X-ray diffraction patterns arise because many tiny “wave sources” within a crystal interfere, and we can model this by adding their complex amplitudes with the correct phases. Consider a monochromatic wave characterised by a (real) amplitude $A\ge0$ and a phase $\phi$. We define the complex amplitude $\tilde{A}$ as
$$\tilde{A} = A e^{i\phi},$$
so that the (real) amplitude of the wave is the magnitude (absolute value) of $\tilde{A}$, and $\phi$ is its phase. Thus, $\tilde{A}$ conveniently encodes both magnitude and phase in a single complex number. In this problem set, we define a dimensionless intensity by the squared magnitude of the total complex amplitude:
$$I = |\tilde{A}|^2 = A^2.$$
Common experimental and geometric proportionality factors, such as detector response, incident-beam normalisation, and common propagation factors, are absorbed into this definition. By contrast, the single-electron scattering amplitude $f_0$ is retained explicitly as the amplitude scale for scattering from a single point electron.
When a crystalline material is exposed to an incident wave, the wave is diffracted by the crystal lattice and the diffracted parts interfere with one another. The intensity of the resulting diffracted wave can be calculated by adding the complex amplitudes of the individual diffracted waves, taking into account the phase differences between them, and then computing the squared magnitude of the resulting total complex amplitude.
Diffraction primarily arises from interactions with electrons, and contributions from heavier particles such as the nuclei are typically negligible. The amplitude of the wave diffracted by a single point electron depends only on $R = |\mathbf{R}|$, the distance from the electron to the detector. Since $R $ is much larger than the dimensions of the sample, its variation across the sample can be neglected. Therefore, we can assume that the amplitudes of the individual diffracted waves are constant, so that the total diffracted wave can be accurately determined by only accounting for the phase differences between individual diffracted waves.
Let $\mathbf{k}_i$ and $\mathbf{k}_f$ denote the wavevectors of the incident and diffracted waves, respectively. An incident plane wave with wavevector $\mathbf{k}_i$ is diffracted into a wave with wavevector $\mathbf{k}_f$. The momentum transfer is defined as
$$\mathbf{q} = \mathbf{k}_f - \mathbf{k}_i$$
and their magnitudes are assumed to be equal, because the wavelength is unchanged:
$$\mathit{k}\equiv|\mathbf{k}_i| = |\mathbf{k}_f| = \frac{2\pi}{\lambda}.$$
Consider two point electrons located at positions $\mathbf{r}_1$ and $\mathbf{r}_2$, and define $\mathbf{r}\equiv \mathbf{r}_2-\mathbf{r}_1$. A detector is located at $\mathbf{P}$, and we define $\mathbf{R}\equiv \mathbf{P}-\mathbf{r}_1$. A plane wave $E_i(\mathbf{r})\propto e^{i\mathbf{k}_i\cdot\mathbf{r}}$ is incident on the two electrons, and the diffracted wave is observed in the far field along the direction $\mathbf{k}_f$. In part A, the common time-dependent factor $e^{-i\omega t}$ is not included, since only relative spatial phases are relevant.
A1 0.60 Under the far-field approximation, $R\equiv|\mathbf R|\gg|\mathbf r|$, keep only the leading-order term in $ |\mathbf r|/R$ and write the outgoing geometric path difference, $\Delta L_{\rm out}\equiv |\mathbf{P}-\mathbf{r}_1|-|\mathbf{P}-\mathbf{r}_2|$, in terms of $\mathbf r$ and $\mathbf k_f$ (or equivalently $\mathbf k_f/k_f$).
A2 0.40 Using your result from A.1 and accounting for the position-dependent phase of the incident wave at $\mathbf{r}_1$ and $\mathbf{r}_2$, find the phase difference between the two diffracted contributions at the detector, expressed in terms of $\mathbf{q}$ and $\mathbf{r}$. The phase difference is defined as $\Delta\phi\equiv\phi_1-\phi_2$, where $\phi_1$ and $\phi_2$ are the phases of the contributions from the electrons at $r_1$ and $r_2$.
A3 0.60 Express the total complex amplitude of the diffracted wave from these two electrons in terms of $\mathbf{q}$ and $\mathbf{r}$. You may ignore any overall common phase factor, since it does not affect the intensity. Assume that the real amplitude of a diffracted wave from a single point electron is a constant $f_0$, independent of position.
Consider two point-like electrons located at positions $\mathbf{r}_1$ and $\mathbf{r}_2$, with $\mathbf{r}\equiv \mathbf{r}_2-\mathbf{r}_1$. A beam of wavelength $\lambda_0$ (so $k=2\pi/\lambda_0$ and $\omega=2\pi c/\lambda_0$) illuminates the electrons, and the diffracted wave is observed in the far field along $\mathbf{k}_f$. We model the incident field as a plane wave with a time-dependent random phase,
$$E_i(\mathbf{r},t)=A\,\exp\!\left[i\left(\mathbf{k}_i\!\cdot\!\mathbf{r}-\omega t+\phi(t)\right)\right],$$
where $\phi(t)$ is piecewise constant and undergoes random phase jumps at regular times after intervals of duration
$$t_0 \equiv \frac{L_0}{c},$$
with $L_0$ the (given) longitudinal coherence length. At the beginning of each interval of length $t_0$, $\phi(t)$ takes a new independent value uniformly distributed on $[0,2\pi)$. Let $I_0$ denote the (time-averaged) intensity at the detector that would be obtained from a single electron in the same geometry. In this problem, the detector is assumed to measure a time-averaged intensity. That is, it does not resolve the individual random phase jumps. Instead, it records the average of the instantaneous intensity over a time much longer than the phase-jump interval $t_0$. The time-averaged intensity is denoted $\langle I\rangle_t \equiv \left\langle |E(t)|^2 \right\rangle_t .$ Here $\langle \cdots \rangle_t$ denotes an average over many phase-jump intervals.
