the
temperature dependence of C., of the alkali metals in terms of the mornents
þr, o-e g by a procedure described by Barron, Berg and lulorrison (1957). In
particular, the values of U_, were determined. In another study of Cr, '
Martin (1965b) noted two fea-tures of C*, at high temperatures. One is a
positive anharnonic contribution occurring in the temperature range Oo/3
to Or, , the other is an additional positive contribution starting at
temperatures about 50K below the melting point. The first phenomenon is
conmon to all alkali metals, the second in all but lithium where the
experinental data is inconclusive. It is possible to relate the results of
the analyses of C*, by lvlartin to the parameters of the one particle
potential, namely o, y and 6 These will be considered in turn beginning
with 0
111.
TABLE 6. B Values of ct for lithiun, sodium and potassium
derived fron the mean inverse square phonon
frequency V_z deduced fron the temperature
dependence of the specific heat Cv
M
Parameter Lithiun
(a.n.u) 6.94r
at 0K [A] x 1026 (sec2 ) 0.129
at oK tBl x 102t ¡sec') 0.129
0.69
0.74
0.s27
0.s27
0.506
0.506
0.422
Sodi.unt
22.9898
0. 750
0.745
0.318
0.320
1.01
I .55
0.289
0.297
0.287
0. 289
0.264
Potassium
39.r02
2.20
2.r9
0. 184
0.18s
I .36
2.28
0.151
0.151
0.151
0.151
0. 151
u
u
-2
-2
MU- I at OK tAl (eVÄ. '? I-2
0.559
MU-l at 0K-2 tBl (eV,{-2 ) 0.559
\u at L47K
& at r47K x to4 (deg.-l )
M1r-1at 293K tAl (eVÄ-'1-2
Ml-_trat 293K tBl (eV,{-'?¡
cx IAI (eVÄ'-'?¡293
cr tBl (eVÄ-2 I293
oexpt (evÅ-2 ¡
TI2.
The values of U_" were cleduced with the aid of a result of Hlang
(1954) v;hich nay be expressed as
æ -nC TdT SNkBf(n+1) 6(n)Ur_r, (1<n<4) (6 .4. 1)
.r,(h)0
where Crr(h) is the harmonic part of C*, (see equatj-on 1.5.5), T is
the absolute tenperature, f(n+1) is the Gamma function and 6(n) is
the Riemann Zeta function. Two rnethods, which will be referred to by rrA'r
and rrBrt, were adopted by Martin in the derivation of Ur-, . In nethod A
it was assuned that C., is purely harmonic for T < Oo/2 In method B
it rr¡as assumed that C*, has an anharmonic component li.near in temperature
for T > 0Þ/6 For the frequency moment of interest, namely V_r, it
will be seen that the difference in results is sma11. The value of U_,
hras converted to a harmonic pararreter cx via a conbination of equations
5.4.1,5.4.2 and 5.4.3.
However the values of Þ_, published by Nfartin (1965a) refer to
the crystal volume at 0K. To obtain the roorn teÍrperature value of l_,
the volume dependence of Ye and Xv in the temperature range 0K to 293K
was taken into account by adopting the values at I47K in the equation
Mu:'^ (OK) = Mr:'^ (2s3K) lr-2\n(147K)x,, (147K) .2931 (6.4.2)-2 -2
To rnaintain consistency witir the analysis of Cr, , Martinrs values of
y_ and X were used in preference to those of Schouten and Swenson (1974).G ,'V
to extrapolate U:|r(0K) to 293K. The values of YG(147K) and Xv(147K)
were interpolated fron the values of Yc and Xv at 90K and 293K
The results of these calcul.ations are given in Table 6.7 together with
values of relevant parameters.
The agreenent of e;perimental with calculated values of o is
excellent for potassium. In fact Corvley et aL. (1966) have shown that the
113.
value of U derived from the density function g deduced from-2
frequencies measured by inelastic neut,ron scattering at 9K coincides
with Martin's value at OK.
For sodium the cliscrepancy between theory and experiment is - B%
As Martinrs value of f_, extrapolated to 90K is rt¡ithin 0.59o of the
value derived by lVoods, Brockhouse; l4arch, Stewart and Borvers (1962) fron
the dispersion curves obtained by inelastic neutron scattering at 90K ,
it appears that the discrepancy may be attributed at least in part to
errors in extrapolating the value of U_, at 0K to 293K . Glyde (1974)
has calculated tire (X-ray) Debye temperature of sodiun using a density
function g(o) at 293K derived from the phonon frequencies of Glyde
and Taylor (1972). At 293K Gtyders resuLt is O(X-ray) = 13BK which
means that o = 0.25 eVÃ-2 and is irr satisfactory agreenent with the
experimental value (see Table 5.9) .
However there is a difference - L6,o between observed and calculated
values of o for lithir¡n. Martin (1965a) has pointed out that there is
some doubt in the values of U' of lithium as no specific heat data below
90K were available for the body-centred cubic phase. In view of the
results for sodium and potassiun it appears that this is indeed the case.
To the best of this authorrs knowledge there have been no improved
calculations of U for lithir-rm.-2
Th.e discussion of the one particle potential now turns to a
consideration of the isotropic anharmonic parameter y The first of the
two positive anharmonic contributions to C., nentioned earlier (see page 110)
is given by the empirical relation
cv cv( h) 3NkBA (T-T' ) (6.4.3)
114.
where T' * l20K for the three alkali metals of interest (ltfartin, 1965a).
A is a positive number and may be related to y as follols.
The specific heat C _ is defined as
(6.4.4)
where the intemal energy of a nonatomic crystal of N atoms is
C c#r
E = þo,t+(Q)(see for example HoLt et aL,, 1970) and 0 is given by
(6.4.s)
(6.4:6)
(6 .4 .8)
(6.4 . s)
(6 .4 . 10)
0 N0 + > 0 (u,) + ) 0 (u,,u,). r -r 2-L-J1'r<J0
(see equation 6.1.1). As the ceI1 cluster theory. in Section 6.2 is
incomplete it is necessary, in order to proceed, to nake the approxination
0 = Nó + )v rgj) (6.4.7)'o ' oPPr
which is an improvement of
> 0 (u=).ì-a1'
as the observed distribution function t (u) is reproduced via(
t(u. ) = Q-r | "*p(-ß0(u., u^.. .u ))dg^. ..dg
-I- - J*= - ---I -2 -N -2 -N
þ -N0 +0
(see equation 6.5.5) , that is,
t(u )-t
<0>
exp (- ßVor" (u, ) ) / exp (- ßVo" "
(u, ) ) dg.,
Thus
where
N0- + N<VOPP0
<V )= Vor" (r)exp(-ßVo"" (g))ag / exp(-ßVo", (g))dg (6.4.11)OPP I
Field (1974) has evaluated this average and the result for 1SC
cV
3NkB (1-10¡y/o'z)t"r) (6.4.12)
115.
Çornparison of equation 6 ,4.3. with equation 6.4,I2 yields
y = -A(T-Tr)/(lokBTcr-2) (6.4.13)
The temperatures T I , coefficients A and corresponding values
of y are given in Table 6.9. It nust be pointed out, however, that
Martinrs values of A refer to the crystal volume at 0l( Shukla and
Ta¡'1o" (1974) have calculated the anharmonic contribution to C and
expressed it in the forn
C., - Crr(h) = SNA(V)kBT (6.4 .I4)
h/here A(V) is a function of crystal volume V Thus the corresponding
value 'of Y is given by
Y = -A(V)/(10k80-2) (6.4.15)
and is presented in Table 6.9 for sodium and potassium for values of A(V)
at OK and 293K. It can be shown that a value for Y of -8 x lO-aeVÄ-a
for sodiun is virtually unobservable in an X-ray diffraction experiment
and the value of -1.4 x 1g-r "y[-+ in potassium is of the order of
magnitude observed experimentally. Thus the negative sign of Y
corresponding to a positive anharmonic contribution to C-. is consistent
with experiment.
The existence of a negative y component neans that the one
particle potential has a turning point. If for the moment the anisotropic
ô-tern is overlooked then
vo"" (r) = þrt + yu4 (6.4.16)
which has turning point tap atL
".p = [ - çul aÐ)'' (6 .4 .I7)
and the height of the rvell at "ap is given by
v("tp) = -a2 /16\ (6.4.18)
116 .
TABLE 6.9 Values of y deduced from an anharmonic
contribution to the specific heat.
