Small-Angle, Elastic Scattering from Atoms
Electrons scattered by the electrostatic potential of the nucleus,screened by the electron cloud
Most important for TEM imaging & diffraction!
-elastic-coherent
(constant phase shift)(no significant energy loss)
-low-angle (forward)
Head-on elastic collision in 1-D
mM m
E
(at rest)
1 fE E
1 fE E 2 0fE
2 0fE Two possible outcomes:
forward scattering
back scattering
1 2
1 2f fE E E
COMmv v
m M
//elastic
1 f COM COM
vv v v v M m v
M m
//forward
//back
center-of-mass motion:
v
Grazing-incidence elastic collision forward scattering
M m : E1 f E, E2 f 0
-Less-massive electron forward scattered in grazing collision-Almost no kinetic energy transferred to atom-Electron energy essentially unchanged coherent
electron atom elastic : E E1 f E2 f
Nearly head-on elastic collision backscattering
-Less-massive electron backscattered in head-on collision-Kinetic energy transferred to atom-Electron loses energy incoherent
electron atom
M m : E1 f Eelastic : E E1 f E2 f
Plane waves: sinusoidal form
, cos 2 pt A t r k r v
k k 1
wavenumber:
We could write a plane wave as :
momentum: amplitude: phase: wave vector: phase velocity
: frequency: time
p
A
ft
kv
, cos 2t A f t r k r
or
20, e i f tt k rr 0 eiA
2, e i f tt k rr
1A
0 0 1
e cos sini i Euler relation:
Plane waves: complex exponential form
For an incident plane wave, we often normalize and pick the phase:
OperatorsConsider a plane wave:
hp k
20, e i f tt k rr
E hf
hp
p
ˆ , , ,t h t i t p r k r r
ˆ ˆ ˆx y z
x y z
ˆ , , ,E t hf t i tt
r r r
//de Broglie
//from photoelectric effect
//momentum operator
//gradient
//energy operator
ˆ i p
E it
Schrodinger equation
//Schrodinger equation
The SE uses energy to relate the time and space dependences of the wave function.
For a free, non-relativistic particle:2 2
2
0 0
ˆˆ2 2pHm m
//Hamiltonian operator
ˆ , ,H t E t r rObserve:
2 2 22
2 2 2x y z
2 2
2
0 0
ˆˆ2 2pH U Um m
r r
In a potential:
2
02pEm
//plane wave is an energy eigenfunction
20, e i f tt k rr
//Laplacian
ˆ , ,H t i tt
r r
//plane wave
Energy eigenstates
The time-independent part satisfies the time-independent SE:
20, e ei f t iE tt k rr r
Plane wave:
These are energy eigenstates, i.e., states with constant energy.
ˆ e iE tH r e iE tE r
H E r r
ˆ , ,H t i tt
r rSchrodinger Equation:
20 e i k rrSo our plane wave is described completely as:
Time-independent Hamiltonian-acts on r only:
Energy operator-acts on t only:
Spherical waves
Surface area of a sphere:
I 2 *1r2
2e ikr
r
No unique direction of wavevector
Intensity= PowerArea
Intensity of wave function probability density
24 r
Atomic Scattering FactorSpherical Scattered Wave
2e ii
k rr 2e ikr
sc fr
r
f : atomic scattering (form) factor
incident wave: scattered wave:
Amplitude depends on scattering angle.
Units of Length
Elastic scattering: k k k
Increases with Z
Smoothly varying functions of scattering angle.
Weak Phase-Object Approximation
e cos sin 1if i i i i sci i i
Assume the only effect of scattering is a slight change in phase of the wave function:
f r i r isc r Weak-phase scattering by atoms:
2
2 ee ikr
if i f
r
k rr
final initial
//Scattered wave is spherical
90° phase shift
small angle
Solid-angle projections
sind d d
Differential solid angle:2
02 sind d d
Annular differential solid angle:
Differential cross section
1sin
d dd d d
Differential Cross Section:
Axial symmetry (atoms):
12 sin
ddd d
Common Form:
scattering cross-section per unit solid angle
2
0sin dd d d
d
d dd d
2 sin dd dd
Total cross-section forms
2 sin ddd
ddd
Total scattering into angles greater than :
0
0
2 sin ddd
ddd
Total scattering into angles less than :
2
0 0sin
tot
tot
d
ddd
dd dd
0
0
2 sintot
tot
ddd
ddd
Azimuthal symmetry:
Cross-section problems
0
2 sind dd
dd
d
d
12 sin
d dd d
cos2
d Ad
0
2
0
3
0
3
2 cos sin2
4 sin cos2 2
4 sin3 2
4 sin3 2
A d
A d
A
A
Example:Example:
sin2tot
sin2 sin 2
1 cos2 sin 2 2
8 sin2
tot
tot
tot
d dd d
dd
One Type of Problem: Another Type of Problem:
tot Note:
Interpreting differential cross-section
Uniform, hard sphere
Top viewSide view
Scattering cross-section: hard sphere (I)
180 2 90 2
cos cos 2
sin 22
b R Rdb Rd
d db b d
2
2 sin sin2
d d Rd d
sind d d
sind b dbd d
2cos 2 sin 2sin 2 4
d R R Rd
2 22
0 0
sin4tot
R d d R
2tot R
Scattering into angles less than
as expected!
Relate angles:
Differential elements:
Differential cross-section:
In terms of only:
Azimuthal () sum (annulus):
0
2
0
202
0
2
sin2
cos cos2 2
1 cos2
ddd
Rd
R R
R
0 30 60 90 120 150 180
Scattering Cross-Section: Hard Sphere (II)
0 30 60 90 120 150 180
2
2R
0
dd
2R
0
tot
largest solid angle
tot
Electric current in scattered wave (I)
* *
2sc sc sc sc sciej
m
2 *, ,sc sc sc scn t e t e r r
* * *,sc sc sc sc sc sc scn t e et t t t
r
22 21
2 2i
t i m m
2 * * 2
* *
,2
2
sc sc sc sc sc
sc sc sc sc
in x t et m
iem
** 2 *
2i
t t m
,sc scn x t jt
//electron "concentration"
//time rate of change
//use Schodinger's eqn.
//result
//continuity eqn. //current
20 e i
i k rr
2
0e ikr
sc fr
r
20
1volumescn r
02 2
ˆ ˆscsc
dI j djr d r d
r r
0scdI j d
20 0j e v
202
ˆscjj fr
r
2d fd
//incident plane wave
//scattered wave
//units
//incident current density
//scattered current density
//electric current in scattering cross-sectional area
//scattered current density
Electric current in scattered wave (II)
2
01 eˆˆ2
ikr
scdfif ik i
r d r
r θ
//gradient
Scattering Amplitude
2 2jj j
k k r
2
1e j
Ni
jj
f F
k k rk k
Phase difference:
Consider the interference between the incident wave and a scattered wave:
Continuous medium:
The scattering amplitude is the Fourier transform of the target scattering strength.
2
1, e j
Ni
jj
f F
k k
Scattering amplitude:
2 3e if F d r k k r
rk k r
Rewrite:
ˆ ˆj j k k r
Path-length difference:
Scattering Amplitude (Atomic Form Factor)
k k
k k
2e if F d k k r
rk k r r
2
0
sin 44
4r
srf s r F r dr
sr
Atom at origin, spherically symmetric:
Define:
F F rr
//scattering parameter
//scattering amplitude
s
sin 22
s
k k
2sin 21 1 1 2cos
k k
Relate to scattering angle:
Computing Atomic Scattering Factors
X-ray scattering factor is Fourier transform of electron density.
2
0
sin 44
4X r
srf s r r dr
sr
Electron scattering factor is proportional to the Fourier transform of the electrostatic potential of the atom:
22
2 0
sin 484e r
srmef s r r drh sr
The two are closely related: 2X
e
Z f sf s
s
2
00 0
44 4
r
r
Ze er r r drr r
2
2 meF r rh
//scattering strength at radius r
//scattering amplitude at s
//atomic potential
nucleus electron density
Electrostatic Potential of a Neutral Atom
r Ze40r
Bare Nuclear Potential:
r Zeff r e
40rScreened Nuclear Potential:
“Effective” Charge:
0
0
e4
r rZerr
Model of Screened Nuclear Potential:
023
04 1 e
re r renc r r
Z r d r r r dr Z
r
r
eeff encZ r Z Z r
Enclosed Electron Charge:
2
2 220 0
2 14 1
eZe mf s
h s r
0
0
e4
r rZerr
2
2 20 0
lim8e
Ze mf sr h s
// Form factor for unscreened (bare) nucleus
Rutherford (Thomas-Fermi) Model
0
0
4 sin 21r
sin 2
s
2 2
2 2 20 0
18 sin 2 sin 2e
Ze mfh
// Form factor
Assume:
In terms of scattering angle:
Rutherford Model (II)
2
2 2 0sin sin2 2
e
c
ZfR
2
2 0
0sin
2
e
c
ZfR
Screening keeps form factorfinite at origin:
2
0
12 137
ehc
2 2
20
1 18 4 4.2 nmc
e m mcR h hc
//fine structure constant
0
10.01 nm
Zr
Evaluating Form Factors
3 or 4
2
1expe i i
if s a b s c
Theoretically calculated potentials have been fit to functions of the form:
Doyle & Turner, Acta Cryst. (1968) A 24, 390Note: point-charge correction added to ionic potential to eliminate infinities at origin
@E=0
Bragg’s Law
2 sin Bd n
2 sin Bd
The n is optional:
Crystal Structure Factor
2
atoms2
mim
mF f q e q dq
Sum of atomic form factors for constituent atoms with appropriate phase factors for lattice positions
sin2 Bq
q
sin2
Bqs
q k - kExample:
1k
k k
2 B
