Diffraction

5.1 Fourier series

5.2 Fourier transform

5.3 Tetragonal reciprocal lattice

5.4 CsCl

5.5 Double hexagonal close-packed

5.6 Hydrogen atomic form factor

5.7 Structure factor

5.8 Chromium

5.9 LEED

5.10 CsCl diffraction

5.11 Nickel

5.12 Ewald's sphere

5.13 Diffraction peaks

5.14 ZnO

5.15 Aluminum

5.16 Powder diffraction

5.17 Fourier series of cylinders repeated on a Bravais lattice

5.18 Reciprocal lattice vectors

5.19 Interplanar separation

5.20 Ewald's sphere

5.21 Volume of a conventional unit cell

5.22 Diffraction

5.23 Ewald's sphere

5.24 Diamond

5.25 x-ray diffraction

5.26 Protein structure

Every periodic function has a Bravais lattice and a corresponding reciprocal lattice. The relations between the primitive lattice vectors in real space and in reciprocal space are,

\[ \begin{eqnarray} \vec{b}_1=2\pi \frac{\vec{a}_2\times\vec{a}_3}{\vec{a}_1\cdot\left(\vec{a}_2\times\vec{a}_3\right)}&\hspace{2cm}&\vec{a}_1=2\pi \frac{\vec{b}_2\times\vec{b}_3}{\vec{b}_1\cdot\left(\vec{b}_2\times\vec{b}_3\right)} \\ \vec{b}_2=2\pi \frac{\vec{a}_3\times\vec{a}_1}{\vec{a}_1\cdot\left(\vec{a}_2\times\vec{a}_3\right)}&\hspace{2cm}&\vec{a}_1=2\pi \frac{\vec{b}_3\times\vec{b}_1}{\vec{b}_1\cdot\left(\vec{b}_2\times\vec{b}_3\right)} \\ \vec{b}_3=2\pi \frac{\vec{a}_1\times\vec{a}_2}{\vec{a}_1\cdot\left(\vec{a}_2\times\vec{a}_3\right)}&\hspace{2cm}&\vec{a}_3=2\pi \frac{\vec{b}_1\times\vec{b}_2}{\vec{b}_1\cdot\left(\vec{b}_2\times\vec{b}_3\right)} \end{eqnarray} \]

The inner product of the primitive lattice vectors in real space and the primitive lattice vectors in reciprocal space satisfy,

$$\vec{a}_i\cdot\vec{b}_j=2\pi\delta_{ij},$$

where $\delta_{ij}$ is the Kronecker delta.

A reciprocal lattice vector can be written in terms of the primitive lattice vectors in reciprocal space,

\[ \begin{equation} \vec{G}=\nu_1\vec{b}_1+\nu_2\vec{b}_2+\nu_3\vec{b}_3, \end{equation} \]

where $\nu_1, \nu_2, \nu_3$ are integers. The reciprocal lattice vectors can be used to write a periodic function $f(\vec{r})$ as a Fourier series,

\[ \begin{equation} f(\vec{r})=\sum\limits_{\vec{G}}f_{\vec{G}}e^{i\vec{G}\cdot\vec{r}}. \end{equation} \]

Here $f_{\vec{G}}$ are complex coefficients called the structure factors.

For the exam you should be able to determine the reciprocal lattice vectors of any Bravais lattice and know that the reciprocal lattice of an orthorhombic lattice with lattice constants $(a,b,c)$ is also an orthorhombic lattice $(2\pi /a,2\pi /b,2\pi /c)$ and the reciprocal lattice of fcc is bcc and the reciprocal lattice of bcc is fcc.

5.1 Fourier series

The function $f(x,y)$ has a periodicity of $a$ in the $x$-direction and $b$ in the $y$-direction,

\[ \begin{equation} f(x,y)=3\sin \left(\frac{2\pi x}{a}\right)\cos \left(\frac{2\pi y}{b}-1\right). \end{equation} \]

This function can be expressed as a sum of complex exponentials,

\[ \begin{equation} f(x,y)=\sum\limits_{\vec{G}}f_{\vec{G}}e^{i(G_xx+G_yy)}. \end{equation} \]

Calculate Fourier coefficients, $f_{\vec{G}}$.

5.2 Fourier transform

Draw the Fourier transform of the function,

\[ \begin{equation} f(x,y)=3\cos (5x) +2\sin (3y). \end{equation} \]

You need to draw the amplitudes $f_{\vec{G}}$ at the positions of the reciprocal lattice vectors in reciprocal space. The amplitudes may be complex.

