7.2 Reflection of an Unpolarized Beam from the Surface of a Bulk Magnetic Sample

An unpolarized incident electron beam is [10,11] an incoherent superposition of pure states representing all directions of the incident spin. Each such direction can be represented by its spherical polar angle co-ordinates $(\theta{},\phi{})$. That is to say, the incident beam contains a flux of electrons

$$F_1\mathrm{d}\theta \mathrm{d}\phi = A\sin\theta\mathrm{d}\theta \mathrm{d}\phi$$ (19)

with polarization direction between $\theta$ and $\theta{}+\mathrm{d}\theta$, and between $\phi$ and $\phi{}+\mathrm{d}\phi$. The flux of such electrons in the reflected beam will, therefore, be

$$F_2\mathrm{d}\theta \mathrm{d}\phi = \vert r_{ij}\vert^2F_1\mathrm{d}\theta \mathrm{d}\phi \textrm{.}$$ (20)

The reflection from the surface of a bulk sample is to be modelled as a single reflection, of amplitude reflection coefficient $r_{01}$, in a situation where $V_0$, $B_0$, and therefore $x_0$, are all zero. In this case,

$$r_{01} = -\frac{1}{4}x_1+\frac{1}{8}x_1^2+O(x_1^3)\textrm{,}$$ (21)

and

$$\vert r_{01}\vert^2 = \frac{1}{16}x_1^2-\frac{1}{16}x_1^3+O(x_1^4)\textrm{,}$$ (22)

assuming that $r_{01}$ is real.

If the spherical polar representation $(\theta_i,\phi_i)$ is used for the direction of the magnetic flux density in region $i$, then

$$\cos{}S_i = \sin\theta_i\cos\phi_i\sin\theta\cos\phi{}+\sin\... ...sin\phi_i\sin\theta\sin\phi{}+\cos\theta_i\cos\theta\textrm{.}$$ (23)

Therefore,

$$x_i = y_i+z_i(\sin\theta_i\cos\phi_i\sin\theta\cos\phi{}+\si... ...\phi_i\sin\theta\sin\phi{}+\cos\theta_i\cos\theta{})\textrm{.}$$ (24)

The polarization of the pure state represented by $(\theta{},\phi{})$, in the Cartesian co-ordinate system associated with this spherical polar system, is

$$\mathbf{P}(\theta{},\phi{}) = (\sin\theta\cos\phi{},\sin\theta\sin\phi{},\cos\theta{})\textrm{,}$$ (25)

and the average polarization of the reflected beam is

$$\mathbf{P} = \frac{\int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}\mathbf{P}(\thet... ...heta{}=0}^{\pi}\int_{\phi=0}^{2\pi}F_2\mathrm{d}\theta \mathrm{d}\phi }\protect$$

$$= \frac{\int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(\sin\theta\cos\... ..._{\phi=0}^{2\pi}\vert r_{01}\vert^2F_1\mathrm{d}\theta \mathrm{d}\phi }\protect$$

$$= \frac{\int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(\sin\theta\cos\... ...{1}{16}x_1^3+O(x_1^4)\right)\sin\theta\mathrm{d}\theta \mathrm{d}\phi }\protect$$

$$= \frac{(I_2-I_6,I_3-I_7,I_4-I_8)+O(\{y_1,z_1\}^4)}{I_1-I_5+O(\{y_1,z_1\}^4)}\textrm{.}$$ (26)

The crucial integrals are

$$I_1 = \int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}x_1^2\sin\theta\mathrm{d}\theta \mathrm{d}\phi \protect$$

$$= \int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(y_1+z_1(\sin\theta_1\... ...+\cos\theta_1\cos\theta{}))^2\sin\theta\mathrm{d}\theta \mathrm{d}\phi \protect$$

$$= \int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(y_1^2\sin\theta{}\protect$$

$$+2y_1z_1(\sin\theta_1\cos\phi_1\sin^2\theta\cos\phi{}+\sin\theta_1\sin\phi_1\sin^2\theta\sin\phi{}+\cos\theta_1\sin\theta\cos\theta{})\protect$$

$$+z_1^2(\sin^2\theta_1\cos^2\phi_1\sin^3\theta\cos^2\phi{}+2\sin^2\theta_1\sin\phi_1\cos\phi_1\sin^3\theta\sin\phi{}\cos\phi{}\protect$$

$$+2\sin\theta_1\cos\theta_1\cos\phi_1\sin^2\theta\cos\theta\cos\phi{}+\sin^2\theta_1\sin^2\phi_1\sin^3\theta\sin^2\phi\protect$$

