SCATTERING BY AN ATOM

Classical theory  Thomson Scattering Equation

I = Io (e4/m2c4R2) [(1+ cos2)/2]

(e4/m2c4R2) this term explains why the nucleus plays no part in x-ray diffraction  unmodified scattering

Also have quantum effect  Compton modified scattering

Detailed calculation of both of these, requires WAVE MECHANICAL TREATMENT.

We can simply use some of the rules of wave mechanics  calculate scattering from an atom.

Note:

(1) correct wave mechanical treatment for scattering from an electron shows that the sum of the
intensities of modified and unmodified scattering  classical intensity per electron

(2) classical treatment correctly predicts the polarization of the modified and unmodified scattering.

(3) Can conveniently express scattering intensity in ELECTRON UNITS. An intensity in ELECTRON
UNITS = Intensity from given sample / classical intensity from single electron

 consider scattering from several centers, e.g. a group of electrons confined to a small volume (e.g. an
atom).
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Electrons clustered around O, (e.g. nucleus), position of each electron .r.t. O is represented by a vector
rn.

Consider scattering at a point of observation P, at a large distance R from the electrons, in direction given
by unit vector S.

With reference to the wavefront through O, the instantaneous value of the field in the incident beam that
acts on the electron n is
o = E0 cos (2t - 2X1/)

 We can use this to obtain the magnitude and phase of scattered wave at P, due to electron n.
 multiply by the classical factor for an electron and take into account the total path length X1 + X2
Note: path lengths and differences become important now since we will be considering the scattering
from >1 electron (in the Thomson case path differences are not important since we only have 1 electron).

NB: Induced accelerations and induced scattered fields are all normal to the page (source, O and P are
coplanar, and Eo is perpendicular to page, therefore the sin  term =1.
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 Take phase into account, i.e. total
Remember =(qa sin )/(c2R) path length

n = e (e o /m) 1/c2X2

= (e2/(m c2X2)) Eo cos[2t - 2/ (X1 + X2)]

Consider that the source of x-rays and the point P are both at distances from O which are very large c. f.
rn   we can assume plane wave conditions

 in the denominator
X2  R
and in the cosine X1 + X2  rn  So + R - rn  S
exact Approx.
(actually X2.S=R-rn.S, but X2cos=X2 when  small)
= R - (S - So )  rn

 n= (e2/m c2x2) Eo cos[2t - 2/ (R - (S - So )  rn)]

 COMPLEX EXPONENTIAL FORM
n= (e2/m c2R) Eo e2i[t - (R - (S - So )  rn)/]
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 = (e2/m c2R) Eo e2i[t - (R/)] e(2i/) (S - So )  rn
SUM OF INSTANTANEOUS FIELDS AT P
 = (e2/m c2R) Eo e2i[t - (R/)] e(2i/) (S - So )  rn
Sum contributions for all
electrons located at the ends
of each rn

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Note: regarding approximation
From trigonometry (X2)2 = R2 + rn2 - 2 R rn cos

We can write,
(X2)2 / R2 = 1 + rn2 / R2 - 2 rn cos /R
X2 = R [1 + (rn2 / R2 - 2 rn cos /R)] 1/2
(1 + x)n = 1 + nx + n(n -1)x2/2! Etc

so that for R/rn >> 1, we can expand the square root and retain only the 1st 2 terms 
X2 = R [1 + 1/2 (rn2 / R2 - 2 rn cos /R)]

For R/rn >> 1, we can neglect the term in rn2/r2, and the path difference r - x2  rn cos = rn  s

This is equivalent to assuming the 2 scattered rays are parallel.

Typically R  0.1 m, and  approximation is valid for rn  105 Å = 10 m i.e. for samples where
coherent contributions to the amplitude arise from parts of sample by < 105 Å.

 now apply results of WAVE MECHANICAL treatment

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We consider each electron to be spread out into a diffuse cloud of charge ---- characterized by a charge
density  (in electron units)

 dv is the ratio of the charge in volume dv to the charge on 1 electron (in quantum mechanical
description,  would be the product of the electron wave function, e, with its complex conjugate e*)

Obviously   dv = 1 for each electron (i.e. total charge density = 1 electron unit)

 wavemechanical treatment says that the amplitude of unmodified scattering from the element  dv
=  dv x (amplitude of classical scattering from free electron)

 get total amplitude of unmodified scattering ---- integrate over volume occupied by the electron,
making allowances for the phase of the contribution from each element  dv.

