introduction to diffraction i-ii

8/8/2019 Introduction to Diffraction I-II

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Scattering from Two

CentersInterference

b

r

1

2

ko

incident wave

scattered waves


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Total Wave Function

,For


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Differential Cross Section

Define , an outgoing wave in the radial direction

and which is the change in the wave vector

then

When there a many scatterers

the scattering amplitude is

and the differential scattering cross section is

.


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Bragg ScatteringBy all thats right I should introduce the notions of coherent

and incoherent scattering at this point. But in theinterest of getting to the major points, I temporarily skipover this.

We now consider the interference effects due to the

structure of the material. Let the atoms be arranged on aregular 3-dimensional lattice, a crystal

are constant vectors called the

lattice basis vectors,

are a set of integersand


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A Cubic Crystal Lattice

x1

D

x2

x3

D

D


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Bragg ScatteringFor the moment, I will assume that all the atoms are

identical:

The sum on each ji extends over those values that

correspond to differences in the positions of two nuclei.

That is, we sum over all ji such that


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Bragg Scattering

We can evaluate the sum on j in the approximation that it

extends from minus infinity to plus infinity. It is a well known

identity that the sum of exponentials is a delta function:

where s is an integer, so.

What does this mean?


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The Laue ConditionsIfQ is equal to a vector

then clearly the conditions required by the

-functions are all satisfied, and for such a valueofQ, the cross section is singular.


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The Laue Conditions

The vectors

have special significance in the theory of crystals and arecalled reciprocal lattice vectors. The full set of these

represents a regular lattice, the reciprocal lattice. The

basis vectors of the reciprocal lattice are the set of vector

products above.

The scattering is singular when the wave vector change Q

is equal to a reciprocal lattice vectorthese are the Laue

Conditions,

Q = 2s


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Reflection Condition in

Reciprocal Space

k'ko

q

2!

!k'

reflecting plane

reflection vector


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Braggs Law

It is easy to see in the diagram that

so that according the the Laue condition

The reciprocal lattice vectors s1,s2,s3 are perpendicular toplanes passing through atomic positions in the crystal lattice.The magnitudes of the s is equal to the spacing of atomicplanes in the crystal lattice,


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Braggs Law

The magnitudes of the s is equal to the spacing of atomicplanes in the crystal lattice, (This is easiest to understandin a cubic crystal when all the ds are equal and perpendicular.)

Here, n is the common multiple of s1, s2, and s3

(if there is any other than n = 1) and p,q,r are the Millerindices of the planes corresponding to s1,s2,s3 ,so that{s1,s2,s3}= {np,nq,nr}.

Therefore becausefrom planes p,q,r

, one has, finally, for reflection

which is Braggs famous law of diffraction..


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Introduction to Diffraction Theory

II

Department of Physics

University of Rome Tor Vergata6 February 2006

J. M. CarpenterArgonne National Laboratory


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Diffraction Results for a Few

Cases

Here I work out results of diffraction for a few specific cases:

Polycrystals - ideal powder or polycrystal

differential cross section

total scattering

Static fluids - glasses and liquids

From here on, the wavevector change is q, not Q; they are the same just a

change in notation to save time.


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Diffraction from PowdersAn ideal powder or polycrystalline solid: many perfectly

randomly oriented crystallites, each small, yet largeenough to exhibit scattering as though from a large singlecrystal.

The cross section for this case is obtained by averaging overall the orientations ofs

The three-dimensional delta-function is


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q

!

"

#s= #

s$#

The angles that define the direction ofs

Carrying out the integral produces the result

in which note that the wavevector change appears as a scalar.


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Powder Diffraction Differential

Cross Section

d!d"

(Q)

Q


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Total Cross Section for a

PowderWhile were at this point, lets calculate the total

scattering cross section for a polycrystalline material.

The total cross section is the integral of the differentialcross section over the directions of k


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Coordinate systemko

!'

"'

k'

The coordinate system for expressing the integration over the

directions of the final scattering vector k


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The cross section depends on the direction ofk through the

momentum transferq

Substituting

The delta-function in terms of the new variable x is


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Total Scattering Cross Section

Because the range ofx is

the delta function is satisfied only when

that is, or,

This last form corresponds to the Bragg condition for exactbackward scattering, 2= 180.


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Total Scattering Cross Sectionfor Polycrystals and Powders

A compact form results in terms of the Heaviside

function

where the Heaviside function is


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Total Scattering Cross Section

for Polycrystals

ko

!total

0

The cross section consists of a series of steps at values

of k that correspond to backscattering from properly

oriented crystallitesthe Bragg edges.

The figure

includes the

absorption cross

section for

smallest ks andaccounts for

thermal motion

(Debye-

Waller Factor)

at high ks.


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Static Fluids

Now consider systems in which the atoms are fixed, but

their positions are not regularly arranged, as for example

the hypothetical model of a fluid in which the atoms do not

movethe static approximation or glasses, closer to

this model. The treatment is sufficiently general to includescattering in crystals for which the positions are exactly

known as discussed just previously.


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Static Fluids

Beginning again with the general expression

in which now the atomic position coordinates ri are at generallocations, not on a lattice, we note that

so that, taking the sum over atoms as equivalent to

averaging over all the atoms in the system


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Static FluidsDefine a pair density function g(r)

then

This is the Zernike-Prins formula, first derived for X-rayscattering. It is also related to the more general time-

dependent pair correlation function G(r,t), on which more

will be said later.


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Isotropic SystemsWhen the system is isotropic,

(not the same function because the one on the left is a

function of a vector argument, while that on the rightis a function of a scalarcomputers know the difference).

Then

The cross section (i.e. Q times it) is the Fourier sine

transform of the pair density function. Carrying out

the inversion of the transform numerically gives rg(r).)


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Pair Correlation Functions

Pair correlation functions derived from both x-ray and

neutron data characterize the short-to-medium range

order in liquids and glasses. It is important to note

that the procedure requires measurements carried

out to large values of Q so that the Fourier inversionconverges with good precision.

Rather recently scientists have applied the pair density

function (PDF) analysis to crystalline powders, forwhich the pair correlation functions consist of sharp

peaks rather than diffuse lines.


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introduction to diffraction i-ii

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