The Phonon_simple McStas Component

9.6 The Phonon_simple McStas Component

A sample for phonon scattering based on cross section expressions from Squires, Ch.3. Possibility for adding an (unphysical) bandgap.

Identiﬁcation

Author: Kim Lefmann
Origin: Risoe
Date: 04.02.04

Description

Single-cylinder shape. Absorption included. No multiple scattering. No incoherent scattering emitted. No attenuation from coherent scattering. No Bragg scattering. fcc crystal n.n. interactions only One phonon branch only -> phonon polarization not accounted for. Bravais lattice only. (i.e. just one atom per unit cell)

Algorithm: 0. Always perform the scattering if possible (otherwise ABSORB) 1. Choose direction within a focusing solid angle 2. Calculate the zeros of (E_i-E_f-hbar omega(kappa)) as a function of k_f 3. Choose one value of k_f (always at least one is possible!) 4. Perform the correct weight transformation

Input parameters

Parameters in boldface are required; the others are optional.


Name	Unit	Description	Default



radius	m	Outer radius of sample in (x,z) plane
yheight	m	Height of sample in y direction
sigma_abs	barns	Absorption cross section at 2200 m/s per atom
sigma_inc	barns	Incoherent scattering cross section per atom
a	Å	fcc Lattice constant
b	fm	Scattering length
M	a.u.	Atomic mass
c	meV/Å\(^{-1}\)	Velocity of sound
DW	1	Debye-Waller factor
T	K	Temperature
target_x	m	position of target to focus at . Transverse coordinate	0
target_y	m	position of target to focus at. Vertical coordinate	0
target_z	m	position of target to focus at. Straight ahead.	0
target_index	1	relative index of component to focus at, e.g. next is +1	0
focus_r	m	Radius of sphere containing target.	0
focus_xw	m	horiz. dimension of a rectangular area	0
focus_yh	m	vert. dimension of a rectangular area	0
focus_aw	deg	horiz. angular dimension of a rectangular area	0
focus_ah	deg	vert. angular dimension of a rectangular area	0
gap	meV	Bandgap energy (unphysical)	0
e_steps_low	1	Amount of possible intersections beneath the elastic line	50
e_steps_high	1	Amount of possible intersections above the elastic line	50

A simple phonon sample

This component models a simple phonon signal from a single crystal of a pure element in an fcc crystal structure. Only one isotropic acoustic phonon branch is modelled, and the longitudinal and transverse dispersions are identical with the velocity of sound being \(c\). Other physical parameters are the atomic mass, \(M\), the lattice parameter, \(a\), the scattering length, \(b\), the Debye-Waller factor, DW, and the temperature, \(T\). Incoherent scattering and absorption are taken into account by the cross sections \(\sigma _\textrm {abs}\) and \(\sigma _\textrm {inc}\).

The sample can have the form of a cylinder with height \(h\) and radius \(r_0\), or a box with dimensions \(w_x, h_y, t_z\).

Phonons are emitted into a speciﬁc range of solid angles, speciﬁed by the location \((x_t, y_t, z_t)\) and the focusing radius, \(r_0\). Alternatively, the focusing is given by a rectangle, \(w_\textrm {focus}\) and \(h_\textrm {focus}\), and the focus point is given by the index of a down-stream component, target_index.

Multiple scattering is not included in this component.

A usage example of this component can be found in the Neutron site/tests/Test_Phonon instrument from the mcgui.

9.6.1 The phonon cross section

The inelastic phonon cross section for a Bravais crystal of a pure element is given by Ref. [Squ78, ch.3 ]

\begin {eqnarray} \frac {d^2\sigma '}{d\Omega dE_\textrm {f}} &=& b^2 \frac {k_\textrm {f}}{k_\textrm {i}} \frac {(2\pi )^3}{V_0}\frac {1}{2M} \exp (-2W) \nonumber \\ &\times & \sum _{\tau ,q,p} \frac {(\boldsymbol {\kappa } \cdot \textbf {e}_{q,p})^2} {\omega _{q,p}} \left \langle n_{q,p} + \frac {1}{2} \mp \frac {1}{2} \right \rangle \delta (\omega \pm \omega _{q,p}) \delta (\kappa \pm \textbf {q}-\tau ) , \end {eqnarray}

where both annihilation and creation of one phonon is considered (represented by the plus and minus sign in the dispersion delta functions, respectively). In the equation, \(\exp (-2W)\) is the Debye-Waller factor, DW and \(V_0 \) is the volume of the unit cell. The sum runs over the reciprocal lattice vectors, \(\tau \), over the polarisation index, \(p\), and the \(N\) allowed wave vectors q within the Brillouin zone (where \(N\) is the number of unit cells in the crystal). Further, \(\textbf {e}_{q,p}\) is the polarization unit vectors, \(\omega _{q,p}\) the phonon dispersion, and the Bose factor is \(\langle n_{q,p} \rangle = (\hbar \exp (|\omega _{q,p}|/k_\textrm {B}T)-1)^{-1}\).

We have simpliﬁed this expression by assuming no polarization dependence of the dispersion, giving \(\sum _{p} (\boldsymbol {\kappa } \cdot \textbf {e}_{q,p})^2 = \kappa ^2\). We assume that the inter-atomic interaction is nearest-neighbour-only so that the phonon dispersion becomes: \begin {equation} d_1(\textbf {q}) = c_1/a \sqrt {z-s_q} , \end {equation} where \(z=12\) is the number of nearest neighbours and \(s_q=\sum _\textrm {nn} \cos (\textbf {q} \cdot \textbf {r}_\textrm {nn})\), where in turn \(\textbf {r}_\textrm {nn}\) is the lattice positions of the nearest neighbours.