Pararneter
TI (K)
Lithiurn Sodium Potassiun Derivation
11.5 r25 Martin (1965a)
0.6 1 .69 L.2I Martin (1965a)
-6 .5 -8.4 -1.9 equatio:r 6 .4 .12
r40
Ax10a atOK
Corresponding value ofy x 103 at 293K (evÂ-a ¡
AxlOa atOK
Corresponding value ofy x 103 at 293K (ev.A-a ¡
0.42 0 .88
-s.4 -2.3
Shukla Ç Taylor,(rs7 4)
equation 6.4.15
A x 104 at 2g3K
Corresponcling value ofy x 103 at 293K (eVÅ-a ¡
0.10 0.53
0.81 -r.4
Shukla Ç Taylor,(rs74)
equation 6.4.15
If this well height is identified as the activation encrgy E"
fornation of a vacancy at a lattice site then the nagnitude of
estimated fron the relation
Y -ú lrcea
for the
y nay be
(6 .4 . 1e)
Field (I974) has calculated Y
including sodiuin and potassium.
by'this method for several elements
His results are
Y
v
-8.4 x 10-3 eVA-a for sodiun
and -3.4 x 10-3 eV,{-a for potassiun.
Lt7.
In view of the simplicity of the ¡node1, the agreement in the case of
potassiun is good and suggests an explanation of the sign of the ô-tern.
The formation of a vacancy is accompanied by a relaxation of
neighbouring atoms about the hole in the lattice. Detailed calculations
have been carried out by Tortens and Gerl (1969), Rao (1975) and Das, Rao
and Vashista (1975). The relaxation is of nearly the same nagnitude for
lithium, sodium and potassium: 5% to 7% of the unrelaxed separation of
atorn and vacancy inward for neatest neighbours and about Seo to 49o outward
for the next nearest neighbours. If A(n) is the displacement of the
atom at the origin due to a defect at n then the total static displacement
d of an atom at the origin is
g = Ð c(n) A(n) $-4.20)n
where c(n) = { å åfr:iíl"t' " derect at n
and it is assumed that the defect displacement fields A(n) superimpose.
Krivoglaz (196I) has shown that these static displacenents are equivalent
to a temperature factor exp(-L(Q)) given by
exp(-L(q)) - < exp(i Q. d) > (6.4.2L)
L (g) = c' ) (l-cos (q . A (n) ))n
and cr is the concentration of defects which are assumed to be randomly
distributed.
In alkali metals the existence of vacancies has been confirmed by
the observation of an additional positive increase in Cv - 50K before
the rnelting point. Martin (1965b) has attributed this contribution
entirely to the thernal generation of holes in the lattice and deduced
that the concentrations of rnonovacancies in sodiun and potassium are
I x 10-3 and 1.4 x 10-3 respectively at their nelting points. This
interpretation is consistent with the work of Feder and Charbnau (1966)
118
wlìo neasured the lengtl'r and lattice parameter expansion of sodium and
obtained a vacancy concentration o1' 7.5 x 10-a at the rnelting point.
The tenperature factor exp(-L(Q)) was calculated for sodium
and potassj.um for tl're vacancy concentrations given by Martin. The
displacement fields r^/ere taken from Das et aL. (1975). It was shown that
exp (-L(330) ) > exp (-L(411) )
and exp(-L(43r)) > exp(-L(510)¡ which is
consistent with exper:iment (see Section 5.2) but for all observable
reflections exp(-L(Q)) - 0.99 and the calculated anisotropy is an order
of rnagnitude smaller than that observed experimentally. Flowever, the
vacancy migration energy in sodium and potassium is - 0.07eV (see for
example Torrens and Ger1, 1969). It is nolv suggested that the large
lattice relaxation coupled with a hi gh mobility of lattice holes could
perturb the one particle potential to the extent observed. Thus the
existence of a mobile anisotropic displacement field q superinposed on
the vibrational displacement field is postulated to account for the sign
and nagnitude of 6
The thermal generation of lattice vacancies nay be expected to
lower the norrnal node frequencies by reducing the average restoring forces
on atoms displaced fron their lattice sites (Flynn , 1972). In this way
the harmonic parameter o would decrease with tenperature faster than
(1-2y.¡.rT) It is possible that such an effect is responsible for the
anomalous behaviour of the Debye-lllal1er factor of sodium in the range
310K to 371K (see page 83). However there is no evidence of this
phenomenon, which foreshadows the breakdown of the crystal lattice, in the
high tenperature measurements on lithium and potassium. ltrenzl and Mair
(1975) have recently measured the Debye-ltraller factor of gallium up to
r 19.
the nelting point and have shown that their results nay be accounted
for by conventional quasi-harmonic theory even at the nelting point.
In this respect the behaviour of sodiun near its rnelting point is
exceptional.
Perhaps the nost interesting feature of the properties of
lithiun, sodiurn and potassiun at their nelting points is the ratio of the
amplitude of vibration in the nearest neighbour direction to the nearest
neighbour separation. According to the Lindenann criterion (Ubbel,ohde, 1965)
a solid melts when this ratio reaches some critical value. For the
metals under discussion this ratio is 10% which is good agteement with
the value of lleo of Shapiro (1970) who estimated < u'n > at the melting
point from a density function g(o) derived fron the elastic constants.
It is pointed out that this ratio is structure dependent. For face-centred
cubic crystals Shapirors result is 7%
6.5 The Monte-Carlo Method Applied to the Lattice Dynanics
of Alkali Metals.
The availability in recent years of a reliable interionic
potential in alkali metals (see Section 1.2) lr.as meant that it is possible
to investigate thermodynanic properties of these metals by the classical
computer simulation Monte-.Carlo nethod of Metropolis, Rosenbluth, Rosenbluth,
Teller and Teller (1955). This chapter concludes with a brief description
of this technique and its application to the lattice dynanics of
potassium and sodium.
The procedure consists of arranging a given number M3 (say) of
atoms on a body-centred cubic lattice embedded in a cube with periodic
boundary conditions. The mininum size of the systen is deternined by the
properties of the interionic potential 0 The maxinurn size is linited
by storage and nachine tine. If O is any physical Property of the
r20.
systern, then the ensemble average is givcn by
< 0> a (6.4.1)
Each of the M3 atoms is rnoved in succession ¡tccording to the fornula
f u*(-oo¡ o #'g
un (i) À 6n (6 .4.2)
where n = x, y or z and i labels the tattice site; Çn is a random
number between -1 and I ; and À is the naxinun allowed displacenent.
In general each atom is constrained to its own Wigner-Seitz cell to preveÌìt
premature nelting of the system. Equation 6.4.2 ensures that all volume
elements within a cube of side ù, celtred on the lattice site have equal
probability of being occupied after any nove. After each move the change
AO in energy of the systen is conputed. If A0 < 0 the move is aliowed.
If AO > 0 the ¡nove is allowed with probability exp(-ß^o), that is, a
random ¡rumber xcl 0, 1 ] is chosen and only if x < exp(-ßlO¡ is the move
allowed. After each step j the value 0U) of the observable of interest
is computed whether the atom was noved or n\lt. It was shown by I'4etropolis
et aL. (1953) that
< 0> lin j -rj+æ
o(i ) (6.4.3)
This method has been applied by Cohen and Klein (1975) to calculate ( u2 )
for potassium and by Cohen, Klein, Duesberry and Taylor (L976) to calculate
(u2 >
and will be discussed first. The results for sodium are as follows
u'n
0.100s Æ
and < ua )/1 ,f , = 1.72
j
at 293K. Conbining these ¡esults with the approximation
tzr.
< ,ro >/< .rt - ${t/o')o.tderived from equations I.4.6 and 1.4.9 , it can be shown that
0 = Q.2505 eV.{-2293
and y =-0.01 eVÅ-a
The value of o is consistent with experirnent (see Table 5.9). Cohen293
and Klein (1975) have calculated ( u2 >
at 160K and 30BK Their results rnay be expressed as
ct = 0.20 t 0. 02 eYK2
at 160K and
cl, = 0.09 t 0.10 eV.{-2
at 308K The corresponding values derived from the present work are
0.166 1 0.002 eVÃ-2 at 160K and 0.149 + 0.002 eVÂ-2 at 30BK
In view of the unc.ertainty (IOu"¡ in the results of Cohen and Klein
(1975) it is possible that the difference between experimental and
theoretical values of o for potassium nay be attributed to statistical
fluctuations in the Monte-Carlo calculations.
Nevertheless the Monte-Car1o nethod is a feasible approach to
the lattice dynamics of alkali netals. Furthermore it is suggested that
the dynanics of a real crystal could be sinulated by introducing holes into
the structure at the outset of the calculation. Vacancy nigration nay be
represented by the exchange of ltrigner-Seitz cells of an atom and a hole
at the nearest neighbour site. A calculation of < u'n > and ( (u.¡)a)
in the principal directions (see Section 1.4) will generate values of cr,
y and ô of a crystal in which the atoms interact via an anharmonic
potential of the type described in Section L.2.
, N t5
I22.
