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Chapter 4
Elastic constants and phase transition in Sulphamic Acid single crystal
The clu.stic pro~~ei-fies of orthorhombic Sulphuil7ic ucid crystril 110i~e been
preserited in thi.s chapter. The measurements ofelasf ic consfa17ts by ultrusonic
PEO technique ond tonperature variution of elustic conslunfs oiler /he runge
300K-JOOK iicn~r heen undertaken. All the nine eltrstic coi7.sttr17t.s ivei.e ilierisured
rwitrg /l'LO /c.chiiic[i~c by /i?easuring velocity i,? riifirunl .vyiiii~ret,?' rIireclior7s.
Slci:/iic.e ,111ot.c phuse velocify, s1ou~nes.s. Young's niodzil~i.~, li17enr
conij~re~sibili~y huve been ))lade and it reveuis the ani.sotro~~j~ in elasric
11ro11w'.iie.s. DS(' i~~eusure/~zents on phase trunsilion in this crystal at u l leq~ s lo~v
heu/i/cg !.cite u1.e pi.esenter1.
lIlii.siic I r , i , . ~ r o i i i \ (iir.1 1'11ii.sr /rii,r,si~io,~ I I ~ I , , .s:,!g/r (.,:ysio/
-. . - -- -. - - I43
Elastic constants and phase transition in Sulphamic Acid single crystal
4.1 Introduction
Sulphalnic acid is a very interesting crystal with chemical formula
NH;SOj possessing orthorhonlbic symlnetly and exhibiting Piezo-electric and
Non-liricnr optical properties [4.1]. NLO crystal finds wide application in
optical iiarmonic generation, optical modulation, telecommunication, computer
and optical signal processing etc. Only little inforlllation about cryslallograpliic
and iphysical properties of Sulphamic acid can be found in the literature. X ray
diffrac~ion technique [4.2, 4.31, IR [4.4] Neutron diffraction scattering [4.5,4.6],
Raman [4.5], l'hermal expansion studies [4.1], Thermal analysis [4.7] and
Thermo-elastic properties [4.1] by resonance ultrasonic technique of Sulphaniic
acid have been reported in the literature. Earlier reports reveals that it
crystallircs in the Orthorhombic symmetry [4.2] with space group Pbca with
lattice parameter j4.21 a = 8 . 1 1 5 4 b = 8.066A, c = 9 . 2 j j A The structure of the
clystal has been studied using X ray diffraction technique [4.2- 4.31. I t has eight
molecules per unit cell. Both determinations are in agreement with respect to
the general si~apc and disposition of the dipolar ion structure for the lnolecule in
thc crystal, but differ by as much as 0.1 A in solile of the atomic positional paranicters. Both authors have stated that molecule cxists in the cryslal as the
zwittcrio~i (N\'f13'S03-).
Ilaman 14.51 and infrared study [4.4] also supported the zwitterion
structure. The Ralnan spectra in the low frequency region indicatcd that the ion
is essentially tetrahedral. The infrared study of Sulphamic acid was well
interpreted in terlils of zwitterion structure. But Philip el al. [4.21] analyzed the
Surface Enhanced Ranian Scattering (SERS) and FTlR spectra of Sulpliamic
acid and suggested a structure between that of zwitterions and molecular form.
A possible hydrogen bonding system was discussed by Osaki [4.2], who lcd to
the conclusion that the NHj+group must bond to five-oxygen atoms resulting in
one single and two bifurcated hydrogen bonds. The asymmetry of the quasi
tetrahedral [NHjSO,]- ion suggests an urusual frequent formation of acentric
species having polar properties like piezo- electric and nonlinear optic effects,
which might be suited for technical applications.