So now we consider that we have charge elements  dv at positions r, instead of electrons  use our
earlier expression, replace the sum by an integral, i.e. we assume a continuous distribution of electron
density for a single electron – write expression similar to our earlier one, except for a single electron (i.e.
not n electrons) and introduce an integral term over our charge distribution

n = (e2/m c2R) Eo e2i[t - (R/)]  e(2i/) (S - So )  r  dv

instantaneous value of electric field for the unmodified scattering due to 1 of the electrons in an atom.

The quantity represented in the integral is called fe, THE SCATTERING FACTOR PER ELECTRON
fe =  e(2i/) (S - So )  r  dv

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fe is the ratio of the amplitude of unmodified scattering from one electron, to that scattered by an electron
according to classical theory.

 fe is the amplitude of unmodified scattering per electron, expressed in electron units.

For closed groups of inner electrons, we can assume spherical symmetry for the charge distribution

i.e.  =  (r) function of r only

Take the origin at the center of the atom  expression for fe

Clearly (S - So)  r = (2sin)(r) cos


fe =

 Integrate fe = over 

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ESM 512, Structure of Materials Lecture Notes
For atom containing several electrons, the amplitude of unmodified scattering per atom is the sum of the
amplitudes per electron.
i.e.
f = n fen = n

f is the ATOMIC SCATTERING FACTOR (or FORM FACTOR)

f = amplitude of unmodified scattering / atom expressed in electron units

Electron units ---- amplitude in units of the amplitude from a single electron according to classical theory.
i.e. f =

To compute f we need the radial dependence of the electron density in the atom n n(r).

For any atom f is a function of (sin/) (since it is a function of (4r sin/)).

For sin/ small, the last term cancels, leaving
n 4 r2 n (r) dr = Z, the total number of electrons in an atom.
(4 r2 n (r) = charge on surface of
r dependent charge sphere of radius r (in units of electron
Surface area of density
sphere charge))

So therefore, f approaches Z at small values of sin/.

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ESM 512, Structure of Materials Lecture Notes
Qualitatively
At 2 = 0, an atom containing Z electrons scatters a wave of amplitude = Z  (amplitude scattered
by single electron)

This is because the waves scattered by all the electrons are in phase  can add amplitudes.

This is not so for all other scattering directions  we have to take into account path differences between
waves scattered by different electrons.

Forward scattered waves ---- in phase on XX' (same distance traveled before + after scattering)

In the other direction, we have a path difference (CB - AD) ---- out of phase along YY’ , this path
difference being less than .

 partial interference between waves scattered by A + B  amplitude in AD direction is less than that in
forward direction.

Our ATOMIC SCATTERING FACTOR describes the "scattering EFFICIENCY" in a given direction.
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 2 = 0 f=Z

f decreases as  increases, since the waves scattered by individual electrons  more + more out of phase

Note:
At fixed , f will be smaller for shorter wavelengths, (path differences will be larger c.f. .  greater
interference, i.e. destructive).

We know that f decreases as sin/ increases.

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ESM 512, Structure of Materials Lecture Notes
Note:

For the INCOHERENT, Compton modified scattering we have different behavior----

The intensity of INCOHERENT relative to COHERENT scattered radiation, increases as the

proportion of LOOSELY BOUND ELECTRONS increases.
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ESM 512, Structure of Materials Lecture Notes
 intensity of Compton modified radiation, increases as Z decreases.

If also increases as sin/ increases, i.e. varies in opposite way to coherent (consequence of one of the
wave mechanical results, for each electron, considered individually (Intensity unmodified) + (Intensity
modified) = (Intensity of classical scattering per electron))

Note in this simple treatment of the ATOMIC SCATTERING FACTOR, we have 2 main assumptions

(1) X-ray  is much smaller than any of the absorption edge wavelengths in the atom
(2) The electron distribution in the atom has spherical symmetry.

With regard to (1), if this is not satisfied a so called "DISPERSION CORRECTION" IS NECESSARY
(TO CORRECT THE SO CALLED EFFECT OR ANOMALOUS DISPERSION)
We can write f = fo + f' + if''

f: corrected scattering factor

fo: tabulated value

f' and f'' are real + imaginary parts of the dispersion correction.

The angular dependence of f' and f'' is much smaller than that of fo.

