8.5 KiB
(dynamic_structure_factor)=
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 {math}\mathbf{Q} is the scattering vector defined as
{math}\mathbf{Q} = \mathbf{k} - \mathbf{k}' with incident wave vector
{math}\mathbf{k} and final wavevector {math}\mathbf{k}'. Similarly,
{math}\omega=1/\hbar (E-E') where {math}E and {math}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. {math}\Delta(\mathbf{Q-q}) is defined so that
{math}\Delta(\mathbf{Q-q})=1 with {math}\mathbf{Q}-\mathbf{q}=\mathbf{G} and
{math}\Delta(\mathbf{Q-q})=0 with
{math}\mathbf{Q}-\mathbf{q} \neq \mathbf{G} where {math}\mathbf{G} is any
reciprocal lattice vector. Other variables are refered to {ref}formulations
page. Note that the phase convention of the dynamical matrix given
{ref}here <dynacmial_matrix_theory> is used. This changes the representation
of the phase factor in {math}F(\mathbf{Q}, \mathbf{q}\nu) from that given in
the book "Thermal of Neutron Scattering", but the additional term
{math}\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, {math}\bar{b}_j is the average scattering length
over isotopes and spins. For inelastic X-ray scattering, {math}\bar{b}_j is
replaced by atomic form factor {math}f_j(\mathbf{Q}) and {math}k'/k \sim 1.
Currently only {math}S(\mathbf{Q}, \nu, \omega)^{+1\text{ph}} is calcualted
with setting {math}N k'/k = 1 and the physical unit is
{math}\text{m}^2/\text{J} when {math}\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__":
phonon = phonopy.load("phonopy_disp.yaml")
# 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