# Paper:Periodic lattices of arbitrary nano-objects: modeling and applications for self-assembled systems

This is a summary/discussion of the results from:

This paper describes the modeling of x-ray and neutron scattering data for lattices of nano-objects. The presented formalism enables both simulation or fitting of small-angle scattering data for periodic heterogeneous lattices of arbitrary nano-objects. Generality is maximized by allowing for particle mixtures, anisotropic nano-objects and definable orientations of nano-objects within the unit cell. The model is elaborated by including a variety of kinds of disorder relevant to self-assembling systems: finite grain size, polydispersity in particle properties, positional and orientation disorder of particles, and substitutional or vacancy defects within the lattice. The applicability of the approach is demonstrated by fitting experimental X-ray scattering data. In particular, the article provides examples of superlattices self-assembled from isotropic and anisotropic nanoparticles which interact through complementary DNA coronas.

# Math

## Formalism

Randomly oriented crystals give scattering intensity:

\begin{alignat}{2} I_0(q) & = C \langle |F(\mathbf{q})|^2 S_0(\mathbf{q}) \rangle \\ & = C P(q) \left\langle \frac{|F(\mathbf{q})|^2}{P(q)} S_0(\mathbf{q}) \right\rangle \\ & = C P(q)S_0(q) \end{alignat}

Where the structure factor is defined by an orientational average (randomly oriented crystal(s)):

$S_0(q) \equiv \left\langle \frac{|F(\mathbf{q})|^2}{P(q)} S_0(\mathbf{q}) \right\rangle$

We can compute a structure factor for a crystal-like material by considering an ideal lattice factor:

$Z_0(q) = c \sum_{ \{hkl\} }^{m_{hkl} } \frac{1}{ q_{hkl}^2 } \left|\sum_{j=1}^{n_c} F_j(M_j \cdot \mathbf{q}_{hkl}) e^{2\pi i(x_jh+y_jk+z_jl)} \right|^2 e^{-\sigma_D^2q_{hkl}^2a^2} L_{hkl}(q-q_{hkl})$

Where c is a constant, and L is the peak shape and the $\sigma_D$ term is the Debye-Waller factor. So that the structure factor is:

$S_0(q) = \frac{Z_0(q)}{P(q)} = \frac{c}{ q^2 P(q) } \sum_{ \{hkl\} }^{m_{hkl} } \left|\sum_{j=1}^{n_c} F_j(M_j \cdot \mathbf{q}_{hkl}) e^{2\pi i(x_jh+y_jk+z_jl)} \right|^2 e^{-\sigma_D^2q_{hkl}^2a^2} L_{hkl}(q-q_{hkl})$

And the intensity is then:

$I_0(q) = \frac{c}{ q^2 } \sum_{ \{hkl\} }^{m_{hkl} } \left|\sum_{j=1}^{n_c} F_j(M_j \cdot \mathbf{q}_{hkl}) e^{2\pi i(x_jh+y_jk+z_jl)} \right|^2 e^{-\sigma_D^2q_{hkl}^2a^2} L_{hkl}(q-q_{hkl})$

The form factor amplitude is computed via:

\begin{alignat}{2} F(\mathbf{q}) & = \int\limits \rho(\mathbf{r}) e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}V \\ \end{alignat}

Or, for an object of uniform density within the volume V:

\begin{alignat}{2} F(\mathbf{q}) & = \Delta \rho \int\limits_V e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}\mathbf{r} \\ \end{alignat}

The (isotropic) form factor intensity is an average over all possible particle orientations:

\begin{alignat}{2} P(q) & = \left\langle |F(\mathbf{q})|^2 \right\rangle \\ & = \int\limits_{S} | F(\mathbf{q}) |^2 \mathrm{d}\mathbf{s} \\ & = \int_{\phi=0}^{2\pi}\int_{\theta=0}^{\pi} | F(-q\sin\theta\cos\phi,q\sin\theta\sin\phi,q\cos\theta)|^2 \sin\theta\mathrm{d}\theta\mathrm{d}\phi \end{alignat}

For a mixture of particles, the measured isotropic form factor intensity is a weighted sum of the various constituents:

\begin{alignat}{2} P_{\mathrm{total}}(q) & = \sum_{j=1}^{n_c} c_j P_{j}(q) \end{alignat}

Where the $c_j$ are scaling factors (concentrations) for the constituents.

## Background Scattering

In actual measurements, the intensity has a background:

\begin{alignat}{2} I_{\mathrm{meas}}(q) & = P(q)S_{\mathrm{ideal}}(q) + pq^{-\alpha} + c \end{alignat}

Where c is a constant background, p is a constant prefactor, and $\alpha$ describes the scaling of the q-dependent background (which dominates at small q). If one experimentally obtains the structure factor by doing: $S_{\mathrm{meas}}(q)=I_{\mathrm{meas}}(q)/P(q)$ this implies:

\begin{alignat}{2} S_{\mathrm{meas}}(q) & = \frac{ I_{\mathrm{meas}}(q) }{ P(q) } \\ & = \frac{ P(q)S_{\mathrm{ideal}}(q) + pq^{-\alpha} + c }{ P(q) } \\ & = S_{\mathrm{ideal}}(q) + \frac{pq^{-\alpha} + c}{P(q)} \end{alignat}

### Background from Form Factor

In some cases, the background scattering comes from the (isotropic) form factor of the particles. For example if clusters are in solution alongside the free particles. In such cases the structure factor will tend towards 1 form large q:

$\lim_{q\rightarrow \infty} S_{\mathrm{meas}}(q) = S_{\mathrm{ideal}}(q) + \frac{P(q)}{P(q)} = 0 + 1$

Note that most structure-factors decay as q6, so a background that takes this into account would be:

\begin{alignat}{2} \mathrm{background} = p_1 q^{-\alpha} + p_2 q^{-6} \end{alignat}

# Derivation

## Intensity

We revisit the standard derivation of scattering intensity for an ensemble of scatterers, recasting the expressions into a form useful for lattices of multiple nano-components. We begin with the general expression for scattering intensity, which is simply an ensemble average of intensities (square of the total scattering amplitude) where we explicitly sum over every scatterer in the sample:

\begin{alignat}{2} I(\mathbf{q}) & = \left\langle \left| \sum_{n=1}^{N} \rho_{n} e^{i \mathbf{q} \cdot \mathbf{r}_n } \right|^2 \right\rangle \\ \end{alignat}