$$+2\sin\theta_1\cos\theta_1\sin\phi_1\sin^2\theta\cos\theta\sin\phi+\cos^2\theta_1\sin\theta\cos^2\theta{}))\mathrm{d}\theta \mathrm{d}\phi \protect$$

$$= 4\pi{}y_1^2+\frac{4\pi{}z_1^2}{3}\textrm{,}$$ (27)

$$I_2 = \int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}x_1^2\sin^2\theta\cos\phi\mathrm{d}\theta \mathrm{d}\phi \protect$$

$$= \int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(y_1+z_1(\sin\theta_1\... ...a_1\cos\theta{}))^2\sin^2\theta\cos\phi\mathrm{d}\theta \mathrm{d}\phi \protect$$

$$= \int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(y_1^2\sin^2\theta{}\cos\phi{}\protect$$

$$+2y_1z_1(\sin\theta_1\cos\phi_1\sin^3\theta\cos^2\phi{}+\sin\thet... ...eta\sin\phi{}\cos\phi{}+\cos\theta_1\sin^2\theta\cos\theta{}\cos\phi{})\protect$$

$$+z_1^2(\sin^2\theta_1\cos^2\phi_1\sin^4\theta\cos^3\phi{}+2\sin^2\theta_1\sin\phi_1\cos\phi_1\sin^4\theta\sin\phi{}\cos^2\phi{}\protect$$

$$+2\sin\theta_1\cos\theta_1\cos\phi_1\sin^3\theta\cos\theta\cos^2\phi{}+\sin^2\theta_1\sin^2\phi_1\sin^4\theta\sin^2\phi\cos\phi\protect$$

$$+2\sin\theta_1\cos\theta_1\sin\phi_1\sin^3\theta\cos\theta\sin\ph... ...eta_1\sin^2\theta\cos^2\theta\cos\phi))\mathrm{d}\theta \mathrm{d}\phi \protect$$

$$= \frac{8\pi{}y_1z_1\sin\theta_1\cos\phi_1}{3} \textrm{,}$$ (28)

$$I_3 = \int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}x_1^2\sin^2\theta\sin\phi\mathrm{d}\theta \mathrm{d}\phi \protect$$

$$= \int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(y_1+z_1(\sin\theta_1\... ...a_1\cos\theta{}))^2\sin^2\theta\sin\phi\mathrm{d}\theta \mathrm{d}\phi \protect$$

$$= \int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(y_1^2\sin^2\theta{}\sin\phi\protect$$

$$+2y_1z_1(\sin\theta_1\cos\phi_1\sin^3\theta\sin\phi\cos\phi{}+\si... ...1\sin^3\theta\sin^2\phi{}+\cos\theta_1\sin^2\theta\cos\theta\sin\phi{})\protect$$

$$+z_1^2(\sin^2\theta_1\cos^2\phi_1\sin^4\theta\sin\phi\cos^2\phi{}+2\sin^2\theta_1\sin\phi_1\cos\phi_1\sin^4\theta\sin^2\phi{}\cos\phi{}\protect$$

$$+2\sin\theta_1\cos\theta_1\cos\phi_1\sin^3\theta\cos\theta\sin\phi\cos\phi{}+\sin^2\theta_1\sin^2\phi_1\sin^4\theta\sin^3\phi\protect$$

$$+2\sin\theta_1\cos\theta_1\sin\phi_1\sin^3\theta\cos\theta\sin^2\... ...a_1\sin^2\theta\cos^2\theta\sin\phi{}))\mathrm{d}\theta \mathrm{d}\phi \protect$$

$$= \frac{8\pi{}y_1z_1\sin\theta_1\sin\phi_1}{3} \textrm{,}$$ (29)

$$I_4 = \int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}x_1^2\sin\theta\cos\theta\mathrm{d}\theta \mathrm{d}\phi \protect$$

$$= \int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(y_1+z_1(\sin\theta_1\... ...a_1\cos\theta{}))^2\sin\theta\cos\theta\mathrm{d}\theta \mathrm{d}\phi \protect$$

$$= \int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(y_1^2\sin\theta\cos\theta{}\protect$$

$$+2y_1z_1(\sin\theta_1\cos\phi_1\sin^2\theta\cos\theta\cos\phi{}+\... ...1\sin^2\theta\cos\theta\sin\phi{}+\cos\theta_1\sin\theta\cos^2\theta{})\protect$$

