## 2 Method of Angular Correlations
The angular pair correlations can be related to their angular momentum depompostions \(B_{l}\) by  
\[C_{2}(q,q^{\prime},\Delta\phi)=\sum_{l}F_{l}(q,q^{\prime},\Delta\phi)B_{l}(q, q^{\prime}), \tag{5}\]  
where  
\[F_{l}(q,q^{\prime},\Delta\phi)=\frac{1}{4\pi}P_{l}[\cos\theta(q)\cos\theta(q^{ \prime})+\sin\theta(q)\sin\theta(q^{\prime})\cos(\Delta\phi)], \tag{6}\]  
where  
\[\theta(q)=\pi/2-\sin^{-1}\left[q/(2\kappa)\right] \tag{7}\]  
and \(P_{l}\) is a Legendre polynomial. Likewise, it can be shown that the triple correlations defined by (2) can be written as [14]  
\[C_{3}(q,q^{\prime}\Delta\phi) = \sum_{l}F_{l}(q,q^{\prime},\Delta\phi)T_{l}(q,q^{\prime}), \tag{8}\]  
where  
\[T_{l}(q,q)=\sum_{l_{1},l_{2},m_{1},m_{2},m}^{l_{max}}(-1)^{m} \bigg{[}\frac{(2l+1)(2l_{2}+1)(2l_{1}+1)}{4\pi}\bigg{]}^{1/2}\times\] \[\bigg{(} \begin{array}{ccc}l_{1}&l_{2}&l\\ 0&0&0\end{array} \bigg{)}\bigg{(} \begin{array}{ccc}l_{1}&l_{2}&l\\ m_{1}&m_{2}&m\end{array} \bigg{)}I_{l_{2}m_{2}}(q)I_{l_{1}m_{1}}(q)I_{lm}^{*}( q)=\] \[\sum_{l_{1},l_{2},m_{1},m_{2},m}^{l_{max}}(-1)^{m}G(l_{1}m_{1};l_ {2}m_{2};lm)I_{l_{2}m_{2}}(q)I_{l_{1}m_{1}}(q)I_{lm}^{*}(q), \tag{9}\]  
where \(G\) is a Gaunt coeffcient [18; 19] and  
\[I_{lm}(q)=\int I(q,\theta,\phi)Y_{lm}^{*}(\theta,\phi)d\Omega. \tag{10}\]  
Although, in general, the \(I_{lm}(q)\)'s are complex quantities, the quantities we are concerned with this paper, namely, the \(I_{l0}(q)\)'s, are real, and all \(m\) values are zeros for the azimuthal symmetry. For a flat Ewald sphere, the quantities \(B_{l}(q,q^{\prime})\) and \(T_{l}(q,q^{\prime})\), which contain the structural information, may be found assuming that \(\cos\left(\theta(q)\right)\) and \(\cos\left(\theta(q^{\prime})\right)\) tend towards zero and \(\sin\left(\theta(q)\right)\) and \(\sin\left(\theta(q^{\prime})\right)\) tend toward unity. In this case, the quantities \(B_{l}(q,q^{\prime})\) and \(T_{l}(q,q^{\prime})\) that contain true structural information about the molecule can be found, for instance, (to within an arbitrary scaling factor) from the integrals  
\[B_{l}(q,q)=\frac{2l+1}{2}\int C_{2}(q,q,\Delta\phi)P_{l}(\cos\Delta\phi)sin \Delta\phi d\Delta\phi \tag{11}\]  
and  
\[T_{l}(q,q)=\frac{2l+1}{2}\int C_{3}(q,q,\Delta\phi)P_{l}(\cos\Delta\phi)sin \Delta\phi d\Delta\phi \tag{12}\]  
over the measured quantities \(C_{2}\), and \(C_{3}\). It should be stressed that, even in the case of a curved Ewald sphere the quantities containing the structural information about the particle, the \(B_{l}(q,q)\) and \(T_{l}(q,q)\) may be found by matrix inversion of Equations (5) and (8). Due to its expected azimuthal symmetry, nanorice particles have diffraction volumes characterized by spherical harmonic expansion coefficients of only azimuthal quantum numbers \(m=0\). In this case, the only relevant spherical harmonic expansion coefficients, \(I_{l0}\), are real. Thus, the computations in this case (\(q^{\prime}=q\)) may be performed entirely with real coefficients and only the diagonal parts of \(B_{l}(q,q^{\prime})\) and \(T_{l}(q,q^{\prime})\) are adopted. It has been shown that these equations may be solved easily for particular common symmetries. It was pointed out by [PERSON] and [PERSON] [20] that viruses tend to possess mainly icosahedral and helical symmetry. We have shown how to solve these equations for both icosahedral [7] and helical [21] cases, and are working on extending the method to particles that deviate a little from exact symmetry, using a form of perturbation theory [22]. In the case of a helical virus, we exploited the fact that if the virus is oriented with a helix axis parallel to the \(z\)-axis, the diffraction volume has exact cylindrical symmetry up to a certain resolution and may therefore be characterized by azimuthal quantum number \(m=0\). Another case includes where the diffraction volume could be characterized by an \(m=0\) quantum number that is a diffraction volume of a nanorice particle.