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Characterization: Classification: Long range ordering periodicity unit cell Symmetry 7 crystal systems 230 space groups Structural information: Unit cell Miller indices ( h, k, l ) d spacing Relative intensity atomic positions. - PowerPoint PPT Presentation
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Characterization: Classification: Long range ordering periodicity unit cell
Symmetry 7 crystal systems 230 space groups
Structural information: Unit cell Miller indices (h, k, l) d spacing
Relative intensity atomic positions
2
X-ray CrystallographyIntroductionCrystal Diffraction
Diffraction Structure
Structure Amplitudes, Fhkl
Atomic scattering factors
Fourier transfer Fhkl (x, y, z)
Least squares refinement
Structure properties --- Distance / angles ; packing etc.
Structure data base
3
Data measurement: h, k, l, dhkl , nhkl , Ihkl ( F2
hkl )
Phase determination heavy atom method; direct method; multiple scattering
Fourier transformation: F2 (r); reciprocal real space
Least squares refinement: (r) xi, yi, zi, ui
Structural model: bond distances; bond angles; atomic thermal vibration etc.
Structural Analysis:
4
d
l
Bragg Diffraction
5
Structure Factor Calculation
90
180
270
360
450540
90
180
270
450540
90
180
270
360
450540
0 0 0
X1 f1cos X2 f2cosXtotal (f1f2)cos
X1 X2
Xtotal
f1 f2
P1 P2f1 f2 Resultant
90
180
450
90
180
270
450
90
180
360
450
0 0 0 X1 X2
Xtotal
540 540 540
360
X1 f1cos(1) X2 f2cos(2)
f1
f2
F
f1
P1
f2
P2
1
2
Resultant
Combination of Wave
Direction (h, k, l)
Amplitude Fhkl
Phase
Xtotal f1cos(1) f2cos(
2) cos (f1cos1 f2cos2)
sin (f1sin1 f2sin2) LetXtotal F cos( )
F cos cos F sin sin
f2sin2
f1sin1
A’f2sin2
f1sin1
Ff2
f1
1
2
B’
F A’ iB’ F ei
1 1 2 2
n
i ii=1
A'=f cos +f cos
= F cosα= f cosDirection (nhkl)
Amplitude fi
Phase i
F sinαB'= =tanα
A' F cosα
-1 B'α=tan
A'
1 1 2 2
n
i ii=1
B'=f sin +f sin
= F sinα= f sin
(A)2 (B’)2 F2cos2 F2sin2 F2
6
Structure Amplitude (Factor)
DetectorR
X1
rnS0
S
X2
1 n 0
2 n
1 2 n 0
X = r S
X = R - r S
X + X = R - r S - S
7
Pathlength difference bet. atom n at rn and the origin at 0
2
02
1exp 2
h nP n
e Ef i t R r
mc R
11
2
0 0
i vt x
E e
h
n n n n
ha kb lc
r x a y b z c
Electric field at rn :
Electric field at P (defection) :
E0 : electric field amplitude of the incident beam at rn
c
nnnnh lzkyhxr
hnn rSSrRXX )( 021
8
2θ
0a
lcSS
kbSS
lkhhaSS
SSLet hkl
)(
)(
,,)(
//
0
0
0
0
integer
0
~S
S~
aS ~~0
aS ~~
9
D
F
E’
G
E
’
’
a
mth order
mth order
Zeroth order
Incident beam
10
Fh
44
if e
33
if e 22
if e
11
if e
If centrosymmetric and no anomalous scatter: Bh=0; α=0 orπ
Ah
Bh
hh
i
j
rij
calh
jhj
iBA
FiF
eF
ehfF
r
jh
sincos
)(
)(2
2
11
2 2
1
cos 2
sin 2
tan
h
h
ih h
h j j j jj
h j j j jj
hh
hh
h
F F e
A f hx ky lz
B f hx ky lz
F A B
B
A
h
12
Centric case with non-anomalous scatterers
hkl hk l
hkl hk l
I I
F F
0
0
hkl hk l
hkl hk l
hkl
A A
B B
B
or
Zpt. charge
atomic sphere (fixed atom at ri) T=0K
with thermal vibration T as function of sin( )
h h
h h
A A
B B
2sin( )
B : Thermal parameter
B = 0
B > 0
"'sin0 fifff ii
13
Centric case with anomalous scatterers
NA : No. of atom types in an asymmetric unit
NE : No. of symm. elements of the space group
Ri : sym. operatorh : (h, k, l)xj : (xj, yj, zj)
2/
2exp2
)2exp(
NE
iji
NA
ji
NE
iji
NA
jjh
xRhif
xRhifF
i jijjj
i jij jh
i jij j
i jijjjh
jjjj
xRhff
xRhfB
xRhf
xRhffA
fifff
2
2
2
2
0
0
0
sin
cos
sin
cos
'
"
"
'
"'if
then
when at (000)1
14
Considering nuclear thermal vibration
12
2
TTBiso
)( sin
)(exp jjjh rhifF 2thermal vibration electron density smearing
if isotropic spherically symmetric
As a point scatterer
Å2
factoretemperaturToffunctionaisT
