Visual Inspection of the Data

For any given film thickness, incident beam current, and channeltron, I find that the most salient visual feature of the data (chapter 5) is the sign of the gradient of each electron arrival rate against beam energy curve: the electron arrival rate decreases with increasing energy. The second most salient feature is the sign of the curvature of each curve: the graph is convex $\cup$. This shape is as one would expect from equation 2.10, which is derived from the new theory in chapter 2.

The third most salient visual feature of the data is the direction, in which the electron arrival rate, for the majority of beam energies, changes on moving from magnetization up to magnetization down^5.1. My visual estimates of this direction are presented in table 5.3.

Table 5.3: Visually Estimated Directions, in Which the Electron Arrival Rates Change on Moving from Magnetization up to Magnetization down

Film thickness	$I_b/\,\mathrm{\mu{}A}$	Channeltron 1 Direction	Channeltron 2 Direction
0	$2\pm{}0.1$	Down	Down
0	$4\pm{}0.1$	Down	Down
0	$6\pm{}0.1$	Down	Down
1	$2\pm{}0.1$	Down	Down
1	$4\pm{}0.1$	Down	Down
1	$6\pm{}0.1$	Down	Down
2	$2\pm{}0.1$	Up	Up
2	$4\pm{}0.1$	Up	Up
2	$6\pm{}0.1$	Up	Up
3	$2\pm{}0.1$	Down	Down
3	$4\pm{}0.1$	Down	Down
3	$6\pm{}0.1$	Down	Down
4	$2\pm{}0.1$	Down	Down
4	$4\pm{}0.1$	Down	Down
4	$6\pm{}0.1$	Down	Down
5	$2\pm{}0.1$	Down	Up
5	$4\pm{}0.1$	Up	Up
5	$6\pm{}0.1$	Up	Up
6	$2\pm{}0.1$	Down	Down
6	$4\pm{}0.1$	Down	Down
6	$6\pm{}0.1$	Down	Down
7	$2\pm{}0.1$	Up	Up
7	$4\pm{}0.1$	Up	Up
7	$6\pm{}0.1$	Up	Up

This distribution of changes in electron arrival rate is interesting, because, from the new theory in chapter 2, and indeed from any theory, in which the process of changing the magnetization direction affects the arrival rates at the channeltrons only through the reflected electron beam's spin polarization, one would expect the changes in the two channeltrons' arrival rates, on reversing the magnetization direction, to be opposite. This has (table 5.3) been the case in only one of the twenty-four sets of conditions of film thickness and beam current measured.

It is suggested that, in terms of the diagrammatic view of the scientific method presented in ``Bayesian Interpolation'' [85], and in ``Bayesian Methods for Adaptive Models'' [52], this section, unlike the rest of this chapter, belongs to the `decide whether to create new models,' stage, rather than to the `assign preferences to the alternative models' stage ^5.2. Only some qualitative pointers to theories that might better match the contents of table 5.3 will be provided; the construction of enough quantitative detail to permit testing of new theories, in the light of data, will be left as an exercise for any reader who wishes to undertake it.

The author suggests three possible forms for new theories, in the light of table 5.3.

1. The precision of the positioning of the sample may be insufficient to prevent variation, of the kind observed in section 5.4, in the acceptance probability at the front of the polarimeter (the $a$ of appendix 5.3.2.) In support of this explanation, it is noted that, since the position of the sample is only reset on changes of magnetization or of film thickness, one would expect, if it were the correct explanation, that the direction of change of the count rate would be independent of channeltron number and beam current, depending only on film thickness, as is (table 5.3) indeed observed in almost all cases. However, there are two strong arguments against this explanation. Firstly, the precision with which the sample was positioned was chosen on the basis of empirical evidence (section 5.4) about the precision needed to prevent this effect. Secondly, the overall proportions of ``up'' and ``down'' directions of change of electron arrival rate, on magnetization reversal, are (table 5.3) unequal, by an amount that, depending on the details of the explanation, may be statistically significant, and militate against any explanation that does not involve a true magnetic effect.
2. The experiment may not be performed quickly enough to prevent drift effects, of the kind observed in section 5.5 between the measurements for different magnetization directions on the same film thickness^5.3, or the drift effects may be related to switching on and off the channeltron, rather than to a particular time-scale. In support of this explanation, it is noted that, as long as the drift effects relate to switching on and off the channeltrons, which occurs only at changes of magnetization direction or film thickness, one would expect, if it were the correct explanation, that the direction of change of the count rate would be independent of channeltron number and beam current, depending only on film thickness, as is (table 5.3) indeed observed in almost all cases. However, there are three strong arguments against this explanation. Firstly, the speed, with which the experiments were performed, was chosen on the basis of empirical evidence (section 5.5) about the speed needed to prevent drift effects; of course, this argument does not apply, if the drift effects relate to switching on and off the channeltrons, rather than to a particular time-scale. Secondly, the overall proportions of ``up'' and ``down'' directions of change of electron arrival rate, on magnetization reversal, are (table 5.3) unequal, by an amount that, depending on the details of the explanation, may be statistically significant, and militate against any explanation that does not involve a true magnetic effect. Thirdly, if this explanation is correct, and the drift effects relate to a particular time-scale, rather than to switching on and off the channeltrons, then, given that the time-scale for changing channeltrons is only a factor of $\sim{}2$ shorter than that for reversing magnetization direction, one would not expect the observed (table 5.3) high degree of independence of channeltron number, in the direction of change of the electron arrival rate.
3. A stray magnetic field, either from the sample itself, or from some ferromagnetic part of the sample holder (figure 4.5,) may be deflecting the electron beam in such a way as to change the acceptance probability at the front of the polarimeter (the $a$ of appendix 5.3.2.) In support of this explanation, it is noted that one would expect, if it were the correct explanation:
   • that the direction of change of the count rate would be independent of channeltron number and beam current, depending only on film thickness, as is (table 5.3) indeed observed in almost all cases, and
   • in contrast to any effects that are not genuinely magnetic, that the overall proportions of ``up'' and ``down'' directions of change of electron arrival rate, on magnetization reversal, would be unequal, as they are (table 5.3,) by an amount that, depending on the details of the explanation, may be statistically significant.

