Order 1238142: Condensed matter

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Lect-7-Diffraction.pdf

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Higher Brillouin Zones

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Electronic Waves in Crystals in Three

Dimensions

Recall first Brillouin Zone (BZ) of

FCC Lattice

Can plot the dispersion

relation, E(k),

(“electronic band structure”)

of diamond which has an

fcc lattice with a diatomic basis.

Displaying a 3-D spectrum by plotting several single-line cuts through

reciprocal space.

Notice lowest band is quadratic at centre of BZ i.e.

Is the “effective mass” (more on that later) Recall, for a completely

free electron

Reciprocal Lattice as Fourier Transform

From Handout 3

The Reciprocal lattice can be though of as being a Fourier transform of the

direct lattice.

Consider one dimension, direct lattice

given as Rn=an with density of lattice points/atoms represented by delta-

function

Now =1 if i.e. if it is a point on the reciprocal lattice

if then terms oscillate and come to zero

Taking the Fourier transform gives

In two or three dimensions we have below where D represents the dimension

The Fourier Transform is a tool that breaks a waveform (a function or signal) into an alternate representation,

characterized by sine and cosines. The Fourier Transform shows that any waveform can be re-written as the

sum of sinusoidal functions.

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Reciprocal Lattice as Fourier Transform

From Handout 3 Is a D-dimensional delta function, e.g.,

e.g. electron density at atomic positions in the lattice

then

Now, let any point in space r be written as the sum of a lattice point R and a vector x within the unit cell r

Structure Factor Important for

later – Scattering!

Wave Scattering by Crystals X-ray diffraction

Neutron diffraction

Generic scattering experiment

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Diffraction of Waves by Crystals

• The structure of a crystal can be determined by studying the diffraction pattern of a beam of radiation incident on the crystal.

• Beam diffraction takes place only in certain specific directions, much as light is diffracted by a grating.

• By measuring the directions of the diffraction and the corresponding intensities, one obtains information concerning the crystal structure responsible for diffraction.

X-RAY

• X-rays were discovered in 1895 by the German

physicist Wilhelm Conrad

Röntgen and were so

named because their nature

was unknown at the time.

• He was awarded the Nobel prize for physics in 1901.

Wilhelm Conrad Röntgen

(1845-1923)

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X-RAY PROPERTIES

• X ray, invisible, highly penetrating electromagnetic radiation of much shorter wavelength (higher frequency) than visible light. The wavelength range for X rays is from about 10-8 m to about 10-11 m, the corresponding frequency range is from about 3 × 1016 Hz to about 3 × 1019 Hz.

X-ray diffraction • The crystals can be used to diffract X-rays (von Laue, 1912).

Laue condition/equation or

conservation of crystal momentum

G/2

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X-ray diffraction, von Laue description

source of x-rays, electrons, neutrons or whatever spherical waves;

The crystal far away, so use the plane waves

scattered wave k’ different direction but same length (since elastic) also kinematic:

only one scattering process

The Ewald construction

• Draw (cut through) the reciprocal lattice.

• Draw a k vector corresponding to the

incoming x-rays which

ends in a reciprocal

lattice point.

• Draw a circle around the origin of the k vector.

• The Laue condition is fulfilled for all vectors k’

for which the circle hits a

reciprocal lattice point.

Laue condition if G is a rec. lat. vec.

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Relation to lattice planes / Miller indices

The vector

is the normal vector to the lattice planes

with Miller indices (m,n,o)

The Bragg description (1912): specular reflection

• The Bragg condition for constructive interference holds for any number of layers

and this only works for

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X-ray diffraction: the Bragg description

• The X-rays penetrate deeply and many layers contribute to the reflected intensity

• The diffracted peak intensities are therefore very sharp (in angle)

Fermi’s Golden Rule

(know from your quantum mechanics)

Transition rate for particle scattering from k to k’ due to some potential V is:

(From Handout 4)

The matrix element is:

Can see this is just the Fourier transform of the potential. L is the linear dimension of the sample, and normalizes the wavefunctions

If the sample is periodic, then is a reciprocal lattice vector –

to see this write:

where R is a lattice vector position and x is a position within the unit cell.

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Look at working

Fermi’s Golden Rule

And using assumption potential is periodic

we get

As we found earlier, for such summations of the form , if k is a point of

the reciprocal lattice then is unity, but if k not a point on the reciprocal

lattice then the summands oscillate and sum comes out to be zero.

Thus, which is the Laue equation/condition which is a statement of the conservation of crystal momentum

When the waves leave

the crystal they should have

which is just the conservation of energy

Structure factor

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Equivalence of Laue and Bragg condition

The spacing between lattice planes is

From geometry we have

Suppose Laue condition satisfied: ,

is the wavelength

Can rewrite Laue condition as:

Now dot product with

Laue and Bragg condition equivalent

Equivalence of Laue and Bragg condition cont.

If there is a reciprocal lattice vector G,

there is also nG; and we would get

which is for reflecting of the

spacing

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Scattering Amplitude

Recall we had

called the Structure Factor

The intensity of scattering off the lattice planes defined by the

Reciprocal lattice vector (hkl) is proportional to the square of the

Structure factor at this reciprocal lattice vector

Neutrons and X-rays

Neutrons – uncharged, scatter from nuclei. Scattering potential

short-ranged, approximate as a delta-function

position of atom in the unit cell

Is the atomic form factor, representing scattering strength, and

Is proportional to the “nuclear scattering-length”

Then,

And substituting into the expression for the structure factor we have

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X-rays – scatter from the electrons in a system, take V(x) to be

Proportional to the electron density

Neutrons and X-rays

Where is the atomic number of atom j (number of electrons) and is a

short-ranged function (few angstroms, size of atom)

And substituting into the expression for the structure factor (effectively taking

the Fourier transform) we have

Where is the form factor (roughly proportional to but has some

dependence on magnitude of G the Fourier transform)

Neutron and X-rays

- Neutron scattering - can easily detect H atoms, can

distinguish atoms with similar atomic numbers (unlike

X-ray scattering)

- Neutrons have spin, so can detect whether electrons

in the unit cell have spins pointing up or down (unlike

X-ray scattering)

- X-rays can have smaller sample sizes;

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Structure Factor

fj form factor

Experiments measure the intensity

Consider some structure factors

Can be described as a simple cubic with a basis given by

Systematic Absences or Selection Rules

Consider pure Cs – replace the Cl in CsCl with another Cs atom

Simple cubic lattice with same basis as above:

Lets see

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Systematic Absences or Selection Rules

Simple cubic with a basis:

“selection rules”

Systematic Absences

Adv - check

this is true for homework

Zincblende structure – fcc with a basis: Zn at [0,0,0] [1/4,1/4,1/4] Consider it as a cubic lattice with basis of 8 atoms in the conventional unit cell

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x-ray/neutron diffraction in practice

Powder Diffraction

reciprocal lattice point

compare positions

with expected positions

calculate intensity

and compare intensity

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advanced X-ray diffraction • The position of the spots gives information about the

reciprocal lattice and thus the Bravais lattice.

• An intensity analysis can give information about the basis. • Even the structure of a very complicated basis can be

determined (proteins...) every spot

crystallize

protein

remember

Advanced x-ray sources: synchrotron radiation

• A highly collimated and monochromatic beam is needed for protein crystallography.

• This can only be provided by a synchrotron radiation source.

SPring-8, Japan Australian Synchrotron

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ANSTO: Australian

Centre for Neutron

Scattering