Reciprocal space can be divided into Brillouin zones. The first Brillouin zone is the set of points closer to the origin in reciprocal space than to any other reciprocal lattice vector. It is analogous to the Wigner-Seitz cell in real space. All points outside the first Brillouin zone can be reached by a vector in the first Brillouin zone plus a reciprocal lattice vector. You should be able to construct the first Brillouin zone of any reciprocal lattice.

5.3 Tetragonal reciprocal lattice

(a) Determine the reciprocal lattice of a tetragonal crystal a = b = 5 Å, c = 8 Å.

(b) Make a sketch to illustrate the unit cell in the real space and the first Brillouin zone in reciprocal space. Specify the lengths of the sides of the Brillouin zone. Label the directions in real space and reciprocal space.

Diffraction occurs when waves strike a periodic structure and the wavelength of the waves is shorter than the periodicity of the structure. Under these conditions, some of the waves will continue in the direction $\vec{k}$ of the primary beam and some will be scattered elastically to directions $\vec{k}'$ where the scattering vector is a reciprocal lattice vector,

\[ \begin{equation} \Delta \vec{k} = \vec{k}' -\vec{k} = \vec{G}. \end{equation} \]

This is called the diffraction condition and it is often used to determine crystal structures. In the experiment, a crystal is put in the primary beam of an x-ray diffractometer. For elastic scattering $|\vec{k}|=|\vec{k}'|=2\pi /\lambda$. By detecting the angles at which diffraction peaks are observed, it is possible to calculate the reciprocal lattice vectors from the diffraction condition. If many reciprocal lattice vectors are determined, it is possible to deduce the primitive lattice vectors in reciprocal space,

\[ \begin{equation} \vec{G}=\nu_1\vec{b}_1+\nu_2\vec{b}_2+\nu_3\vec{b}_3\hspace{1.5 cm}\nu_1,\nu_2,\nu_3 = \cdots ,-2,-1,0,1,2,\cdots \end{equation} \]

From the primitive lattice vectors in reciprocal space, it is possible to calculate the primitive lattice vectors in real space (the formulas are given at the top of this page). The primitive lattice vectors in real space determine the Bravais lattice and the volume of the unit cell in real space. The atoms in the basis can be determined by comparing the intensities of the diffraction peaks to the structure factors that appear in the Fourier series for the electron density,

\[ \begin{equation} n\left(\vec{r}\right)= \sum\limits_{\vec{T}} \sum\limits_j n_j\left(\vec{r}-\vec{r}_j+\vec{T}\right)=\sum\limits_{\vec{G}}n_{\vec{G}}e^{i\vec{G}\cdot\vec{r}}. \end{equation} \]

Here $\vec{T}$ are the translation vectors of the Bravais lattice, $n_j\left(\vec{r}\right)$ is the electron density of atom $j$ and $n_{\vec{G}}$ are the structure factors. The intensity of the diffraction peak $\vec{G}$ is proportional to $|n_{\vec{G}}|^2$. Since most electrons are core electrons, the electron density in a crystal is sharply peaked around the nuclei. A simple approximation for the electron density of a unit cell of a crystal is,

\begin{equation} n(\vec{r})=\sum \limits_j Z_j\delta(\vec{r}-\vec{r}_j). \end{equation}

Where $Z_j$ is the atomic number of atom $j$ and $\vec{r}_j$ is the position of atom $j$. In this case, the structure factors are,

\begin{equation} n_{\vec{G}}=\sum \limits_j Z_j\exp (-i\vec{G}\cdot\vec{r}_j), \end{equation}

The electron density of an atom can be described more accurately using the atomic form factors listed in the International Tables for Crystallography. The structure factor is calculated from the atomic form factors $f(G)$ as,

\begin{equation} n_{\vec{G}} = \sum\limits_j f_j\left(G\right)e^{-i\vec{G}\cdot\vec{r}_j} = \sum\limits_j f_j\left(G\right)\left(\cos\left(\vec{G}\cdot\vec{r}_j\right)-i\sin\left(\vec{G}\cdot\vec{r}_j\right)\right). \end{equation}

To determine how the atoms are arranged in the basis, typically you have to guess the arrangement of the atoms, calculate the resulting structure factors, and them compare them to the intensities of the diffraction peaks.

