Framework of the Models

Both models are based on the new theory of polarized electron reflection from a bulk sample presented in chapter 2. The use of a model intended for a bulk sample is justified on the basis that the inelastic mean free path of electrons, of the energies used in this experiment, is [83,51] no more than $\sim{}1\,\mathrm{nm}$, with the result that even rather thin films will appear, when examined with an electron beam, to be bulk materials.

Therefore, when the beam current into the electron gun is $F$, the rate of electron incidence on the sample is $F/e$, and the rate of electrons leaving the sample, in the reflected beam (figure 3.1,) is, from integration of equation 2.26,

$$G = \left(\frac{e^2V^2}{16E_b^2\cos^4I}+\frac{e^2\hbar^2B^2}{192m_e^2E_b^2\cos^4I}\right)\frac{F}{e}\textrm{.}$$ (5.19)

The polarization of the reflected beam is, from equation 2.8,

$$P = -\frac{4e^2\hbar{}m_eV_1B_1}{12e^2m_e^2V_1^2+e^2\hbar^2B_1^2}\textrm{.}$$ (5.20)

The electrostatic potential, $V$, and the magnetic flux density, $B$, in the sample, are adjustable parameters of the ``main'' model, while in the ``null'' model, only $V$ is adjustable, $B$ being fixed at zero. In fact, since there are eight cobalt film thicknesses, including zero thickness (table 5.1,) there are multiple parameters of this type, $V_j$ and $B_j$, with the film thicknesses indexed by integers $j$, from $0$ to $7$ inclusive; for thickness $0$, $B_0$ is fixed at zero even for the ``main'' model. These then lead to electron leaving rates $G_j$ and reflected polarizations $P_j$.

Since the entrance hole of the polarimeter (figure 3.1) is of finite width, it is possible that not all of the electrons in the reflected beam will enter the polarimeter. Therefore, an acceptance probability $a$ is defined, so that the rate of electrons entering the polarimeter is $aG_j$. The entrance hole and grids are assumed to be non-polarizing, with the result that the polarization of the beam entering the polarimeter is $P_j$.

The thorium foil then scatters electrons towards the two channeltrons. The rate of electron arrival at channeltron $1$ will be, by equation 3.4,

$$f_a^{(1, j)} = \Gamma{}(1+SP_j)aG_j\textrm{,}$$ (5.21)

and that at channeltron $2$ will be, by equation 3.5,

$$f_a^{(2, j)} = \Gamma{}(1-SP_j)aG_j\textrm{,}$$ (5.22)

where the spin-averaged scattering probability $\Gamma$ and the Sherman function $S$ are characteristics of the thorium foil. Calibration values of $\Gamma$ and $S$ for a Mott polarimeter, of identical design to that used for these experiments, have been provided by other workers [6], and those for electrons with energy $20\,\mathrm{keV}$ are reproduced in figures 5.6 and 5.7. Although the energy of the electrons on striking the thorium foil, in the experiments presented in this thesis, is $(20.5\pm{}0.00725)\,\mathrm{keV}$, not $20\,\mathrm{keV}$, the systematic error introduced by this energy difference [6] is negligible, compared with the quantization error of the author's readings from the published graphs (section 5.3.4.)

Figure 5.6: Graph of spin-averaged scattering probability $\Gamma$ against energy loss window $W$, from the calibration data for the compact retarding-potential Mott polarimeter, provided in ``High-efficiency retarding-potential Mott polarization analyzer'' [6]. The error bars represent the standard deviations associated with the quantization of the author's readings from the published graph.

Figure 5.7: Graph of Sherman function $S$ against energy loss window $W$, from the calibration data for the compact retarding-potential Mott polarimeter, provided in ``High-efficiency retarding-potential Mott polarization analyzer'' [6]. The error bars represent the sums in quadrature of the standard deviation associated with the quantization of the author's readings from the published graph, and the error quoted on the published graph.

The published calibration data are for electron energies, on striking the thorium foil, of $20\,\mathrm{keV}$ and $25\,\mathrm{keV}$, and are presented as a function of the energy loss window $W$, defined by the potential at the retarding grids (chapter 3.) The author has chosen to include the published data for a $20\,\mathrm{keV}$ energy, on striking the thorium foil, in the data set (section 5.3.4,) from which inference is to proceed, and to characterize the thorium foil by by twelve adjustable parameters, $\Gamma_k$ and $S_k$, where $k$ runs through integers from $0$ to $5$ inclusive,

$$\Gamma = \sum_{k=0}^5\Gamma_kW^k\textrm{,}$$ (5.23)

and

$$S = \sum_{k=0}^5S_kW^k\textrm{.}$$ (5.24)

These fifth order Taylor expansions are an extension, to calibration data that include relatively large energy loss windows, of the spirit of the first and second order Taylor expansions in ``Use of thorium as a target in electron-spin analyzers'' [94]. The fifth order has been chosen to provide a number of adjustable parameters, in each Taylor expansion, that is the largest integer less than or equal to half the number of calibration data points, in line with MacKay's [93,90] recommendation for the number of parameters in neural network models.

Channeltron $i$ (figure 3.2,) used in single-electron counting mode in these experiments, has a detection efficiency $\eta_i$, adjusted by a time-dependent perturbation $p_{ij}$ to allow for drift (section 5.5.) Since the speed of the present experiments has been chosen so that the drift should not be significant within the time-scale of measurements on a single film thickness, each channeltron has just one $p_{ij}$ value for each film thickness; the subscript $j$, as before, indexes the film thickness. Therefore, the electron detection rate at the channeltron $i$, from film thickness $j$, is predicted to be

$$f_p^{(i, j)} = p_{ij}\eta_if_a^{(i, j)}\textrm{.}$$ (5.25)

The $p_{ij}$ and $\eta_i$ values are further adjustable parameters of the model.