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

2018-06-02 Adam Washington