Dynamical Diffraction

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Dynamical Diffraction. Boyu Liu Mar. 30 th. The main principle of Dynamical scattering: An electron beam can be strongly scattered by a set of planes of atoms.

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### Dynamical Diffraction

Boyu Liu

Mar. 30th

The main principle of Dynamical scattering: An electron beam can be strongly scattered by a set of planes of atoms.

This diffracted beam can then be rediffracted by a second set of planes. This repeated scattering between the diffracted beams and the direct beam is the persistent topic of this part.

So we call this repeated diffraction “dynamical diffraction”.

Kinematical Diffraction

Dynamical Diffraction

Scattering equation for a unit cell

Time-independent Schrodinger equation

We want to know the intensities of every diffracted beams.

So we need to know the behavior of a electron which is living in a crystal.

So we need to solve the Schrodingerequations, a wave function which describes the electrons. And the solution is the so called Bloch Waves.

But the Bloch Waves are rather mathematical and pure theoretical physics. It’s difficult to comprehend.

So we need a pictorial representation to draw a picture for the Bloch Waves, then for the diffracted beams. It is the so called dispersion surface.

Diffracted beams, Bloch Waves and dispersion surface are the three topics in today’s report.

Our aim is:

Obtain the intensity of the diffracted beam:

From the amplitude of the electron beam scattered by a unit cell

Σ is over all i atoms in the unit cell. θis the angle between the diffracted beam and incident beam. f(θ)is the atomic scattering factor. r is the distance from a point on the bottom of the specimen to the scattering center. ri defines the position of an atom in the unit cell.

A single unit cell:

We sum all the atoms in the unit cell and rename this sum as F(θ), the structure factor of the unit cell.

We sum all the unit cells in the specimen. φg is the amplitude in a diffracted beam. rn is the position of each unit cell. ξg is characteristic length. (mistake in 13.3)

Specimen:

The intensity is

We call this characteristic distance as extinction distance. It gives us a way of thinking about nearly all diffraction-contrast phenomena. It depends on the lattice parameters (through Vc), the atomic number (through Fg), and the kV used (through λ).

The “totally wave function”

Notebook

Using “two-beam approximation” to simplify the “wave function” by considering only on diffracted beam G.

1. two-beam approximation

Using the concept of the amplitude in a diffracted beam:

and replacing a by the short distance dz

Because of the dynamical process, the change in φg depends on the magnitude of both φg and φ0, and χO-χD is the change in wave vector as the φg beam scatters into the φ0 beam, similarly in the χD-χO.

Darwin-Howie-Whelan equations

r and s are both parallel to z

The main idea is that, The change in the amplitude of the direct beam and the diffracted beam depends on both beams.

φg and φ0 are dynamically coupled and are constantly changing.

Let’s kick the ξ0 out of our equations

Do the same to φ0 we got

φ0and φ0(sub) only differ by a phase factor, we will ignore the difference in calculating intensities since the amplitude is important rather than phase.

These two equations can be combined to give the second-order differential equation for φ0

z and s are geometric parameters, the nature of the material only enters through ξg.

Let’s go continuously to get the φg

Do you still remember our aim?

To obtain the intensity in the diffraction beam:

Notebook

• two-beam approximation;
• Two beams so two γ, w = sξ = cotβ, and γ(1)+γ(2)=s;

If we can solve the Howie-Whelan equations, then we can predict the intensities in the directbeam and diffractedbeam in two-beam case.

The solutions must have the form

then

If we want to know φ0, we should determine the amplitude C0 and the γ in phase. Since z is a distance in realspace, then γ must be a distance in reciprocalspace.

Let’s introduce new parameter w = sξ = cotβ.

Then we restrict the absolute magnitude of φg and φ0, then

and

The ratio of the amplitudes of the diffracted and direct beams, Cg to C0, depends on γ, and hence on s, the excitation error, and hence on how close the specimen is to the Bragg orientation.

Do you still remember our aim?

To obtain the intensity in the diffraction beam:

In next part we need to use the concept of BlochWaves. So we have to introduce BlochWave briefly.

So what’s b?

So what’s A?

The solutions to the Schrodinger equation (wave function) under periodic condition are Bloch waves.

Notebook

Notebook

• two-beam approximation;
• Two beams so two γ, w = sξ = cotβ, and γ(1)+γ(2)=s;
• two-beam approximation;
• Two beams so two γ, w = sξ = cotβ, and γ(1)+γ(2)=s;
• Each Bloch wave depends on only one k value

can be written as a product of a plane-wave ( ) and a Bloch

function which has the same period as its crystal lattice.

