# Polarisation/Magnetic Scattering

Models which define a scattering length density parameter can be evaluated as magnetic models. In general, the scattering length density (SLD = $$\beta$$) in each region where the SLD is uniform, is a combination of the nuclear and magnetic SLDs and, for polarised neutrons, also depends on the spin states of the neutrons.

For magnetic scattering, only the magnetization component $$\mathbf{M_\perp}$$ perpendicular to the scattering vector $$\mathbf{Q}$$ contributes to the magnetic scattering length.

The magnetic scattering length density is then

$\beta_M = \dfrac{\gamma r_0}{2\mu_B}\sigma \cdot \mathbf{M_\perp} = D_M\sigma \cdot \mathbf{M_\perp}$

where $$\gamma = -1.913$$ is the gyromagnetic ratio, $$\mu_B$$ is the Bohr magneton, $$r_0$$ is the classical radius of electron, and $$\sigma$$ is the Pauli spin.

Assuming that incident neutrons are polarized parallel $$(+)$$ and anti-parallel $$(-)$$ to the $$x'$$ axis, the possible spin states after the sample are then:

• Non spin-flip $$(+ +)$$ and $$(- -)$$
• Spin-flip $$(+ -)$$ and $$(- +)$$

Each measurement is an incoherent mixture of these spin states based on the fraction of $$+$$ neutrons before ($$u_i$$) and after ($$u_f$$) the sample, with weighting:

$\begin{split}-- &= (1-u_i)(1-u_f) \\ -+ &= (1-u_i)(u_f) \\ +- &= (u_i)(1-u_f) \\ ++ &= (u_i)(u_f)\end{split}$

Ideally the experiment would measure the pure spin states independently and perform a simultaneous analysis of the four states, tying all the model parameters together except $$u_i$$ and $$u_f$$.

If the angles of the $$Q$$ vector and the spin-axis $$x'$$ to the $$x$$ - axis are $$\phi$$ and $$\theta_{up}$$, respectively, then, depending on the spin state of the neutrons, the scattering length densities, including the nuclear scattering length density $$(\beta{_N})$$ are

$\beta_{\pm\pm} = \beta_N \mp D_M M_{\perp x'} \text{ for non spin-flip states}$

and

$\beta_{\pm\mp} = -D_M (M_{\perp y'} \pm iM_{\perp z'}) \text{ for spin-flip states}$

where

$\begin{split}M_{\perp x'} &= M_{0q_x}\cos(\theta_{up})+M_{0q_y}\sin(\theta_{up}) \\ M_{\perp y'} &= M_{0q_y}\cos(\theta_{up})-M_{0q_x}\sin(\theta_{up}) \\ M_{\perp z'} &= M_{0z} \\ M_{0q_x} &= (M_{0x}\cos\phi - M_{0y}\sin\phi)\cos\phi \\ M_{0q_y} &= (M_{0y}\sin\phi - M_{0x}\cos\phi)\sin\phi\end{split}$

Here, $$M_{0x}$$, $$M_{0x}$$, $$M_{0z}$$ are the x, y and z components of the magnetization vector given in the laboratory xyz frame given by

$\begin{split}M_{0x} &= M_0\cos\theta_M\cos\phi_M \\ M_{0y} &= M_0\sin\theta_M \\ M_{0z} &= -M_0\cos\theta_M\sin\phi_M\end{split}$

and the magnetization angles $$\theta_M$$ and $$\phi_M$$ are defined in the figure above.

The user input parameters are:

 sld_M0 $$D_M M_0$$ sld_mtheta $$\theta_M$$ sld_mphi $$\phi_M$$ up_frac_i $$u_i$$ = (spin up)/(spin up + spin down) before the sample up_frac_f $$u_f$$ = (spin up)/(spin up + spin down) after the sample up_angle $$\theta_\mathrm{up}$$

Note

The values of the ‘up_frac_i’ and ‘up_frac_f’ must be in the range 0 to 1.

Document History

2015-05-02 Steve King
2017-11-15 Paul Kienzle