If more quantitative theories are constructed, based on the first two explanations, they would be expected to have significantly more adjustable parameters than the models in appendix 5.3.2, and therefore to take a prohibitive amount of CPU time to fit to the data using a Monte Carlo method. With reluctance, because it would have the consequence of ignoring evidence, which is relevant to a problem of parameter estimation and model comparison under consideration, long-term^5.4, it might, therefore, be necessary to set aside the existing experimental results (chapter 5,) and instead assess the existing models (chapter 2, appendix 5.3.2,) in the light of future experimental results, with the experimental method adapted to position the sample more precisely, to be quicker, or to avoid switching on and off the channeltrons.

However, if more quantitative theories are constructed, based on the, apparently more plausible, third explanation, they will provide an exciting possibility for using electron beams to probe magnetic surfaces, without relying on spin polarization effects. The data in chapter 5 would be immediately suitable for use in this process, but future experimental data could be gathered without the practical difficulties involved in Mott polarimetry. If, despite this, demand for measurements of the spin polarization of reflected beams continues to exist, it may be worthwhile to know that Lind [65] discovered that the stray field around a sample could be substantially reduced, by briefly applying a magnetic field, smaller than that used to magnetize the sample and in the opposite direction, after the sample was magnetized.

It is also, of course, possible that the future of magnetism measurements with electron beams will take a course that the author has not foreseen; either one for which the measurements in chapter 5 are useful, or one for which they are not useful.

In passing, it is noted that although the author is familiar with several examples (section 2.10) [1,4,62,63,64,25,65,26,51] of published data, obtained using Mott polarimeters, on the spin polarization of reflected, diffracted, transmitted, inelastic, and secondary electron beams from magnetic and non-magnetic materials, these all present the data in a processed form, similar to that of section 5.2, rather than in the raw form of chapter 5. This is understandable, given the length constraints of papers in collections, journal articles, and theses, but regrettably, renders it impossible to apply reasoning of the kind above to these data.

Although the primary purpose of this section was to make qualitative suggestions, it will probably be useful to make a quantitative estimate of the deflection that a stray magnetic field produces, in a reflected electron beam (figure 5.3,) as a check on the plausibility of this suggestion.

Figure 5.3: Deflection of an Electron Beam by a Stray Magnetic Field

The classical equations of motion of an electron, in a stray magnetic flux density, in the $z$ direction, $B$, are

$$\frac{\mathrm{d}^2x}{\mathrm{d}t^2} = \frac{eB}{m}\frac{\mathrm{d}y}{\mathrm{d}t}\textrm{,}$$ (5.1)

and

$$\frac{\mathrm{d}^2y}{\mathrm{d}t^2} = -\frac{eB}{m}\frac{\mathrm{d}x}{\mathrm{d}t}\textrm{,}$$ (5.2)

where $t$ is the time, $-e$ is the charge on an electron, and $m$ is the mass of an electron.