5.4 CsCl

A simple approximation for the electron density of an atom is the atomic number times a delta function $Z\delta (\vec{r})$. The atomic number $Z$ is the number of electrons that an atom has. In this approximation, the electron density of a crystal is,

$n(\vec{r}) = \sum \limits_{j,l,m,n}Z_i\delta(\vec{r}_j+l\vec{a}_1+m\vec{a}_2+n\vec{a}_3)$,

where $j$ sums over the atoms in the unit cell and the translation vector $\vec{T}_{lmn}=l\vec{a}_1+m\vec{a}_2+n\vec{a}_3$ repeats the unit cell everywhere in the crystal. Here $Z$ is the atomic number. It is the number of electrons in the atom.

(a) Write down the general expression for a 3-D periodic function in terms of a Fourier series.

(b) CsCl has a simple cubic Bravais lattice. The atomic numbers are $Z_{Cs} = 55$, $Z_{Cl} = 17$. What are the structure factors for $G_{000}$ and $G_{100}$?

5.5 Double hexagonal close-packed

The figure below shows a crystallographic unit cell of the double hexagonal close-packed (dhcp) structure of the element americium. The Bravais lattice is hexagonal with $\vec{a}_1 = a\hat{x}, \vec{a}_2 =\frac{a}{2}\hat{x} +\frac{\sqrt{3}a}{2}\hat{y}, ~\text{and}~ \vec{a}_3 = c\hat{z}$. The basis is (0 0 0), (2/3 2/3 1/4), (0 0 1/2) and (1/3 1/3 3/4).

(a) Find the reciprocal lattice vectors $\vec{G}$. Describe in words and sketch the reciprocal lattice.

(b) Sketch the first Brillouin zone. Give values for the important dimensions.

(c) Find the structure factors associated with the points 100, 001, and 120 of reciprocal lattice. Use the atomic form factor app to determine the atomic form factors. Check you results using the structure factor app.

5.6 Hydrogen atomic form factor

For the hydrogen atom in its ground state, the electron density is $n(r) = (\pi a_0^3)^{-1}\exp(-2r/a_0)$, where $a_0$ is the Bohr radius. Integrating this expression over all space yields 1 indicating that there is one electron.

\begin{equation} \int \limits_0^{\infty} 4\pi r^2 n(r)dr=1 \end{equation}

The atomic form factor is the three-dimensional Fourier transform of the electron density in [1,-1] notation. Show that the atomic form factor is $f_G = 16/(4 + G^2a_0^2)^2$.

5.7 Structure factor

A simple cubic crystal has one atom per unit cell. The electron density in a unit cell can be described by a Gaussian $n(r) = A \exp\left(-a^2r^2\right)$ Å-3. Here $a = 5\times 10^{10}$ and $r$ is measured in meters. The lattice constant is 0.2 nm. What is the ratio of the structure factors $S(111)/S(000)$?

The integral of $n(r)$ over all space is the number of electrons in this atom.

$$Z = A\int\limits_0^{\infty} 4\pi r^2 \exp(-a^2r^2)dr = \frac{\pi\sqrt{\pi}A}{a^3}.$$

The atomic form factor $f(q)$ is the Fourier transform of $n(r)$. You can find the Fourier transform of a Gaussian function in the table at the bottom of the page on Fourier transforms,

$$ f(q) = \frac{\pi\sqrt{\pi}A}{a^3} \exp\left(-\frac{q^2}{4a^2}\right)=Z \exp\left(-\frac{q^2}{4a^2}\right).$$

5.8 Chromium

Chromium forms a bcc lattice with a lattice constant of $a=2.91$ Å. There is one atom in the basis $\vec{B}_1 = (0,0,0)$. The primitive lattice vectors are,

$$\vec{a}_1=\frac{a}{2}(\hat{x}+\hat{y}-\hat{z}),\quad \vec{a}_2=\frac{a}{2}(-\hat{x}+\hat{y}+\hat{z}),\quad\vec{a}_3=\frac{a}{2}(\hat{x}-\hat{y}+\hat{z}).$$

It is also possible to describe this crystal using the cubic conventional lattice vectors,

$$\vec{a}_1=a\,\hat{x},\quad \vec{a}_2=a\,\hat{y},\quad \vec{a}_3=a\,\hat{z},$$

and two atoms in the basis $\vec{B}_1 = (0,0,0),\,\vec{B}_2 = (0.5,0.5,0.5)$. The labeling of the diffraction peaks with Laue indices $hkl$ are different for the two descriptions.