Therefore, the Bloch function can be expressed as a Fourier series,

Combining these definitions gives

We have two γ,

then two Co and Cg,

then two b(r)

A(1)and A(2) tell us the relative contributions of each Bloch wave

Each of these Bloch-wave functions could be a wave in the crystal ---each one depends on only one k value.

The “totally wave function”

The blue one must be the φg term because it depends on g. At the top of the specimen (r=0), φ0 is unity and φg is zero. Then

In two-beam case

From these three equations we could get

Where k(1) and k(2) are electron waves, K is equal to kD-kI

It is the first time we see the value of φg, we nearly reach the destination

Notebook

Comparing the exponential terms we know that k-K=γ, so

• two-beam approximation;
• Two beams so two γ, w = sξ = cotβ, and γ(1)+γ(2)=s;
• Each Bloch wave depends on only one k value;
• k-K=γ, Δk=k(1)-k(2)=γ(1)-γ(2)=√(s2+1/ξg2)

By using these two formulas,

We got

cos α-cos β=-2sin[(α+β)/2]·sin[(α-β)/2]

We can make this equation look more familiar by defining an effective excitation error, where

This is the really important equation for us.

Finally, we reach the destination.

Ig∝ sin2 and thus I0∝cos2.

Ig and I0 are both periodic in both t and seff.

As φg increases and decreases, φ0 behaves in a complimentary manner.

Dynamical Diffraction

Notebook

Our aim is:

• two-beam approximation;
• Two beams so two γ, w = sξ = cotβ, and γ(1)+γ(2)=s;
• Each Bloch wave depends on only one k value;
• k-K=γ, Δk=k(1)-k(2)=γ(1)-γ(2)=√(s2+1/ξg2);
• Dispersion relations: wavelength and frequency, wave vector and energy;

Obtain the dispersion relation of the electron of crystal in TEM

From the Time-independent Schrodinger equation

Dispersion relation describes the interrelation of wave properties like wavelength and frequency, hence wave vector and energy.

This analysis leads directly to one of the most important concepts used to understand images of defects in thin foils: it explains the origin of the extinction distance, ξg, so again you must persevere.

Let’s start in Time-independent Schrodinger equation:

Notebook

• two-beam approximation;
• Two beams so two γ, w = sξ = cotβ, and γ(1)+γ(2)=s;
• Each Bloch wave depends on only one k value;
• k-K=γ, Δk=k(1)-k(2)=γ(1)-γ(2)=√(s2+1/ξg2);
• Dispersion relations: wavelength and frequency, wave vector and energy;
• E is more larger than V, Ug is the amplitude of V in the Fourier series.

kinetic energy. The electron has a kinetic energy due to the acceleration which is given in the gun.

potential energy. When the electron passes through the crystal, it will have a potential energy due to the periodic potential associated with the atoms in the crystal.

E is the accelerating voltage, -E=100 kV – 1 MV.

total energy.

Since is periodic, we can express it as a Fourier series.

The potential inside the crystal, and it has special properties because we are only considering crystalline materials

The basic property of a crystal is that its inner potential is periodic.

The solutions to the Schrodinger equation (wave function) under periodic condition are Bloch waves.

Each Bloch wave ↔k(j), there will be more than one Bloch wave for a particular physical situation.

can be written as a product of a plane-wave ( ) and a Bloch function which has the same period as its crystal lattice.

Therefore, the Bloch function can be expressed as a Fourier series,

A(1)and A(2) tell us the relative contributions of each Bloch wave

Combining these definitions gives

We have two γ,

then two Co and Cg,

then two b(r)

If we have just two beams excited, O and G, the “wave function” is

Each Bloch wave is a sum over all the points in reciprocal space.

Each Bloch wave depends on every g, and conversely, each g beam depends on every Bloch wave.

Thanks Bethe for the wonderful work on the original analysis of electron diffraction.

(Thanks L. Huang for this two-beams diffraction pattern. )

Notebook

• two-beam approximation;
• Two beams so two γ, w = sξ = cotβ, and γ(1)+γ(2)=s;
• Each Bloch wave depends on only one k value;
• k-K=γ, Δk=k(1)-k(2)=γ(1)-γ(2)=√(s2+1/ξg2);
• Dispersion relations: wavelength and frequency, wave vector and energy;
• E is more larger than V, Ug is the amplitude of V in the Fourier series.
• , so as Ug;

In TEM

Therefore

Do you still remember our aim? To obtain the dispersion relation of the electron of crystal in TEM:

Do you still remember our aim? To obtain the dispersion relation of the electron of crystal in TEM:

Let’s simplify the situation by using two-beams approximation. So only two beams exit, the amplitudes are C0 and Cg. And let’s simplify the crystal condition by only allowing two components exsist in its Fourier series, Up and U-p.

make

make

Since k(j) can point in any direction, the dispersion relation defines a surface, known as the dispersion surface, which is just the locus of all allowed k(j) vectors for a particular fixed energy.

