Phys 446:

Solid State Physics / Optical

Properties

Fall 2015

Lecture 3 Andrei Sirenko, NJIT 1

Solid State Physics

Lecture 3
(Ch. 2)

Last week:
• Crystals, Crystal Lattice, Reciprocal Lattice,
Diffraction from crystals

• Today:
• Scattering factors and selection rules for
diffraction
• HW2 discussion

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 The Bragg Law
Conditions for a sharp peak in the
intensity of the scattered radiation:
1) the x-rays should be specularly
reflected by the atoms in one plane
2) the reflected rays from the
successive planes interfere constructively
The path difference between the two x-rays: 2d·sinθ 
the Bragg formula: 2d·sinθ = mλ
The model used to get the Bragg law are greatly oversimplified
(but it works!).
– It says nothing about intensity and width of x-ray diffraction peaks
– neglects differences in scattering from different atoms
– assumes single atom in every lattice point
– neglects distribution of charge around atoms
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Diffraction condition and reciprocal lattice

Von Laue approach:
– crystal is composed of identical atoms placed
at the lattice sites T
– each atom can reradiate the incident radiation
in all directions.
– Sharp peaks are observed only in the
directions for which the x-rays scattered from
all lattice points interfere constructively.

Consider two scatterers separated by a lattice vector T.

Incident x-rays: wavelength λ, wavevector k; |k| = k = 2/; k'k   T  2m
Assume elastic scattering: scattered x-rays have same energy (same λ) 
wavevector k' has the same magnitude |k'| = k = 2/  k  k'
k k' 
 

Condition of constructive interference: k'k  T  m or  k k'

Define k = k' - k - scattering wave vector

Then k = G , where G is defined as such a vector for which G·T = 2m
We got k = k' – k = G  |k'|2 = |k|2 + |G|2 +2k·G  G2 +2k·G = 0
2k·G = G2 – another expression for diffraction condition
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 Ewald Construction for Diffraction
Condition and reciprocal space

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Geometric interpretation of Laue condition:

2k·G = G2 

– Diffraction is the strongest (constructive interference) at the

perpendicular bisecting plane (Bragg plane) between two reciprocal
lattice points.
– true for any type of waves inside a crystal, including electrons.
– Note that in the original real lattice, these perpendicular bisecting
planes are the planes we use to construct Wigner-Seitz cell
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 Geometric interpretation of Laue condition:
2k·G = G2 

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Summary
 Various statements of the Bragg condition:
2d·sinθ = mλ ; k = G ; 2k·G = G2
 Reciprocal lattice is defined by primitive vectors:

 A reciprocal lattice vector has the form G = hb1 + kb2 + lb3

It is normal to (hkl) planes of direct lattice
 Only waves whose wave vector drawn from the origin
terminates on a surface of the Brillouin zone can be diffracted
by the crystal First BZ of bcc lattice First BZ of fcc lattice

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 Solid State Physics
Lecture 3 (continued)
(Ch. 2)

Atomic and structure factors

Experimental techniques:
Neutron and electron diffraction

Lecture 3 Andrei Sirenko, NJIT 9

Diffraction process:
1) Scattering by individual atoms
2) Mutual interference between scattered rays
Scattering from atom
2
Consider single electron. Plane wave u  Aei ( k r t ) k  k 

A
Scattered field: u '  f e ei ( kR t ) fe – scattering length of electron
R R – radial distance

Two electrons: u'  fe

A ikR
R

e 1  e ikr 
or, more generally
A ikR ik r1
u'  fe
R
e e 
 e ik r2 
A ikR
u '  f e e  e ik rl similar to single electron with
many electrons:
Lecture 3
R l f  f
Andrei Sirenko, NJIT
e ik rl e 
l
10

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 2

e
intensity: 2 ik rl
I ~ f  fe
l

2
this is for coherent scatterers. If random then I ~ Nf e
f e  1  cos 2 2  / 2  re
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Scattering length of electron:

e2 1
classical electron radius re   2.8 10 15 m
4 0 mc 2

In atom, f e  eik rl  f e  n(r )e ik rl d 3 r

l n(r) – electron density

f a   n(r )e ik rl d 3r - atomic scattering factor (form factor)

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Atomic scattering factor (dimensionless) is determined by

electronic distribution.
If n(r) is spherically symmetric, then

sin Δk  r 
r0

f a   4r 2 n(r ) dr
0
Δk  r

in forward scattering k = 0 so f a  4  r 2 n(r )dr  Z

Z - total number of electrons

Atomic factor for forward scattering is equal to the atomic Z number

(all rays are in phase, hence interfere constructively)

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 Scattering from crystal

crystal scattering factor: f cr   eik rl   f al e ik R l

l l
Rl - position of lth atom, fal - corresponding atomic factor

rewrite f cr  F  S

F   f aj e
ik s j - structure factor of the basis,
where
summation over the atoms in unit cell
j

S   e ik R l
c
- lattice factor, summation over all
and
unit cells in the crystal
l

Where R l  R lc  s j

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Since k = G,
S   e iG R l   e i 2m  N
c

the lattice factor becomes

l l

f cr  F  N  N  f aj e
iG s j
Then scattering intensity I ~ |fcr|2 where
j

G = Ghkl = hb1 + kb2 + lb3 if sj = uja1 + vja2 + wja3

F   f aj e   f aj e
i ( u j a1  v j a 2  w j a 3 )( hb1  kb 2  lb 3 ) 2i ( hu j  kv j  lw j )
Then
j j

structure factor structure factor

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 F
F F F F

F (h, k , l )  f (exp 0)  1 fa

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Example: structure factor of bcc lattice (identical atoms)

F   f aj e
2 i ( hu j  kv j  lw j )
structure factor
j
Two atoms per unit cell: s1 = (0,0,0); s2 = a(1/2,1/2,1/2)


F  f a 1  ei ( h  k  l ) 
 F=2fa if h+k+l is even, and F=0 if h+k+l is odd
Diffraction is absent for planes with odd sum of Miller indices

For allowed reflections in fcc lattice h,k,and l are all even or all odd
4 atoms in the basis.
What about simple cubic lattice ?

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Lecture 3 Andrei Sirenko, NJIT 17

hk l

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 F (h, k , l )
F (h, k , l )  f [1  exp(i ( h  k )  exp(i (h  l )  exp(i (k  l )]
F (h, k , l )
F
F

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 Low Energy Electron Diffraction (LEED)

= h/p = h/(2mE)1/2
E = 20 eV    2.7Å;
200 eV  0.87 Å
Small penetration depth (few tens of Å)
– surface analysis

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Reflection high Energy Electron Diffraction (RHEED)

• Glancing incidence: despite the high energy of the electrons

(5 – 100 keV), the component of the electron momentum
perpendicular to the surface is small
• Also small penetration into the sample – surface sensitive technique
• No advantages over LEED in terms of the quality of the diffraction
pattern
• However, the geometry of the experiment allows much better
access to the sample during observation of the diffraction pattern.
(important if want to make observations of the surface structure
during growth or simultaneously with other measurements
• Possible to monitor the atomic layer-by-atomic layer growth of
epitaxial films by monitoring oscillations in the intensity of the
diffracted beams in the RHEED pattern.

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MBE and Reflection high Energy Electron Diffraction (RHEED)

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Real time growth control by

Reflection High Energy Electron Diffraction (RHEED)
Growth start

RHEED Intensity (arb. un.)

(BaTiO3)8 (BaTiO3)8 (BaTiO3)8

(SrTiO3)4 (SrTiO3)4 (SrTiO3)4
Growth end

Ti shutter open

Ba shutter open Sr shutter open

0 200 400 600 800 1000 1200 1400

Lecture 3 Andrei Sirenko, NJIT Time (sec.) 26
110 azimuth

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 Neutron Diffraction
• = h/p = h/(2mE)1/2 mass much larger than electron 
  1Å  80 meV Thermal energy kT at room T: 25 meV
called "cold" or "thermal' neutrons
• Don't interact with electrons. Scattered by nuclei
• Better to resolve light atoms with small number of electrons, e.g.
Hydrogen
• Distinguish between isotopes (x-rays don't)
• Good to study lattice vibrations
Disadvantages:
• Need to use nuclear reactors as sources; much weaker intensity
compared to x-rays – need to use large crystals
• Harder to detect
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Summary
 Diffraction amplitude is determined by a product of several
factors: atomic form factor, structural factor
 Atomic scattering factor (form factor):
reflects distribution of electronic cloud. f a   n(r )e ik rl d 3 r
sin Δk  r 
In case of spherical distribution r0

f a   4r 2 n(r ) dr
0
Δk  r
Atomic factor decreases with increasing scattering angle

F   f aj e
 Structure factor 2i ( hu j  kv j  lw j )

j
where the summation is over all atoms in unit cell
 Neutron diffraction – "cold neutrons" - interaction with atomic
nuclei, not electrons
 Electron diffraction – surface characterization technique
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