CHAPTËR 7
CONCLUSIONS AND DISCUSSION
The proba.bility distribution function for atonic displacenents
in potassiun and lithium has been described by a one particle potential
mode1. Perhaps the rnost important characteristic of this potential which
r{as expressed to fourth order in the displacement u = (u--, ur, ur) as
V (u)OPP I"t + yu4 + 6(u*a 3 +--SilJ4 4
+. uuv
+
is the harmonic paraneter o which is well-defined in spite of the large
vibration anplitudes observed in alkali netal crystals. At the nelting
point of potassium, for example, the root mean square amplitude of
vibration u*,^ is 0.8f4. At this displacement, the ratio of the
anharmonic part of Vo"" (Ð, that is, vo"" (r) I lr-r l=r*^o" - i orl*" ,
to the harrnonic part, I cru].,- , flây be shown to be - 5% for the paranetersT RMs
given in Table 5.8. Thus the fourth order isotropic and anisotropic
components of V--- represent perturbations of an essentially harnonic
potential well.
There is an important distinction between the one particle
potential model and Einstein nodels in which the vibration of any atom is
influenced only by the other atoms of the crystal at their lattice sites.
In a real crystal there is a relaxation of neighbouring atoms in such a
way as to mininize the change in crystal energy ^0
associated with the
displacement of any atom fron its equilibriun position. Although the
distribution function t given by
It(Ð exp(-ßvop,(g))/ exp(-ßvop, (g))dg
r23.
i-s a function of a single variabl€ ü , the individual" displacenents e1
of the N atorns of a crystal are not indepencient, that is,
t(ur) P(u ,-t -2
uI du.-1
N
ulII' L=¿
but the density function P is not separable and
P(u ... u:... u-.) + t t(u,)--l -1 i=1
--1 -
Thus the assunption that
{ã2ruì1rotherwise
i=j(u.. u. )-r_ -J
is trnrealistic and leads to calculated values of oü which consistently
over:estimate the actual observed values.
A cel1-cluster expansion of tlie configurational part Õ -of the
energy of a crystal was used to derive an anaiytical expression relating
G directly to the interatonic potential 0 For a rnodel in which only
nearest neighbour interactions vrere considered, the result is
or(1 + ,1'22/3c[
where 0E
J
and 0-^, and 0-, are the radial and tangential force constantsJ 'TJ
respectively of the z nearest neighbours and q is the correlation
factor. However, to account conpletely for correlation it is necessary to
extend the treatment of Section $.3 beyond first order in correlation
coefficients. The observed values for o of the alkali metals were
reconciled to the force constants via the density function g(ut)
I¡r the limit of high temperatures o represents the mean invetse
square frequency l_, of the quasi-harmonic phonons. If M is the atomic
I3 l(0" * 2þìj
r24.
mass then
O - T'IU-1-2
where
u = (1/3N)-2
o-'?g(o) | dtrT
and the density function g(o) I is appropriate to the temperature T of
ínterest. Satisfactory u*t""*"la of values of U-, , derived from the
specific heat, with values derived fron o¿ has been noted for sodium and
potassiun. The lithiun situation is unresolved as a reliable value of
u is unavailable.-2
It was pointed out in Section L.2 that of the inert gases only
in helium are the vibration anplitudes larger than in the alkali metals
under discussion. The interatomic potential in helium is given by the
Lennard-Jones potential which is of the form
t2 6
0 (r)
where o represents the spatial scale of the interactj-on a.nd e its
strength. This potential has a minimun at r = r = /uo In the case of0
He3 , which is body-centred cubic, it.can be shown that ro= 2.86,& (Guyer,
1969). However the kinetic energ'y, associated with the localization of a
He3 atom within a l\rigner-Seitz cel1 cf dinensions corresponding to a
nearest neighbour separation in the solid state of 2.86I^, is sufficient
to make thc total crystal energy positive and renders the stÎucture unstable.
The actual nearest neighbour separation in a body-centred cubic He3
crystal is 3.77Ë^. The corresponding classical radial force corìstant 0r
given by
d R2
4e[ (þ (qJ l
0rR=3.774
125.
is negative and the associated phonon frequencies imaginary. The actual
phonon frequencies in helium are determined by self-consistent rather
than cla-ssj-cal force constants (see Section 6.1). 0n the other hand, in
alkali rnetals the nearest neighbour of any atom is very close to, if not
ãt, the mininum of the interatomic potential well (see Figure 1.4). The
quasi-harmonic force constants, rvhich are the derivatives of þ for the
interatomic separations app::opriate to the tenperature, accurately
describe the phonon frequencies which in turn represent o
An unexpected feature of the one particle potential is its
anisotropy. The time averaged Einstein potential, unlike Vo"" , is
higher in the nearest neighbout directions than in the next nearest
neighbour directions. It is.now proposed that the observed ô term in
Vo"" is a measure of the difference between a real crystal and a perfect
crystal in which there is a 1 : I correspondence between atoms and
lattice sites. There is a defect displacement field d given by
g. = Ec(n)A(n)n
The high nobility of vacancies in alkali rnetals means that the field d
is not static. The resultant distortion of the distribution function t(u)
is equivalent to a positive sign for the value of ô Cooper and Rouse
(1976) have shown via neutron diffraction that the ions in the face-centred
cubic alkali halide KCl vibrate preferentially towards their nearest
neighbours. In that crystal the nearest neighbour of any ion is of
opposite charge and the result of Cooper and Rouse has been attributed to
attractive Coulonb forces between anion and cation. The vacancy formation
energy for K+ and Cl- ions is - 2eV and the nigration energy - 1eV
(Flynn, 1972). The corresponding values for fornation and migration energies
of alkali netals are - leV and - 0.07eV respectively (see Section 6.4).
I26
In effect any vacancies in KCl at room temperature are t'frozenrr in
the structure and it is therefore unlikely that their effects contribute
significantly to ô for that crystal.
The interpretation of ô in terms of crystal defects does
however suggest the following eraeriment. If a crystal of KCf is heated
in potassium vapour the crystal becomes coloured. Alkali atoms in the
gas are absorbed by the crystal and fit into the normal K+ ion sites. A
corresponding nunber of negative ion vacancies are formed and the exc.ess
electrons bound at such sites are responsible for the colour (see for
example Flynn, Ig72). It is suggested that the relaxation of K+ ions
about the vacancies, or V centres, would increase the magnitude of the
6 conponent of the one particle potential of K* ions whereas the. ô
component of Cl- ions would renain unchanged. On the other hand if the
KCI crystal were heated in chlorine gas with Cl atoms taking up Cl
sites in the cïystal the opposite effect should be observed.
Extrapolation of the rneasured ô to zero concentration of artificially
introduced defects might well settle any uncertainty in the interpretation
ofô
Intuitively it is evident that the optimum conditions for the
observation of anharmonic components of Vo", are high temperatures where
vibration arnplitudes are 1arge. However these large vibration anplitucles,
particularly in the vicinity of the nelting point, mãY be more interesting
in themselves than the anharmonic potential which determines them. Guyer
and Zane (1969) define a quantum solid as one in which u*r" is a large
fraction of the nearest neighbour separation trr' Their criterion is
ulrRrvrs' nn 20%
and under these conditions it is possible for neatest neíghbour pairs of
atoms to exchange lattice sites. This exchange phenomenon has been realised
r27.
in crystalline helium (Guyer, 1969). At their melting points alkali netal
crystals qualify as quantum solids as
uFt
rnn
LTeo
(see Section 5.4) .
In the cel1 cluster theory of lattice dynanics described in
Chapter 6, each atom was confined to its own ltligner-Seitz cell. Originally
this cell model had been enployed in the thecry of liquids. The nost
serious criticism of the approach, as applied to the liquid state, was
that it underestinated the entropy of a liquid. In his generalized cell-
cluster nodel of a liquid, Þ Boer (1954) has described a liquid in terms
of clusters of atoms in which each aton of a cluster has access to the
entire volume of that cluster. The entropy associated with this
accessibility of an aton to the entire volune of the cluster is the
commtrnal entropy. It is now proposed that the exchange of lattice sites
at large vibration anplitudes may be the mechanisrn underlying the transition
from solid to liquid. The probability distribution furrction t(u) j.s no
longer confined to one Wigner-Seitz cel1 and assumes a lo¡ig range nature
as the solid ne1ts.
lhe dynanics of the exchange process in heliun has bcen studied
by Guyer and Zane (1969). Sr-rppose that an atom and its nearest neighbour
are labelled by rr1'r and tt2tt and their lattice sites denoted b.¡ R, and
B, The probability, prr, of a tunnelling process in which aton I tunnels
through the potential barrier of the lattice nedium fron the site *, to
R-2
at the same time as atom 2 twrnels to R-t
rE ( ,r *t,nnnM-S
P rr. = exp(- lÉIìn * ê ) /< ,Í >)
is given by
I28.
rt¡here the parameter o (see page 124) accounts for the correlation in
the motion of the tlo atoms and ( u' >
vibration corresponding to the single particle ground state Gaussian
wavefuncti.on of atoms 1 and 2 For simplicity the parameter o will
be ignored and the result written as
exp (- f;{trrr.,/r*n )' )
For alkali metals at their nelting points, t(g) is approximately Gaussian
and assurning that the above resttlt is applicable p comes out to be
p 10-11t2
The tunnelling frequency ,, - to P, where oo is the Debye frequency.