The $\rho_{n}$ is the scattering contribution of scatterer $n$, whose interpretation depends on the kind of scattering (e.g. nuclear scattering length in the case of neutron scattering). We now split the summation into a triple-summation by conceptually dividing the sample into $N_n$ sub-cells, where each sub-cell contains $N_j$ particles (nano-objects), and each particle contains $N_p$ scatterers (e.g. electrons). The position vector is now denoted $\mathbf{r}_{njp}$ to emphasize that it points to scatterer $njp$ (pth scatterer of particle j in sub-cell n):

\begin{alignat}{2} I(\mathbf{q}) & = \left\langle \left| \sum_{n=1}^{N_n} \sum_{j=1}^{N_j} \sum_{p=1}^{N_p} \rho_{njp} e^{i \mathbf{q} \cdot \mathbf{r}_{njp} } \right|^2 \right\rangle \\ \end{alignat}

Note that at this stage we have not sacrificed generality, since in principle each sub-cell could contain only a single scatterer. However as we shall see shortly, this formulation is convenient as it simplifies considerable in cases where particular sub-cells reoccur throughout the sample. We now decompose the position vector into a component that points to the sub-cell, $\mathbf{r}_n$, a component that points from the origin of the sub-cell to the center-of-mass of the particle j, $\mathbf{r}_j$, and a component that points from that center-of-mass to the final position $\mathbf{r}_p$:

$\mathbf{r}_{njp}$ = $\mathbf{r}_n$ + $\mathbf{r}_j$ + $\mathbf{r}_p$

So:

\begin{alignat}{2} I(\mathbf{q}) & = \left\langle \left| \sum_{n=1}^{N_n} e^{i \mathbf{q} \cdot \mathbf{r}_{n} } \sum_{j=1}^{N_j} e^{i \mathbf{q} \cdot \mathbf{r}_{j} } \sum_{p=1}^{N_p} \rho_{p} e^{i \mathbf{q} \cdot \mathbf{r}_{p} } \right|^2 \right\rangle \\ \end{alignat}

The third summation is a well-known quantity: the per-particle form factor amplitude. We thus define:

$F_{j}(\mathbf{q}) \equiv \sum_{p=1}^{N_p} \rho_{p} e^{i \mathbf{q} \cdot \mathbf{r}_{p} }$

For convenience we also define:

$\mathcal{U}(\mathbf{q}) \equiv \sum_{j=1}^{N_j} F_{j}(\mathbf{q}) e^{i \mathbf{q} \cdot \mathbf{r}_{j} }$

We now consider the form of the scattering intensity for a crystal-like lattice of particles. In such a case, $\mathcal{U}(\mathbf{q})$ is effectively the form factor of the unit-cell, and the sum over n is a sum over the $N_n$ unit cells. We also convert from $I(\mathbf{q})$ to $I(q)$ by including in the $\langle...\rangle$ an average over grains at all possible orientations (power-like sample).

\begin{alignat}{2} I(q) & = \left\langle \left| \sum_{n=1}^{N_n} \mathcal{U}_n(\mathbf{q}) e^{i \mathbf{q} \cdot \mathbf{r}_{n} } \right|^2 \right\rangle \\ \end{alignat}

For a perfect crystal, all $\mathcal{U}_n$ are identical. As is well-known,[Warren, Guinier] the sum over identical unit cells serves to define a peak shape. A small number of unit cells interfere constructively to give a broad peak centered at the reciprocal-lattice spacing, whereas a progressively larger lattice produces a progressively sharper peak. The peak positions are defined by the symmetry of the lattice. We thus convert to a sum over Miller indices:

\begin{alignat}{2} I(q) & = \frac{N_n}{\Omega q^{d-1}} \sum_{ \{hkl\} }^{m_{hkl} } \left | \mathcal{U}(\mathbf{q}_{hkl}) \right |^2 L(q-q_{hkl}) \\ & = c Z_0(q) \end{alignat}

Where $m_{hkl}$ is the multiplicity of the reflection $hkl$ which appears in reciprocal-space at $\mathbf{q}_{hkl}$. The L is a peak-shape function. A variety of peak shapes are commonly used. A particularly versatile form is that which allows mixing of Gaussian and Lorentzian character.

The $\Omega q^{d-1}$ is a Lorentz factor for a $d$-dimensional lattice, where $\Omega$ is a solid angle. For a three-dimensional lattice $4 \pi q_{hkl}^2$ is the surface area, in reciprocal-space, over which the intensity from the $hkl$ reflection is uniformly spread due to the orientational averaging. We let $c = N_n/\Omega$ and note that in practice $c$ is used as a scaling factor to account for a variety of effects. For instance scattering intensity scales with scattering volume (intersection between the incident beam and the sample), or with concentration of scattering elements for solution scattering. We also define $Z_0(q)$ to be the lattice factor. We note, however, that traditional lattice factors consider only the positions of objects (e.g. atoms) in the unit cell, and do not include any form factors. In our case, however, we cannot extract the $F_j$ from $Z_0$ since we wish to allow for arbitrary and distinct form factors for each particle in the lattice.

$Z_0(q) = \frac{1}{q^2} \sum_{ \{hkl\} }^{m_{hkl} } \left | \sum_{j=1}^{N_j} F_{j}(M_j \cdot \mathbf{q}_{hkl}) e^{2\pi i (x_j h + y_j k + z_j l) } \right |^2 L(q-q_{hkl})$

The $x_j$, $y_j$, and $z_j$ are fractional coordinates within the unit-cell, and we introduce $M_j$ as a rotation matrix to account for the relative orientation of particle $j$ within the unit-cell.