$$+z_1^2(\sin^2\theta_1\cos^2\phi_1\sin^3\theta\cos\theta\cos^2\phi... ...2\theta_1\sin\phi_1\cos\phi_1\sin^3\theta\cos\theta\sin\phi{}\cos\phi{}\protect$$

$$+2\sin\theta_1\cos\theta_1\cos\phi_1\sin^2\theta\cos^2\theta\cos\phi{}+\sin^2\theta_1\sin^2\phi_1\sin^3\theta\cos\theta\sin^2\phi\protect$$

$$+2\sin\theta_1\cos\theta_1\sin\phi_1\sin^2\theta\cos^2\theta\sin\... ...cos^2\theta_1\sin\theta\cos^3\theta{}))\mathrm{d}\theta \mathrm{d}\phi \protect$$

$$= \frac{8\pi{}y_1z_1\cos\theta_1}{3}\textrm{,}$$ (30)

$$I_5 = \int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}x_1^3\sin\theta\mathrm{d}\theta \mathrm{d}\phi \protect$$

$$= \int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(y_1+z_1(\sin\theta_1\... ...+\cos\theta_1\cos\theta{}))^3\sin\theta\mathrm{d}\theta \mathrm{d}\phi \protect$$

$$= \int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(y_1^3\sin\theta\protect$$

$$+3y_1^2z_1(\sin\theta_1\cos\phi_1\sin^2\theta\cos\phi{}+\sin\theta_1\sin\phi_1\sin^2\theta\sin\phi{}+\cos\theta_1\sin\theta\cos\theta{})\protect$$

$$+3y_1z_1^2(\sin^2\theta_1\cos^2\phi_1\sin^3\theta\cos^2\phi{}+2\sin^2\theta_1\sin\phi_1\cos\phi_1\sin^3\theta\sin\phi{}\cos\phi{}\protect$$

$$+2\sin\theta_1\cos\theta_1\cos\phi_1\sin^2\theta\cos\theta\cos\phi{}+\sin^2\theta_1\sin^2\phi_1\sin^3\theta\sin^2\phi\protect$$

$$+2\sin\theta_1\cos\theta_1\sin\phi_1\sin^2\theta\cos\theta\sin\phi{}+\cos^2\theta_1\sin\theta\cos^2\theta{})\protect$$

$$+z_1^3(\sin^3\theta_1\cos^3\phi_1\sin^4\theta\cos^3\phi{}+3\sin^3\theta_1\sin\phi_1\cos^2\phi_1\sin^4\theta\sin\phi\cos^2\phi{}\protect$$

$$+3\sin^2\theta_1\cos\theta_1\cos^2\phi_1\sin^3\theta\cos\theta\co... ...}+3\sin^3\theta_1\sin^2\phi_1\cos\phi_1\sin^4\theta\sin^2\phi\cos\phi{}\protect$$

$$+6\sin^2\theta_1\cos\theta_1\sin\phi_1\cos\phi_1\sin^3\theta\cos\... ...3\sin\theta_1\cos^2\theta_1\cos\phi_1\sin^2\theta\cos^2\theta\cos\phi{}\protect$$

$$+\sin^3\theta_1\sin^3\phi_1\sin^4\theta\sin^3\phi{}+3\sin^2\theta_1\cos\theta_1\sin^2\phi_1\sin^3\theta\cos\theta\sin^2\phi{}\protect$$

$$+3\sin\theta_1\cos^2\theta_1\sin\phi_1\sin^2\theta\cos^2\theta\si... ...cos^3\theta_1\sin\theta\cos^3\theta{}))\mathrm{d}\theta \mathrm{d}\phi \protect$$

$$= 4\pi{}y_1z_1^2\textrm{,}$$ (31)

$$I_6 = \int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}x_1^3\sin^2\theta\cos\phi\mathrm{d}\theta \mathrm{d}\phi \protect$$

$$= \int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(y_1^3\sin^2\theta\cos\phi\protect$$

$$+3y_1^2z_1(\sin\theta_1\cos\phi_1\sin^3\theta\cos^2\phi{}+\sin\th... ...3\theta\sin\phi\cos\phi{}+\cos\theta_1\sin^2\theta\cos\theta\cos\phi{})\protect$$

$$+3y_1z_1^2(\sin^2\theta_1\cos^2\phi_1\sin^4\theta\cos^3\phi{}+2\sin^2\theta_1\sin\phi_1\cos\phi_1\sin^4\theta\sin\phi{}\cos^2\phi{}\protect$$

