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

This is a summary/discussion of the results from:

• Yager, K.G.; Zhang, Y.; Lu, F.; Gang, O. "Periodic lattices of arbitrary nano-objects: modeling and applications for self-assembled systems" Journal of Applied Crystallography 2014, 47, 118–129. doi: 10.1107/S160057671302832X

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:

${\begin{alignedat}{2}I_{0}(q)&=C\langle |F(\mathbf {q} )|^{2}S_{0}(\mathbf {q} )\rangle \\&=CP(q)\left\langle {\frac {|F(\mathbf {q} )|^{2}}{P(q)}}S_{0}(\mathbf {q} )\right\rangle \\&=CP(q)S_{0}(q)\end{alignedat}}$

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:

$Z_{0}(q)=c\sum _{\{hkl\}}^{m_{hkl}}{\frac {1}{q_{hkl}^{2}}}\left|\sum _{j=1}^{n_{c}}F_{j}(M_{j}\cdot \mathbf {q} _{hkl})e^{2\pi i(x_{j}h+y_{j}k+z_{j}l)}\right|^{2}e^{-\sigma _{D}^{2}q_{hkl}^{2}a^{2}}L_{hkl}(q-q_{hkl})$

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:

$S_{0}(q)={\frac {Z_{0}(q)}{P(q)}}={\frac {c}{q^{2}P(q)}}\sum _{\{hkl\}}^{m_{hkl}}\left|\sum _{j=1}^{n_{c}}F_{j}(M_{j}\cdot \mathbf {q} _{hkl})e^{2\pi i(x_{j}h+y_{j}k+z_{j}l)}\right|^{2}e^{-\sigma _{D}^{2}q_{hkl}^{2}a^{2}}L_{hkl}(q-q_{hkl})$

And the intensity is then:

$I_{0}(q)={\frac {c}{q^{2}}}\sum _{\{hkl\}}^{m_{hkl}}\left|\sum _{j=1}^{n_{c}}F_{j}(M_{j}\cdot \mathbf {q} _{hkl})e^{2\pi i(x_{j}h+y_{j}k+z_{j}l)}\right|^{2}e^{-\sigma _{D}^{2}q_{hkl}^{2}a^{2}}L_{hkl}(q-q_{hkl})$

The form factor amplitude is computed via:

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

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

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

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

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

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

${\begin{alignedat}{2}P_{\mathrm {total} }(q)&=\sum _{j=1}^{n_{c}}c_{j}P_{j}(q)\end{alignedat}}$

Where the $c_{j}$ are scaling factors (concentrations) for the constituents.

Background Scattering

In actual measurements, the intensity has a background:

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

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:

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

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:

${\begin{alignedat}{2}\mathrm {background} =p_{1}q^{-\alpha }+p_{2}q^{-6}\end{alignedat}}$

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:

${\begin{alignedat}{2}I(\mathbf {q} )&=\left\langle \left|\sum _{n=1}^{N}\rho _{n}e^{i\mathbf {q} \cdot \mathbf {r} _{n}}\right|^{2}\right\rangle \\\end{alignedat}}$

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):

${\begin{alignedat}{2}I(\mathbf {q} )&=\left\langle \left|\sum _{n=1}^{N_{n}}\sum _{j=1}^{N_{j}}\sum _{p=1}^{N_{p}}\rho _{njp}e^{i\mathbf {q} \cdot \mathbf {r} _{njp}}\right|^{2}\right\rangle \\\end{alignedat}}$

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:

${\begin{alignedat}{2}I(\mathbf {q} )&=\left\langle \left|\sum _{n=1}^{N_{n}}e^{i\mathbf {q} \cdot \mathbf {r} _{n}}\sum _{j=1}^{N_{j}}e^{i\mathbf {q} \cdot \mathbf {r} _{j}}\sum _{p=1}^{N_{p}}\rho _{p}e^{i\mathbf {q} \cdot \mathbf {r} _{p}}\right|^{2}\right\rangle \\\end{alignedat}}$

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).

