polarization holograms and diffraction anisotropy in amorphous chalcogenides
Diffraction: Real Sample (From Chapter 5 of Textbook 2, Chapter 9 of reference 1,) Different sizes,...
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Transcript of Diffraction: Real Sample (From Chapter 5 of Textbook 2, Chapter 9 of reference 1,) Different sizes,...
![Page 1: Diffraction: Real Sample (From Chapter 5 of Textbook 2, Chapter 9 of reference 1,) Different sizes, strains, amorphous, ordering Diffraction peaks.](https://reader036.fdocuments.us/reader036/viewer/2022062802/56649e8f5503460f94b93c8c/html5/thumbnails/1.jpg)
Diffraction: Real Sample(From Chapter 5 of Textbook 2, Chapter 9 of reference 1,)
Different sizes, strains, amorphous, ordering Diffraction peaks
![Page 2: Diffraction: Real Sample (From Chapter 5 of Textbook 2, Chapter 9 of reference 1,) Different sizes, strains, amorphous, ordering Diffraction peaks.](https://reader036.fdocuments.us/reader036/viewer/2022062802/56649e8f5503460f94b93c8c/html5/thumbnails/2.jpg)
t = m
d hkl
+
+ … …
012
m
Constructive Interference
Bhkl
Bhkl
Bhkl
d
d
d
sin)3(23
sin)2(22
sin2
+ + .
);sin(2 Bhkld );sin()2(2)(2 Bhkld
);sin()3(2)(3 Bhkld …
Destructive interference:extra path difference (plane 0 and plane m/2): /2
![Page 3: Diffraction: Real Sample (From Chapter 5 of Textbook 2, Chapter 9 of reference 1,) Different sizes, strains, amorphous, ordering Diffraction peaks.](https://reader036.fdocuments.us/reader036/viewer/2022062802/56649e8f5503460f94b93c8c/html5/thumbnails/3.jpg)
);sin()2/(2))(2/( Bhkldmm 2/)2/( m
BBB cossincossin)sin(
<< 1 cos ~ 1 and sin ~ .);cos)(sin2/(22/)2/( BBhkldmm
BhklBhkl mddmm cos2/sin)2/(2)2/( = t
Bt
cos2
Broadening (): FWHMThickness dependent (just like slits)
![Page 4: Diffraction: Real Sample (From Chapter 5 of Textbook 2, Chapter 9 of reference 1,) Different sizes, strains, amorphous, ordering Diffraction peaks.](https://reader036.fdocuments.us/reader036/viewer/2022062802/56649e8f5503460f94b93c8c/html5/thumbnails/4.jpg)
Bt
cos
2B
2(B-) 2(B+)
Scherrer’s formula
Bt
K
cos
K: in general = 0.9 but very close to1 depends on the crystal shape
![Page 5: Diffraction: Real Sample (From Chapter 5 of Textbook 2, Chapter 9 of reference 1,) Different sizes, strains, amorphous, ordering Diffraction peaks.](https://reader036.fdocuments.us/reader036/viewer/2022062802/56649e8f5503460f94b93c8c/html5/thumbnails/5.jpg)
2 *hkld
2B
/sin2/1 **Bhklhkl d d
*hkld
B
BBhkl t
cos2
cos2cos2*
d
thkl /1* d
The size of the reciprocal lattice point 1/t
![Page 6: Diffraction: Real Sample (From Chapter 5 of Textbook 2, Chapter 9 of reference 1,) Different sizes, strains, amorphous, ordering Diffraction peaks.](https://reader036.fdocuments.us/reader036/viewer/2022062802/56649e8f5503460f94b93c8c/html5/thumbnails/6.jpg)
Interference function:
B intensity ≠0; 2(s – s0)/≠G.