(a) Calculate the interplanar distance $d_{100}$, the structure factor $S_{100}$, and the intensity $I_{100}=S_{100}^2$ of the $100$ reflection using the conventional unit cell.

(b) Calculate the the interplanar distance $d_{110}$, the structure factor $S_{110}$, and the intensity $I_{110}$ of the $110$ reflection using the conventional unit cell.

(c) Calculate the the interplanar distance $d_{100}$, the structure factor $S_{100}$, and the intensity $I_{100}$ of the $100$ reflection using the primitive unit cell.

(d) Compare the Laue indices, interplanar distances, and intensities of the two descriptions.

The atomic form factor for Cr is plotted below.

You should be able to define: powder diffraction, neutron diffraction, LEED, and the Ewald sphere.

5.9 LEED

Explain how this LEED pattern could be used to determine the 2D Bravais lattice of atoms at a surface.

5.10 CsCl diffraction

This image shows the CsCl crystal structure. The lattice parameters of the conventional unit cell are given in the figure.

(a) What is the closest distance between a Cs atom and a Cl atom?

(b) Draw the (110) plane of CsCl.

(c) What are the primitive lattice vectors in reciprocal space?

$\vec{b}_1=$

$\vec{b}_2=$

$\vec{b}_3=$

(d) What is the length of the reciprocal lattice vector $\vec{G}_{111}$?

The atomic form factors for Cs and Cl are:

(e) The horizontal axis is the scattering vector $\vec{q}= \vec{k}'-\vec{k}$. What is the structure factor of $\vec{G}_{111}$?

5.11 Nickel

Determine the lattice parameter of Ni (fcc) from the fact that the Bragg angle of the (220) reflection is 38.2° and the wavelength of the X-rays is 1.54Å. The netplanes are indexed using the conventional unit cell.

5.12 Ewald's sphere

X-rays with an energy of 8 keV are used to analyze a simple orthorhombic lattice ($a=$ 0.6 nm, $b=$ 0.8 nm, $c=$ 0.4 nm). The x-rays propagate in the (001) plane and also the detector scans in the (001) plane. Draw Ewald's sphere in the (001) plane. Label the axes with the [100] and [010] directions.

Draw the reciprocal lattice together with the wave vector of the primary beam $\vec{k}$ as well as the wave vector of the scattered beam $\vec{k}'$ for the reflection 320.

5.13 Diffraction peaks

When electrons, X-rays, neutrons, or helium are diffracted by a crystal, the same diffraction peaks are observed in each case. However, the intensities of the peaks depend on the type of beam that is diffracted. Why do all of these beams produce the same diffraction peaks and why are the intensities different for the different beams?

5.14 ZnO

Zinc oxide (ZnO) crystallizes in three crystal structures: rocksalt, zincblende, and wurtzite. An x-ray diffraction measurement determines that the Bravais lattice is fcc and that the (111) peak is larger than the (200) peak. The diffraction peaks are indexed using the conventional unit cell. What is the crystal structure?

Get the atomic form factors from http://lampz.tugraz.at/~hadley/ss1/crystaldiffraction/atomicformfactors/formfactors.php

5.15 Aluminum

The powder diffraction data for aluminum is shown below.

      Aluminum
      CELL PARAMETERS:    4.0496   4.0496   4.0496   90.000   90.000   90.000
      SPACE GROUP: Fm3m      
      X-RAY WAVELENGTH:     1.541838
      Cell Volume:     66.409
      Density (g/cm3):      2.698
      MAX. ABS. INTENSITY / VOLUME**2:      34.61439413    
      RIR:      4.177
      RIR based on corundum from Acta Crystallographica A38 (1982) 733-739
               2-THETA      INTENSITY    D-SPACING   H   K   L   Multiplicity
                38.50        100.00        2.3380    1   1   1         8
                44.76         47.49        2.0248    2   0   0         6
                65.16         28.01        1.4317    2   2   0        12
                78.30         30.71        1.2210    3   1   1        24
                82.52          8.74        1.1690    2   2   2         8
================================================================================

What do the columns 2-THETA, INTENSITY, D-SPACING, H K L, mean?

Based on the powder diffraction data, determine the primitive reciprocal lattice vectors for aluminum.

5.16 Powder diffraction

(a) Explain the difference between powder diffraction and x-ray diffraction on single crystals.