Table.?. 1 The general information about !he crystal can be tabulated as follows
Parameters I Neutrcn scattering X ray Diffraction l a
b
c
Space group
Molecular volu~ne
8 036 A 8.074 A
8 025 A 5.020 A
9 236 A 9 236 A
Calculated density
Neutron diffraction studies on th: crystal structure of Sulphamic acid
were carried out by Sass [4.6]. It confirn~t:d the zwitterion form of the molecule
but differs from that of X ray studies [4.2]. The study of the deformation
electron density in Sulphamic acid at 7E K by X ray and Neutron diffraction
technique was conducted by Bats el a1.[1.5]. Since the hydrogen bonds are so
disposed throughout the structure, Bats e t a / . suggested that there exists no well-
defined cleavage in any direction of the crystal.
2.165g1nicc
Michaut el 01. [4.20] canied out the temperature dependence of the ESR
spectra of the radical S03WH2 trapped in gamma irradiated Sulphamic acid single
crystal. His study described the potential well hindering the reorientations of the
N'H~ group.
2.167glnicc
0.7106 A h I.CO8 8, A'
I'lic ihcriiio elastic constant T,j, and anisotropy of the thcrmal ex l~~ns ion
ri,, and iir~~ncisen tcnsor G were studied by Haussiihl el cri. l4.31 and thcy fc>und
tilac Sulphamic acid possesses large value of T,, since i t has strong Hydrogen bond.
Maussiihl studied the anonlalous behavior of thennal expansion [4.3] of certain
si~lphamates. Studies of Rapp [4.7] by DSC suggested that Sulphamic acid exhibits
a first order phase transition in the temperature range of about 450 K.
Aim of the present study is to investigate the phase transition of Sulphamic
acid above roorn tempelatwe using PEO technique. Measurenlents of elastic
constants using ultrasonic Pulse Echo Overlap technique are proveci to be excellent
probe to investigate phase transition. The elastic propelties of Sulphanic Acid is well
studied by measuring ultrasonic velocity in the crystal in certain specified
crystallographic directions and evaluating the elastic stiffiiess constant, compliance
constant, Poisson's ratio. The surface plots of Phase velocity, Slowness, Yow~g's
modulns and Linear colnpressibility in a-b, a-c and b-c planes are made.
4. 2 Experimental Tecl~nique
4. 2.1 San~ple preparation
Large single crystals of Sulphamic acid of size (45~35x12) mm3 have
been grown from supersaturated aqueous solution of the salt by slow
evaporation technique over a period of 60-65 days. The temperature of the bath
was maintained constant at 305 K. The arrangement of moleculcs in unit cell of
the crystal is as shown in Figure 4.1. This is made with the help of the computer
programme "Atoms". The photograph of the grown crystal is dcpicted in Figure
4.2. 'l'he morphology of the crystal is shown in Figure 4.3. It is drawn with the
aid of the computer programme "Shape".
At 305 K thc soliibility is 21.7gm/lDOn~l.I-lz0.l'l~e solubility cut-ve is as
s h o w ~ ~ in Figure 1.4. Thc solubility increases slowly with rise o i tcnlperature
and reaches saturation value around 35'C.
Figure 4.4 So1ubil;ty curve of Sulpharnic acid crystal
Table. 4.2 Comparison of computed interfa.:ial angles of the Sulpharnic acid crystal wfth measured values.
Crystal faces Interfacial angles between faces
Cclnputed
100- T I T
010- 101 149.85
The interfacial angles are me~sured using an accurate contact
goniometer. By knowing the lattice pararreters, crystal system and space group
one can construct a stereographic plot by using the computer progranllnc
'Jcrystal'. Thc natural fices of the sample have been identified by the rnetliod
as discussed in Section 2.2.3 The stereograms of the crystal are depicted in
I:igurcs [4.5(a)-4.5(c)J.
