Formalism II - Neutron Cross section

mcdiff uses the standard formalism to calculate the elastic neutron scattering cross section, which can be obtained from the elastic part of the double differential cross section:


\begin{displaymath}
\frac{d^2\sigma}{d\Omega dE'}=N\frac{k'}{k}\left( \frac{\h...
...mathbf Q,\omega)+
N\frac{k'}{k}S_{\rm nuc}(\mathbf Q,\omega)
\end{displaymath} (24)

In equation (25) $\frac{d^2\sigma}{d\Omega dE'}$ denotes the double differential cross section, $N$ the number of atoms, $k$ and $k'$ the wave vectors of the incoming and scattered neutron, respectively. $\gamma=\frac{g_n}{2\hbar}$ is the gyromagnetic ratio of the neutron and $e^2/m_ec^2=2.82$ fm is the classical electron radius, $\mathbf Q=\mathbf k-\mathbf k'$ the scattering vector, $\hbar \omega=E-E'=\frac{(\hbar\mathbf k)^2}{2m_n}-\frac{(\hbar\mathbf k')^2}{2m_n}$ the energy transfer and the $S$ are the nuclear and magnetic Van Hove scattering functions. Both are a sum of elastic and inelastic contributions:


$\displaystyle S_{\rm nuc}$ $\textstyle =$ $\displaystyle S_{\rm nuc}^{\rm el}+S_{\rm nuc}^{\rm inel}$ (25)
$\displaystyle S_{\rm mag}$ $\textstyle =$ $\displaystyle S_{\rm mag}^{\rm el}+S_{\rm mag}^{\rm inel}$ (26)

The nuclear elastic scattering can be split into a coherent and an incoherent part:


\begin{displaymath}
S_{\rm nuc}^{\rm el}=S_{\rm nuc}^{\rm el,inc}+S_{\rm nuc}^{\rm el,coh}
\end{displaymath} (27)

The nuclear elastic coherent scattering function is given by a product of lattice factor and nuclear structure factor NSF:


$\displaystyle S_{\rm nuc}^{\rm el,coh}$ $\textstyle =$ $\displaystyle \delta(\hbar \omega)
\left ( \frac{(2\pi)^3}{v_0}\sum_{\mathbf \t...
...r b_{d'} e^{-i\mathbf Q(\mathbf B_d-\mathbf B_{d'})} e^{-W_d-W_{d'}} %
\right )$ (28)
  $\textstyle =$ $\displaystyle \delta(\hbar \omega)
\left ( \frac{(2\pi)^3}{v_0}\sum_{\mathbf \tau } \delta(\mathbf Q-\mathbf \tau ) \right )
\frac{1}{N_B}\vert NSF\vert^2$ (29)

$N_B$ denotes the number of atoms in the basis and the sums run over all reciprocal lattice vectors $\mathbf \tau $ and over all atoms $d=1,\dots,N_B$ in the unit cell (unit cell volume is $v_0$). $\bar b_d$, $\mathbf B_d$ and $W_d$ are the coherent scattering length, the unit cell position vector and the Debye Waller factor of the atom $d$ ( $W_d= \langle (\mathbf Q.\mathbf u_d)^2 \rangle/2=\langle u_{\rm iso}^2 \rangle Q^2/2=B_{\rm iso}Q^2/16\pi^2=
B_{\rm iso} sin^2 \Theta / \lambda^2$), respectively.

Similar, the magnetic elastic scattering function can be written as a product:


$\displaystyle \sum_{\alpha\beta}(\delta_{\alpha\beta}-\hat \mathbf Q_{\alpha} \hat \mathbf Q_{\beta})S_{\rm mag}^{\rm %
el,\alpha\beta}$ $\textstyle =$ $\displaystyle \delta(\hbar \omega)
\left ( \frac{(2\pi)^3}{v_0}\sum_{\mathbf \t...
...f Q_{\alpha} \hat \mathbf Q_{\beta})
\frac{1}{2\mu_B}F_d(Q) M_{d\alpha} \right.$  
    $\displaystyle \left. \frac{1}{2\mu_B}F_{d'}(Q) M_{d'\beta}
e^{-i\mathbf Q(\mathbf B_d-\mathbf B_{d'})} e^{-W_d-W_{d'}} \right )$ (30)
  $\textstyle =$ $\displaystyle \delta(\hbar \omega)\left ( \frac{(2\pi)^3}{v_0}\sum_{\mathbf \ta...
...delta(\mathbf Q-\mathbf \tau ) \right )
\frac{1}{N_B} \vert\vec{\rm MSF}\vert^2$ (31)

In dipole approximation the products $F_d(Q) M_{d\alpha}$ are evaluated as follows (see also appendix J):