(b) A powder of a crystalline material is investigated by x-ray powder diffraction using a wavelength of 0.229 nm. The crystalline material has a simple tetragonal crystal structure with lattice constants $a =$ 0.25 nm and $c =$ 0.3 nm. At which angles $2\theta$ are diffraction peaks expected?

5.17 Fourier series of cylinders repeated on a Bravais lattice

Cylinders with the cylindrical axis in the $z$ direction are arranged on a Bravais lattice. The cylinders have a radius $R$, a length $L$, and do not overlap. Consider a function $f(\vec{r})$ that has a value $C$ inside the cylinders and is zero outside the cylinders. The goal is to calculate the Fourier series for this function.

A three-dimensional periodic function can be described with a Fourier series of the form,

$$f(\vec{r})=\sum \limits_{\vec{G}} f_{\vec{G}}\exp \left(i\vec{G}\cdot\vec{r}\right),$$

where the Fourier coefficients are given by,

$$f_{\vec{G}} = \frac{1}{V_{\text{uc}}}\int\limits_{\text{unit cell}}f(\vec{r})\exp (-i\vec{G}\cdot \vec{r})d\vec{r}.$$

Here $V_{\text{uc}}$ is the volume of a unit cell (not the volume of a cylinder). The coefficient $f_{\vec{G}}$ is proportional to the Fourier transform of a single cylinder, evaluated at $\vec{G}$. For this problem, it is convenient to work in the [1,-1] notation where,

$$F_{1,-1}\left(\vec{k}\right)= \int f\left(\vec{r}\right)e^{-i\vec{k}\cdot\vec{r}}d\vec{r}.$$

A single cylinder centered at the origin can be defined as, $CH\left(z+\frac{L}{2}\right)H\left(\frac{L}{2}-z\right)H(R-\sqrt{x^2+y^2})$ where $H(x)$ is the Heaviside step function. From the table of Fourier transforms, we see that the Fourier transform of the square pulse function in the $z$ direction is,

$$\mathcal{F}\{H\left(z+\frac{L}{2}\right)H\left(\frac{L}{2}-z\right)\} = \frac{2\sin(Lk_z/2)}{k_z},$$

and the Fourier transform of the disc function is,

$$\mathcal{F}\{H(R-\sqrt{x^2+y^2})\} = \frac{2\pi R}{\sqrt{k_x^2+k_y^2}} J_1 \left(R\sqrt{k_x^2+k_y^2}\right),$$

where $J_1(x)$ is the first order Bessel function of the first kind. Now we come to a subtle point about Fourier transforms. The convolution theorem says that the Fourier transform of the product of two functions is the convolution of their transforms. The proof of the convolution theorem can be found here. This holds if the two functions have the same variable for their arguments, $\mathcal{F}\{f_1(x)f_2(x)\}=\mathcal{F}\{f_1\}*\mathcal{F}\{f_2\}$. However, if the arguments of the functions are different, the transform of their product is just the product of their transforms, $\mathcal{F}\{f_1(x) f_2(y)\}=\mathcal{F}\{f_1(x)\}\mathcal{F}\{f_2(y)\}$.

To show this, consider the Fourier transform of a product function $f_1(x)f_2(y)f_3(z)$,

$$\int \limits_{-\infty}^{\infty}f_1(x)f_2(y)f_3(z)e^{-i\vec{k}\cdot\vec{r}}d^3r \\= \int \limits_{-\infty}^{\infty}\int \limits_{-\infty}^{\infty}\int \limits_{-\infty}^{\infty}f_1(x)f_2(y)f_3(z)e^{-i(k_xx+k_yy+k_zz)}dxdydz\\=\int \limits_{-\infty}^{\infty}f_1(x)e^{-ik_xx}dx\int \limits_{-\infty}^{\infty}f_2(y)e^{-ik_yy}dy\int \limits_{-\infty}^{\infty}f_3(z)e^{-ik_zz}dz.$$

The Fourier transform of a cylinder is then,

$$C\frac{2\sin(Lk_z/2)}{ k_z}\frac{2\pi R}{\sqrt{k_x^2+k_y^2}} J_1 \left(R\sqrt{k_x^2+k_y^2}\right).$$

Use this expression to write the Fourier series of the function $f(\vec{r})$. Why is it convenient to use the [1,-1] notation in this case?