This dispersion relation may be modiﬁed with a small eﬀort, since it is given as a separate c-function attatched to the component.

To calculate \(d\sigma /d\Omega \) we need to transform the q sum into an integral over the Brillouin zone by \(\sum _q \rightarrow N V_\textrm {c} (2\pi )^{-3} \int _\textrm {BZ} d^3\textbf {q}\). The \(\boldsymbol {\kappa }\) sum can now be removed by expanding the q integral to inﬁnity. All in all, the partial diﬀerential cross section reads

\begin {eqnarray} \frac {d^2\sigma '}{d\Omega dE_\textrm {f}} (\boldsymbol {\kappa },\omega ) &=& N b^2 \frac {k_\textrm {f}}{k_\textrm {i}} \frac {1}{2M} \int \frac {\hbar \kappa ^2}{\hbar \omega _q} \left \langle n_{q}+\frac {1}{2}\mp \frac {1}{2} \right \rangle \delta (\omega \pm \omega _{q}) \delta (\boldsymbol {\kappa }\pm \textbf {q}) d^3\textbf {q} \nonumber \\ &=& N b^2 \frac {k_\textrm {f}}{k_\textrm {i}} \frac {\hbar ^2 \kappa ^2}{2M \hbar \omega _q} \left \langle n_{\kappa }+\frac 12\pm \frac 12 \right \rangle \delta (\hbar \omega \pm d_1(\kappa )) . \label {e:phonon-pdcross} \end {eqnarray}

9.6.2 The algorithm

All neutrons, which hit the sample volume, are scattered into a particular range of solid angle, \(\Delta \Omega \), like many other components. One of the diﬃcult things in scattering from a dispersion is to take care to fulﬁll the dispersion criteria and to ﬁnd the correct weight transformation.

In Phonon_simple, the following steps are taken:

If the sample is hit, calculate the total path length inside the sample, otherwise leave the neutron ray unchanged.
Choose a scattering point inside the sample
Choose a direction for the ﬁnal wave vector, \(\hat {\textbf {k}}_\textrm {f}\) within \(\Delta \Omega \).
Calculate possible values of \(k_\textrm {f}\) so that the dispersion relation is fulﬁlled for the corresponding value of \(\textbf {k}_\textrm {f}\). (There is always at least one possible \(k_\textrm {f}\) value [Bac75].)
Choose one of the calculated \(k_\textrm {f}\) values.
Propagate the neutron to the scattering point and adjust the neutron velocity according to \(k_\textrm {f}\).
Calculate and apply the correct weight factor correction, see below.

9.6.3 The weight transformation

Before making the weight transformation, we need to calculate the probability for scattering along one certain direction \(\Omega \) from one phonon mode. To do this, we must integrate out the delta functions in the cross section (??). We here use that \(\hbar \omega _q = \hbar ^2 (k_i^2 - k_f^2) / (2 m_\textrm {N})\), \(\kappa = \textbf {k}_\textrm {i} - k_\textrm {f}\hat {\textbf {k}}_\textrm {f}\), and the integration rule \(\int \delta (f(x)) = (df/dx)(0)^{-1}\). Now, we reach \begin {equation} \label {eq:phononcross} \left (\frac {d\sigma '}{d\Omega }\right )_j = \int \frac {d^2\sigma '}{d\Omega dE_\textrm {f}} dE_\textrm {f} = N b^2 \frac {k_\textrm {f}}{k_\textrm {i}} \frac {\hbar ^2 \kappa ^2}{2M d_1(\kappa _j) J(k_{\textrm {f},j})} \left \langle n_{\kappa }+\frac 12\pm \frac 12 \right \rangle . \end {equation}

where the Jacobian reads \begin {equation} J = 1 - \frac {m_\textrm {N}}{k_\textrm {f} \hbar ^2} \frac {\partial }{\partial k_\textrm {f}} \left ( d_1(\kappa ) \right ) . \end {equation}

A rough order-of-magnitude consideration gives \(\frac {k_{\textrm {f},j}}{k_\textrm {i}}\approx 1\), \(J \approx 1\), \(\langle n_{\kappa }+\frac 12\pm \frac 12 \rangle \approx 1\), \(\frac {\hbar ^2\kappa ^2}{2M d_1(\kappa )} \approx \frac {m}{M}\). Hence, \(\left (\frac {d\sigma }{d\Omega }\right )_j \approx N b^2 \frac {m}{M}\), and the phonon cross section becomes a fraction of the total scattering cross section \(4 \pi N b^2\), as it must be. The diﬀerential cross section per unit volume is found from (9.27) by replacing \(N\) with \(1/V_0\).

The total weight transformation now becomes \begin {equation} \label {eq:phonon_mult} \pi _i = a_\textrm {lin} l_\textrm {max} n_\textrm {s} \Delta \Omega b^2 \frac {k_{\textrm {f},j}}{k_\textrm {i}} \frac {\hbar ^2 \kappa }{2 V_0 M d_1(\kappa ) J(k_{\textrm {f},j})} \left \langle n_{\kappa }+\frac 12 \pm \frac 12 \right \rangle , \end {equation} where \(n_s\) is the number of possible dispersion values in the chosen direction.

The Test_Phonon test/example instrument exists in the distribution for this component.

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