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Chem 253, UC, Berkeley What we will see in XRD of simple cubic, BCC, FCC?

Position Intensity

Chem 253, UC, Berkeley

Structure Factor: adds up all scattered X- from each lattice points in n iKd j Sk  e j1 K  ha  kb  lc     d j  x a y b z c

2 I(hkl)  Sk

1 Chem 253, UC, Berkeley X-ray scattered from each primitive cell interfere constructively when:

eiKR 1 2d sin  n

For n-atom basis:

sum up the X-ray scattered from the whole basis

Chem 253, UC, Berkeley

  ' k d k  d di R j  '  K  k  k

Phase difference: K (di  d j ) The amplitude of the two rays differ: eiK(di d j )

2 Chem 253, UC, Berkeley

The amplitude of the rays scattered at d1, d2, d3…. are in

the ratios : eiKd j

The net ray scattered by the entire cell:

n iKd j Sk  e j1

2 I(hkl)  Sk

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For simple cubic: (0,0,0)

iK0 Sk  e 1

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For BCC: (0,0,0), (1/2, ½, ½)…. Two point basis

1    2 iK ( x  y  z ) iKd j iK0 2 Sk  e  e  e j1 1 ei (hk l) 1 (1)hkl

S=2, when h+k+l even S=0, when h+k+l odd, systematical absence

Chem 253, UC, Berkeley

For BCC: (0,0,0), (1/2, ½, ½)…. Two point basis

S=2, when h+k+l even S=0, when h+k+l odd, systematical absence

(100): destructive

(200): constructive

4 Chem 253, UC, Berkeley

Observable peaks h2  k 2  l 2 Ratio

SC: 1,2,3,4,5,6,8,9,10,11,12.. Simple cubic BCC: 2,4,6,8,10, 12….

FCC: 3,4,8,11,12,16,19, 24…. a dhkl  h2  k 2  l 2 2d sin  n

Chem 253, UC, Berkeley

FCC

S=4 when h+k, k+l, h+l all even (h,k, l all even/odd)

S=0, otherwise

i (hk ) i (hl) i (lk ) Sk 1 e  e  e

5 Chem 253, UC, Berkeley

Observable diffraction peaks h2  k 2  l 2 Ratio SC: 1,2,3,4,5,6,8,9,10,11,12..

Simple BCC: 2,4,6,8,10, 12, 14…. cubic (1,2,3,4,5,6,7…)

FCC: 3,4,8,11,12,16,19, 24…. a dhkl  h2  k 2  l 2 2d sin  n

Chem 253, UC, Berkeley

Observable diffraction peaks h2  k 2  l 2 Ratio a d  hkl 2 2 2 SC: 1,2,3,4,5,6,8,9,10,11,12.. h  k  l

BCC: 2,4,6,8,10, 12, 14…. (1,2,3,4,5,6,7…) 2d sin  n

FCC: 3,4,8,11,12,16, 19, 24…. 2 sin 2   (h2  k 2  l 2 ) 4a2

6 Chem 253, UC, Berkeley What we will see in XRD of simple cubic, BCC, FCC?

2 sin  A 2  0.5 (BCC); 0.75 (FCC) sin  B

a dhkl  h2  k 2  l 2

Chem 253, UC, Berkeley Ex: An element, BCC or FCC, shows diffraction peaks at 2: 40, 58, 73, 86.8,100.4 and 114.7. Determine:(a) ?(b) Lattice constant? (c) What is the element? 2theta thetasin 2  h2  k 2  l 2 (hkl) 40 20 0.117 1 (110) 58 29 0.235 2 (200) 73 36.5 0.3538 3 (211) 86.8 43.4 0.4721 4 (220) 100.4 50.2 0.5903 5 (310) 114.7 57.35 0.7090 6 (222)

a =3.18 Å, BCC,  W

7 Chem 253, UC, Berkeley Reading : West Chapter 5

Polyatomic Structures

n  K  ha  kb  lc S  f (k)ei Kd j k  j     j1 d j  x a y b z c

fj: atomic scattering factor  No. of electrons n n iKd j 2i(hxkylz) Sk   f je   f je j1 j1

Chem 253, UC, Berkeley

Simple Cubic Lattice

Caesium (CsCl) is primitive cubic Cs (0,0,0) Cl (1/2,1/2,1/2) i (hkl) Sk  fCs  fCle

What about CsI?