5.18 Reciprocal lattice vectors

Determine the reciprocal lattice of a monoclinic crystal $a = 5$ Å, $b = 8$ Å, $c = 25$ Å, and $\alpha = 98$°. Make a sketch to illustrate the unit cell in the real space and in reciprocal space. Give the lengths of the primitive reciprocal lattice vectors.

5.19 Interplanar separation

Consider a plane hkl in a crystal lattice.
(a) Prove that the reciprocal lattice vector $ \vec{G} = h\vec{b}_1+k\vec{b}_2+l\vec{b}_3 $ is perpendicular to the $(hkl)$ plane.
(b) The distance between two adjacent net planes is $ d_{hkl}=\frac {2\pi}{|\vec{G}|} $. Show for a simple cubic lattice that $ d_{hkl}^2 = \frac {a^2} {h^2+k^2+l^2} $.

5.20 Ewald's sphere

X-ray diffraction is performed on an orthorhombic crystal ($a=$2 nm, $b=3$ nm, $c=1$ nm). The primary x-ray beam propagates within the (001) plane and also the detector scans the (001) plane. The problem is solvable in two dimensions, i.e., Ewald's sphere becomes a circle.

Which reflections are visible in the diffraction pattern when the primary x-ray beam propagates
  (a) in the [010] direction and has a wavelength of 2.08 nm
  (b) in the [120] direction and has a wavelength of 2.28 nm

Hint: Please solve the problem graphically. The first part of the problem can be solved by drawing only the reciprocal lattice. For the second part of the problem, the real lattice as well as the reciprocal lattices have to be drawn, for simplification use the same origin for both lattices. The reason you need the real space lattice in (b) is that the direction of the x-ray beam is given in real space. An arrow from the origin to the 120 Bravais lattice vector in real space is the direction of the beam.

5.21 Volume of a conventional unit cell

A x-ray diffraction experiment is performed on a simple orthorombic crystal. The lengths of the reciprocal lattice vectors from a powder diffraction experiment are given below. What is the volume of the primitive unit cell in real space of this crystal?

5.22 Diffraction

A crystal has an orthorhombic structure with $ a =$ Å, $ b = $ Å and $ c = $ Å.

(a) Calculate the lengths of reciprocal primitive lattice vectors.

$|\vec{b}_1|$ = Å-1, $|\vec{b}_2|$ = Å-1, $|\vec{b}_3|$ = Å-1.

(b) A neutron beam with a kinetic energy of eV is diffracted from this sample. Neutrons at this energy can be treated non-relativistically, i.e. $ E_{kin} = \frac {\hbar ^2 k^2} {2m} $ where $ m = 1.67 \times 10^{-27} $ kg. Calculate the Bragg angle for the (111) peak.

$\theta$ = rad.

(c) What is the smallest possible nonvanishing Bragg angle for this crystal?

$ \theta_{\text{min}} $ = rad.

5.23 Ewald's sphere

Diffraction can occur whenever the diffraction condition, $\vec{k}' -\vec{k} =\vec{G}$, is satisfied. Here $\vec{k}$ is the wave vector of the incoming waves, $\vec{k}'$ is the wave vector of the scattered wave, and $\vec{G}$ is a reciprocal lattice vector. The diffraction condition can be rewritten as $\hbar\vec{k}' -\hbar\vec{k} =\hbar\vec{G}$. This is a statement of the conservation of momentum. For elastic scattering, energy is conserved so the wavelength of the scattered photon is the same as the incoming photon, $|\vec{k}|=|\vec{k}'|$. Diffraction can only occur for $2|\vec{k}| \gt |\vec{G}|$ so there are only a finite number of diffraction peaks observable. The diffraction condition can be visualized with an Ewald construction as illustrated below. You draw the incoming wavevector $\vec{k}$ so that it ends at the origin of reciprocal space and draw a sphere around the beginning of this vector with a radius $|\vec{k}|$. When you rotate the crystal it will rotate the points in reciprocal space and whenever a reciprocal lattice point lies on the sphere, the diffraction condition will be satisfied. In the simulation below, the crystal is aligned so that the $(hkl)$ plane is parallel to the beam. The crystal is then rotated around the normal to the $(hkl)$ plane. Since $\vec{G}_{hkl}$ is normal to the $(hkl)$ plane, the reciprocal lattice points are rotated around $\vec{G}_{hkl}$. When diffraction occurs, both $\vec{k}$ and $\vec{k}'$ will lie on the Ewald sphere. The angle between $\vec{k}$ and $\vec{k}'$ is $2\theta$. Press the and buttons to rotate the crystal.