[rare earth ($g_J\neq0$ in sipf file):] $F_d(Q) M_{d\alpha}=g_J \mu_B\langle J_{d\alpha} \rangle_{T,H}\left [\langle j_0(Q) \rangle + \frac{2-g_J}{g_J}\langle j_2(Q) \rangle \right ]$ [transition metals ($g_J=0$ in sipf file) given (only) magnetic moments $\langle Ma \rangle, \langle Mb \rangle, \langle Mc \rangle$ in mcdiff.in:] $F_d(Q) M_{d\alpha}= \langle M_{d\alpha} \rangle_{T,H}\left [\langle j_0(Q) \rangle \right ]$ (i.e. a quenched orbital moment is assumed) [transition metals ($g_J=0$ in sipf file) given $\langle Ma \rangle, \langle Mb \rangle, \langle Mc \rangle$ and $\langle Sa \rangle, \langle La \rangle, \langle Sb \rangle, \langle Lb \rangle, \langle Sc \rangle, \langle Lc \rangle$ in mcdiff.in:] $F_d(Q) M_{d\alpha}= \mu_B\langle 2 S_{d\alpha} \rangle_{T,H}\langle j_0(Q) \ran...
... \rangle_{T,H}\left [ \langle j_0(Q) \rangle + \langle j_2(Q) \rangle \right ] $

Here $g_J$, $F_d(Q)$, $\langle S_{d\alpha} \rangle_{T,H}$ , $\langle L_{d\alpha} \rangle_{T,H}$ and $\langle J_{d\alpha} \rangle_{T,H}$ denote the Landé factor, the form factor, the expectation value of the spin-, orbital- and total angular momentum operator of the atom $d$ in the unit cell, respectively.

Program mcdiff calculates the scattering angle $2\Theta$ of a reflection $\mathbf Q=h\mathbf a^*+k \mathbf b^*+l \mathbf c^*$ according to


\begin{displaymath}
\sin(\Theta)=\lambda \frac{\vert\mathbf Q\vert}{4\pi}
\end{displaymath} (32)

Nuclear elastic coherent and magnetic intensities are calculated according to:


\begin{displaymath}
I^{nuc}_{hkl}=\vert\frac{\rm NSF}{N_B}\vert^2 \exp(-\frac{{\rm OTF}\times Q^2}{8\pi^2}) \times {\rm LF}
\end{displaymath} (33)


\begin{displaymath}
I^{mag}_{hkl}=\frac{3.65}{4\pi}\vert\frac{\vec{\rm MSF}}{N_...
... \exp(-\frac{{\rm OTF}\times Q^2}{8\pi^2}) \times {\rm %
LF}
\end{displaymath} (34)


$\displaystyle {\rm OTF}$ $\textstyle =$ $\displaystyle {\rm ... Overall Temperature Factor (}B_{\rm iso}),{\rm OTF}.Q^2/(8\pi^2) =\langle (\mathbf Q.\mathbf %
u)^2 \rangle = \langle u^2 \rangle Q^2$ (35)
$\displaystyle {\rm LF}$ $\textstyle =$ $\displaystyle {\rm ... Lorentzfactor}$ (36)
  $\textstyle =$ $\displaystyle \sin^{-2}(2\Theta){\rm ... powder flat sample}$ (37)
  $\textstyle =$ $\displaystyle \sin^{-1}(2\Theta)\sin^{-1}(\Theta){\rm ... powder cylindrical sample}$ (38)
  $\textstyle =$ $\displaystyle \sin^{-1}(2\Theta){\rm ...single crystal sample}$ (39)
  $\textstyle =$ $\displaystyle d^3 {\rm ... TOF powder cylindrical sample log scaled d pattern}$ (40)
  $\textstyle =$ $\displaystyle d^4 {\rm ... TOF powder cylindrical sample linear scaled d pattern}$ (41)
$\displaystyle {\rm NSF}$ $\textstyle =$ $\displaystyle {\rm ... nuclear structure factor}$ (42)
  $\textstyle =$ $\displaystyle \sum_{d} \bar b_d e^{i\mathbf Q \mathbf B_d} e^{-W_d}$ (43)
$\displaystyle \vec{\rm MSF}$ $\textstyle =$ $\displaystyle {\rm ... magnetic structure factor}$ (44)
  $\textstyle =$ $\displaystyle \sum_{d}\frac{1}{2\mu_B}F_d(Q) \vec M^{\perp}_{d} e^{i\mathbf Q\mathbf B_d} e^{-W_d}$ (45)

Mind: no absorption and extinction corrections are calculated.

Martin Rotter 2017-01-10