l3ulk samples havc been cut using a slow speed diamond wliccl saw
(niodcl-650. Soutll lJay technology, USA). The diamond saw consists of a thin
metal disc wit11 tiiicro sized diamond powder embedded on the outer edge. The
blade fixed to n rotation mechanism is driven by a speed controlled motor. 'l'iie
crystal to be cut is glued to a precision movable arm with a goniorneter and
counter weight. Once the directions are carefully adjusted with respect to the
blade. the arm can be lowered so that the crystal rests on the rotating blade
edge. 'l'lic blade is continuously cooled and cleaned by a coolant, whicli is kept
below tile tray. One advantage of this diamond wheel saw is that samplcs can bc
cut easily to havc parallel faces. This is very important for ultrasonic work
because the non-parallelism will bring in a non- exponentially decaying echo
pattern and thereby additional errors in velocity measurements. Saniples with
pairs ui' parallel planes perpendicular to [loo], [OIO], [OOI], [ I 101, [OI I] and
1 1 011 tlirectioris lia\:c been prepared for ultrasonic wave velocity measureinents.
All cuttings arc made very accurately. The error due to misorientation is below
i 0.5". No clcavage is present in any direction of the crystal.
I'hc edges o f the samples have been polished carefully using cerium
oxide powder to optical reflection level so as to ensure proper bonding of the
transducer to the sample surface. Polishing of the sample and cleaning of the
transduccr are vcry important for successful bonding. To make the bond thin,
thc transducer lias to be held pressed to the sample under spring action.
A r r ~ m p m m l qfmoleculrs in rhe Ul~i l cell of'Sulpha~nic ocid crysl~d rrhou~ cr uxis
-
4.2.2 1)cnsity tneasurcments
'fhe density ol'tl~e material is measured by Archimedes' principle by lii~cli11~
the loss of weight of the solid in liquid. Carbon tetrachloride is used in this
measurement. The density of CC4 is 1.67gdcc. It is measured to be 2.162gnlicc. In
the elastic stiff~ess constant measurement density has vital role [C, = p v2]. So it should be measured with great accuracy. Results of this study are in well ageernent
wit11 calculated valuc lorn X-ray diffraction study [2.167gldcc][4.3] a11d that fi-om
neutron ditrraction study [2.165gn/cc][4.5].
4.2.3 Velocity nieasurements
The ultrasonic velocities are measured using the PEO technique [4.10].
The details of measurement technique are by Papadakis [4.1 I]. A MATEC
modcl 7700 pulse modulator and receiver system with its associated subunits
has been used for the velocity measurements. X- Cut transduccrs of resonant
frequency 10 MHz and 6 mm in diameter are used for the measurement of
longitudinal velocity and Y-Cut transducer of resonant frequency 10 MHz and
6mm diametcr are used for the measurement of transverse velocity. Large
number of clear cchoes indicated that the grown samples are free from defects.
The same transducer has been used to detect the echoes generated by successive
reflection of the waves from the rear end of the sample. Absolute velocities at
room temperature (303 K) have been measured for the selected direction and
modes. The McSkirnin At criterion [4.12-141 has been applied to correct the
phase lag introduced by the bonding medium on the RF echoes. The
temperature variation of the velocity of longitudinal and shear waves
propagating along the various directions in the crystal (Figures 4.6-4.8) have
been deternlincd in the range 300K-400 K by keeping the salnple mounted on a
suitable holder in a temperature controlled chamber. Thc rate of temperature
change in all the measurements is in the range of 0.5 to 1 K per minute. The
variation of velocity with tetllperature beyond 400 K was not invcsligatecl
because of bonding problems. The thermal expansion has been neglected while
measitring lhc variation of ultrasonic wave velocities with temperature.
Eiusrlc cwmrrrs a d phase amsl(luns In &lphmlc Add single t~ysrof
X
Figuri? 4.5 (8) Stereographic prajedbn of SuEphamic acid about a-exis
.&
Figurn 4.5 (b) stemgraphic pmjtxtion Sulphamk acid about &xis
Hmfk canmAl caod Ekw t a w s t a n in cWphmi~ Add sin& m n b
1
Fgure4.5(c) Sfereographk projestion of the Sulphamic acid crystal about c-axis
4.3 Results and Discussio~~s
1.3.1 Measurcmcnts of elastic constants
NH,SO,, being an Orthorhombic crystal, has the following ninc second
order elastic stiffness constants C I I , C22, C33, C44, Cjj, Cb6, Ci2. C i 3 and Cz;
('fable 4.3). l'he diagonal elastic constants C I I , C 22, C33, C44. Csj and C66 have
direct relationship with tlic ultrasonic mode velocity given by C,, = p ~ 2 . Tile
relationships between clastic constants for relevant ultrasonic wave velocities
for the orthorhonibic system are reported in literature [4.8]. The off diagonal
elastic constants can be found out from the following equation. The elastic
constant C l z can bc calculated by measuring the velocity perpendicular to a-b
planes; here tlic angle is measured from a- axis.
