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