For alkali metals oo 10r 3 sec- t hence ,, 102 sec- 1 Although'
0, <<
this respect the solid has acquired some of the character of a liquid.
It is pointed out that there are other processes such as vaca.ncy
nigration whereby atoms may exchange lattice sites (see for example
Peterson, 1968) It is the success of the Lindemann criterion which
suggests that the direct exchange of lattice sites at large vibration
arnplitudes is responsible for nelting. The structural dependence of the
ratio u_-__ : r__ at the nelting point is perhaps related to the totalRMS nn
number of exchange processes per second given by (þrlooprz where t^,is the number of nearest neighbour pairs in the crystal. In body-centred
cubic crystals z=B conpared with z=L2 in face-centred cubic crystals
where the ratio t*nr=/tr' at the melting point is - 7% conpared with
Ijeo for body-centred cubic crystals (see Section 6.4).
Melting is of course a complex phenomenon and the Lindenann
criterion is the empirical result of macroscopic statistical averaging.
p2
r29.
Arguments presented here might well lead to a re-statenent of at least
sorne of the factors contributìng to the Lindemann criterion.
APPENDIX 1
RECORD OF EXPERIMENTAL DATA
The experinental data Ec(oi) are presented in Tables 4.1.2
to 4.1.9 in which calculated intensities are denoted by Ec(cilUJ
where j refers to the model used in deriving Ec(ci) (see Table 5.1)
A sunnary of these tables is given in Table 4.1.1
TABLE 4.1.1 Summary of Tables A.1.2 to 4.1.9
Element Ref. in thesis
potassiun P. 68; 72
potassium p. 70, 74
lithiun
lithium
iithiun
lithiurn
lithiun
lithiun
p. 78
p. 78
p. 78
p. 78
p. 79
p. 80
Crystal
I
2
3
3
3
3
4
4
Data Set
I
2
3A
34.
3B
3B
4
4
Table
^.I.2
4.1.3
4.1.4
4.1 .5
A.1.6
A.t.7
A.1.8
4.1.9
Trial Value
L.34
L.34
1.80
r.34
I.s4
1.93
I .93
0.86
0.86
0.86
0. 86
0.86
0.86
1n
in
in
in
in
in
in
in
in
in
in
in
in
model 1
nodel 2
nodel 3
model I
nodel 2
model 3
nodel 4
model I
nodel 1
model 1
nodel 1
nodel 1
model 2
ofY
TABLE Ã,I.2 Observed and calculated intensitiesfor data set 1
(hkl) i
))t
222
)))D',4-
222
222
222
222
222
222
222
222
222
222
222
222
222
222
222
411
4LT
4IL
4LT
4IL
4LI
T.t-
225
230
236
24r
,'247
252
25B
263
268
273
278
283
288
296
297
298
303
508
3r3
236
24r
247
252
2s8
263
o1].
24r4
2r72
4672
196 B
1767
1659
2968
TTIB
9866
6742
I208
3450
L047
9¿[3
884
854
853
823
794
530
478
4s6
4r4
369
335
Ec (oi)
r46L32
r31472
L26766
113180
I 06978
1 0040 B
99636
86240
805 76
72658
73r44
57476
63366
57070
s3492
51718
5r662
49826
48046
32078
28953
26470
25088
223L5
20302
Ec¡ci¡ [11
1 33078
12567 I
].1888 B
I12L92
106055
10051 9
9466I
895 73
8491 1
80206
7s981
7L954
6 8119
62706
61670
60989
57679
54551
51536
30023
274s5
25I69
2s095
2TLO7
t9373
c(ci) [2]
I377II
L29576
t22t25
rr4790
1 080 79
TOLB24
9s665
90r40
Bs 091
80008
7 5456
7TL3O
67022
6L248
60I47
s9422
5s913
s2586
49434
3Ð742
27999
2s566
23365
2t265
r9438
c(ci) [3]
Is4r70
72646s
1 19384
Ir2389
I 0s 966
99960
94027
88689
83795
78853
7 4416
701 BB
66L62
60484
5 9399
s 8684
s5219
sL926
48BOO
31205
28432
25964
2372s
21s 81
19710
E E
TABLE A.I.2 Continued.
(hk1) T.1
268
273
278
283
2BB
293
297
303
308
3r3
207
213
2r9
225
230
236
24r
247
252
258
263
268
273
278
283
288
lo11
E E [1 Ec (ci) c(ci) 12 E t3llI c(oi)
r874r
165 90
r4933
13639
r2768
1Is2B
10398
9639
8794
7946
70863
622T5
57853
54065
51084
4928L
46488
42603
38462
357s9
33476
5090s
28287
25s 99
23526
20892
c (ci)
411
4TL
41r
4TT
4IT
4rl
41r
4II
4IL
411
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
400
310
274
247
344
2TT
190
t72
159
14s
131
I17I
1117
966
B93
864
L223
1293
867
635
591
553
510
734
423
389
345
L7828
163\3
r4993
r3770
I2638
1 159r
10811
9730
890s
BL44
69053
63s77
58510
54032
49934
46246
42680
39475
36s40
s3700
31198
28949
26724
2476s
22934
2I225
L7BI7
16234
I 48s9
13s91
L2423
LT347
I 0548
9446
8609
7840
72IOI
66L62
606s 9
55808
sr379
47406
43s74
4OI4T
37006
5598s
3r330
28953
266r0
24554
2264L
2OB6L
I8047
16418
15000
13689
T24BO
1136s
1 0556
93 91
8522
.7'742
7L807
66028
60647
55 883
51515
47s80
437 7r
40345
37206
34169
31494
29090
26713
24624
22674
2A857
TABLE ^.I
.2 Continued
(hk1). T.1
293
297
298
305
308
3L3
273
2L9
225
230
236
24r
247
252
258
263
268
273
278
283
288
289
297
298
303
Ec
Eo1a
Ec(oi)
194B8
1 7905
L73T4
'16584
15264
13865
48 700
4s498
40392
35115
32899
31106
2783r
2s634
23200
2T164
r92s9
186r8
158s6
1481 8
L3633
L3047
I 1608
1 1456
10663
(ci) [1]
r9629
I 8431
I 8141
167s4
1546r
L42s7
43396
39605
36288
33280
30s98
2BO28
2s740
23662
2L677
1 9951
18380
16858
1s530
L4299
131s8
L2940
LL304
LI2T9
r0220
E [3
T9L64
1 7895
17589
T6124
r4764
r3502
455 15
41444
37877
34637
3r747
28976
26s07
24267
22120
20246
18577
r6943
15s18
I42OL
12983
L27SL
LIOZ3
r0922
9866
)l2l c (ci) l(c ct
400
400
400
400
400
400
330
330
330
330
330
330
330
330
330
330
530
330
s30
330
330
330
350
330
330
322
296
3L4
3L2
2s2
229
804
752
667
580
543
s14
460
423
511
424
518
318
262
245
226
266
I92
189
176
r9207
17969
T7677
T6214
r4922
13696
4s059
409s 9
37382
34I48
31273
28s27
2609r
23886
2t780
L9947
LB32O
16729
15346
r4069
T289I
12667
1 0999
I 0901
9884
TABLE 4.I.2 Continued
(hk1) T.1
308
233
233
233
233
¿55
233
233
233
233
233
260
260
260
260
260
260
260
260
260
260
260
260
296
159
TB76
L297
783
540
527
239
L67
131
l02
57
t444
864
51s
1010
327
2L3
t44
96
51
72
53
15
960
ol.t Ec (oi)
9651
113549
78527
47423
32680
31898
L44s3
1008s
5 781
6r66
2546
874ts
s2289
51196
?,219r
I 9795
L2909
8716
551 8
3062
3409
7320
922
581 19
c(ci) [1]
9585
12224L
76874
48058
31913
31538
13958
9169
580s
6132
2682
92184
55 r. s2
32476
2OBI7
20257
L2733
8330
s1 88
3047
3309
L293
882
62359
c (ci) [2]
9036
1 25 805
78900
493s7
32682
32150
r4329
946r
6052
6362
2834
92973
55430
32685
20880
20368
12823
8399
5274
3147
3394
1366
942
60879
c(ci¡ [31
8986
L22883
78162
495I4
33r64
32623
1 4808
9845
6334
6656
2985
9T428
55 200
32860
2t20I
2C663
13096
8622
5422
3228
3489
1392
955
60L20
E E E1
330
222
32L
400
330
4TT
332
422
510
43L
52L
222
32I
400
330
4tr
420
332
422
5lc
43t
s2L
440
222
TABLE ^.I.2
Continued
(hk1) i c (oi)
32466
1806s
TL456
1 0666
6069
4048
2272
1130
1 355
4s5
323
225
176
48226
277r2
14024
8728.