${\begin{alignedat}{2}I(q)&=\left\langle \left|\sum _{n=1}^{N_{n}}{\mathcal {U}}_{n}(\mathbf {q} )e^{i\mathbf {q} \cdot \mathbf {r} _{n}}\right|^{2}\right\rangle \\\end{alignedat}}$

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:

${\begin{alignedat}{2}I(q)&={\frac {N_{n}}{\Omega q^{d-1}}}\sum _{\{hkl\}}^{m_{hkl}}\left|{\mathcal {U}}(\mathbf {q} _{hkl})\right|^{2}L(q-q_{hkl})\\&=cZ_{0}(q)\end{alignedat}}$

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, including X, X, X. 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:

${\begin{alignedat}{2}I(\mathbf {q} )&=\left|{\mathcal {U}}(\mathbf {q} )\sum _{n_{1}=1}^{N_{1}}e^{i\mathbf {q} \cdot n_{1}\mathbf {a} }\sum _{n_{2}=1}^{N_{2}}e^{i\mathbf {q} \cdot n_{2}\mathbf {b} }\sum _{n_{3}=1}^{N_{3}}e^{i\mathbf {q} \cdot n_{3}\mathbf {c} }\right|^{2}\\\end{alignedat}}$

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

${\begin{alignedat}{2}\sum _{n_{1}=0}^{N_{1}-1}e^{i\mathbf {q} \cdot n_{1}\mathbf {a} }&=1+(e^{i\mathbf {q} \cdot \mathbf {a} })^{1}+(e^{i\mathbf {q} \cdot \mathbf {a} })^{2}+...+(e^{i\mathbf {q} \cdot \mathbf {a} })^{N_{1}}\\&={\frac {1-(e^{i\mathbf {q} \cdot \mathbf {a} })^{N_{1}}}{1-(e^{i\mathbf {q} \cdot \mathbf {a} })}}\end{alignedat}}$

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

${\begin{alignedat}{2}\left|\sum _{n_{1}=0}^{N_{1}-1}e^{i\mathbf {q} \cdot n_{1}\mathbf {a} }\right|^{2}&=\left({\frac {1-(e^{i\mathbf {q} \cdot \mathbf {a} })^{N_{1}}}{1-(e^{i\mathbf {q} \cdot \mathbf {a} })}}\right)\left({\frac {1-(e^{-i\mathbf {q} \cdot \mathbf {a} })^{N_{1}}}{1-(e^{-i\mathbf {q} \cdot \mathbf {a} })}}\right)\\&={\frac {1-e^{-i\mathbf {q} \cdot \mathbf {a} N_{1}}-e^{+i\mathbf {q} \cdot \mathbf {a} N_{1}}+e^{0}}{1-e^{-i\mathbf {q} \cdot \mathbf {a} }-e^{+i\mathbf {q} \cdot \mathbf {a} }+e^{0}}}\\&={\frac {2-2\cos(\mathbf {q} \cdot \mathbf {a} N_{1})}{2-2\cos(\mathbf {q} \cdot \mathbf {a} )}}\\&={\frac {\sin ^{2}(\mathbf {q} \cdot \mathbf {a} N_{1}/2)}{\sin ^{2}(\mathbf {q} \cdot \mathbf {a} /2)}}\end{alignedat}}$

Combining the results:

${\begin{alignedat}{2}I(\mathbf {q} )&=\left|{\mathcal {U}}(\mathbf {q} )\right|^{2}{\frac {\sin ^{2}(\mathbf {q} \cdot \mathbf {a} N_{1}/2)}{\sin ^{2}(\mathbf {q} \cdot \mathbf {a} /2)}}{\frac {\sin ^{2}(\mathbf {q} \cdot \mathbf {b} N_{2}/2)}{\sin ^{2}(\mathbf {q} \cdot \mathbf {b} /2)}}{\frac {\sin ^{2}(\mathbf {q} \cdot \mathbf {c} N_{3}/2)}{\sin ^{2}(\mathbf {q} \cdot \mathbf {c} /2)}}\\&=\left|{\mathcal {U}}(\mathbf {q} )\right|^{2}L_{s}(\mathbf {q} \cdot \mathbf {a} ,N_{1})L_{s}(\mathbf {q} \cdot \mathbf {b} ,N_{2})L_{s}(\mathbf {q} \cdot \mathbf {c} ,N_{3})\end{alignedat}}$

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:

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

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

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

and consider Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{q}} expressed in terms of these reciprocal vectors:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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.