offunctionaisfj
:sin
sin
2
2
invibrationofamplitudesquaremeanu
uBiso
2
228
15
Tanisotropic thermal vibration as an ellipsoid
3×3 matrix 6 elements uij
11 12 13
21 22 23
31 32 33
u u u
u u u
u u u
symmetric: u12= u21; u23 = u32; u13 = u31
based on a, b, c,-axis
16
1
2
3
0 00 00 0
u
uu
i ju diagonize
eigen value eigen function
u3
u1
u2
(three principle axes of the ellipsoid)
**12
2*211
122
11
**12
2*211
24
1exp
2exp
22exp
exp
bhkaBahB
hkh
bhkauahu
uT kijh
ii
aCi
~1i
i
aCi
~2
ii
aCi
~3
Thermal ellipsoid
17
In case of ab-plane mirror plane sym:
U11 , U22 , U33 , U12 U13=U23=0
x → x
y → y
z → -z
U11 U22 U33 U12 U13 U23
U12=U12
U13 →-U13 ; U23 →-U23 U∴ 13=U23=0
1 0 0
0 1 0
0 0 1
Constraint in Thermal vibration
Cm ~
18
Systematic Absences
ijh xihfF 2expP21 for any atom i at x y z R1 x -x y+1/2 -z R2 x
cos 2 sin 2
cos 2 1 2 sin 2 1 2
h iF f hx ky lz i hx ky lz
hx k y lz i hx k y lz
02
10
100
010
001
0
0
0
100
010
001
21 ;; RR
19
G0k0 cos2ky cos2[ky (k/2)]
i {sin2ky sin2[ky (k/2)]}
cos2ky cos2kycosk sin2kysink
i sin2ky i sin2kycosk i cos2kysink
2(cos2ky i sin2ky)
0
when k 2n
when k 2n1
i.e. for 0k0 reflections
222
kkykyfG ih coscos
222
kkykyi sinsin
When 0 i.e. h 0 and I 0
Let hx + lz
20
Systematic Absences(space group extinction from translational sym. elements)
abscentn
hhR
presentnhhRIf
hRxiT
hxxh
xRih
xRxxihT
j
j
j
jhkl
j
j
jj
NE
jjhkl
2
12
2
2
2
2
1
1
1
1
)(
)(exp
exp
;exp
scalarsince
21
nlkkld
nll
nlhlhn
nllhc
nkk
absent
40
41
41
0
100
010
001
400
41
0
0
100
001
010
4
120
21
0
21
100
010
001
120
21
0
0
100
010
001
1200
0
21
0
100
010
001
2
1
1
/
/;
;
;
;
;
22
12
21
0
21
12
12
12
21
21
021
0
210
21
21
100
010
001
12
21
21
21
100
010
001
nlhhklB
nlk
nkh
nlh
hkl
F
nlkhhklI
CenteringLattice
;
;
23
nll
nkk
nhh
nknhhka
nhnllhc
nknlklc
nkkhklcenterC
conditionsccaCEx
200
200
200
220100
220010
220001
2
;
;
24
Difference in phase
2
X-rayBeam Atom
Atomic Scattering
25
Atomic Scattering
If the atom is a point charge (compared w.r.t the wavelength), it scatters as Z (atomic number)
Scattering amplitude amplitude scattered by atom Eatom
amplitude scattered by free e- Ee(factor)
ρ(r) : electron density around nucleusdq = ρ(r) dV q: charge V : volume
e
atom
E
Ef
d
r
26
calculatedbecanfchosenisnfwaveonce
er
drkr
krrr
e
kwhere
drdrikrre
f
zfSSeiSSwhen
r
drdvrSSrSS
dvrSS
ie
rdf
e
dvf
r
""'""
)(
sin)(
sin
sincosexp)(
//..
cossin
sincos)(
exp)(
;
2
0
2
2
00
200
0
4
4
221
0
2
2
2
0 0
27
Int’l Table of X-ray CrystallographyVol CΨ mostly from HF p.477
sinθ/λ
fZ
2
2
0
3
21
1
10
0
sin
:
af
ea
Hfor
H
a
r
electroninnerofejectionf
forcebindingelectronfieldmagneticexternalwhenf
fifff
DispersionAnomalous
cibaparameters
ceaf
ii
bi
i
":
:
"'
,,
sinsin
0
0
24
12
0
49
For example
Analytical form : Vol C.p500.
28
f0 for atoms from Z = 1 to 90
29
coefficients of analytical form in atomic scattering factors
30
31
Gd
f”
f’
1705 1710 1715
30
20
10
0
10
20
30
(Å)
30
20
10
0
10
20
30
184 185
(Å)
Sm
f”
f’
Anomalous Scattering
30 20 10 00
10
20
30
Gd f”
f’ 30 20 10
10
20
Sm
f’
30
f”
00
(a) (b)
(c) (d)
Fig. :Anomalous scattering terms f’ and f” for :(a) gadolinium near the L3 edge ; (b) samarium near the L3 edge
Fig. :Plot in the complex plane of f’ i f” :(c) gadolinium near the L3 edge ; (d) samarium near the L3 edge
32
Atomic f’ and f”
33
Relationship between Fhkl & ρuvw
dvere
fdvere
df rii
ri
hh 22 11
Crystal scattering general form with continuous ρ(u,v,w)
Corresponds to the intensity of h,k,l refln.
reciprocal space
corresponds to the electron density at u,v,w position
direction space
dvlwkvhuiuvwF
dveuvwF
v
hkl
v
ui
hh
2exp
~~2~
uvwF FT
h~
lwkvhuilkhlkh eF
Vuvw 2
,,,,
1
Atomic scattering