With regard to (2),

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ESM 512, Structure of Materials Lecture Notes
 The lack of complete spherical symmetry in the electron distribution in C, allows weak Bragg
reflections from diamond which would otherwise not exist.

Prof. Michael Dudley, Dept. of Materials Science and Engineering, SUNY at Stony Brook 279
ESM 512, Structure of Materials Lecture Notes
Equation for scattering factor, assuming spherical symmetry

f = n fen = n 4 r2 (r) [sin(4r sin/)/(4r sin/)] dr

Obviously need expression for (r).

EXAMPLE: NEUTRAL LITHIUM

Each of the 3 electrons can be represented by a "hydrogen - like" expression of the form

 = e-(2r/a) /a3

where for each K electron a = aK = 0.2 Å and for the L electron a = aL = 1.6 Å

substitute, perform integration

 fLi = 2 feK + feL

Where feK = 1/ [1 + (2aKsin/)2]2 feL = 1/ [1 + (2aLsin/)2]2

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ESM 512, Structure of Materials Lecture Notes
ANALYTICAL EXPRESSION FOR SCATTERING FACTOR
FORM IS USEFUL FOR COMPUTER PROGRAMING

f (sin/) = ai exp (-bi sin2/2) + c ……

ai , bi and c are tabulated in "International Tables for X-ray Crystallography" Vol IV, pp 99-101

Original Quantum Mechanical calculations which are fitted are classified according to the different types of
wavefunctions used:
RHF - Relativistic Hartree – Fock
For neutral atoms, except H:
Doyle, P.A and Turner, P.S. Acta Cryst. A24, 390, (1968)
Coulthard, M.A. Proc. Phys. Soc, 91, 44 (1967).
*RHF - Relativistic Hartree – Fock
Cromer, D.T and Waber, J.T. Unpublished work, (1968).
Mann, J.B. Unpublished work.

HF – Nonrelativistic Hartree – Fock

For H and for ions below Rb:
Cromer, D.T. and Mann, J.B, Acta Cryst. A24, 321, (1968).

*DS – Modified relativistic Dirac-Slater

For ions above Rb:
Cromer, D.T and Waber, J.T. Unpublished work, (1968).

SDS Stewart, Davidson and Simpson

For H
Stewart, R.F., Davidson, E.R., and Simpson, W.T., J. Chem. Phys., 42, 3175, (1965).

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SCATTERING BY A UNIT CELL
 We want intensity of diffracted beam from a CRYSTAL
 consider coherent scattering from all the atoms in the crystal.

Since the crystal is composed of a repetition of the basic building block, the unit cell, it is enough to
consider the way in which the arrangement of atoms within a single unit cell affects the diffracted
intensity.

Waves scattered by the atoms within the unit cell are not necessarily in phase except in the forward
direction.

Consider first the variation in atom position in 1 Dimension; i.e. compute phase difference between wave
scattered by an atom at the origin and another atom where position is variable in the x-direction.

Section of orthogonal unit cell. Diffraction from (h00)

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ESM 512, Structure of Materials Lecture Notes
Bragg Law is satisfied;

Path difference between 2' and 1', 2'1', is

2'1' = MCN = 2 dhoo sin = 

From definition of Miller Indices

dhoo = AC = a/h

consider effect of atom B, at distance x from A (considering scattering in the same direction i.e. that for
which Bragg Law is satisfied).

Path difference between 3' and 1', 3'1', is less than  (i.e. 2'1'), by simple proportion
3'1'= RBS = AB/AC () = (x/(a/h)) ()

We can express the phase difference as an angle (instead of a multiple of )

1  360  2 (radians)

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path difference =  
phase difference, , in radians is
 = (/) 2

 phase difference between wave scattered by atom B and that by A at origin, is,
3'1'= (3'1'/) (2) = 2 hx/a

Specify atom B by its fractional coordinate u = x/a

 3'1'= 2 h u

 extend to 3 Dimensions

consider atom B with coordinates x, y, z

Fractional coordinates x/a, y/b, z/c = u, v, w respectively.

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Phase difference between wave scattered by B, and that scattered by A at the origin, for h k l reflection is:

 = 2 (h u + k v + l w)

(applies to unit cell of any shape)

Note these 2 waves will differ in phase, but they also may differ in amplitude if atom B is different from
A.
 Scattering from a unit cell, reduces to problem of adding waves of different phase and amplitude
together.