 Primitive lattice vectors:

 $\vec{a}_1=$ $\hat{x}+$ $\hat{y}+$ $\hat{z}$ [Å] 
 $\vec{a}_2=$ $\hat{x}+$ $\hat{y}+$ $\hat{z}$ [Å]
 $\vec{a}_3=$ $\hat{x}+$ $\hat{y}+$ $\hat{z}$ [Å]

 The x-rays scan the $hkl$ plane.
$h=$ $k=$ $l=$ 

X-ray wavelength λ = [Å]
= 1.540598 Å
= 2.28975 Å
= 0.7093165 Å


Reciprocal lattice points in the in the ( ) plane

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Primitive reciprocal lattice vectors

$\vec{b}_1=$ [ , , ] [Å-1]
$\vec{b}_2=$ [ , , ] [Å-1]
$\vec{b}_3=$ [ , , ] [Å-1]

$\phi$

Use this simulation to determine the diffraction angle $2\theta$ for the 3 2 1 reflection of spinel using Cu K$\alpha$ radiation. You will need to specify an $hkl$ plane where $\vec{G}_{hkl}$ is perpendicular to $\vec{G}_{3\,\overline{2}\,\overline{1}}$, otherwise the point $3\,\overline{2}\,\overline{1}$ won't appear (hint: 3 - 2 - 1 = 0). Rotate the crystal to get the 3 2 1 reflection on the Ewald sphere and read off the angle. The choice of the $hkl$ plane is not unique. Show that you can get the same angle for another choice of $\vec{G}_{hkl}$.

5.24 Diamond

X-rays with a wavelength of 0.5 Å are used to analyze a crystal with a diamond crystal structure with a lattice constant of 3 Å.

The diffraction peaks are labeled using the conventional (cubic) unit cell. If the primary beam is directed in the positive $x$-direction, what are $\vec{k}$, $\vec{k}'$ and $\vec{G}$ for the 212 reflection?

$\vec{k}=$ $\hat{k}_x +$ $\hat{k}_y +$ $\hat{k}_z $ 1/m

$\vec{G}$ = $\hat{k}_x +$ $\hat{k}_y +$ $\hat{k}_z $ 1/m

$\vec{k}'=$ $\hat{k}_x +$ $\hat{k}_y +$ $\hat{k}_z $ 1/m

5.25 x-ray diffraction

$\Delta k_x$

$\Delta k_y$

$\Delta k_z$

Intensity

0

0

0

1

-1.5E10

0

0

0.229

0

-1.5E10

0

0.229

0

0

-1.5E10

0.229

0

0

1.5E10

0.229

0

1.5E10

0

0.229

1.5E10

0

0

0.229

-1.5E10

-1.5E10

0

0.667

-1.5E10

0

-1.5E10

0.667

-1.5E10

0

1.5E10

0.667

-1.5E10

1.5E10

0

0.667

0

-1.5E10

-1.5E10

0.667

0

-1.5E10

1.5E10

0.667

0

1.5E10

-1.5E10

0.667

0

1.5E10

1.5E10

0.667

1.5E10

-1.5E10

0

0.667

1.5E10

0

-1.5E10

0.667

1.5E10

0

1.5E10

0.667

1.5E10

1.5E10

0

0.667

-1.5E10

-1.5E10

-1.5E10

0.192

-1.5E10

-1.5E10

1.5E10

0.192

-1.5E10

1.5E10

-1.5E10

0.192

1.5E10

1.5E10

1.5E10

0.192

1.5E10

-1.5E10

-1.5E10

0.192

1.5E10

-1.5E10

1.5E10

0.192

1.5E10

1.5E10

-1.5E10

0.192

1.5E10

1.5E10

1.5E10

0.192

The table to the right shows the intensity of diffraction peaks that were measured in an x-ray diffraction experiment on a single crystal. The three components of the scattering vector $\Delta k$ are given in units of [1/m]. The intensities have been normalized to their largest value.

(a) What are the primitive lattice vectors of this crystal in real space?

(b) What Bravais lattice does this crystal have?

(c) How can you estimate the number of atoms in the basis? How many atoms do you estimate are in the basis of this crystal?

(d) How could you determine what atoms are in the basis and how they are arranged?

(e) Knowing how the atoms are arranged in the basis, how could you determine the point group of this crystal?

5.26 Protein structure

How is the shape of a protein molecule determined by x-ray diffraction?