### Lattice Factor

In order to see how the lattice factor (a sum over $hkl$ Miller indices) arises from the sum over unit cells, consider a crystal shaped like a parallelepiped with edge-length $N_1 a$, $N_2 b$, and $N_3 c$, where $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ are the vectors that define the unit cell. Note that selecting a parallelepiped simplifies the exposition, but the sequence of arguments would be the same for any crystal shape or unit cell shape. The intensity is now a sum over the three dimensions of the crystal:

\begin{alignat}{2} I(\mathbf{q}) & = \left| \mathcal{U}(\mathbf{q}) \sum_{n_1=1}^{N_1} e^{i \mathbf{q} \cdot n_1 \mathbf{a} } \sum_{n_2=1}^{N_2} e^{i \mathbf{q} \cdot n_2 \mathbf{b} } \sum_{n_3=1}^{N_3} e^{i \mathbf{q} \cdot n_3 \mathbf{c} } \right|^2 \\ \end{alignat}

Each summation takes the form of a geometric progression. Re-indexing to start from zero, we obtain:

\begin{alignat}{2} \sum_{n_1=0}^{N_1-1} e^{i \mathbf{q} \cdot n_1 \mathbf{a} } & = 1+(e^{i \mathbf{q} \cdot \mathbf{a} })^1 + (e^{i \mathbf{q} \cdot \mathbf{a} })^2 + ... + (e^{i \mathbf{q} \cdot \mathbf{a} })^{N_1} \\ & = \frac{1-(e^{i \mathbf{q} \cdot \mathbf{a} })^{N_1}}{1-(e^{i \mathbf{q} \cdot \mathbf{a} })} \end{alignat}

Each sum is multiplied by its complex conjugate, resulting in terms like:

\begin{alignat}{2} \left| \sum_{n_1=0}^{N_1-1} e^{i \mathbf{q} \cdot n_1 \mathbf{a} } \right|^2 & = \left( \frac{1-(e^{i \mathbf{q} \cdot \mathbf{a} })^{N_1}}{1-(e^{i \mathbf{q} \cdot \mathbf{a} })} \right)\left( \frac{1-(e^{-i \mathbf{q} \cdot \mathbf{a} })^{N_1}}{1-(e^{-i \mathbf{q} \cdot \mathbf{a} })} \right) \\ & = \frac{ 1 - e^{-i \mathbf{q} \cdot \mathbf{a}N_1 } - e^{+i \mathbf{q} \cdot \mathbf{a}N_1 } + e^{0} }{ 1 - e^{-i \mathbf{q} \cdot \mathbf{a} } - e^{+i \mathbf{q} \cdot \mathbf{a} } + e^{0} }\\ & = \frac{2-2 \cos(\mathbf{q} \cdot \mathbf{a}N_1)}{2-2 \cos(\mathbf{q} \cdot \mathbf{a})} \\ & = \frac{\sin^2(\mathbf{q} \cdot \mathbf{a}N_1/2)}{\sin^2(\mathbf{q} \cdot \mathbf{a}/2)} \end{alignat}

Combining the results:

\begin{alignat}{2} I(\mathbf{q}) & = \left|\mathcal{U}(\mathbf{q})\right|^2 \frac{\sin^2(\mathbf{q} \cdot \mathbf{a}N_1/2)}{\sin^2(\mathbf{q} \cdot \mathbf{a}/2)} \frac{\sin^2(\mathbf{q} \cdot \mathbf{b}N_2/2)}{\sin^2(\mathbf{q} \cdot \mathbf{b}/2)} \frac{\sin^2(\mathbf{q} \cdot \mathbf{c}N_3/2)}{\sin^2(\mathbf{q} \cdot \mathbf{c}/2)} \\ & = \left|\mathcal{U}(\mathbf{q})\right|^2 L_s(\mathbf{q} \cdot \mathbf{a},N_1) L_s(\mathbf{q} \cdot \mathbf{b},N_2) L_s(\mathbf{q} \cdot \mathbf{c},N_3) \end{alignat}

The function:

$L_s(x,N) = \frac{\sin^2(Nx/2)}{\sin^2(x/2)}$

defines a peak when $x=2 \pi$ (or multiple thereof). The width of the peak is defined by $N$: for small values of $N$, the peak is broad. As $N$ increases, the peak becomes narrower, in the limit approaching a delta function. Physically, this corresponds to the constructive interference between the unit cells of the crystal: as more cells interfere constructively, the reciprocal-space peak becomes sharper. Although the function $y$ has oscillations beyond the central lobe, these are usually ignored and the peak-shape, $L$ described using a Gaussian, Lorentzian, or other singly-peaked approximation. The width of the peak function is then converted into an effective crystal grain size, or correlation length, using a Scherrer analysis. Note also that $N$ in $L_s$ controls the peak height, and we thus extract it as a prefactor while converting to the normalized peak shape $L$. In order to re-introduce the periodicity, in reciprocal space, of the peak function $L_s$, we note that $L_s$ has maxima when the following relations are satisfied:

\begin{alignat}{2} \mathbf{q} \cdot \mathbf{a} & = 2 \pi h \\ \mathbf{q} \cdot \mathbf{b} & = 2 \pi k \\ \mathbf{q} \cdot \mathbf{c} & = 2 \pi l \\ \end{alignat}

Where $h$, $k$, and $l$ are integers. We define reciprocal-space vectors:

\begin{alignat}{2} \mathbf{u} = \frac{\mathbf{b}\times\mathbf{c}}{\mathbf{a}\cdot\mathbf{b}\times\mathbf{c}} \\ \mathbf{v} = \frac{\mathbf{c}\times\mathbf{a}}{\mathbf{a}\cdot\mathbf{b}\times\mathbf{c}} \\ \mathbf{w} = \frac{\mathbf{a}\times\mathbf{b}}{\mathbf{a}\cdot\mathbf{b}\times\mathbf{c}} \\ \end{alignat}

and consider $\mathbf{q}$ expressed in terms of these reciprocal vectors:

\begin{alignat}{2} \mathbf{q} & = (\mathbf{q}\cdot\mathbf{a})\mathbf{u} + (\mathbf{q}\cdot\mathbf{b})\mathbf{v} + (\mathbf{q}\cdot\mathbf{c})\mathbf{w} \end{alignat}

Combining with the three Laue equations yields:

\begin{alignat}{2} \mathbf{q}_{hkl} & = (2 \pi h)\mathbf{u} + (2 \pi k)\mathbf{v} + (2 \pi l)\mathbf{w} \\ & = 2 \pi(h\mathbf{u} + k \mathbf{v} + l \mathbf{w}) \\ & = 2 \pi \mathbf{H}_{hkl} \end{alignat}

Where $\mathbf{H}_{hkl}$ is a vector that defines the position of Bragg reflection $hkl$ for the reciprocal-lattice. Now that we are considering only the positions where scattering appears in reciprocal-space, we can convert the intensity to a sum over the reciprocal-space positions of the peaks.