${\begin{alignedat}{2}I(\mathbf {q} )&=N_{1}N_{2}N_{3}\sum _{\{hkl\}}\left|{\mathcal {U}}(\mathbf {q} _{hkl})\right|^{2}L(\mathbf {q} -\mathbf {q} _{hkl})\end{alignedat}}$

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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathbf{q}_{hkl}} . For powder-like samples we introduce an orientation average:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_j} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_j} , and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z_j} , for the three axes of the unit cell:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{alignat}{2} \mathbf{r}_j & = x_j \mathbf{a} + y_j \mathbf{b} + z_j \mathbf{c} \end{alignat} }

Such that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \rho(\mathbf{r})} , the form factor given in equation [XXXX] is computed by integrating over all space:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta \rho} , and the form factor is simply:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q=0} , we expect:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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} }

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(q) \equiv \left\langle\left|\mathcal{U}(\mathbf{q})\right|^2\right\rangle }

We have also defined Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S_0(q)} to be an ideal structure factor, which as before can be converted into a sum over lattice peaks:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1-\beta(q))} scattering contribution from disorder:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle S\left( q \right) = 1 + S_0(q) - \beta(q) }

Calculation of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta(q)}

The average form factor intensity can be computed via:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta} is:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_R<0.1} ).[doi:10.1063/1.441443]

Approximation for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta(q)}

For large q, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(q)} scales approximately as:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(q) \approx q^{-4} \sum_{j} \Delta \rho_j^2 V_j^2 }

The Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta(q)} ratio is affected by polydispersity. For monodisperse particles, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta(q)=1} . For particle size distributions of finite width Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta(q) \approx e^{-\sigma_R^2 R^2 q^2} \equiv \hat{\beta}(q)}

For particles of radius Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R} .

For spheres (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle d=3} ) at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q=0} , it has been shown [doi doi:10.1063/1.446055] that:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta(0) = \frac{\left\langle R^3\right\rangle^2}{\left\langle R^6\right\rangle} }

Forms for Individual Particles

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{Z(q)}{P(q)} = \frac{Z_0(q)}{P(q)}G(q) + \beta(q)(1-G(q)) }

This alters the scattering intensity to:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N_j} particles in the lattice are identical, we can simplify the scattering expressions. For monodisperse particles, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \beta(q)=1} and:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (1-G(q))} diffuse scattering. Allowing for particle polydispersity gives a result that matches [Forster et al.].

Sources

The paper builds upon work presented in:

Basics

• Michael Kotlarchyk and Sow‐Hsin Chen Analysis of small angle neutron scattering spectra from polydisperse interacting colloids J. Chem. Phys. 1983, 79, 2461 doi:10.1063/1.446055
• Georg Pabst, Michael Rappolt, Heinz Amenitsch, and Peter Laggner Structural information from multilamellar liposomes at full hydration: Full q-range fitting with high quality x-ray data Phys. Rev. E 2000, 62, 4000–4009 doi: 10.1103/PhysRevE.62.4000

Formalism

• Supplementary Information of: Matthew R. Jones, Robert J. Macfarlane, Byeongdu Lee, Jian Zhang, Kaylie L. Young, Andrew J. Senesi, and Chad A. Mirkin DNA-nanoparticle superlattices formed from anisotropic building blocks Nature Materials 2010, 9, 913-917 doi: 10.1038/nmat2870
• S. Förster, A. Timmann, M. Konrad, C. Schellbach, A. Meyer, S.S. Funari, P. Mulvaney, R. Knott, J. Scattering Curves of Ordered Mesoscopic Materials Phys. Chem. B 2005, 109 (4), 1347–1360 DOI: 10.1021/jp0467494