Add waves scattered by all the atoms of the unit cell (including the one at the origin)
 use exponential notation
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Amplitude of each wave is given by the appropriate value of “f” for the scattering atom, and the value of
sin/ involved.

The phase of each wave is given by

 = 2 (h u + k v + l w)
For h k l reflection, u v w fractional coordinates of atom.

We can express a scattered wave as

A ei = f e2i(h u + k v + l w)
The resultant wave, scattered by all the atoms of the unit cell is the STRUCTURE FACTOR (describes
how atom arrangement, u v w affects the scattered beam).

Structure Factor, F, is obtained by adding together all the waves scattered by the individual atoms.
e.g. if unit cell contains atoms
Fractional Coordinates. Atomic Scattering Factors
1 u1 v1 w1 f1
2 u2 v2 w2 f2
3 u3 v3 w3 f3

N uN vN wN fN
Structure Factor for h k l reflection is

F = f1 e2i(h u1 + k v1 + l w1) + f2 e2i(h u2 + k v2 + l w2) + f3 e2i(h u3 + k v3 + l w3)

Or Fhkl = fn e2i(h un + k vn + l wn)

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Summation over all the N atoms of the unit.

F in general is a complex number, expresses both amplitude and phase of resultant wave.

F  gives the amplitude of the resultant wave in term of the amplitude of the wave scattered by a single
electron (i.e. in "electron units" )

F  = amplitude of wave scattered by all atoms in unit cell / amplitude (classical) of wave scattered by 1
electron

Intensity of beam diffracted by all the atoms in the unit cell in the direction predicted by Bragg Law is
proportional to F 2

F 2 = F F* (F*: complex conjugate)

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CALCULATION OF THE STRUCTURE FACTOR
e.g. (1) consider unit cell containing only one atom at the origin  fractional coordinates 0, 0, 0

F = f e2i(0) = f
F2 = f 2
F2 is independent of h k l (same for all reflections)

(2) Base centered orthorhombic

2 atoms of the same kind per unit cell, located at 0 0 0 and 1/2 1/2 0

F = f e2i(0) + f e2i(h/2 + k/2)

= f [1 + ei(h + k) ]

note eni = (-1)n = real number (n = integer)

 in this case F is real and we don't have to multiply by the complex conjugate to get F 2.
 if h + k are both even or both odd (ie "unmixed" )
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 then the sum is always even
 ei(h +k ) = +1
 F = 2f h and k unmixed
2 2
F =4f
However if h and k are "mixed" (one even, one odd)
 ei(h +k ) = -1
F=0
F2 = 0

l has no effect on F F111 = F112 = F113= F114

F021 = F022 = F023 etc
F101= F102= F103 = 0

(3) Body centered orthorhombic cell (take for any body centered cell)
2 atoms of same kind located at 0 0 0 and 1/2, 1/2, 1/2

F = f e2i(0) + f e2i(h/2 + k/2 + l/2)

= f [1 + ei(h + k +l) ]

F = 2f for (h + k + l) even
F2 = 4 f 2

F=0 for (h + k + l) odd

2
F =0
(remember earlier e.g. of non existence of 001 reflection)
(4) Face Centered Cubic Cell

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 Contains 4 atoms of same kind (0, 0, 0) (1/2, 1/2, 0) (1/2, 0, 1/2) (0, 1/2, 1/2)

F = f e2i(0) + f e2i(h/2 + k/2 ) + f e2i(h/2 + l/2 ) + f e2i(k/2 + l/2 )

= f [1 + ei(h + k) + ei(h + l) + ei(k + l)]

 if h, k, and l are unmixed

then (h + k), (h + l), (l + k) are even integers

F = 4f unmixed indices
F2 = 16 f2

if h, k, l are mixed,  sum of 3 exponentials is -1

F=0
F2 = 0

(5) NaCl Structure

4 Na and 4 Cl atoms per unit cell

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 Na 000 1/2 1/2 0 1/2 0 1/2 0 1/2 1/2
Cl 1/21/21/2 0 0 1/2 0 1/2 0 1/2 0 0

F = fNa [1 + ei(h + k ) + ei(h + l) + ei(l + k )] + fCl [ ei(h + k ) + ei(h + kl) + ei(l + k )]

Factoring etc.

F = [1 + ei(h + k ) + ei(h + l) + ei(l + k )] [fNa + fCl ei(h + k + l ) ]

 define rules for allowable reflections

see Cullity

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