We set the determinant of the coefficients equal to 0

The inner potential of the crystal is <20V while the energy of the electrons from TEM gun is >100k eV. So the |k+p| and |k| are both very close to K, so the difference is important.

We can rename p to g to make it look more familiar.

Do you still remember our aim? To obtain the dispersion relation of the electron of crystal in TEM:

Finally we obtained the dispersion relation of the electron of crystal in TEM:

Good newsfor everyone

I decided to skip this section.

If we encounter problems

in further learning as follows,

We can come back to this section.

Two-beam case

Notebook

• two-beam approximation;
• Two beams so two γ, w = sξ = cotβ, and γ(1)+γ(2)=s;
• Each Bloch wave depends on only one k value;
• k-K=γ, Δk=k(1)-k(2)=γ(1)-γ(2)=√(s2+1/ξg2);
• Dispersion relations: wavelength and frequency, wave vector and energy;
• E is more larger than V, Ug is the amplitude of V in the Fourier series.
• so as Ug;

The electrons are in the vacuum, Ug=0, the dispersion equation has two solutions.

gap

The electrons are in the crystal, Ug≠0, the dispersion equation resembles a hyperbola.

A gap exists between the two branches, caused by the Ug. Ug is not zero because we have a periodic array of atoms.

We define the lines M1B and M2B and M1 and M2 to be tie lines because they tie together points on the difference branches of the dispersion surface.

Each of these tie lines is normal to the surface which produces it.

Notebook

• two-beam approximation;
• Two beams so two γ, w = sξ = cotβ, and γ(1)+γ(2)=s;
• Each Bloch wave depends on only one k value;
• k-K=γ, Δk=k(1)-k(2)=γ(1)-γ(2)=√(s2+1/ξg2);
• Dispersion relations: wavelength and frequency, wave vector and energy;
• E is more larger than V, Ug is the amplitude of V in the Fourier series.
• so as Ug;
• Each of these tie lines is normal to the surface which produces it.

Notebook

• two-beam approximation;
• Two beams so two γ, w = sξ = cotβ, and γ(1)+γ(2)=s;
• Each Bloch wave depends on only one k value;
• k-K=γ, Δk=k(1)-k(2)=γ(1)-γ(2)=√(s2+1/ξg2);
• Dispersion relations: wavelength and frequency, wave vector and energy;
• E is more larger than V, Ug is the amplitude of V in the Fourier series.
• so as Ug;
• Each of these tie lines is normal to the surface which produces it.

Notebook

• two-beam approximation;
• Two beams so two γ, w = sξ = cotβ, and γ(1)+γ(2)=s;
• Each Bloch wave depends on only one k value;
• k-K=γ, Δk=k(1)-k(1)=γ(1)-γ(2)=√(s2+1/ξg2);
• Dispersion relations: wavelength and frequency, wave vector and energy;
• E is more larger than V, Ug is the amplitude of V in the Fourier series.
• so as Ug;
• Each of these tie lines is normal to the surface which produces it.

So we have two spots at O and two spots at G. The wedge specimen split the spots at O and G.

Notebook

• two-beam approximation;
• Two beams so two γ, w = sξ = cotβ, and γ(1)+γ(2)=s;
• Each Bloch wave depends on only one k value;
• k-K=γ, Δk=k(1)-k(1)=γ(1)-γ(2)=√(s2+1/ξg2);
• Dispersion relations: wavelength and frequency, wave vector and energy;
• E is more larger than V, Ug is the amplitude of V in the Fourier series.
• so as Ug;
• Each of these tie lines is normal to the surface which produces it.

An inclined planar defect

When a defect is present, energy is transferred from one Bloch wave to the other along the tie line; this is the interband scattering. The defect splits the spots.

2G sphere

G sphere

O sphere

G sphere

2G sphere

BZB

BZB

BZB

BZB

BZB

BZB

BZB

BZB

The original reason for introducing the intensity of is that what comes into our eyes is the intensity of the electron beams. The difference of intensity in every position on the screen forms the contrast and the diffraction pattern.

So we need to know the behavior of electrons which are traveling in our crystal.

The original reason for introducing the concept of Bloch waves is that only Bloch waves can exist in a periodic potential, no beams in the crystal.

But the concept of Bloch waves is difficult to be accepted and understood.

The original reason for introducing the concept of dispersion surfaceis that it gives us a visualized picture of the Bloch waves, the k, and the energy.