\begin{alignat}{2} I(\mathbf{q}) & = N_1N_2N_3 \sum_{ \{hkl\} }\left|\mathcal{U}(\mathbf{q}_{hkl})\right|^2 L(\mathbf{q}-\mathbf{q}_{hkl}) \end{alignat}

Where the $\{hkl\}$ denotes the indices of the reciprocal-space lattice. This allows us to generalize to an arbitrary lattice type. The effect of lattice symmetry is to define the positions of the reciprocal-space peaks via the $\{hkl\}$ indices and the corresponding rules for $\mathbf{q}_{hkl}$. For powder-like samples we introduce an orientation average:

\begin{alignat}{2} I(q) & = \left\langle I(\mathbf{q}) \right\rangle \\ & = \left\langle N_n \sum_{ \{hkl\} }^{m_{hkl}}\left|\mathcal{U}(\mathbf{q}_{hkl})\right|^2 L(\mathbf{q}-\mathbf{q}_{hkl}) \right\rangle \\ & = \frac{N_n}{\Omega q^{d-1}} \sum_{ \{hkl\} }^{m_{hkl}}\left|\mathcal{U}(\mathbf{q}_{hkl})\right|^2 L(q-q_{hkl})\\ \end{alignat}

Where $\Omega q^{d-1}$ is a Lorentz factor accounting for the spherical averaging. For a three-dimensional lattice, the surface area of spherical averaging in reciprocal space is $4 \pi q^2$, so the sum of all lattice peaks at that $q$ must be divided by that area. Finally, we express position within the unit cell in terms of our new reciprocal-space definitions, by defining fractional coordinates $x_j$, $y_j$, and $z_j$, for the three axes of the unit cell:

\begin{alignat}{2} \mathbf{r}_j & = x_j \mathbf{a} + y_j \mathbf{b} + z_j \mathbf{c} \end{alignat}

Such that:

\begin{alignat}{2} \mathbf{q}_{hkl}\cdot\mathbf{r}_j & = 2 \pi(h\mathbf{u} + k \mathbf{v} + l \mathbf{w})\cdot(x_j \mathbf{a} + y_j \mathbf{b} + z_j \mathbf{c}) \\ & = 2 \pi (h x_j + k y_j + l z_j) \end{alignat}

Which leads to the following expression for the intensity:

\begin{alignat}{2} I(q) & = \frac{N_n}{\Omega q^{d-1}} \sum_{ \{hkl\} }^{m_{hkl}}\left|\mathcal{U}(\mathbf{q}_{hkl})\right|^2 L(q-q_{hkl})\\ & = \frac{N_n}{\Omega q^{d-1}} \sum_{ \{hkl\} }^{m_{hkl} } \left | \sum_{j=1}^{N_j} F_{j}(\mathbf{q}_{hkl}) e^{2\pi i (x_j h + y_j k + z_j l) } \right |^2 L(q-q_{hkl}) \end{alignat}

## Form Factors

For an arbitrary distribution of scattering density, $\rho(\mathbf{r})$, the form factor given in equation [XXXX] is computed by integrating over all space:

$F_{j}(\mathbf{q}) = \int \rho_j(\mathbf{r}) e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}V$

For a particle of uniform density and volume V, we denote the scattering contrast with respect to the ambient as $\Delta \rho$, and the form factor is simply:

$F_{j}(\mathbf{q}) = \Delta \rho \int\limits_{V} e^{i \mathbf{q} \cdot \mathbf{r} } \mathrm{d}V$

For monodisperse particles, the average (isotropic) form factor intensity is an average over all possible particle orientations:

\begin{alignat}{2} P_j(q) & = \left\langle |F_j(\mathbf{q})|^2 \right\rangle \\ & = \int\limits_{\phi=0}^{2\pi}\int\limits_{\theta=0}^{\pi} | F_j(-q\sin\theta\cos\phi,q\sin\theta\sin\phi,q\cos\theta)|^2 \sin\theta\mathrm{d}\theta\mathrm{d}\phi \end{alignat}

In later sections, where we introduce distributions in, e.g., particle size, the above expression would average over those properties. Note that for $q=0$, we expect:

\begin{alignat}{2} F(0) & = \int\limits_{\mathrm{all\,\,space}} \rho(\mathbf{r}) e^{0} \mathrm{d}\mathbf{r} = \rho_{\mathrm{total}} \\ & = \Delta \rho \int\limits_{V} e^{0} \mathrm{d}\mathbf{r} = \Delta \rho V \end{alignat}

And:

\begin{alignat}{2} P(0) & = \left\langle \left| F(0) \right|^2 \right\rangle & = \Delta \rho^2 V^2 \end{alignat}

As expected, scattering intensity scales with the square of the scattering contrast and the particle volume. For multi-component lattices, this has the effect of greatly emphasizing larger particles. For instance, a 2-fold increase in particle diameter results in a $(2^3)^2 = 64$-fold increase in scattering intensity. In lattices where one particle is much larger (or has much higher scattering length density), the smaller particles can be neglected.

## Particle Distributions

In order to account for distributions in the particle properties (size, shape, orientation, etc.), we recast the scattering intensity into a form that highlights the variance of inter-particle scattering. We assume that the per-particle properties are not correlated to their positions, so that we can write the summation over n as:

\begin{alignat}{2} I(q) & = \left\langle \left| \sum_{n=1}^{N_n} \mathcal{U}_n(\mathbf{q}) e^{i \mathbf{q} \cdot \mathbf{r}_{n} } \right|^2 \right\rangle \\ & = \left\langle \sum_{n=1}^{N_n} \sum_{n^{\prime}=1}^{N_n} \left\langle \mathcal{U}_{n}(\mathbf{q}) \mathcal{U}_{n^{\prime}}^{*}(\mathbf{q}) \right\rangle \times e^{i \mathbf{q} \cdot (\mathbf{r}_{n}-\mathbf{r}_{n^{\prime}}) } \right\rangle \\ \end{alignat}

Where the inner angle brackets are an average over the particle distributions. The average can be written:

$\left\langle \mathcal{U}_{n}(\mathbf{q}) \mathcal{U}_{n^{\prime}}^{*}(\mathbf{q}) \right\rangle = \left| \left\langle \mathcal{U}(\mathbf{q}) \right\rangle \right|^2 + \left[ \left\langle\left|\mathcal{U}(\mathbf{q})\right|^2\right\rangle - \left|\left\langle \mathcal{U}(\mathbf{q})\right\rangle\right|^2 \right] \delta_{nn^{\prime}}$

Where the term in square brackets is effectively a variance. This introduces a variance term into the scattered intensity:

\begin{alignat}{2} I(q) & = \left\langle \sum_{n=1}^{N_n} \sum_{n^{\prime}=1}^{N_n} \left| \mathcal{U}(\mathbf{q}) \right|^2 e^{i \mathbf{q} \cdot (\mathbf{r}_{n}-\mathbf{r}_{n^{\prime}}) } \right\rangle + \left[ \left\langle\left|\mathcal{U}(\mathbf{q})\right|^2\right\rangle - \left|\left\langle \mathcal{U}(\mathbf{q})\right\rangle\right|^2 \right] \\ & = \left\langle\left|\mathcal{U}(\mathbf{q})\right|^2\right\rangle \left[ \frac{1}{\left\langle\left|\mathcal{U}(\mathbf{q})\right|^2\right\rangle}\left\langle \sum_{n=1}^{N_n} \sum_{n^{\prime}=1}^{N_n} \left| \mathcal{U}(\mathbf{q}) \right|^2 e^{i \mathbf{q} \cdot (\mathbf{r}_{n}-\mathbf{r}_{n^{\prime}}) } \right\rangle + \frac{\left\langle\left|\mathcal{U}(\mathbf{q})\right|^2\right\rangle}{\left\langle\left|\mathcal{U}(\mathbf{q})\right|^2\right\rangle} - \frac{\left|\left\langle \mathcal{U}(\mathbf{q})\right\rangle\right|^2}{\left\langle\left|\mathcal{U}(\mathbf{q})\right|^2\right\rangle} \right]\\ & = P(q) \left[ S_0(q) + 1 - \beta(q) \right] \\ & = P(q) S(q) \end{alignat}

Note that the outer angle brackets included an orientation average that has been included into the averages for our new definitions:

$\beta(q) \equiv \frac{\left|\left\langle \mathcal{U}(\mathbf{q})\right\rangle\right|^2}{\left\langle\left|\mathcal{U}(\mathbf{q})\right|^2\right\rangle}$
$P(q) \equiv \left\langle\left|\mathcal{U}(\mathbf{q})\right|^2\right\rangle$

We have also defined $S_0(q)$ to be an ideal structure factor, which as before can be converted into a sum over lattice peaks:

\begin{alignat}{2} S_0(q) & = \frac{1}{P(q)} \left\langle \sum_{n=1}^{N_n} \sum_{n^{\prime}=1}^{N_n} \left| \mathcal{U}(\mathbf{q}) \right|^2 e^{i \mathbf{q} \cdot (\mathbf{r}_{n}-\mathbf{r}_{n^{\prime}}) } \right\rangle \\ & = \frac{c}{q^2 P(q)} \sum_{ \{hkl\} }^{m_{hkl} } \left | \sum_{j=1}^{N_j} F_{j}(M_j \cdot \mathbf{q}_{hkl}) e^{2\pi i (x_j h + y_j k z_j l) } \right |^2 L(q-q_{hkl}) \\ & = \frac{cZ_0(q)}{P(q)} \end{alignat}

And have defined an apparent structure factor, which includes a $(1-\beta(q))$ scattering contribution from disorder:

$S\left( q \right) = 1 + S_0(q) - \beta(q)$

### Calculation of $\beta(q)$

The average form factor intensity can be computed via:

\begin{alignat}{2} P(q) & = \left\langle\left|\mathcal{U}(\mathbf{q})\right|^2\right\rangle \\ & = \left\langle \left| \sum_{j=1}^{N_j} F_{j}(\mathbf{q}) e^{i \mathbf{q} \cdot \mathbf{r}_{j} } \right|^2 \right\rangle \\ & = \left\langle \sum_{j=1}^{N_j} \left| F_{j}(\mathbf{q}) \right|^2 + \sum_{j\neq k} F_{j}(\mathbf{q}) F_{k}^{*}(\mathbf{q}) e^{i \mathbf{q} \cdot (\mathbf{r}_{j}-\mathbf{r}_{k}) } \right\rangle \\ & \approx \left\langle \sum_{j=1}^{N_j} \left| F_{j}(\mathbf{q}) \right|^2 + 0 \right\rangle \\ & = \sum_{j=1}^{N_j} \left\langle \left| F_{j}(\mathbf{q}) \right|^2 \right\rangle \\ & = \sum_{j=1}^{N_j} P_{j}(q) \\ \end{alignat}

Where we have again assumed a decoupling between the particle distributions, which are averaged over by the angular brackets, and the particle positions within the unit cell. This form of $P(q)$ corresponds to what would be measured experimentally: for a lattice of nano-objects that is dissociated, the measured scattering will be an incoherent sum of the isotropic form factor contributions of each particle type, weighted by the relative occurrence of that particle type. The numerator of $\beta$ is:

\begin{alignat}{2} \left|\left\langle \mathcal{U}(\mathbf{q})\right\rangle\right|^2 & = \left|\left\langle \sum_{j=1}^{N_j} F_{j}(\mathbf{q}) e^{i \mathbf{q} \cdot \mathbf{r}_{j} } \right\rangle\right|^2 \\ & = \left| \sum_{j=1}^{N_j} \left\langle F_{j}(\mathbf{q}) \right\rangle e^{i \mathbf{q} \cdot \mathbf{r}_{j} } \right|^2 \\ & = \sum_{j=1}^{N_j} \left| \left\langle F_j(\mathbf{q}) \right\rangle \right|^2 + \sum_{j\neq k} \left\langle F_{j}(\mathbf{q}) \right\rangle \left\langle F_{k}^{*}(\mathbf{q}) \right\rangle e^{i \mathbf{q} \cdot (\mathbf{r}_{j}-\mathbf{r}_{k}) } \\ & \approx \sum_{j=1}^{N_j} \left| \left\langle F_j(\mathbf{q}) \right\rangle \right|^2 \end{alignat}