8257
3073
1703
827
1059
311
2L9
Ec¡ci¡ [11
34776
T8724
1 1s06
11002
6s05
4084
2355
I22B
1408
466
309
20r
177
s4s3 1
29674
15461
9385
B90s
31 99
1796
B94
104 9
328
215
c(ci) [2]
3 3815
t8272
1119s
r0743
6382
4025
2358
1264
r433
500
338
225
200
52586
28504
r4922
9056
8609
3r34
L792
923
106 9
5s5
239
E -.-t3lc (cr. _) '- '
33663
18205
LL227
IO738
6373
4015
2326
T2LB
1395
466
310
203
17?,
st926
28306
14764
B9E6
8522
3076
L729
861
101 3
3L7
209
s2r
400
330
4TT
420
332
422
510
43L
52r
440
433
550
222
32r
400
330
4i1
332
422
510
43r
52L
440
T.1
296
296
296
296
296
296
296
296
296
296
296
296
296
508
308
308
308
308
308
308
308
508
308
308
536
298
189
176
191
67
46
19
3T
T3
16
t7
I2
797
458
620
t44
L73
51
70
27
43
22
6
o11
E E
TABLE 4.1.3 Observed and calculated intensities for data set 2
(hk1) T1
297
297
297
297
297
297
297
297
297
297
297
297
297
297
302
302
302
302
302
342
302
302
302
302
302
o1IE E [2 Etll E l l3 E [4]c (c-i )1 c (oi) c (ci) c (ci) c (ci)
222
32r
400
330
4IL
420
332
422
510
43r
52I
440
433
550
222
32I
400
330
411
420
332
422
s10
43L
52t
635
346
183
110
105
62
59
23
11
13
4
6
5
I
566
318
1s9
98
92
52
33
19
9
L2
5
28099
t5292
B1 16
485 8
4659
2762
1734
I023
507
593
192
154
90
74
25044
14079
7042
4342
40s2
2307
1459
843
407
s22
150
25698
14202
7 434
4656
4375
2587
1 656
94r
463
560
178
72r
81
68
242s0
I3257
683 1
4262
3987
2337
1491
837
402
493
rs2
277s8
L4947
7664
4729
tr442
2603
1662
946
469
568
187
130
89
75
25938
13810
6969
4287
4008
2331
14 8s
837
406
498
160
26397
146s6
7726
4844
456 1
2706
17 35
991
493
592
190
130
87
t5
24735
t3s67
7027
4387
411 I
2416
1542
869
42r
513
160
27272
1 4988
7844
4885
4602
272I
17 40
995
497
s97
195
L34
90
76
2543I
1 3805
7098
4403
4r27
24r7
1541
870
424
516
L64
TABLE 4.1.3 Continued
(hkl). Ec tor.J
113
66
237L7
13186
6462
3 961
3717
203s
1318
760
s44
46L
148
95
53
22843
12051
6043
3810
3492
1905
7r92
70s
295
410
108
Ec ¡cl1 [1
r04
69
23083
L2502
6355
39s3
3684
2r44
r364
7s7
556
443
L33
91
60
2T99L
11802
5 918
3671
3408
1 969
r249
686
3L6
398
717
E .-12c [c]_J '
T12
76
2248r
729r0
6426
3944
3672
2t22
135 0
754
359
446
L4L
99
67
23r27
I208L
595 I
363s
3369
1 935
r239
680
318
400
t24
Ec¡ci1 [51
109
73
23398
I2699
6477
4030
3760
2L92
r394
776
367
455
138
'94
62
22L49
1189 7
5975
3707
3442
I 990
r263
694
32I
403
119
E ..t4c. [c1J '
T13
76
23954
I2866
6s 14
4028
3759
110rLIÙJ
1589
77'r
370
456
t4r
97
65
22580
I20OT
5 983
3690
3428
I977
L254
692
323
404
r22
440
433
222
327
400
330
4LT
420
332
422
510
43t
527
440
433
222
32r
400
330
4LL
420
332
422
510
43r
52r
T.1
s02
302
307
307
307
307
307
307
307
307
307
307
307
307
307
311
311
311
511
311
311
311
511
311
311
311
3
6
536
298
r46
90
84
46
30
I7
B
10
5
3
6
516
272
r37
B6
79
43
27
16
8
9
3
-l\J.a l
TABLE 4.1 .3 Continued. . .
(hkr) i
440
222
32r
400
530
4rl
420
55¿
422
510
43r
52L
440
222
32I
400
330
4TL
420
332
422
510
43r
52r
440
T.1
311
315
315
315
315
315
315
315
515
315
315
315
315
320
320
320
320
320
320
s20
s20
320
320
32C
320
otL
11
481
260
r26
79
7I
39
25
t4
7
8
9
11
459
24L
118
72
67
34
23
L4
5
I
5
8
trc [o1_)
97
21296
11510
5581
3499
3r54
T72T
1108
601
279
374
113
B9
20308
10656
s228
3189
2977
r526
1000
s94
239
333
98
68
trc Icr__)
BO
20922
TII22
5499
3402
3144
1804
TL42
620
260
357
r02
70
L9923
L0493
51 14
3157
2905
1655
1045
s62
248
320
90
61
[1] Ec (ci)
B7
2TBTT
IL284
546r
3240
3083
L760
ILLT
613
28r
358
109
77
20590
10551
s034
3075
2825
1603
10 18
553
249
32L
96
68
l2l Ec¡ci¡ [31
81
20929
TTT2T
5494
3400
314I
TBO2
I 140
619
279
356
I02
70
19791
10405
5056
3L2L
2869
1632
1051
555
243
315
88
60
Ec¡c11 [4J
B4
2t242
11 168
5477
3370
31 15
T783
TI29
616
280
55/
105
72
19998
1 0401
5 018
3082
2833
16 10
1018
s49
244
315
90
63
TABLE A.1.3 Continued
(hk1) T.I
324
324
324
32.4
324
s24
324
324
324
324
324
324
328
328
328
328
328
328
328
328
328
328
328
333
333
333
432
222
12L
66
61
31
20
I3
5
7
7
4
585
202
95
59
s4
29
18
10
5
6
3
354
184
86
o11
t"c (oi)
1910 7
9801
4762
2923
269s
L382
897
522
220
294
B2
s7
I 7035
8926
4203
2629
2372
L27 6
790
426
192
2s3
73
15665
81s3
3 818
E.¡.i1 [r
1 8945
9882
474s
2923
2677
1514
9s4
507
2L9
287
7B
54
18031
9316
4407
27LT
2471
1588
873
459
193
257
68
17737
8768
4084
Ec¡ci1 [2
19404
9846
4628
2824
2582
r457
924
498
220
287
B4
60
18304
9198
4260
2598
2364
L326
842
449
194
2s8
74
17236
8s76
3911
Ec¡ci1 [51
18681
97L2
4639
) Qq,1
2612
r47 4
929
492
2ro
277
75
51
L7646
9074
4258
26L9
2380
L332
838
438
r82
24s
64
r6637
8457
5896
E f ÁL z .- lrc [cIJ '
18788
9663
4sB2
2809
2s68
I44B
914
487
2l-r
277
78
54
17666
8986
4t87
2564
233r
I 305
823
433
r83
245
67
16575
8335
38 13
l1
222
32r
400
330
4TI
420
332
422
510
43r
s2I
440
222
32I
400
350
41r
420
332
422
510
43r
s2l
222
32I
400
TABLE 4.1.3 Continued
rhkll.
530
4tt
332
422
510
43L
T.I
333
33s
333
333
333
333
c (ci)
2385
2158
764
404
T7I
23r
l2l Ec (ci)
2393
2T6I
754
589
L57
2L5
tsl Ec (ci)
2334
2r07
738
384
158
2I5
o1I E E tll Ec (ci)
2s08
2274
796
4r4
170
230
t4l
55
49
16
7
t2
9
c (oi)
2433
2756
7L2
4r3
r67
226
' The temperatures Ti given in this table have been corrected
via equation 2.5.1 (see page 40) and are 297K, 302K, 307K, 311K,
315K, 320K, 324K, 328K and 333K to the nearest degree (as given
in Table 4.1.3) but for the purpose of conputing the intensities
Ec¡ci¡ [jl were actually taken to be 297.0K, 302.2K, 306.6K, 310.9K,
315.3K, 519.6K, 324.0K, 328.3K and 332.7K respectively. It is
pointed out that the correction given by equation 2.5.1 was applied to
data set 2 on1y. The values of Ti given in all other tables in
Appendix 1 are the same as those used in calculating the intensities
Ec(ci; Ij 1
TABLE A. 1 .4 Observed and calculated intensitiesfor 248K data at data set 5A
(hk1) i
200
2tI
220
510
222
32L
400
350
411
420
332
422
510
43r
T1
248
248
248
248
248
248
248
248
248
248
248
248
248
248
2853
1647
976
s84
355
2LL
r29
88
82
49
37
22
32
26
L5t473
90104
s3610
32794
L9842
11805
7504
7339
4586
2959
IB35
108 7
TI62
or.]-
E c(oi)
26181 I
1s1151
89544
53577
326L5
19398
11812
7473
7527
4535
2948
L867
t0s2
TL75
E tllc (ci)
256283
TABLE A.1 .5 Observed and calculated intensities forthe 296K data of data set 3A
(hkl). Tt_
296
296
296
296
296
296
296
296
296
296
296
296
296
296
296
296
296
o1]-
Ec(oi)
22696L
T25I2B
69053
æ663
22L73
r2629
6989
4440
415 B
2447
1500
874
489
499
160
159
83
Ec (ci) t1l
200
2LI
220
310
222
32I
400
330
4LI
420
332
422
510
43L
s2L
440
433
55 99
3087
1704
954
s47
312
172
110
103
60
37
22
24
T2
5
T2
T2
22s605
L25368
7 0015
58871
22572
r2766
6956
4259
4089
24TL
1500
864
454
s14
170
L12
73
TABLE 4.1.6 Observed and calculated intensitiesfor the 248K data of data set 3B
(hk1) T1
248
248
248
248
248
248
248
248
248
1o1I Ec(oi)
256614
L47392
88591
3L572
TL243
7069
7TL4
276I
1 758
Ecçci1 [11
253502
148916
88111
3 1638
L7447
7r3S
70s2
2762
L7L4
200
2IT
220
222
400
330
47r
332
422
2453
1409
847
,3þz
166
68
6B
46
43
TABLE 4.I.7 Observed and calculated intensitiesfor the 296K data of data set 3B
(hkl) i T1
296
296
296
296
296
296
296
296
296
296
296
296
296
296
296
296
296
o11
Ec (oi)
220866
L2LS23
68279
37764
2L733
12087
6650
4229
4022
2333
1433
838
422
507
167
113
70
Ec (ci) t1l
200,
21I
220
310
222
s2L
400
330
4LL
420
332
422
510
43r
521
440
433
55 76
3068
L724
953
549
505
Í68
IO7
103
59
36
2L
L7
L3
L3
15
7
2 185 15
121s06
67860
37612
21924
L2385
6695
4r35
3945
2332
L464
840
433
500
L64
109
7L
TABLE 4.1.8 Observed and calculated intensitiesfor the 293K data of data set 4
rhkll .' -L T.1
293
293
293
293
293
293
293
293
293
293
293
293
293
293
293
293
293
o1l_
Ec (oi)
66579
375L6
2LOL6
1 1830
6779
s766
2206
2240
T3T4
799
442
26s
267
87
63
39
38
Ec (ci)
66087
36973
20702
11895
6770
3798
2263
22L8
r302
788
460
258
274
95
60
55
37
t1l
2IT
220
310
222
32L
400
330
411
420
332
422
510
43r
s2L
440
530
43s
1079
608
34r
L92
110
61
36
36
2t
13
7
5
5
3
3
1
I
TABLE 4.1.9 0bserved and calculated intensitiesfor data set 4.
(hk1) i T o1.I
Ec (oi)
66579
37516
2LOI6
118 30
6779
3766
2206
2240
T3L4
799
442
263
267
87
63
39
38
ss424
53905
31.301
29144
27L24
25L70
23899
E 12lc (ci)
68399
37560
207I8
L1732
66 16
3697
2I89
2L50
T26L
763
449
255
270
97
62
38
40
35297
33L54
31L25
29207
27392
25678
24059
2L1
220
310
222
32L
400
330
4rl
400
332
422
510
43r
s2L
440
530
433
220
220
220
220
220
220
220
293
293
293
293
293
293
293
293
293
293
293
293
293
293
293
293
293
303
313
323
333
343
353
363
1187
669
375
2TL
t2L
67
39
40
23
14
B
5
5
3
3
1
1
708
678
610
583
542
503
478
TABLE 4.1.9 Continued
(hk1) T.1
373
383
393
403
413
423
1o1
1Ec (oi)
22178
20788
192 1B
1B06B
17023
15884
Ec(ci1 [2i
2253r
21089
t9779
I 8448
17242
16106
220
22A
220
220
220
220
444
4L6
s84
361
340
318
The parameters used to derive Ec(ci)
are as follows:
ct, = 0.438 eV,{-2293
Y - -4.6 x 10-2 eV'{-a
with
k5
ô 2.0 x 10-2 eVffa
1.19 x 106
and
R 2.59eo
l.2l in Tab Ie A. I . 9
BI B LIOGRAPHY
I Nurnbers in brackets after each reference indicate the text page where
the reference is cited ]
Arakatzu, B. and Scherrer, P. 1930, Flelv. Phys. Acta 3, 428, lZ01
Arndt, U.hl. and Wi11is, B.T.M. 1966, "Singfe Crystal Diffractomer,ry',
Cambridge University Press. [50]
Aslrcroft, N.W. 1968, J. Phys. C : Solid State Phys. L,232, [10, 95]
Ãzaroff , L.V. Kaplow, R., Kato, N ., Weiss, R.J., Wilson, A.J.C. and
Young, R.A. 1974, 'rX-Ray Diffraction", McGraw-Hi11. [35, 38, 44]
Barker, J.A. 1963, rrThe International Encyc.lopedia of Physical
Chemistry and Chemical Physicsrr, Topic 10, 'rThe Fluid State'r,
edited by J.S. Rawlinson, Vol. 1, 'rLattice Theories of the Liquid
Staterr, Pergarnon Press. [89]
Barrett, C.S. 1956, Acta Cryst. 9, 67I. 12)
Barron, T.H.K., Berg, W.T. and Morrison, J.A. 1957, Proc. Roy. Soc.
^242,478. [110]
BaLron, T.H.K. , Leadbetter, A.J. and lt{orrison, J.A. 1964, Proc. Roy.
Scjc. 4279, 62. [22]
Barron, T.H.K., Leadbetter, 4.J., Morrison, J.A. and Salter, L.S. 1963,
'rlnelastic Scattering of Neutrons in Solids and Liquids 1",
Vienna: I.A.E.A. [22]
Bearden, J.A. and Burr, A.F. 1967, Rev. Mod. Phys. 39, I25. [3]
Beg, M.M. and Ì.iielsen, M. 1976, Phys. Rev. 814, 4266 [90]
Benesch, R. ànd Smith, V.H. 1970, J. Chen. Phys. 53, L466. lITl
Bertoni, C.M., Bortolani, V., Calandra, C. and Nizzoli, F. 1974,
J. Phys . F : l4etal Phys . 4, 19. [10]
Blackman, M. 1955, 'rEncyclopedia of Physics" , YoI. 7, Part 1,
rrCrystal Physics 1'r edited by S. Flugge, Springer-Verlag. [1, 231
Bohm. B. and Klemn, W. 1939. Z, anorg. Chem. 243, 69. l2l
Born, M. and Huang, K. 1954, rrDynarnical Theory of Crystal Latticesrr,
Clarenclon Press. [6, 10]
Brovman, E.G. and Kagan, Yu. 1967, Soviet Phys, JETP 25, 365. tB]
Buerger, M.J. 1960, 'tCrystal-structure analysis'r, John Wiley and Sons.
144, 471
Burbank, R. D. 1965, Acta Cryst . 18, 88. [57]
Busing, W.R. and Levy, H.A. 1967, Acta Cryst. 22, 457. [34)
Buyers, l1I.J.L. and Cowley, R.A. 1969, Phys. Rev. 180, 755. [10]
Chester, G.V. 1961, Adv. in Phys. 10, 357. tB]
Cochran, lV. 1963, Proc. Roy.Soc. ^276,
308. t9]
Cochran, W. 1969, Acta Cryst. A25, 95. [50, 52]
Cochran, I{. 1973, rrThe Dynanics of Atons in Crystalsrr, Edward
Arnold. [10]
Cohen, M.H. 1962, J. Phys. Radiun 23, 643, [8]
Cohen, S.S. and Klein, M.L. 1975, Phys. Rev. BlZ, 2984. U20, I2Il
Cohen, S.S., Klein, M.L., Duesberry, M.S. and Taylor, R. 1976, J. Phys.
