diffraction: real sample (from chapter 5 of textbook 2, chapter 9 of reference 1,) different sizes,...
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Diffraction: Real Sample(From Chapter 5 of Textbook 2, Chapter 9 of reference 1,)
Different sizes, strains, amorphous, ordering Diffraction peaks
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
);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)
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
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
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
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
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
Telling the same thing:Size of reciprocal lattice points (nodes): point 1/t
Mosaic Structure
http://ees2.geo.rpi.edu/spear/SpearGraduateFunding.html
B - < < B + .
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
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).
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
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
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.
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
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
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
uuxux dgfV
Naa )()()(2
From intensity measurement Fourier transformS(G) - 1 g(x)
xxGG xG degV
NSI i
N )(1)()( V: volume
Example:
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.
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