The elastic constant Cz3 is measured by propagating the sound with
tlic velocity normal to the b-c plane. The angle is measured from b-axis
l h e elastic constant C l j can be measured by propagating the waves
perpendicular to a-c plane where angle 0 is measured from c-axis
where s = sin 0 c = cos 0 and 8 is the angle of rotation for I-espective axes. The
angles are 44.83 ", 48.93 and 4 1.25 "for fab, fb, and f, and are measured from the a, b,
kind c axcs respcctivcly. 7'11~11 vlo , is the velocity of the wave in [I 101, v l l in the [OI I] and
\ I ? in the [I01 / direction, a ~ i d p is tlic density of the sample (= 2.162gmlcc. for Sulfal~iic
t~cid).
Elcisfic coiuiorils u~ id Pilase rru~lsiriui~ i , ~ S~ii/~iiciiiiic Acid sir~gle C ~ s l a l 161
01' the 18 propagation modes, velocity ~iieasuremellts of 12 modes are
sut'ticie~it to evaluate all the nine second- order elastic constants with cross
checks possible using the remaining modes. Considering all experimental
uncertainties, the absolute accuracy of elastic constant value is estiiiiated to be
bcttcs than 0.2% for diagonal elastic constants and 1% for off diagonal elastic
const:~nts. 111 all velocity measurements, the correct overlap identification is
made and McSkirnin At criterion [4.12, 4.131 for bond correctioll has been
applied using computer programme [4.15]. By measuring uitraso~iic velocity in
the Sulphamic acid crystal in certain specified crystallographic directions, the
anisotropy of elastic properties of the crystal is studied and the elastic stiffness
constants, compliance constants and Poisson's ratios [4.16-4.181 are evaluated
(Table 4.4). There are nine values of compliance and constants wliich are tlie
components obtained from the matrix inverse of elastic constants.
Table 4.3 Measured velocities and elastic constants of Sulphamic acid crystal at 300K
Velocity Elastic Direction of' ( Direction of 1 measured 1 rollstallt V-C. , 1 1 ' I N o I*lodo a i o poIarizatio~i Vlmls) C..(GPa) relationsh~p
The abbreviation used have the following meaning: L-Longitudinal, T-Transverse. QL-quasi- longitudinal QT-quasi-transverse s = sin 8 c = cos 0 and 0 is the angle of rotation with the respective axes. Tlic angles are 44.83 ,48.93 and 41.25 for tlie f,, fhc atid f,, and are ~nieasurcd fro111 tlic ;I, b, and c axes respectively Then v is tlie velocity of propagation of respzctivc lnodz and p is the density of the sample (=2.162gm/cc. for sulfaiiiic acid).
I62 ( ' I?opier 4
Also. here arc six values for Poissc~n's ratio in 01-thol.hombic crystal.