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(hkl) CsCl CsI (100)  (110)  (111)  (200)  (210)  (211)  (220)  (221)  (300)  (310)  (311) 

Chem 253, UC, Berkeley FCC Lattices (NaCl)

Na: (0,0,0)(0,1/2,1/2)(1/2,0,1/2)(1/2,1/2,0)

Add (1/2,1/2,1/2)

Cl: (1/2,1/2,1/2) (1/2,0,0)(0,1/2,0)(0,0,1/2)

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i (hk l) i (hk ) i (hl) i (lk ) Sk  [ f Na  fCle ][1 e  e  e ]

S=4 (fNa +fCl) when h, k, l, all even;

S=4(fNa-fCl) when h, k, l all odd S=0, otherwise

Chem 253, UC, Berkeley

(hkl) NaCl KCl (100) (110) (111)  (200)  (210) (211) (220)  (221) (300) (310) (311) 

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Chem 253, UC, Berkeley Lattice

11 Chem 253, UC, Berkeley i (hk ) i (hl) i (lk ) S fcc 1 e  e  e

FCC: Diamond Lattice S=4 when h+k, k+l, h+l all even (h,k, l all even/odd) SSi(hkl)=Sfcc(hkl)[1+exp(i/2)(h+k+l)]. S=0, otherwise

SSi(hkl) will be zero if the sum (h+k+l) is equal to 2 times an odd integer, such as (200), (222). [h+k+l=2(2n+1)]

SSi(hkl) will be non-zero if (1)(h,k,l) contains only even numbers and (2) the sum (h+k+l) is equal to 4 times an integer. [h+k+l=4n] Shkl will be non-zero if h,k, l all odd: 4( fsi  ifsi )

Chem 253, UC, Berkeley

Silicon Diffraction pattern

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Chem 253, UC, Berkeley

13 Chem 253, UC, Berkeley

Chem 253, UC, Berkeley Bond charges in covalent solid (1/8,1/8,1/8), (3/8,3/8,1/8),(1/8,3/8,3/8) and (3/8,1/8,3/8). Bond charges form a "crystal" with a fcc lattice with 4 "atoms" per unit cell. origin : (1/8,1/8,1/8). Bond charge position: (0,0,0), (1/4,1/4,0),(0,1/4,1/4) and (1/4,0,1/4).

Sbond-charge(hkl) = Sfcc(hkl)[1+exp(i/2)(h+k)+exp(i/2)(k+l)+exp(i/2)(h+l)].

For h=k=l=2:

Sbond-charge(222)=Sfcc(222)[1+3exp(i/2)4]=4Sfcc(222) non-zero!

n n iKd j 2i(hxkylz) Sk   f je   f je j1 j1

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Nanocrystal X-ray Diffraction

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Finite Size Effect

15 Chem 253, UC, Berkeley Bragg angle:  B in phase, constructive

For 1  B

1 2 Phase lag between two planes: 3 4  

j-1 j At j+1 th plane: j+1 Phase lag:    j   2j-1 2 2j

2j planes: net diffraction at  1 :0

Chem 253, UC, Berkeley

Bragg angle:  B

in phase, constructive

For 2  B

1 2 Phase lag between two planes: 3 4 5  

j-1 At j+1 th plane: j j+1 Phase lag:    j   2j-1 2 2j

2j planes: net diffraction at 2 :0

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How particle size influence the peak width of the diffraction beam.

Full width half maximum (FWHM)

Chem 253, UC, Berkeley

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The width of the diffraction peak is governed by # of crystal planes 2j. i.e. crystal thickness/size

Scherrer Formula: 0.9 t  B2  B2  B2 B cosB M S

BM: Measured peak width at half peak intensity (in radians) BS: Corresponding width for standard bulk materials (large grain size >200 nm)

Readily applied for crystal size of 5-50 nm.

Chem 253, UC, Berkeley

• Suppose =1.5 Å, d=1.0 Å, and =49°. Then for a crystal 1mmindiameter, the breath B, due to the small crystal effect alone, would be about 2x10-7 radian (10-5 degree), or too small to be observable. Such a crystal would contain some 107 parallel lattice planes of the spacing assumed above.

• However, if the crystal were only 500 Å thick,itwould contain only 500 planes, and the diffraction curve would be relatively broad, namely about 4x10-3 radian (0.2°), which is easily measurable.

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