1
0
1
0
1
03322110
1
1
2
2
3
3
)]()[(2
expN
n
N
n
N
ntotal nnn
iFA aaass
)(/)( 3210 bbbHss lkh
Now (s - s0)/ is deviation from the reciprocal lattice:
1
0
1
0
1
0332211321
1
1
2
2
3
3
)]()[(2expN
n
N
n
N
ntotal nnnlkhiFA aaabbb
ijji ab
![Page 7: Diffraction: Real Sample (From Chapter 5 of Textbook 2, Chapter 9 of reference 1,) Different sizes, strains, amorphous, ordering Diffraction peaks.](https://reader036.fdocuments.us/reader036/viewer/2022062802/56649e8f5503460f94b93c8c/html5/thumbnails/7.jpg)
1
0
1
0
1
0332211
1
1
2
2
3
3
)2exp()2exp()2exp(N
n
N
n
N
ntotal nininiFA aaa
The same as the Fraunhofer diffraction for aone dimensional net
1
1111
1
011 sin
sin)1(exp)2exp(
1
1a
aaa
NNini
N
n
332211
3
1
)1()1()1(exp
sin
sin
aaa
a
a
NNNi
NFA
i i
iitotal
2
2222
1
022 sin
sin)1(exp)2exp(
2
2a
aaa
NNini
N
n
3
3333
1
033 sin
sin)1(exp)2exp(
3
3a
aaa
NNini
N
n
![Page 8: Diffraction: Real Sample (From Chapter 5 of Textbook 2, Chapter 9 of reference 1,) Different sizes, strains, amorphous, ordering Diffraction peaks.](https://reader036.fdocuments.us/reader036/viewer/2022062802/56649e8f5503460f94b93c8c/html5/thumbnails/8.jpg)
3
12
222
sin
sin
i i
iitotal
NFAI
a
a
Interference function
For reasonable number ofunit cells, ripples beyondmain peak are very weak
3
122
2
223
22
21 sin
sin
i ii
ii
N
N
FNNN
I
a
a
Some define interference function as
![Page 9: Diffraction: Real Sample (From Chapter 5 of Textbook 2, Chapter 9 of reference 1,) Different sizes, strains, amorphous, ordering Diffraction peaks.](https://reader036.fdocuments.us/reader036/viewer/2022062802/56649e8f5503460f94b93c8c/html5/thumbnails/9.jpg)
Telling the same thing:Size of reciprocal lattice points (nodes): point 1/t
![Page 10: Diffraction: Real Sample (From Chapter 5 of Textbook 2, Chapter 9 of reference 1,) Different sizes, strains, amorphous, ordering Diffraction peaks.](https://reader036.fdocuments.us/reader036/viewer/2022062802/56649e8f5503460f94b93c8c/html5/thumbnails/10.jpg)
Mosaic Structure
http://ees2.geo.rpi.edu/spear/SpearGraduateFunding.html
B - < < B + .
![Page 11: Diffraction: Real Sample (From Chapter 5 of Textbook 2, Chapter 9 of reference 1,) Different sizes, strains, amorphous, ordering Diffraction peaks.](https://reader036.fdocuments.us/reader036/viewer/2022062802/56649e8f5503460f94b93c8c/html5/thumbnails/11.jpg)
Strain
22/cos
2/sinsin2
dd
dd
tan2 2b
b: extra broadening induced by the non uniform strain
tantan
cos2
sin2
cos2 22
d
dd
dd
d
d
Strain peak shift
![Page 12: Diffraction: Real Sample (From Chapter 5 of Textbook 2, Chapter 9 of reference 1,) Different sizes, strains, amorphous, ordering Diffraction peaks.](https://reader036.fdocuments.us/reader036/viewer/2022062802/56649e8f5503460f94b93c8c/html5/thumbnails/12.jpg)
Directly sum over all the scattered waves:
N
nnn
ifA
10 ])[(
2exp)( xsss
Amorphous and partially crystalline samples
Nxxxx ,,,, 321
sdviGxA xG 2exp)( /)( 0ssG
individual atoms electron density (x).
![Page 13: Diffraction: Real Sample (From Chapter 5 of Textbook 2, Chapter 9 of reference 1,) Different sizes, strains, amorphous, ordering Diffraction peaks.](https://reader036.fdocuments.us/reader036/viewer/2022062802/56649e8f5503460f94b93c8c/html5/thumbnails/13.jpg)
individual atoms: x1, x2, …, xn. Density of electron cloud of the nth atom (xn) n(x - xn)
N
nnx1
)( xx
s
N
nn dviGA xxxG 2exp)()(1
snnn
N
n dviGiG xxxxx 2exp)(2exp1
Atomic formfactor
Total density of electron cloud:
xn x
x - xn
N
nn iGfA1
2exp)( xG
sdviGxA xG 2exp)(
sdviA xGGx 2exp)(} Fourier Transform
![Page 14: Diffraction: Real Sample (From Chapter 5 of Textbook 2, Chapter 9 of reference 1,) Different sizes, strains, amorphous, ordering Diffraction peaks.](https://reader036.fdocuments.us/reader036/viewer/2022062802/56649e8f5503460f94b93c8c/html5/thumbnails/14.jpg)
Scattering power of the object: Unit scattering power: NII N /)()( GG
)(GNI
2)( GG AI N
N
nn
N
nnN ififAAI11
* )2exp()2exp()( xGxGGGG
N
nn
N
nn ifif1 1
)2exp()2exp( xGxG
N N
nnnn iff1 1
)](2exp[ xxG
n = n term: N
nf1
2
![Page 15: Diffraction: Real Sample (From Chapter 5 of Textbook 2, Chapter 9 of reference 1,) Different sizes, strains, amorphous, ordering Diffraction peaks.](https://reader036.fdocuments.us/reader036/viewer/2022062802/56649e8f5503460f94b93c8c/html5/thumbnails/15.jpg)
n n term: N(N-1)
)](2exp[)](2exp[ nnnnnnnn iffiff xxGxxG
)2cos()2cos(
)](2cos[2
nnnnnnnn
nnnn
ffff
ff
xGxG
xxG
nnnn
nnnn
xxx
xxx
nn
nnnn
N
nN fffI )2cos()(1
2 xGG
nn
nnN FNFI )2cos()( 22 xGG
Identical atoms or atomic groups with structure factor F.