So:

\begin{alignat}{2} \beta(q) & = \frac{\left|\left\langle \mathcal{U}(\mathbf{q})\right\rangle\right|^2}{\left\langle\left|\mathcal{U}(\mathbf{q})\right|^2\right\rangle} \\ & = \frac{\sum_{j=1}^{N_j} \left| \left\langle F_j(\mathbf{q}) \right\rangle \right|^2}{\sum_{j=1}^{N_j} P_{j}(q)} \\ & = \frac{\sum_{j=1}^{N_j} \left| \left\langle F_j(\mathbf{q}) \right\rangle \right|^2}{P(q)} \end{alignat}

The elimination of inter-particle effects is expected to be negligible for relatively monodisperse systems ($\sigma_R<0.1$).[doi:10.1063/1.441443]

### Approximation for $\beta(q)$

For large q, $P(q)$ scales approximately as:

$P(q) \approx q^{-4} \sum_{j} \Delta \rho_j^2 V_j^2$

The $\beta(q)$ ratio is affected by polydispersity. For monodisperse particles, $\beta(q)=1$. For particle size distributions of finite width $\sigma_R$, it has been shown [Forster] that the ratio has an oscillating and a non-oscillating part. The non-oscillating part, which dictates the overall scaling, is approximately:

$\beta(q) \approx e^{-\sigma_R^2 R^2 q^2} \equiv \hat{\beta}(q)$

For particles of radius $R$.

For spheres ($d=3$) at $q=0$, it has been shown [doi doi:10.1063/1.446055] that:

$\beta(0) = \frac{\left\langle R^3\right\rangle^2}{\left\langle R^6\right\rangle}$

### Forms for Individual Particles

$\begin{array}{l l l} \mathbf{Quantity} & \mathbf{Monodisperse} & \mathbf{Polydisperse} \\ \mathrm{Form \,\,factor \,\,(amplitude)} & F_j(\mathbf{q}) & \left\langle F_j(\mathbf{q}) \right\rangle = \int_{0}^{\infty} F_j(\mathbf{q}, R) h(R) \mathrm{d}R \\ \mathrm{Form \,\,factor \,\,squared} & \left| F_j(\mathbf{q}) \right|^2 = F_j(\mathbf{q}) F_j^{*}(\mathbf{q}) & \left\langle \left| F_j(\mathbf{q}) \right|^2 \right\rangle = \int_{0}^{\infty} \left| F_j(\mathbf{q}, R) \right|^2 h(R) \mathrm{d}R \\ \mathrm{Form \,\,factor \,\,absolute \,\,value} & \left| F_j(\mathbf{q}) \right| = \sqrt{ F_j(\mathbf{q}) F_j^{*}(\mathbf{q}) } & \left\langle \left| F_j(\mathbf{q}) \right| \right\rangle = \int_{0}^{\infty} \left| F_j(\mathbf{q}, R) \right| h(R) \mathrm{d}R \\ \mathrm{Isotropic \,\,form \,\,factor} & \left\langle F_j(\mathbf{q}) \right\rangle_{\mathrm{iso}} = \int_{0}^{2\pi}\int_{0}^{\pi} F_j(\mathbf{q}) \sin\theta\mathrm{d}\theta\mathrm{d}\phi & \left\langle \left\langle F_j(\mathbf{q}) \right\rangle \right\rangle_{\mathrm{iso}} = \int_{0}^{\infty} (\int_{0}^{2\pi}\int_{0}^{\pi} F_j(\mathbf{q},R) \sin\theta\mathrm{d}\theta\mathrm{d}\phi)h(R) \mathrm{d}R \\ \mathrm{Isotropic \,\,form \,\,factor \,\,intensity} & P_j(q) = \left\langle \left| F_j(\mathbf{q}) \right|^2 \right\rangle_{\mathrm{iso}} = \int_{0}^{2\pi}\int_{0}^{\pi} | F_j(\mathbf{q})|^2 \sin\theta\mathrm{d}\theta\mathrm{d}\phi & \overline{P}_j(q) = \left\langle \left\langle \left| F_j(\mathbf{q}) \right|^2 \right\rangle \right\rangle_{\mathrm{iso}} = \int_{0}^{\infty} P_j(q,R) h(R) \mathrm{d}R \\ \mathrm{Beta \,\,numerator\,\,(isotropic)} & \left\langle \left| F_j(\mathbf{q}) \right|^2 \right\rangle_{\mathrm{iso}} = P_j(q) & \left| \left\langle \left\langle F_j(\mathbf{q}) \right\rangle\right\rangle_{\mathrm{iso}} \right|^2 = | \int_{0}^{\infty} (\int_{0}^{2\pi}\int_{0}^{\pi} F_j(\mathbf{q},R) \sin\theta\mathrm{d}\theta\mathrm{d}\phi)h(R) \mathrm{d}R |^2 \\ \mathrm{Beta \,\,(isotropic)} & \beta_j(q) = \frac{ \left\langle \left| F_j(\mathbf{q}) \right|^2 \right\rangle_{\mathrm{iso}} }{ \left\langle \left| F_j(\mathbf{q}) \right|^2 \right\rangle_{\mathrm{iso}} } = 1 & \beta_j(q) = \frac{ \left| \left\langle \left\langle F_j(\mathbf{q}) \right\rangle\right\rangle_{\mathrm{iso}} \right|^2 }{ \left\langle\left\langle \left| F_j(\mathbf{q}) \right|^2 \right\rangle\right\rangle_{\mathrm{iso}} } = \frac{ \left| \left\langle \left\langle F_j(\mathbf{q}) \right\rangle\right\rangle_{\mathrm{iso}} \right|^2 }{ \overline{P}_j(q) } \\ \end{array}$