.I'he equation for Poisson's ratio when ur~iaxial stress is applied along a-
direction is
Siniilarly equations for Poisson's ratio when uniaxial stress is applied along b -
and c- directions are
Table 4.4 Elastic stiffness constants. Elastic ~ompliance constants, and Poisson's ratios of Sulphamic acid crystal at 303 K
Elastic stiffness Elastic corn 1 ance $ 2 .I 1 Poisson~s ratio / constant(GPa) constant(xl0- n N )
Eluslic ciiti.si~~1~1.~ o,,,/ P / ~ ~ , v c i~.o,~siiiwi i,,/i0117l""uic lci(/.\i i,g/e Cry"u1 163
Soine elastic stiffness constants of this crystal measured by this s t ~ ~ d y
(PEO neth hod) show appreciable deviation from those measured sing resonant
ultrasound technique [4.1]. Out of the diagonal constants, the constants C i I
(5%), C 3, (8.7%)) and C j j (9.8%) show deviation above 5%, whereas constants
C22 (7.8%), C 4 ~ (2.1%) andC66 (4%)exhibit deviation below 5%. While off
diagolial constants C l z (43%), CI, (19.8%) and Cz3 (17.1%) have exhibited
large deviation fro111 the RUT values.
4.3.2 'l'emper:iture variation of Elastic constants
The temperature variation of the elastic constants in specified directions
in the clystal arc plotted in [Figures 4.6 - 4.81 in the range 300 K-400 K by
keeping the sample mounted on a suitable holder in a temperature-controlled
chamber. Tlic rate of temperature change in all the measurements is in the range
of 0.5 K to 1 K per minute. The variation of velocity with temperature beyond
400 K was not investigated because of bonding problenls
Figure 4.6 Varfat~on of C33 and C55 of SA with temperature
Figure 4.7 Variation of C,, and C,, of SA with temperature
Figure 4.8 Variation of C,, and I:,, of SA with temperature.
E l a s ~ i ~ coiisiarir.\ ~ 1 i 0 i'iiuse tro,~silion 111 .S~ i I / i /~ t i i i i i c :Ici(/ v i i g i e Ctyslai ~- 165
Elastic crtrut7rrrlie.s it1 tire regiort of325K to 345K
Longitudinal elastic constants C i l , Czl, and trallsversc elastic
constants C44, C;< C6f, were subjected to temperature variation study. It can be
seen from I:igu1.%(4.6 - 4.8) that a number of elastic constants are showing
a~~oi~ia lous belia\,iours in the temperature region 325K-345K. Thc most
pronounced anomalies are shown by the elastic constant Cd4. It shows a dip at
3?5K and pcak at 335K and a small dip at 340K. Tlic constant C C , ~ shows a
small step decrcasc at 345K. The constant C Z ~ shows ~ilinor anomalies in the
rangc 335K-34SK. The constant Css exhibits anomalies in the range 335K-
340ti. The constants Cl and Cj3 do have significant anomalies. The anomaly in
Cq4 is also sho\vn in a cooling experiment and small thermal hysterisis of nearly
2K \vas j. . , ,. obscrved. These anon~alous behaviours of elastic constants of the
crystals can be attributed to a phase transition in the crystal around 335K. The
thel-ma1 expansion has been neglected while measuring the variation of
ultrasonic wave velocities with temperature.
4.3.3 Investigation of phase transition using DSC
'l'he present DSC study on sulphamic acid, in the temperature range
238K-473K has been carried out at a very slow heating rate of l0/min. Annealed
saml)les arc usetl for this study. Differential scanning calorimetric scan (Figure
4.9) shows that there is a clear change in the specific heat appearing as a sharp
dip near 33 1 K (58 '~) . The endothermic thermal anomaly is well correlated with
the suspected phase transition near 335K as suggested from the present
ultrasonic study. The energy involved in the transition is 1.599Jlg. DSC data
also shows a possible transition at 1 2 . 9 3 ~ ~ . But this region was not scanned
using ultrasonic. Transition already reported by Rapp [4.7] at 450K is also
prescnt in our data. In the thermal expansion studies Haussuhl ct al. [4.1]
obscrved anomalies for the sulphamate family members CsNH2S03 and betaine
NHiSO; around 350K.
I:?
Figure 4.9 Differentfal Scanning Calorimcitric spectrum of Sulphamic acid
4.3.4 Surface plots of Phase velocity, !Slowness, Young's modulus ant1 linear compressibility
With the aim of getting an insight into the anisotropy of elastic wavz
propagation in NH3SOj single crystal, the vc:locity surface plots in the a-b, b-c
and a-c plane have been made following a well-known procedure [4.17,4.19].
