# Dynamic structure factor#

From Eq. (3.120) in the book “Thermal of Neutron Scattering”, coherent one-phonon dynamic structure factor is given as

$S(\mathbf{Q}, \nu, \omega)^{+1\text{ph}} = \frac{k'}{k} \frac{N}{\hbar} \sum_\mathbf{q} |F(\mathbf{Q}, -\mathbf{q}\nu)|^2 (n_{\mathbf{q}\nu} + 1) \delta(\omega - \omega_{\mathbf{q}\nu}) \Delta(\mathbf{Q} - \mathbf{q}),$
$S(\mathbf{Q}, \nu, \omega)^{-1\text{ph}} = \frac{k'}{k} \frac{N}{\hbar} \sum_\mathbf{q} |F(\mathbf{Q}, \mathbf{q}\nu)|^2 n_{\mathbf{q}\nu} \delta(\omega + \omega_{\mathbf{q}\nu}) \Delta(\mathbf{Q} + \mathbf{q}),$

with

$F(\mathbf{Q}, \mathbf{q}\nu) = \sum_j \sqrt{\frac{\hbar}{2 m_j \omega_{\mathbf{q}\nu}}} \bar{b}_j \exp\left[ -\frac{1}{2} \langle |\mathbf{Q}\cdot\mathbf{u}(j0)|^2 \rangle \right] e^{i(\mathbf{Q} + \mathbf{q}) \cdot \mathbf{r}_{j0}} \mathbf{Q} \cdot\mathbf{e}(j, \mathbf{q}\nu).$

where $$\mathbf{Q}$$ is the scattering vector defined as $$\mathbf{Q} = \mathbf{k} - \mathbf{k}'$$ with incident wave vector $$\mathbf{k}$$ and final wavevector $$\mathbf{k}'$$. Similarly, $$\omega=1/\hbar (E-E')$$ where $$E$$ and $$E'$$ are the energies of the incident and final particles. These follow the convention of the book “Thermal of Neutron Scattering”. In some other text books, their definitions have opposite sign. $$\Delta(\mathbf{Q-q})$$ is defined so that $$\Delta(\mathbf{Q-q})=1$$ with $$\mathbf{Q}-\mathbf{q}=\mathbf{G}$$ and $$\Delta(\mathbf{Q-q})=0$$ with $$\mathbf{Q}-\mathbf{q} \neq \mathbf{G}$$ where $$\mathbf{G}$$ is any reciprocal lattice vector. Other variables are refered to Formulations page. Note that the phase convention of the dynamical matrix given here is used. This changes the representation of the phase factor in $$F(\mathbf{Q}, \mathbf{q}\nu)$$ from that given in the book “Thermal of Neutron Scattering”, but the additional term $$\exp(i\mathbf{q}\cdot\mathbf{r}_{j0})$$ comes from the different phase convention of the dynamical matrix or equivalently the eigenvector. For inelastic neutron scattering, $$\bar{b}_j$$ is the average scattering length over isotopes and spins. For inelastic X-ray scattering, $$\bar{b}_j$$ is replaced by atomic form factor $$f_j(\mathbf{Q})$$ and $$k'/k \sim 1$$.

Currently only $$S(\mathbf{Q}, \nu, \omega)^{+1\text{ph}}$$ is calcualted with setting $$N k'/k = 1$$ and the physical unit is $$\text{m}^2/\text{J}$$ when $$\bar{b}_j$$ is given in Angstrom.

## Usage#

Currently this feature is usable only from API. The following example runs with the input files in example/NaCl.

import numpy as np
import phonopy
from phonopy.phonon.degeneracy import degenerate_sets
from phonopy.spectrum.dynamic_structure_factor import atomic_form_factor_WK1995
from phonopy.units import THzToEv

def get_AFF_func(f_params):
def func(symbol, s):
return atomic_form_factor_WK1995(s, f_params[symbol])

return func

def run(
phonon, Qpoints, temperature, atomic_form_factor_func=None, scattering_lengths=None
):
# Transformation to the Q-points in reciprocal primitive basis vectors
Q_prim = np.dot(Qpoints, phonon.primitive_matrix)
# Q_prim must be passed to the phonopy dynamical structure factor code.
phonon.run_dynamic_structure_factor(
Q_prim,
temperature,
atomic_form_factor_func=atomic_form_factor_func,
scattering_lengths=scattering_lengths,
freq_min=1e-3,
)
dsf = phonon.dynamic_structure_factor
q_cartesian = np.dot(dsf.qpoints, np.linalg.inv(phonon.primitive.cell).T)
distances = np.sqrt((q_cartesian ** 2).sum(axis=1))

print("# [1] Distance from Gamma point,")
print("# [2-4] Q-points in cubic reciprocal space, ")
print("# [5-8] 4 band frequencies in meV (becaues of degeneracy), ")
print("# [9-12] 4 dynamic structure factors.")
print("# For degenerate bands, dynamic structure factors are summed.")
print("")