### Forms for Lattices

$\begin{array}{l l l} \mathbf{Quantity} & \mathbf{Monodisperse} & \mathbf{Polydisperse} \\ \mathrm{Unitcell \,\,form \,\,factor \,\,(amplitude)} & \mathcal{U}(\mathbf{q}) = \sum_{j=1}^{N_j} F_{j}(\mathbf{q}) e^{i \mathbf{q} \cdot \mathbf{r}_{j} } & \left\langle \mathcal{U}(\mathbf{q}) \right\rangle = \sum_{j=1}^{N_j} \left\langle F_{j}(\mathbf{q}) \right\rangle e^{i \mathbf{q} \cdot \mathbf{r}_{j} } \\ \mathrm{Unitcell \,\,form \,\,factor \,\,squared} & \left| \mathcal{U}(\mathbf{q}) \right|^2 = \mathcal{U}(\mathbf{q}) \mathcal{U}^{*}(\mathbf{q}) & \left| \left\langle \mathcal{U}(\mathbf{q}) \right\rangle \right|^2 = \left | \sum_{j=1}^{N_j} \left\langle F_{j}(\mathbf{q}) \right\rangle e^{i \mathbf{q} \cdot \mathbf{r}_{j} } \right|^2 \\ \mathrm{Unitcell \,\,form \,\,factor \,\,absolute \,\,value} & \left| \mathcal{U}(\mathbf{q}) \right| = \sqrt{ \mathcal{U}(\mathbf{q}) \mathcal{U}^{*}(\mathbf{q}) } & \left| \left\langle \mathcal{U}(\mathbf{q}) \right\rangle \right| = \left | \sum_{j=1}^{N_j} \left\langle F_{j}(\mathbf{q}) \right\rangle e^{i \mathbf{q} \cdot \mathbf{r}_{j} } \right| \\ \mathrm{Isotropic \,\,form \,\,factor} & \left\langle \mathcal{U}(\mathbf{q}) \right\rangle_{\mathrm{iso}} = \sum_{j=1}^{N_j} \left\langle F_j(\mathbf{q}) e^{i \mathbf{q} \cdot \mathbf{r}_{j} } \right\rangle & \left\langle\left\langle \mathcal{U}(\mathbf{q})\right\rangle\right\rangle_{\mathrm{iso}} = \sum_{j=1}^{N_j} \left\langle F_j(\mathbf{q}) e^{i \mathbf{q} \cdot \mathbf{r}_{j} } \right\rangle \\ \mathrm{Isotropic \,\,form \,\,factor \,\,intensity} & P(q) \approx \sum_{j=1}^{N_j} P_{j}(q) & \overline{P}(q) = \left\langle\left\langle\left| \mathcal{U}(\mathbf{q})\right|^2\right\rangle\right\rangle_{\mathrm{iso}} \approx \sum_{j=1}^{N_j} \overline{P}_{j}(q) \\ \mathrm{Beta \,\,numerator\,\,(isotropic)} & \left\langle \left| \mathcal{U}(\mathbf{q}) \right|^2 \right\rangle_{\mathrm{iso}} = P(q) & \left|\left\langle\left\langle \mathcal{U}(\mathbf{q})\right\rangle\right\rangle_{\mathrm{iso}} \right|^2 \approx \sum_{j=1}^{N_j} \left| \left\langle F_j(\mathbf{q}) \right\rangle \right|^2 \\ \mathrm{Beta \,\,(isotropic)} & \beta(q) = 1 & \beta(q) = \frac{\left|\left\langle \mathcal{U}(\mathbf{q})\right\rangle\right|^2}{\left\langle\left|\mathcal{U}(\mathbf{q})\right|^2\right\rangle} = \frac{ \sum_{j=1}^{N_j} \left| \left\langle F_j(\mathbf{q}) \right\rangle \right|^2 }{ \overline{P}(q) } \\ \end{array}$

## Lattice Disorder

Lattice disorder causes deviations in the position of particles from their lattice sites. It has been shown that this kind of disorder replaces the formless lattice-factor intensity as:

$\mathcal{Z} = \mathcal{Z}_0 G(q) + (1-G(q))$

Where (1-G(q)) is the diffuse scattering that arises from disorder, and G is the Debye-Waller factor:

$G(q) = e^{- \sigma_D^2 q^2 a^2 }$

For the lattice factor considered in this work, which includes per-particle form factors, we expect the lattice intensity to be transformed as:

$\frac{Z(q)}{P(q)} = \frac{Z_0(q)}{P(q)}G(q) + \beta(q)(1-G(q))$

This alters the scattering intensity to:

\begin{alignat}{2} I(q) & = P(q) \left[ 1 + \frac{Z(q)}{P(q)} - \beta(q) \right] \\ & = P(q) \left[ 1 + \frac{Z_0(q)}{P(q)}G(q) + \beta(q)(1-G(q)) - \beta(q) \right] \\ & = P(q) \left[ 1 + \frac{Z_0(q)}{P(q)}G(q) - \beta(q)G(q) \right] \\ \end{alignat}

## Single-Particle Lattice

In the case where all the $N_j$ particles in the lattice are identical, we can simplify the scattering expressions. For monodisperse particles, $\beta(q)=1$ and:

\begin{alignat}{2} I(q) & = P(q) \left[ \frac{Z_0(q)}{P(q)}G(q) +(1- G(q)) \right] \\ \end{alignat}

This result is similar to that used previously in the literate [B. Lee], with the explicit addition of the $(1-G(q))$ diffuse scattering. Allowing for particle polydispersity gives a result that matches [Forster et al.].

# Comparing with Experimental Data

## Simulated Intensity

To summarize, the simulated intensity is given by:

\begin{alignat}{2} I(q) & = P(q) S(q) \\ & = P(q) \left[ 1 + \frac{c Z_0(q)}{P(q)}G(q) - \beta(q)G(q) \right] \\ \end{alignat}

Where:

\begin{alignat}{2} P(q) & = \sum_{j=1}^{N_j} P_j(q) & = \sum_{j=1}^{N_j} \left\langle |F_j(\mathbf{q})|^2 \right\rangle \end{alignat}
$Z_0(q) = \frac{1}{q^2} \sum_{ \{hkl\} }^{m_{hkl} } \left | \sum_{j=1}^{N_j} F_{j}(M_j \cdot \mathbf{q}_{hkl}) e^{2\pi i (x_j h + y_j k z_j l) } \right |^2 L(q-q_{hkl})$

## Background Scattering

Experimentally measured scattering may have contributions from effects not accounted for in the present model. For instance, a distribution of aggregate sizes leads to low-q diffuse scattering, and other species in solution, or the solvent itself, may give rise to a constant background. We introduce a background term, $b(q)$ to account for various additional contributions to measured scattering. Experimental data is frequently converted from intensity into a structure factor by dividing by the scattering curve obtained when the constituent particles are free (e.g. the lattice is heated so that the inter-particle links 'melt'). This melted state is an approximation for $P(q)$, but may have a background:

$P_{\mathrm{meas}}\left(q\right) = C_P P(q) + b_{P}(q)$

Where $C_P$ is a constant. The measured intensity also has a background:

\begin{alignat}{2} I_{\mathrm{meas}}(q) & = C_I P(q) S(q) + b_{I}(q) + C_{PI} (P_{\mathrm{meas}}(q)-b_{P}(q))\\ \end{alignat}

Where again the $C$ are constants. The inclusion of non-zero $C_{PI}$ can be used to account for any free particles which may exist in the solution alongside the nano-object lattices. The background can take a variety of forms. We use the form:

$b(q) = p_0 + p_1 q^{-\alpha_1} + p_2 q^{-\alpha_2}$

Where $p_0$ is a constant background, and the other $p$ are pre-factors for q-dependent backgrounds with scaling exponents $\alpha$. Experimentally, the structure factor is estimated by the ratio:

$S_{\mathrm{meas}}(q) = \frac{I_{\mathrm{meas}}(q)}{P_{\mathrm{meas}}(q)}$

To compare to experimental data, we would thus use:

\begin{alignat}{2} S_{\mathrm{meas}}(q) & = \frac{C_I P(q) S(q) + b_{I}(q) + C_{PI} (P_{\mathrm{meas}}(q)-b_{P}(q)) }{C_P P(q) + b_{P}(q)} \\ & = \frac{C_I P(q) S(q)}{C_P P(q) + b_{P}(q)} + \frac{ b_{I}(q) + C_{PI} (P_{\mathrm{meas}}(q)-b_{P}(q)) }{C_P P(q) + b_{P}(q)} \\ & = \frac{C_I P(q) S(q)}{C_P P(q) + b_{P}(q)} + \frac{ b_{I}(q) - C_{PI} b_{P}(q) }{C_P P(q) + b_{P}(q)} + C_{PI} \\ & = \frac{C_I P(q) S(q)}{C_P P(q) + b_{P}(q)} + b_{S}(q) \end{alignat}

### Common Background

If the background in question is common between measurements (e.g. detector noise), one can assume $b_I(q)=b_P(q)=b(q)$, and:

\begin{alignat}{2} S_{\mathrm{meas}}(q) & = \frac{C_I P(q) S(q)}{C_P P(q) + b_{P}(q)} + \frac{ b_{I}(q) - C_{PI} b_{P}(q) }{C_P P(q) + b_{P}(q)} + C_{PI} \\ & = \frac{C_I P(q) S(q)}{C_P P(q) + b(q)} + \frac{ b(q)(1 - C_{PI}) }{C_P P(q) + b(q)} + C_{PI} \\ \end{alignat}

The background can dominate $S_{\mathrm{meas}}(q)$ at high $q$ due to the division by $P(q)$.

## Use Experimental Form Factor 1

To avoid an assumption about the form of $b_{P}(q)$, we can divide the simulated intensity by the experimental $P_{\mathrm{meas}}$:

\begin{alignat}{2} S_{\mathrm{meas}}(q) & = \frac{1}{P_{\mathrm{meas}}(q)} \left[ C_I P(q) S(q) + b_{I}(q) + C_{PI} (P_{\mathrm{meas}}(q)-b_{P}(q)) \right] \\ & = \frac{C_I}{P_{\mathrm{meas}}(q)} P(q)S(q) + \frac{ b_I(q) -C_{PI} b_P(q) }{P_{\mathrm{meas}}(q)} + C_{PI} \\ & = \frac{C_I}{P_{\mathrm{meas}}(q)} P(q)S(q) + b_S(q) \end{alignat}

Where we have defined a background observed in the structure factor, $b_S(q)$. Further rearrangement produces:

\begin{alignat}{2} S_{\mathrm{meas}}(q) & = \frac{C_I}{P_{\mathrm{meas}}(q)} P(q)\left[ 1 + \frac{c Z_0(q)}{P(q)}G(q) - \beta(q)G(q) \right] + b_S(q) \\ & = \frac{C_I}{P_{\mathrm{meas}}(q)} \left[ c Z_0(q)G(q) + P(q)(1 - \beta(q)G(q)) \right] + b_S(q) \end{alignat}

## Use Experimental Form Factor 2

To avoid any theoretical calculation of $P(q)$, one can instead use $P_{\mathrm{meas}}$ throughout, which is advantageous since the experimental form factor inherently includes the appropriate averaging over distributions:

$P(q) = \frac{P_{\mathrm{meas}}\left(q\right)}{C_P} - \frac{b_{P}(q)}{C_P}$

And:

\begin{alignat}{2} S_{\mathrm{meas}}(q) & = \frac{C_I}{P_{\mathrm{meas}}(q)} P(q)\left[ 1 + \frac{c Z_0(q)}{P(q)}G(q) - \beta(q)G(q) \right] + b_S(q) \\ & = \frac{C_I}{P_{\mathrm{meas}}(q)} \left( \frac{P_{\mathrm{meas}}\left(q\right)}{C_P} - \frac{b_{P}(q)}{C_P} \right) \left[ 1 + \frac{c Z_0(q)}{\frac{P_{\mathrm{meas}}\left(q\right)}{C_P} - \frac{b_{P}(q)}{C_P}}G(q) - \beta(q)G(q) \right] + b_S(q) \\ & = \frac{C_I}{C_P} \left( 1-\frac{b_{P}(q)}{P_{\mathrm{meas}} } \right) \left[ 1 + \frac{C_P c Z_0(q)}{P_{\mathrm{meas}}(q) - b_{P}(q)}G(q) - \beta(q)G(q) \right] + b_S(q) \\ & = C_{I/P} \left( 1-\frac{b_{P}(q)}{P_{\mathrm{meas}} } \right) \left[ 1 + \frac{C_{Pc} Z_0(q)}{P_{\mathrm{meas}}(q) - b_{P}(q)}G(q) - \beta(q)G(q) \right] + b_S(q) \\ \end{alignat}

If $b_P(q) \approx 0$:

\begin{alignat}{2} S_{\mathrm{meas}}(q) & = C_{I/P} \left[ 1 + \frac{C_Pc Z_0(q)}{P_{\mathrm{meas}}}G(q) - \beta(q)G(q) \right] + b_S(q) \\ \end{alignat}

# Sources

The paper builds upon work presented in: