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I'm trying to calculate the dynamic structure factor for inelastic neutron scattering, but I get a function which doesn't obey certain symmetries of the crystal lattice and which changes depending on choice of unit cell.

For example, let's say we have atoms A and B in a lattice like this:

ABABAB...
BABABA...
ABABAB...
BABABA...

We choose

AB
BA

for the unit cell, with A at (1/4,3/4) and (3/4,1/4), and B at (1/4,1/4) and (3/4,3/4), in lattice constant units. We can also choose

BA
AB

which should be valid since it's just shifted by half a lattice constant. The dynamic structure factor (eqn 4.88 in Lovesey's neutron scattering book) is:

$H^j_\mathbf{q}(\mathbf{\kappa}) = \sum_d \bar{b}_d e^{-W_d(\mathbf{\kappa})+i \mathbf{\kappa} \cdot \mathbf{d}} (\mathbf{\kappa} \cdot \mathbf{\sigma}^{j}_{d} (\mathbf{q})) (M_d)^{-1/2}$

where $\mathbf{\kappa}$ is the momentum transfer, $\mathbf{q}$ is the wavevector from the center of the Brillouin zone, $j$ enumerates the phonon branches, $d$ enumerates the atoms located at position $\mathbf{d}$ in the unit cell, $\bar{b}_d$ is the neutron scattering length, and $W(\mathbf{\kappa})$ is the Debye-Waller factor. The full one-phonon inelastic coherent cross-section (eqn 4.89 and 4.91) is

$\frac{k^\prime}{k} \frac{(2 \pi)^2}{v_0} \sum_{\mathbf{\tau}} \sum_{j \mathbf{q}} \delta (\mathbf{\kappa} + \mathbf{q} - \mathbf{\tau}) | H^j_\mathbf{q}(\mathbf{\kappa}) |^2 \frac{1}{2 \omega_i (\mathbf{q})} [n_i(\mathbf{q})\delta(\omega+\omega_i (\mathbf{q})) + (1+n_i(\mathbf{q})) \delta(\omega-\omega_i (\mathbf{q}))]$

The problem is that, if I plot $| H^j_\mathbf{q}(\mathbf{\kappa}) |$ for many $\mathbf{q}$, I see that $| H^j_\mathbf{q}(\mathbf{\kappa}) |$ is asymmetric about the (H,0,0)-axis, and this pattern flips about this axis if I switch from the first unit cell chosen above to the second one. It seems obvious to me that the dynamic structure factor should remain the same despite choice of unit cell. Can anyone let me know what I'm doing wrong or misunderstanding?

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One thing to note is that you are not using the primitive cell, which is obtained by taking the lattice vectors joining your A atom to the A atom diagonally down and to the right and from the original A atom diagonally up and to the right, such that the A atoms sit at the lattice points, and the B atom sits in the space between.

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  • $\begingroup$ That's true, but it doesn't explain why I'm getting different values for $ | H^j_{\mathbf{q}} |^2 $ if I reflect the atom positions about $y=0$ in the $x-y$ plane that the atoms are in. $\endgroup$ – user1704042 Apr 27 '16 at 18:32

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