![Page 16: Diffraction: Real Sample (From Chapter 5 of Textbook 2, Chapter 9 of reference 1,) Different sizes, strains, amorphous, ordering Diffraction peaks.](https://reader036.fdocuments.us/reader036/viewer/2022062802/56649e8f5503460f94b93c8c/html5/thumbnails/16.jpg)
Expression in term of electron density
uuN dvidviI )2exp()()2exp()()( uGuuGuG
uudvdvi )](2exp[)()( uuGuu
Define x = u - u;
xuN dvdviI ]2exp[)()()( xGuxuG
xdviP ]2exp[)( xGx udvP )()()( uxux
![Page 17: Diffraction: Real Sample (From Chapter 5 of Textbook 2, Chapter 9 of reference 1,) Different sizes, strains, amorphous, ordering Diffraction peaks.](https://reader036.fdocuments.us/reader036/viewer/2022062802/56649e8f5503460f94b93c8c/html5/thumbnails/17.jpg)
ua dviGFA uuGG 2exp)()(
xaN dviPFI ]2exp[)()( 2 xGxG
uaaa dvP )()()( uxux Patterson Function or
Autocorrelation Function withrespect to atomic density
dviIxP Na ]2exp[)()( xGG
Inverse Fourier transform
N identical atoms or atomic groups: F (common scattering or structure factor)a as the local atomic density instead of electron density
![Page 18: Diffraction: Real Sample (From Chapter 5 of Textbook 2, Chapter 9 of reference 1,) Different sizes, strains, amorphous, ordering Diffraction peaks.](https://reader036.fdocuments.us/reader036/viewer/2022062802/56649e8f5503460f94b93c8c/html5/thumbnails/18.jpg)
![Page 19: Diffraction: Real Sample (From Chapter 5 of Textbook 2, Chapter 9 of reference 1,) Different sizes, strains, amorphous, ordering Diffraction peaks.](https://reader036.fdocuments.us/reader036/viewer/2022062802/56649e8f5503460f94b93c8c/html5/thumbnails/19.jpg)
ij
ja )()( xxx
uuxux dNNfP aaxoa )()()( ,2
autocorrelation with itself
Autocorrelationwith all otheratoms paircorrelation function
where
Pair correlation (distribution) function g(x): related tothe probability of finding the center of a particle a givendistance from the center of another particle
![Page 20: Diffraction: Real Sample (From Chapter 5 of Textbook 2, Chapter 9 of reference 1,) Different sizes, strains, amorphous, ordering Diffraction peaks.](https://reader036.fdocuments.us/reader036/viewer/2022062802/56649e8f5503460f94b93c8c/html5/thumbnails/20.jpg)
uuxux dgfV
Naa )()()(2
From intensity measurement Fourier transformS(G) - 1 g(x)
xxGG xG degV
NSI i
N )(1)()( V: volume
Example:
![Page 21: Diffraction: Real Sample (From Chapter 5 of Textbook 2, Chapter 9 of reference 1,) Different sizes, strains, amorphous, ordering Diffraction peaks.](https://reader036.fdocuments.us/reader036/viewer/2022062802/56649e8f5503460f94b93c8c/html5/thumbnails/21.jpg)
There are free software on the internet that can bedownloaded for extracting PCF. E.g.http://www.pa.msu.edu/~petkov/software.htmlhttp://dmoz.org/Science/Physics/Crystallography/Software/etc.
![Page 22: Diffraction: Real Sample (From Chapter 5 of Textbook 2, Chapter 9 of reference 1,) Different sizes, strains, amorphous, ordering Diffraction peaks.](https://reader036.fdocuments.us/reader036/viewer/2022062802/56649e8f5503460f94b93c8c/html5/thumbnails/22.jpg)
http://francesa.phy.cmich.edu/people/petkov/nano.html
http://www.pa.msu.edu/cmp/billinge-group/programs/PDFgetX/
http://www.ccp14.ac.uk/solution/high_q_pdf/index.html