F : Metal Phys. 6, 337. [120]
Cooper, M.J. and Rouse, K.D. 1968, ^24,
405. [48, 49]
Cooper, M.J. and Rouse, K.D. 1973, Acta Cryst. A?2, 514. [19, 25]
Cooper, M.J. and Rouse, K.D. 1976, Acta Cryst. A32, I7I. [125]
Copley, J.R.D., Macpherson, R.ll1. and Timusk, T. 1969, Phys. Rev. L82,
e6s. [106]
Coppens, P. 1968, Acta Cryst. A4, 253. [36]
Coppens, P . L972 , Israel . J. Chen. 10, 85 . [56]
Coulthard, M.A. 1967, Proc. Phys. Soc. 91, 44. [17]
Cowley, E.R. and Shukla, R.C. I974, Phys. Rer¡. 89, 126I. I36]
Cowley, R.A. 1968, Rep. Prog. Phys. 31, 123. lITl
Cowley, R.A.y Woods, A.D.B. and Do11ing, G. 1966, Phys. Rev. 150,
457. [10, rr2]
Dagens, L. and Perrot, F. 1973, Phys. Rev. B 8, I28L. t5]
Dagens, L., Rasolt, M. and Taylor, R. 1975, Phys. Rev. B 11, 2726.
[9, Figure 1.4]
Das, S.G Rao, P.V S. and Vashishta, P.
5, L35. fLr7,11Bl
J. Sci. Instrum . 27,
1975, J. Phys. Ft
lr{etal Phys.
Davies, P.T. 1950, 338. [53]
Dawson, B. 1964, Acta Cryst. 17, 990. LI2l
Dawson, B. I967a, Proc. Roy. Soc. 4298, 255, ÍI2l
Dawson, B. I967b, Proc. R.oy.Soc. (London) A298,264. [11, 14]
Dawson, B. L975, I'Advances in Structure Research by Diffraction lvfethodsrr,
edited by R. l.,lason and W. Hoppe, Pergamon Press. lL7, 64, 721
Dawson, B., Hurley, A.C. and Maslen, V.W. L967, Proc. Roy. Soc. 4298,
28e. [14]
De Boer, J. 1954, Physica 20, 655. iL27l
Dobrzynski, L. and lr{asri, P. 1972, J . Phys. Chem. Solids 33_,
1603. Ie0]
Dolling, G. and lVoods, A.D.B. 1965, r?Thermal Neutron Scattering'r,
edited by P.A. Egelstaff, Academic Press. tl]
Donovan, B. and Angress, J.l'-. I97 I, 'rLattice Vibrationsrr,
ehapnan and Ha1l . [21]
Doyle, P.A. and Turner, P.S. 1968, Acta Cryst. ^24,
590. [16, 17, 62]
Eisenberger, P., Lam, L., Platznan, P.M. and Schmidt, P. L972'
Phys. Rev. B 6, 3671. t6l
Elcombe, M.lrf . ]-972, J. Phys . C : Solid State Phys. 5, 2702. [106]
Elconbe, l'l.M. and Pryor, A.lV. 1970 , J. Phys. C : Solid State Phys.
3, 492. [106]
Feder, R. and Charbnau, H.P. 1966, Phy5. Rev. I49, 464. [117']
Feynman, R.P., Leighton, R.B. and Sands, M. 1964, rrThe Feynman Lectures
on Physicsrr, Vo1.2, Addison-Wesley Publishing Conpany. [106]
Field, D.W. I97I, rrsome Physical Properties of Crystalline Sodiun :
an X-ray Investigation'r, Thesis, University of Adelaide. II,15,38]
I;ield, D.Í1. I974, Aust. J. Phys. 27, 831. [19, IL4, 116]
Field, D.W. and Bednarz, B. L976, Acta Cryst. ^32,
315. 164, 82, 83]
Field, D.W. and Medlin, E.H. 1974, Acta Cryst. 450, 234. [1, 11]
Flinn, P.A. and Maradudin, A.A. 1962, Ann. Phys. 18, 81. [106]
Flynn, C.P. 1972, 'rPoint Defects and Diffusionrr, Clarendon Press.
[ 118, r25, L26]
Geldart, D.J.W. and Taylor, R. I97Oa, Can. J. Phys . 48, 155. [10]
Geldart, D.J.l\r. and Taylor, R. 1970b, Can. J. Phys. 48, 167, [10]
Gerlich, D. L975, Phys. Rev. B 11, 1365. [71]
Gilat, G. and Nicklow, R.M. 1966, Phys. Rev. I43, 487. [106]
Gillis, N.S., Wertharner, N.R. and Koehler, T.R. 1968, Phys. Rev.
165 , 951 . [91 ]
Glyde, H.R. 1974, Can. J. Phys. 52, 228I. [113]
Glyde, H.R. and Taylor, R. 1972, Phi'5. Rev. B 5, 1206. [113]
Goodnan, J.M. 1968, Phys. Rev. l7I, 64I. t5]
Gugan, D. 1975, Phil . Mag. 3I , 453 . [38]
Guyer, R.A. 1969, Solid State Phys. 23, 4I3. [91, I24, 127]
Guyer, R.A. and Zane, L.I. 1969, Phys. Rev. 188, 445. 1126, I27l
Hamilton, W.C. 1965, Acta Cryst. 18, 5A2. [66]
Handbook of Chemistry and Physics (I97I-2), 52nd Edition, edited by
R.C. Weast, The Chemical Rubber Co. [33]
Harrison, I\I.A. 1966, rrPseudopotentials in the Theory of Metalsr!,
Benjanin. [9,96]
Holt, 4.C., Hoover, W.G., Gray, S.G. and Shortle, D.R. 1970,
Physica 49, 61. [86, 114]
Hooton, D.J. 1955a Phil. Mag. 46, 422. [91]
Hooton, D.J. 1955b, Phil. Mag. 46, 433 [9f]
Hooton, D.J. 1958, Phil. l"Iag. 3, 49. [91]
Hoover, W.G. 1968, J. Chem. Phys. 49, 1981. [109]
Hope, H. 197L, Acta Cryst. ^27,
392. [45]
Hwang, J-L. 1954, J. Chen. Phys. 2?, 154. ÍII2l
International Tables for X-ray Crystallography, L962, Vo1.3,
Kynoch Press. l47l
International Tables for X-ray Crystallography, 1974, Vo1.4,
Kynoch Press. [46; 60]
Jackson, D.P., PoweIl, B.M. and Do11ing, G. 1975, Phys. Lett. 514,87.
Ir 02]
James, R.ltl. 1948, rrThe Optical Principles of the Diffraction of X-rays",
First Edition, G. Bell and Sons. 143, 461
Jastrow, R. 1955, Phys. Rev. 9E, I479. [103]
Jennings, L.D. 1968, Acta Cryst. ^24,
472. [45]
Johnson, C.K. L969, Acta Cryst. ^25,
187. l25l
Joshi, S.K. aäd Rajagopal, A.K. 1968, Solid State Physics 22,
lse. [10]
Kanney, L.B. 1973, J. Low Tenp. Phys. 12, 239, [11]
Kelly, F.M. and lr4acDonald, D.K.C. 1953, Can. J. Phys. 37, 147. 127)
Kennard, E.B. and lllaber, J.T. 1976, Internat. J. Quantum Chen. 10,
261. [3]
Kheiker, D.14., Gorbatyi, L.V. and Lube, E.L. 1969, Soviet Phys. Cryst.
14, 196. L34, 501
Kirkwood, J.G. 1950, J.. Chen. Phys. 18,380. [89]
Kittel, C. 1971, rrlntroduction to Solid State Physicsr', Fourth Edition,
John Wiley and Sons. [5, 9, 10, 7I, 77, 106]
Korhonen, U , Rantavuori, E. and Linkoaho, M. 1971, Ann. Acad. Sci. Fenn.
No 361, 1. [28]A 6,
Krishna Kumar, S.
126, Bsl
and Viswanitra, M.A. 197L, Phys. Lett. 37A., I43
Krivoglaz,
1.
M.A. 1961, Physics of Metals and ltletallography,
[117]
Kurki-Suonio, K. 1968, Acta Cryst. A24, 379. [L2]
L2, No. 4,
Kurki-Suonio, K.
No. 241, I
and Meisalo, V.
1,201
1967, Ann. Acad. Sci. Fenn. 46,
Kushwaha, S.S. and Rajput, J.S. 1975, Phys. Stat. So1. (b) 69,
64s. 172)
LeibfrieC, G. 1965, I'Lattice þnamicsrr, edited by R.F. l{allis,
Pergarnon Press, lITl
Leibfried, G. and Ludwig, W. 1961, Solid State Phys. 12, 275. [20]
Linkoaho, M.V. 1972, Physica Scripta 5, 27I. [28]
Lloyd, P.
Lundqvist,
Mair, S. L. ,
A30,
Mair, S.L.,
1964, Aust. J. Phys. 77, 269. [88]
B.I. and Lyden, C. L97L, Phys. Rev. 84, 3360. t6]
Barnea, 2., Cooper, M.J. and Rouse, R.D. 1974, Acta Cryst.
806. [63]
114s.
