iData_vDOS: multi_phonons: compute the integrated multi-phonon generalised density of states (gDOS) from an initial vibrational density of states (vDOS)
Gw = multi_phonons(gw, Ki, T, m, n)
compute: Density of States -> multi-phonon gDOS terms
The input argument 'gw' should be a vibrational density of states (vDOS) as
obtained from an experiment (e.g. Bedov/Oskotskii estimate), molecular
dynamics, or lattice dynamics. In the so-called incoherent approximation,
the vDOS obtained from incoherent and coherent scattering laws are equal.
This method is equivalent to the 'incoherent' one, but provides the
density-of-states gDOS instead of a scattering law Sinc(q,w).
The result is the 'neutron weighted' (generalised) density of states (gDOS),
which should be compared with:
Gw -> \int q. I(q,w) /Ki/Kf dq ~ \int I(theta, w) sin(theta) d(theta)
where I(q,w) and I(theta,w) are the measured scattering intensity as a function
of the momentum q or scattering angle theta.
In practice, the measured I(q,w) contains both the single phonon and multi-phonon
terms (and potentially multiple scattering).
These should be e.g. multiplied by the neutron scattering bound cross section
'sigma' [barns]. This calculation includes the Debye-Waller factor.
The 1st element of the result 'Gw' is the generalised density of states, as computed
in the Oskotskii formalism. The following terms are the multi-phonon contributions.
The generalised density of states (gDOS) is the sum of the terms in this expansion.
This implementation is in principle exact for an isotropic monoatomic material,
e.g. a liquid, powder, or cubic crystal. This methodology is equivalent to
an iteration of the MUPHOCOR code.
For a poly-atomic material with a set of non-equivalent atoms with relative
concentration Ci, mass Mi and bound scattering cross section sigma_i,
one should use:
sigma = sum_i Ci sigma_i weighted cross section
m = [sum_i Ci sigma_i]/[sum_i Ci sigma_i/Mi] weighted mass
Reference:
H. Schober, Journal of Neutron Research 17 (2014) 109–357
DOI 10.3233/JNR-140016 (see esp. pages 328-331)
V.S. Oskotskii, Sov. Phys. Solid State 9 (1967), 420.
A. Sjolander, Arkiv for Fysik 14 (1958), 315.
W. Reichardt, MUPHOCOR Karlsruhe Report 13.03.01p06L (1984)
syntax:
Gw = multi_phonons(gw, Ki, T, m, n)
Gw = multi_phonons(gw, 'Ki',Ki, 'T',T, 'm',m, 'n',n)
Gw = multi_phonons(gw, 'lambda', lambda)
Missing arguments (or given as [] empty), are searched within the initial density
of states object. Input arguments can be given in order, or with name-value
pairs, or as a structure with named fields.
input:
gw: the vibrational density of states per [meV] [iData]
Ki: incident wavevector [Angs-1]. Ki=2*pi/lambda=0.695*sqrt(Ei)
T: temperature [K]
m: mass [g/mol]
phi: detector angles, min and max [deg]
n: number of iterations in the series expansion, e.g. 5
DW: Debye-Waller coefficient gamma=<u^2> [Angs^2] e.g. 0.005
The Debye-Waller function is 2W(q)=gamma*q^2
The Debye-Waller factor is exp(-2W(q))
'lambda','Ei': additional named arguments to specify the incident energy
output:
Gw: neutron weighted gDOS terms, to be summed [iData_vDOS array]
Tp: p-phonon terms [iData array]
Example: s=sqw_cubic_monoatomic; gw=dos(s);
mp=multi_phonons(gw); gdos=plus(mp); plot( [gw mp ]);
See also: iData_Sqw2D/dos, iData_vDOS/incoherent
(c) E.Farhi, ILL. License: EUPL.