$$+2\sin\theta_1\cos\theta_1\cos\phi_1\sin^3\theta\cos\theta\cos^2\phi{}+\sin^2\theta_1\sin^2\phi_1\sin^4\theta\sin^2\phi\cos\phi\protect$$

$$+2\sin\theta_1\cos\theta_1\sin\phi_1\sin^3\theta\cos\theta\sin\phi\cos\phi{}+\cos^2\theta_1\sin^2\theta\cos^2\theta\cos\phi{})\protect$$

$$+z_1^3(\sin^3\theta_1\cos^3\phi_1\sin^5\theta\cos^4\phi{}+3\sin^3\theta_1\sin\phi_1\cos^2\phi_1\sin^5\theta\sin\phi\cos^3\phi{}\protect$$

$$+3\sin^2\theta_1\cos\theta_1\cos^2\phi_1\sin^4\theta\cos\theta\co... ...3\sin^3\theta_1\sin^2\phi_1\cos\phi_1\sin^5\theta\sin^2\phi\cos^2\phi{}\protect$$

$$+6\sin^2\theta_1\cos\theta_1\sin\phi_1\cos\phi_1\sin^4\theta\cos\... ...sin\theta_1\cos^2\theta_1\cos\phi_1\sin^3\theta\cos^2\theta\cos^2\phi{}\protect$$

$$+\sin^3\theta_1\sin^3\phi_1\sin^5\theta\sin^3\phi\cos\phi{}+3\sin... ...eta_1\cos\theta_1\sin^2\phi_1\sin^4\theta\cos\theta\sin^2\phi\cos\phi{}\protect$$

$$+3\sin\theta_1\cos^2\theta_1\sin\phi_1\sin^3\theta\cos^2\theta\si... ...a_1\sin^2\theta\cos^3\theta\cos\phi{}))\mathrm{d}\theta \mathrm{d}\phi \protect$$

$$= \left(4\pi{}y_1^2z_1+\frac{4\pi{}z_1^3}{5}\right)\sin\theta_1\cos\phi_1\textrm{,}$$ (32)

$$I_7 = \int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}x_1^3\sin^2\theta\sin\phi\mathrm{d}\theta \mathrm{d}\phi \protect$$

$$= \int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(y_1^3\sin^2\theta\sin\phi\protect$$

$$+3y_1^2z_1(\sin\theta_1\cos\phi_1\sin^3\theta\sin\phi\cos\phi{}+\... ...1\sin^3\theta\sin^2\phi{}+\cos\theta_1\sin^2\theta\cos\theta\sin\phi{})\protect$$

$$+3y_1z_1^2(\sin^2\theta_1\cos^2\phi_1\sin^4\theta\sin\phi\cos^2\phi{}+2\sin^2\theta_1\sin\phi_1\cos\phi_1\sin^4\theta\sin^2\phi{}\cos\phi{}\protect$$

$$+2\sin\theta_1\cos\theta_1\cos\phi_1\sin^3\theta\cos\theta\sin\phi\cos\phi{}+\sin^2\theta_1\sin^2\phi_1\sin^4\theta\sin^3\phi\protect$$

$$+2\sin\theta_1\cos\theta_1\sin\phi_1\sin^3\theta\cos\theta\sin^2\phi{}+\cos^2\theta_1\sin^2\theta\cos^2\theta\sin\phi{})\protect$$

$$+z_1^3(\sin^3\theta_1\cos^3\phi_1\sin^5\theta\sin\phi\cos^3\phi{}+3\sin^3\theta_1\sin\phi_1\cos^2\phi_1\sin^5\theta\sin^2\phi\cos^2\phi{}\protect$$

$$+3\sin^2\theta_1\cos\theta_1\cos^2\phi_1\sin^4\theta\cos\theta\si... ...}+3\sin^3\theta_1\sin^2\phi_1\cos\phi_1\sin^5\theta\sin^3\phi\cos\phi{}\protect$$

$$+6\sin^2\theta_1\cos\theta_1\sin\phi_1\cos\phi_1\sin^4\theta\cos\... ...eta_1\cos^2\theta_1\cos\phi_1\sin^3\theta\cos^2\theta\sin\phi\cos\phi{}\protect$$

$$+\sin^3\theta_1\sin^3\phi_1\sin^5\theta\sin^4\phi{}+3\sin^2\theta_1\cos\theta_1\sin^2\phi_1\sin^4\theta\cos\theta\sin^3\phi{}\protect$$

$$+3\sin\theta_1\cos^2\theta_1\sin\phi_1\sin^3\theta\cos^2\theta\si... ...a_1\sin^2\theta\cos^3\theta\sin\phi{}))\mathrm{d}\theta \mathrm{d}\phi \protect$$