Wilkins, S.ltl. 1976, J. Phys. C : Solid State Phys. 9,
[ 1s]
Mararludin, A.A. L974, I'Dynarnical Properties of Solids'r, Vol 1,
edited by G.K. Horton and A.A. Maradudin, North-Hoiland. 17)
Maradudin, A.A. and Flinn, P.A. 1963, Phys. Rev. 129, 2529. lITl
Maradudin, Â.4., l,lontroll, E.W. and Weiss'. G.H. 1963,
Lattice Dynamics in the Harrnonic Approximationrr,
State Physics, Academic Press. f7, 8l
ItTheory of
Suppl. 3 of Solid
Marquardt, W.R. and Trivisonno, J. 1965, J. Phys. Chern. So1ids , 26,
273. [s4]
Martin, D.L. 1965a., Phys. Rev. 139, 4150. 127, 77, 110, LI2, 113, 114]
Martin, D.L. 1965b, "Lattice Dynarnics'f, edited by R.F. Wallis,
Perganon Press. 127, 110, 117]
Martin, J.W. L975, J. Phys. F : Metal Phys. 5, L257. [10]
Mathieson, A. McL. 1969, Acta Cryst. A2S, 264, llzl
Matsubara, T. 1976, J. Phys. Soc. Japan 40, 603. l92l
lr{etropoliS, N., Rosenbluth, 4.W., Rosenbluth, M.N., Te11er, A.H-
TeIler, E. 1953, J. Chen. Phys. 21, 1087. [119, I20]
and
Meyer, J., Dolling,
J. Phys. F :
lrfeyer, J., Dolling,
Metal Phys.
Nakanishi, A.
14ls.
, Vettier, C. and Paureau, J. 1976,
6,1899. 172)
G. , Kalus, J
Metal Phys.
Pankow, G. 1936, HeIv. PhYs. Acta 9,
Perrin, R.C Taylor, R. and Nlarch, N.H
[Figure 1.3, 6,
and Glyde, H.R. 1976, J. Phys. F
8sl
Phys. 2I, 536. tsl
87. 126, 851
L975, J. Phys. F : Metal
98, 99]
G., Schenn, R
6,943. [g+,
Michaelson, H.B. 1950, J. Appl
Miyake, S. 1969, Acta Cryst. A25, 257. ÍI2l
Moon, R.M. and Shull, C.G. 1964, Acta Cryst. 17,' 805. [55]
Nash, H.C. and Smith, C.S. 1959, J. Phys. Chen. Solids 9, 113. [54]
and Matsubara, T.
ls2l
1975, J. Phys. Soc. Japan 39,
Thompson, M.W. and Montgomery, H. 1962, Phil. Mag. 7,
1102)
1966, Phys. Rev. 146, I20. [103]
and Shaw, G.L. 1962, Phys. Rev. I2!i, 546. [90]
Ìielson, R. S. ,
1385,
Nosanow, L.H.
Nosanow, L.H.
Nuffield, E.W
[33]
1966, lrX-ray Diffraction Methodsrr, John l\/iley Ê Sons.
Owen, E.A. and lrJillians, G.I. 1954, Proc. Phys. Soc. ^67,
895. l2l
Phys. 5, 1490.
Peterson, N.L. 1968, Solid State Phys. 22, 409. [128]
Posnjak, J. 1928, J. Phys. Chem: 32, 354, l2l
Price, D.L., Singrvi, K.S.
2983. [93, 99]
and Tosi, M.P. 1970, Phys. Rev. B 2,
Pryor, A.W. 1966, Acta Cryst. 20, 138. [52]
Rao, P.V.S. 1975, J. Phys. F: Metal Phys.5,843. [117]
Rasolt, M. and Taylor, R. 1975, Phys. Rev. B 11, 27,7. [10]
Rouse, K.D. and Cooper, M.J. 1969, Acta Cryst, A25, 615. 147, 50, 51]
Scheringer,
Schnepp, O.
Schouten, D
12,
Sears, V.F.
C. 1973, Acta Cryst. A29, 283. [55]
and Jacobi, N. 1972, Adv. Chem. Phys. 22, 205. [91]
.R. and Swenson, C.A. 1974, Phys. Rev. B 10, 2175.
7r, 73, 84, 85, rr2l
and Khanna, F.C. 1972, Phys. Rev. Lett. 29, 549. [1I, 103]
Seitz, .F. 1940, 'rThe Modern Theory of Solidsr', First Edition,
McGraw-Hi11. t5l
Shapiro, J.N. 1970, Phys. Rev. B!, 3982. [119]
Shoenberg, D
J.I\f .
Shukla, R.C.
Shukla, R.C.
1969, "Physics of Metal-s 1 Electrons", edited by
Zinan, Cambridge University Press. t5]
and Taylor, R. L974, Phys. Rev. 89, 4116. [115, f16]
and Tay1.or, R. 1976, J. Phys. F : Metal Phys. 6, 531. [90]
Shyu, W Singwi, K.S. and Tosi, M.P. 1971, Phys. Rev. B 3, 237. [9]t
Singwi, K.S., Sjolander, A.
B l, 1044. [10, 93,
Tosi, M.P. and Land, R.H. 1970, Phys. Rev
1o0l
Sirota, N.N. 1969, Acta Cryst. ^25,
223, lI2l
SIater, J.C. 1959, rrlntroduction to Chemical Physicsrr, McGraw-Hi11.
17r)
Snith, H.G., DoIling, G., Nicklow, R.M., Vijayaraghavan, P.R.
Itlilkinson, I{.K. 1968, rrNeutron Inelastic Scattering",
and
Syrnposiurn at Copenhagen, Vol I , p.I49, Vienna : I.A.E.A.
Snith, P.A. and Smith, C.S. 1965, J. Phys. Chem. Solids , 26,
Srivastava, R.S. and Singh, K. 1970, Phys. Stat. So1. 39, 25.
Stokes, .
R.H. 1966 , J. Phys. Chen. Solids , 27, 51. l2l
Taylor, R. and Glyde, H.R. 1976, J. Phys. F : Itletal Phys. 6,
Taylor, ltl.J. 1956, J. Chem. Phys. 24,454. [103, 104]
Torrens, I .I'l . and GerI , M. 1969, Phys . Rev. I87 , gI2 . ]LIT ,
I .A.E .4.
[ 10]
27s. [s4]
172l
le1s. l72l
118l
Trivisonno, J. and Smith, C.S. 1961, Acta lr{etall. 9, 1064. [54]
LJbbelohde, A.R
Press.
1965, I'Melting and Crystal Structurerr, Clarendon
[ 11e]
Varna, C.M. 1976, Phys. Rev. B 14, 244. [17]
Von der Lage, F.C.and Bethe, H.A. L947 , Phys. Rev. 7I, 612. [18]
Vosko, S.H. 1964, Phys. Lett. L3, 97 . 18]
Weiss, R.J. 1966, 'rX-ray Determination of Electron Distributionsrr,
North-HoIIand. Í441
Weiss, R.J. 1969, Acta Cryst. ^25,
248. ll2l
Weiss, R.J. and De Marco, J.J. 1965, Phys. Rev. L40, ^1223.
[34]
Wenzl, H. and Mair, G. 1975, Z. Physik B 2L, 95. [118]
Westera, K. and Cowley, E.R. 1975, Phys. Rev. B11, 4008. 187, 109, 110]
Whittaker, E.J.W. 1953, Acta Cryst. 6, 222. 1,44)
l{illis,B.T.I'1. 1969, ActaCryst. A25,277, [18,63, 86]
I{ilson, A. J .C . L942, Nature ]!0, 151 . [56]
lrloods, A.D.B., Brockhouse, B.N., I{arch, R.H., Stewart, A.T. and
Bowers, R. 1962, Phys. Rev. L2B, III2. [113]
Wooster, IlI.A. L962, "Diffuse X-ray Reflections from Crystalsr',
Clarendon Press. [51 , 52]
Yake1, H.L. and Fankuchen, I. 1962, Acta Cryst. 15, 1188. 1,341
Wi11is, B.T.M. and PrYor,
Crystal lographyt',
Yegorov, R.F
Phys.
Young, R.A.
Young, R.A.
Zachariasen,
Zíman, J.M.
A.lV. 1975 , "Thernal Vibrations in
Canbridge University Press. [1 , 241
Kuznetsov, Ye. V., ShirokovskiY,
of Metals and l'{etallography . 39, No
1965, Trans. An. Cfyst. Assoc. 1,
1969, Acta Cryst. A25,55. [34]
IV.H. I965, Acta Cryst, 18, 705. l34l
1964, Adv. Phys. L3, 89. [8]
V.P. and Ganin, G.V. 1975'
.2, 11. [3, Figure 1.2, 4f
42. 1.34)
![Copyright ABRAHAM [2008] - All rights reserved · Tetryonics 52.04 - Alkali Metals Copyright ABRAHAM [2008] - All rights reserved 207 Alka H Metals The alkali metals arc silvcr-](https://img.pdfslide.net/doc/110x75/5f260dded7fdaa4ce9203b58/copyright-abraham-2008-all-rights-reserved-tetryonics-5204-alkali-metals.jpg)