B1 2.30 Using the phase-jump model above, derive the time-averaged total intensity $\langle I\rangle_t$ at the detector due to the two electrons. Your final result should be written in terms of $I_0,$ $\mathbf{q}\equiv \mathbf{k}_f-\mathbf{k}_i$, the separation vector $\mathbf{r}$, the wavelength $\lambda_0$, and the coherence length $L_0$. Assume the detector averages over times much longer than $t_0$.
An electron is often treated as a classical point particle, but in a more realistic model its charge may be regarded as being distributed over a finite spatial region. Consider two idealised charge distributions: (i) an ideal point charge located at $\mathbf r=0$, whose amplitude is $A_1(\mathbf q)=Q_0$, and (ii) an extended Gaussian charge density distribution $\rho_2(\mathbf r)=\rho_0\exp\!\left(-\frac{r^2}{R_0^2}\right)$, where $r = | \mathbf{r} |$.
The constants $Q_0$ and $\rho_0$ are chosen so that the total charge is the same in both cases:
$$Q_0= \int \rho_2(\mathbf{r})\,d^3r.$$
Here $d^3 r$ denotes the volume element in three-dimensional space. In Cartesian coordinates, $d^3r=dxdydz$ and $\int_{\mathbb R^3}$ means integration over all space. Useful identities (may be used without proof):
$$\int_{0}^{\infty} e^{-r^{2}/R_0^{2}}\,4\pi r^{2}\,dr=\pi^{3/2}R_0^{3}, \int_{\mathbb{R}^3} e^{-\alpha r^2}e^{i\mathbf{k}\cdot\mathbf{r}}\, d^3\mathbf r = \left(\frac{\pi}{\alpha}\right)^{3/2} \exp\!\left(-\frac{k^2}{4\alpha}\right), \qquad \alpha>0.$$
Imagine that the surface of a film (i.e., the top atomic layers) is not perfectly flat, but exhibits surface roughness. A common model is to assume that the local film thickness (measured in monolayers) follows a Gaussian distribution. Let $N$ denote the local number of completed monolayers within a lateral coherence area of the beam. We assume that $N$ varies across the surface and is normally distributed with mean $\bar N$ and standard deviation $\sigma$ (both in units of monolayers):
$$P(N)=\frac{1}{\sqrt{2\pi}\sigma} \exp\!\left[-\frac{(N-\bar N)^2}{2\sigma^2}\right].$$
(For the purpose of evaluating averages, you may treat $N$ as a continuous variable.) Let $d$ be the spacing between adjacent atomic layers (monolayers), and let $q_z$ denote the component of the scattering vector $\mathbf q=\mathbf k_f-\mathbf k_i$ normal to the flat surface of the film, i.e. $q_z=\mathbf q\cdot\hat{\mathbf z}$. For an integer number of monolayers $N$, the scattering amplitude is
$$A_N(q_z)=\sum_{n=0}^{N-1} e^{iq_znd} =\frac{1-e^{iq_zNd}}{1-e^{iq_zd}}.$$
When averaging over the Gaussian thickness distribution, we treat $N$ as a continuous variable and use the closed-form expression
$$A_N(q_z)\equiv \frac{1-e^{iq_zNd}}{1-e^{iq_zd}}$$
as the corresponding continuous extension. Assume the measured diffraction intensity is obtained from the coherent average amplitude,
$$I(q_z)\equiv \bigl|\langle A(q_z)\rangle\bigr|^2, \qquad \langle A(q_z)\rangle=\int_{-\infty}^{\infty} P(N)\,A_N(q_z)\,dN.$$
Imagine a thin film with a simple cubic structure grown on a substrate in a layer-by-layer mode, i.e. each monolayer is completed before the next monolayer starts to grow. Let $d$ be the spacing between adjacent atomic layers (monolayers), and let $q_z$ denote the component of $\mathbf{q}=\mathbf{k}_f-\mathbf{k}_i$ normal to the flat surface of the film, i.e., $q_z=\textbf{q}\cdot\hat{\textbf{z}}$. As the film is being grown, the thickness of the film changes, and so does the diffraction intensity at $\textbf{q}=\dfrac{\pi}{d}\hat{\textbf{z}}$. Calculations are performed at the specified out-of-plane momentum transfer $\mathbf{q}$ by coherently summing all monolayer contributions.
E1
2.30
If the film starts to grow at $t = 0$ and the time required to complete one monolayer is $t_0$, obtain the diffraction intensity ratio
$$\frac{I(t = 0.8 t_0)}{I(t = 3.6 t_0)}$$
measured at $\textbf{q}=\dfrac{\pi}{d}\hat{\textbf{z}}$. Assume that during each monolayer growth interval, the fractional coverage of the top layer increases linearly from 0 to 1, so that at time $t=(N+\theta)t_0$, there are exactly $N$ completed monolayers and a fractional coverage $\theta$ of the next monolayer.