The general solutions of these equations of motion are

$$x = -\frac{imC}{eB}\exp\left(\frac{ieBt}{m}\right)+\frac{imD}{eB}\exp\left(-\frac{ieBt}{m}\right)+F\textrm{,}$$ (5.3)

and

$$y = \frac{mC}{eB}\exp\left(\frac{ieBt}{m}\right)+\frac{mD}{eB}\exp\left(-\frac{ieBt}{m}\right)+H\textrm{,}$$ (5.4)

where $C$, $D$, $F$, and $H$ are arbitrary constants. Given the boundary conditions, for the incident beam, that, at $t = 0$, $(x,y) = (-a,a)$ and $\left(\frac{\mathrm{d}x}{\mathrm{d}t},\frac{\mathrm{d}y}{\mathrm{d}t}\right) = (u,-u)$, the arbitrary constants can be set, to give

$$x = \frac{mu}{eB}\left(\cos\left(\frac{eBt}{m}\right)+\sin\left(\frac{eBt}{m}\right)-1\right)-a\textrm{,}$$ (5.5)

and

$$y = \frac{mu}{eB}\left(\cos\left(\frac{eBt}{m}\right)-\sin\left(\frac{eBt}{m}\right)-1\right)+a\textrm{.}$$ (5.6)

Only the electron trajectory, not the behaviour in time, is of interest. Therefore, it would be useful to find a time-independent function of $x$ and $y$. Using the advance knowledge that the trajectory would be a circle, it was decided to seek such a function of the form $(x-X)^2+(y-Y)^2$. The incident electron trajectory was thus discovered to be, where $\mu{}(B) = \frac{mu}{eB}$,

$$(x+\mu{}(B)+a)^2+(y+\mu{}(B)-a)^2 = 2(\mu(B))^2\textrm{,}$$ (5.7)

a circle, of radius $\sqrt{2}\mu(B)$, centred on $(-\mu{}(B)-a,-\mu{}(B)+a)$. The $x$ co-ordinate at which the reflection takes place, i.e. of the intersection of this circle and the sample surface, is

$$b(B) = -\mu{}(B)-a\pm\sqrt{(\mu{}(B))^2+2\mu{}(B)a-a^2}\textrm{.}$$ (5.8)

Of the two intersections, the one of interest is that closest to the starting point $(-a,a)$. Therefore, where $\mu{}(B) < 0$,

$$b(B) = -\mu{}(B)-a-\sqrt{(\mu{}(B))^2+2\mu{}(B)a-a^2}\textrm{,}$$ (5.9)

and where $\mu{}(B) > 0$,

$$b(B) = -\mu{}(B)-a+\sqrt{(\mu{}(B))^2+2\mu{}(B)a-a^2}\textrm{.}$$ (5.10)

The reflected electron trajectory will also be a circle of radius $\sqrt{2}\mu{}(B)$. However, the centre will be geometrically reflected, in the plane $x = b(B)$, to $(\mu{}(B)+a+2b(B),-\mu{}(B)+a)$, i.e. the trajectory is

$$(x-\mu{}(B)-a-2b(B))^2+(y+\mu{}(B)-a)^2 = 2(\mu{}(B))^2\textrm{.}$$ (5.11)

On the assumption that $a' = a$, this trajectory strikes the polarimeter front at an $x$ co-ordinate

$$c(B) = \mu{}(B)+a+b(B)\pm\sqrt{(\mu{}(B))^2-(b(B))^2}\textrm{.}$$ (5.12)

The intersection of interest is the one closest to the starting point $(b(B),0)$, i.e. where $\mu{}(B) < -a$,

$$c(B) = \mu{}(B)+a+b(B)+\sqrt{(\mu{}(B))^2-(b(B))^2}\textrm{,}$$ (5.13)

and where $\mu{}(B) > -a$,

$$c(B) = \mu{}(B)+a+b(B)-\sqrt{(\mu{}(B))^2-(b(B))^2}\textrm{.}$$ (5.14)

The distance that the reflected electron beam is displaced, along the polarimeter front, on reversing the stray field, is

$$d(B) = \sqrt{2}(c(B)-c(-B))\textrm{.}$$ (5.15)

This distance is plotted against the stray magnetic flux density in figure 5.4, using the estimated $a = 100\,\mathrm{mm}$, and a value of $u$ based on an estimated electron kinetic energy of $750\,\mathrm{eV}$.

Figure 5.4: The Distance along the Polarimeter Front that the Reflected Electron Beam Is Displaced, on Reversing the Stray Magnetic Field, against Stray Magnetic Flux Density

The width of the electron beam is of the order of $1\,\mathrm{mm}$, and the width of the polarimeter opening is of order $2\,\mathrm{mm}$. Therefore, any displacement of $\sim{}100\,\mathrm{\mu{}m}$ or more will have a significant effect on the acceptance probability. It is clear, from the graph, that a displacement of this size can be obtained with a realistic flux density.