$$= \left(4\pi{}y_1^2z_1+\frac{4\pi{}z_1^3}{5}\right)\sin\theta_1\sin\phi_1\textrm{,}$$ (33)

and

$$I_8 = \int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}x_1^3\sin\theta\cos\theta\mathrm{d}\theta \mathrm{d}\phi \protect$$

$$= \int_{\theta{}=0}^{\pi}\int_{\phi=0}^{2\pi}(y_1^3\sin\theta\cos\theta\protect$$

$$+3y_1^2z_1(\sin\theta_1\cos\phi_1\sin^2\theta\cos\theta\cos\phi{}... ...1\sin^2\theta\cos\theta\sin\phi{}+\cos\theta_1\sin\theta\cos^2\theta{})\protect$$

$$+3y_1z_1^2(\sin^2\theta_1\cos^2\phi_1\sin^3\theta\cos\theta\cos^2... ...2\theta_1\sin\phi_1\cos\phi_1\sin^3\theta\cos\theta\sin\phi{}\cos\phi{}\protect$$

$$+2\sin\theta_1\cos\theta_1\cos\phi_1\sin^2\theta\cos^2\theta\cos\phi{}+\sin^2\theta_1\sin^2\phi_1\sin^3\theta\cos\theta\sin^2\phi\protect$$

$$+2\sin\theta_1\cos\theta_1\sin\phi_1\sin^2\theta\cos^2\theta\sin\phi{}+\cos^2\theta_1\sin\theta\cos^3\theta{})\protect$$

$$+z_1^3(\sin^3\theta_1\cos^3\phi_1\sin^4\theta\cos\theta\cos^3\phi... ...theta_1\sin\phi_1\cos^2\phi_1\sin^4\theta\cos\theta\sin\phi\cos^2\phi{}\protect$$

$$+3\sin^2\theta_1\cos\theta_1\cos^2\phi_1\sin^3\theta\cos^2\theta\... ...theta_1\sin^2\phi_1\cos\phi_1\sin^4\theta\cos\theta\sin^2\phi\cos\phi{}\protect$$

$$+6\sin^2\theta_1\cos\theta_1\sin\phi_1\cos\phi_1\sin^3\theta\cos^... ...3\sin\theta_1\cos^2\theta_1\cos\phi_1\sin^2\theta\cos^3\theta\cos\phi{}\protect$$

$$+\sin^3\theta_1\sin^3\phi_1\sin^4\theta\cos\theta\sin^3\phi{}+3\sin^2\theta_1\cos\theta_1\sin^2\phi_1\sin^3\theta\cos^2\theta\sin^2\phi{}\protect$$

$$+3\sin\theta_1\cos^2\theta_1\sin\phi_1\sin^2\theta\cos^3\theta\si... ...cos^3\theta_1\sin\theta\cos^4\theta{}))\mathrm{d}\theta \mathrm{d}\phi \protect$$

$$= \left(4\pi{}y_1^2z_1+\frac{4\pi{}z_1^3}{5}\right)\cos\theta_1\textrm{.}$$ (34)

This gives a polarization

$$\mathbf{P} = \frac{(I_2-I_6,I_3-I_7,I_4-I_8)+O(\{y_1,z_1\}^4)}{I_1-I_5+O(\{y_1,z_1\}^4)}\protect$$

$$= \frac{(10y_1z_1-15y_1^2z_1-3z_1^3)(\sin\theta_1\cos\phi_1,\sin\th... ...\theta_1)+O(\{y_1,z_1\}^4)}{15y_1^2+5z_1^2-15y_1z_1^2+O(\{y_1,z_1\}^4)}\protect$$

$$= \frac{(10y_1z_1-15y_1^2z_1-3z_1^3)\mathbf{\hat{B_1}}+O(\{y_1,z_1\... ...1^2}\left(1-\frac{5y_1z_1^2}{3y_1^2+z_1^2}+O(\{y_1,z_1\}^2)\right)^{-1}\protect$$

$$= \frac{2y_1z_1\mathbf{\hat{B_1}}}{3y_1^2+z_1^2}+\frac{(36y_1^2z_1^... ...1^5)\mathbf{\hat{B_1}}}{45y_1^4+30y_1^2z_1^2+5z_1^4}+O(\{y_1,z_1\}^2)\textrm{,}$$ (35)

where $\mathbf{\hat{B_1}}$ is a unit vector, in the direction of the magnetic flux density in region 1.