# Use as iterator
for Q, d, f, S in zip(
Qpoints, distances, dsf.frequencies, dsf.dynamic_structure_factors
):
bi_sets = degenerate_sets(f)  # to treat for band degeneracy
text = "%f  " % d
text += "%f %f %f  " % tuple(Q)
text += " ".join(
["%f" % (f[bi].sum() * THzToEv * 1000 / len(bi)) for bi in bi_sets]
)
text += "  "
text += " ".join(["%f" % (S[bi].sum()) for bi in bi_sets])
print(text)

if __name__ == "__main__":

# Q-points in reduced coordinates wrt cubic reciprocal space
Qpoints = [
[2.970000, -2.970000, 2.970000],
[2.950000, 2.950000, -2.950000],
[2.930000, -2.930000, 2.930000],
[2.905000, -2.905000, 2.905000],
[2.895000, -2.895000, 2.895000],
[2.880000, -2.880000, 2.880000],
[2.850000, -2.850000, 2.850000],
[2.810000, -2.810000, 2.810000],
[2.735000, -2.735000, 2.735000],
[2.660000, -2.660000, 2.660000],
[2.580000, -2.580000, 2.580000],
[2.500000, -2.500000, 2.500000],
]

# Mesh sampling phonon calculation is needed for Debye-Waller factor.
# This must be done with is_mesh_symmetry=False and with_eigenvectors=True.
mesh = [11, 11, 11]
phonon.run_mesh(mesh, is_mesh_symmetry=False, with_eigenvectors=True)
temperature = 300

IXS = True

if IXS:
# For IXS, atomic form factor is needed and given as a function as
# a parameter.
# D. Waasmaier and A. Kirfel, Acta Cryst. A51, 416 (1995)
# f(s) = \sum_i a_i \exp((-b_i s^2) + c
# s = |k' - k|/2 = |Q|/2 is in angstron^-1, where k, k', Q are given
# without 2pi.
# a1, b1, a2, b2, a3, b3, a4, b4, a5, b5, c
f_params = {
"Na": [
3.148690,
2.594987,
4.073989,
6.046925,
0.767888,
0.070139,
0.995612,
14.1226457,
0.968249,
0.217037,
0.045300,
],  # 1+
"Cl": [
1.061802,
0.144727,
7.139886,
1.171795,
6.524271,
19.467656,
2.355626,
60.320301,
35.829404,
0.000436,
-34.916604,
],
}  # 1-
AFF_func = get_AFF_func(f_params)
run(phonon, Qpoints, temperature, atomic_form_factor_func=AFF_func)
else:
# For INS, scattering length has to be given.
# The following values is obtained at (Coh b)
# https://www.nist.gov/ncnr/neutron-scattering-lengths-list
run(phonon, Qpoints, temperature, scattering_lengths={"Na": 3.63, "Cl": 9.5770})

The output of the script is like below.

# [1] Distance from Gamma point,
# [2-4] Q-points in cubic reciprocal space,
# [5-8] 4 band frequencies in meV (becaues of degeneracy),
# [9-12] 4 dynamic structure factors.
# For degenerate bands, dynamic structure factors are summed.

0.009132  2.970000 -2.970000 2.970000  0.990754 1.650964 19.068021 30.556134  0.000000 989.100086 0.000000 61.862136
0.015219  2.950000 2.950000 -2.950000  1.649715 2.748809 19.026010 30.498821  0.000000 359.101167 0.000000 62.419653
0.021307  2.930000 -2.930000 2.930000  2.306414 3.842450 18.964586 30.414407  0.000000 184.887464 0.000000 63.060431
0.028917  2.905000 -2.905000 2.905000  3.122869 5.200999 18.863220 30.273465  0.000000 101.732633 0.000000 63.969683
0.031961  2.895000 -2.895000 2.895000  3.447777 5.741079 18.815865 30.206915  0.000000 83.809696 0.000000 64.363976
0.036526  2.880000 -2.880000 2.880000  3.933076 6.546928 18.738420 30.097099  0.000000 64.884831 0.000000 64.984545
0.045658  2.850000 -2.850000 2.850000  4.895250 8.140375 18.563906 29.845228  0.000000 42.801713 0.000000 66.313660
0.057833  2.810000 -2.810000 2.810000  6.157511 10.217162 18.300255 29.453883  0.000000 28.489244 0.000000 68.206188
0.080662  2.735000 -2.735000 2.735000  8.440395 13.901752 17.738201 28.593810  0.000000 18.900914 0.000000 71.718225
0.103491  2.660000 -2.660000 2.660000  10.558805 17.073109 17.174759 27.604416  0.000000 0.000000 19.251626 73.945633
0.127842  2.580000 -2.580000 2.580000  12.497501 16.203294 19.926554 26.474368  0.000000 0.000000 32.207405 69.110148
0.152193  2.500000 -2.500000 2.500000  13.534679 15.548262 21.156819 25.813428  0.000000 0.000000 78.865720 36.788580