Figures 4.10 [a, b and c] show the ptase velocity surface plots in the
respective planes (a) corresponds to quasi - ongitudinal [QL] mode with higher velocity of propagation (b) and (c) represer t pure shear [PSI and quasi shear
[QS] modes respectively. A greater insigk.t into the elastic anisotropy of a
crystal is obtained by plotting the inverse phase velocity [slowness] surfaces.
Slowness surface plots provide a better pictorial representation of elastic
anisotropy in a crystal. The slowness surface plots [4.17,4.19] for NHj SO?
crystal are plotted in Figures 4.1 1 (a, b and c)
-61300 -6000 -4000 -2000 0 2000 4000 6000
velocity (r/s)
Figure 4.10 (a) Surface plots ofphase velocity along the xy plane
nO00 1 I I I I I -6000 -4000 -2000 0 2000 4000 6000
Phase velocity (r/s)
F~gure 4.10 (b) Surface plots ofphase velocity in the x-z plane
P h a s e velocity- YZ plane
-5000 0 5000 v e l o c i t : ~ (m/s)
Figure 4.10 (c) Surface plots of phase velocity in the y-z plane
-2.10-' 0 2.10-' 410-" Slowness (s/m)
Figure 4.11 (a) Surface plots of slowness in the x-y plane
:- az Plane
1
. .. Slowness (s/.)
Figure 4.11 (6) Surface plots of slowness in the x-z plane
Slovness -YZ plane 1 I I I
F~gore 4.11 (c) Surface plots of slowness in the y-z plane
Young's modulus
Figure 4.12 Surface plots of Young's 61oduli in the x -y, x-z and y-z planes
Figure 4.13 Surface plots of linear conpressibility in the x -y, x-z and y-z planes
The velocity surface plots are unable to describe the anisotropy of
the elastic properties of a crystal completely. Young's ~nodulus [4.17.4.19]
surface plots are very important in this regards. The Young's lnodulus E in
Eluslic i , ~ , i i ~ ~ r i , i i s o,id 1'11ii.sc i r u n s i ~ i o ~ ? i t i S i i / p l ? ~ u u ~ ~ f c ! ~ / . s ! ~ , ~ : / ~ , ~. ( ' ! :~~.v lo / 171
the direction of ~lnit vector ni for an orthorhombic crystal is discussed in
the literature and is given by
?he cross sectionSol' Young's moduli.., surfaces of NH3SO3 plotted in tlie a-b,
b-c aid a-c planes are shown in Figure 4.12. The linear co~npressibility of an
o~?horiiombic ciyst:il in matrix fonn can be written as
'l'he surface plots of linear compressibility [4.17] of HNH2S03 crystal in
thc a-h, b-c and ,I-c planes are shown in (Figure 4.1 3).
The Volun~e compressibility [4.17] Sllkk is an invariant parameter for a
crystal. In matrix notation it is given by
Where S,,'s are t l ~c corresponding compliance constants. Hence Bulk rnodulus
of the crystal is given by
The \olumc coml)ressibility of this crystal is evaluated to be 0 . 4 3 7 x 1 0 ~ ~ ~ ~ ~ ~ n 1 ~
and bulb n~odulus is 22.88 GPa.
I'EO technique has been successfully implemented for evaluating all thc
nine elastic constants, Compliance constants, Poisson's ratios, Bulk l~lodulus
and Volume conlpressibility of Sulphamic acid crystal. The anisotropy in elastic
propcrtics are well studied by the surface plots of Phase velocity, Slowness,
L.incar coinpressihility and Young's modulus.
The variation: of elastic constants C 11, C22, C33, C.I.I. Cj j and C66 in the
temperature range 300K- 400K have shown a clear indication of weak phase
transition around 335K. DSC study also exiibited a weak anomaly near 331 K.
In summary, fro111 the observed anomalies in the elastic constants and from tile
DSC investigation a new weak phase transition in sulphamic acid crystal near
335K is proposed. Further studies are required to establish the nature of this
transition.
References
4.1 E.Haussulil and S. Haussiihl, 2. fur Kristallogr., 210, 269 (1995) )lElrtsiic
~ ~ r ( ~ / ~ e ~ i i e . s i!/' S~tlphatttic acid and sulphomu~es ofNri, K . .. ...
4.2 K.Osaki, H.l'adokoro and i.Nitta, Bull. Chem. Soc. Japan, 28, 524 (1955) / X -
rriy dv$~tctioii un o.ys/al structure of sulpharnic acid crystal
4.3 F.A.Kanda and A.J.King, J . Am. Chem. Soc., 73, 23 15, (195 1) / X-ru~j
rltffr.uciio~i 111i crys~ul struclure of sulfarnic ucid
4.4 . M . V u a g i i ; ~ t and i'..L.Wagner, J . Chem. Pliy., 26, 1,77 (1957) ll'ihrmtiot~cil
spect1.u ~irirl .s~ruciu~e ofsolid sulphai~iic ucid a11d the S I I / / J ~ U I I I L I ~ ~ jot/
4.5 J.W.Bats and I'.Coppens, Acta Crystallogr. B, 33,37 (1977) /A stud!, o f the
e j r t t ~ t i i ~ electron de~uiry in S u l p h a ~ ~ ~ i c acid at 78K by X-ray roi~l tie~itroti
Diifructioti
4.6 R.L Sass: Acta Crystallalogr., 13,320-324 (1960) /A ~ ~ e u t r o n dffr.rrciioti study
oil the cry.str11 struclure of sulpharnic acid
4.7 K.W.Rapp : Kristallographie einiger Amidosulfate ein-und zweiwertiger
kationen.Diplo~~?arbeit Universitat Munchen 1992 iDfferetltial sca~~nirlg
ci110~itrieh.1~ ~i ieasurer~ze~~~s of sulphan~ates
4.10 .I.E.May JI-: IRE. Natl. Conv. Rec., 6 part 2, 134 (1958)
4.1 1 E.I'.Papadnhis: in 'P/~ysical Acoustics' Vol. ,YII Eds. IV.P.Musoti L I I I ~
RIV Tirirrsii~~~ (Academic Press New York 1976) p.227
4.12 tl.J.McSkimin: Acou. Soc. Am., 33, 12 (1961)
4.14 H.J.McSkitnin: irr ~Plry.sica1 Acoustics' Voluri7e I, Part A Erl. WP.Musol7
(Academic Press New York 1964) p.271
4.15 L.Godfrey and J.Philip : Acoust. Letters, 19,111-14,(1995) / A nuliiericui
Teclririque for bond correciion in zrltrasc~~ic nreasureii~en/.s
4.16 J.F.Nye : Physicul properties of crysials, (Oxford university press ,London
1957) p. 143
4.17 A.V.Alex and J.Philip: Material Scielce and Engineering B, 90, 241-245
(2000). / Elnsric j~i.r~/~crtics of di-ariiiii~~niurii hydrogen cifrrrte .single cr~,s/ci/.\~:
Ail ult~.a.so~~ic sllrdy
4.18 N.V.Pcrelomova and M.M.Tagieva : in 'Problems in Crystal P11)~sics' (Mir
Publishers, Moscow. 1983) p.118
4.19 J.P. Musgrave : Ciy.sta1 Acoustics, Infroduction to study of elaslic waves ci~itl
vibration in crysra1.s Holden-Day ( 19:'0 ) p.61
4.20. P.Michaut,J.Roncin and K.Leibler : Ctlemical physics letters, 40,1,28-3 1 (!976)
4.21. D. Philip, A. Eapen and G.Aruldas ; J.Solid State Chemistry, 116,217-223
(1995) /Vibrational and surface ee,ihanced Ra~nuii scalterii7g speciru of
S1rlphan7ic acid.