# Dynamic structure factor¶

This feature is under testing.

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

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

with

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

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 $$\exp(i\mathbf{q}\cdot\mathbf{r})$$ 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.

#!/usr/bin/env python

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

def get_func_AFF(f_params):
def func(symbol, Q):
return atomic_form_factor_WK1995(Q, f_params[symbol])
return func

def run(phonon,
G_points_cubic,
directions,
temperature,
func_AFF=None,
scattering_lengths=None,
n_points=51,
verbose=False):
P = phonon.primitive_matrix

for G_cubic in np.array(G_points_cubic):
G_prim = np.dot(G_cubic, P)
for direction in directions:
direction_prim = np.dot(direction, P)

if verbose:
print("# %s to %s (Primitive: %s to %s)"
% (G_cubic, G_cubic + direction,
G_prim, G_prim + direction_prim))

qpoints = np.array(
[direction_prim * x
for x in np.arange(n_points) / float(n_points - 1)])
phonon.set_band_structure([qpoints])
_, distances, frequencies, _ = phonon.get_band_structure()
# Remove Gamma point because number of bands is different.
qpoints = qpoints[1:]
distances = distances[0][1:]
frequencies = frequencies[0][1:]

if func_AFF is not None:
phonon.set_dynamic_structure_factor(
qpoints,
G_prim,
temperature,
func_atomic_form_factor=func_AFF,
freq_min=1e-3,
run_immediately=False)
elif scattering_lengths is not None:
phonon.set_dynamic_structure_factor(
qpoints,
G_prim,
temperature,
scattering_lengths=scattering_lengths,
freq_min=1e-3,
run_immediately=False)
else:
raise SyntaxError
dsf = phonon.dynamic_structure_factor
for i, S in enumerate(dsf):  # Use as iterator
Q_cubic = np.dot(dsf.Qpoints[i], np.linalg.inv(P))

if verbose:
f = frequencies[i]
bi_sets = degenerate_sets(f)
text = "%f  " % distances[i]
text += "%f %f %f  " % tuple(Q_cubic)
text += " ".join(["%f" % (f[bi].sum() / len(bi))
for bi in bi_sets])
text += "  "
text += " ".join(["%f" % (S[bi].sum()) for bi in bi_sets])
print(text)

if verbose:
print("")
print("")

if __name__ == '__main__':
primitive_matrix=[[0, 0.5, 0.5],
[0.5, 0, 0.5],
[0.5, 0.5, 0]],
unitcell_filename="POSCAR",
force_sets_filename="FORCE_SETS",
born_filename="BORN")
phonon.symmetrize_force_constants()

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

# Gamma-L path i FCC conventional basis
directions_to_L = [[0.5, 0.5, 0.5],
[-0.5, 0.5, 0.5]]
G_points_cubic = ([3, 3, 3], )
n_points = 51
temperature = 300

print("# Distance from Gamma point, 4 band frequencies in meV, "
"4 dynamic structure factors")
print("# For degenerate bands, summations are made.")
print("# Gamma point is omitted due to different number of bands.")
print("")

# With scattering lengths
print("# Running with scattering lengths")
run(phonon,
G_points_cubic,
directions_to_L,
temperature,
scattering_lengths={'Na': 3.63, 'Cl': 9.5770},
n_points=n_points,
verbose=True)
print("")

# With atomic form factor
print("# Running with atomic form factor")
# D. Waasmaier and A. Kirfel, Acta Cryst. A51, 416 (1995)
# f(Q) = \sum_i a_i \exp((-b_i Q^2) + c
# Q is in angstron^-1
# 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-
run(phonon,
G_points_cubic,
directions_to_L,
temperature,
func_AFF=get_func_AFF(f_params),
n_points=n_points,
verbose=True)