Magnetic guidance of charged particles

Many experiments and devices in physics use static magnetic fields to guide charged particles from a source onto a detector, and we ask the innocent question: What is the distribution of particle intensity over the detector surface? One should think that the solution to this seemingly simple problem is well known. We show that, even for uniform guide fields, this is not the case and present analytical point spread functions (PSF) for magnetic transport that deviate strongly from previous results. The"magnetic"PSF shows unexpected singularities, which were recently also observed experimentally, and which make detector response very sensitive to minute changes of position, field amplitude, or particle energy. In the field of low-energy particle physics, these singularities may become a source of error in modern high precision experiments, or may be used for instrument tests, for instance in neutrino mass retardation spectrometers.


Introduction
The motion of charged particles in magnetic fields is a highly developed subject, treated in numerous papers and books. The most frequently investigated case is magnetic focussing, as used in electron microscopes, oscilloscopes, electron spectrometers, particle accelerators, or in mass spectrometers with magnetic sector fields. Electron optics was developed early in the past century [1], mainly for small angular ranges of particle emission Δθ << 1, and for trajectories that describe less than one full orbit of gyration (n < 1), see also [2] and the books quoted therein. The case of a magnetic -ray spectrometer based on one full orbit (n = 1) was treated in [3,4], while a survey on magnetic electron and ion spectrometers is given in [5].
In more recent times, magnetic fields are increasingly being used to simply guide charged particles, like electrons, muons, ions, or other, efficiently from a source to a detector. Such setups are found in magnetic photoelectron imaging [6,7,8], invented in the early 1980ies, molecular reaction microscopes [9,10] (early nineties), retardation spectrometers [11,12,13] (early nineties), time projection chambers [14] (mid-seventies), and in muon [15], neutron [16,17], or nuclear decay [18] spectrometers, to name just a few experiments and surveys. In these applications charged particles are emitted over a wide range of emission angles (0 < θ  /2), and the number of orbits of gyration may vary widely (0 < n < ). This magnetic guidance of charged particles is the topic of the present paper.
For particles emitted from a point source, as sketched in Fig. 1, the distribution function of particle intensity over the detector plane is called a point spread function (PSF). Once a PSF is known, the particle distribution for any type of extended source can be calculated from it.
Pictures like Fig. 1 are found in most introductory textbooks on physics, and one should think that the magnetic PSF for this setup is well known and no longer subject of investigation. But this is not the case, and we find some striking features in this PSF which, to our knowledge, have not been published before, and which may be of interest to a wider community.
Our particular interest is on the role of the magnetic PSF in the field of low-energy particle physics, which field is entering what some call the high-precision era [19,20,21], often searching for 10 −4 effects that might signal new physics beyond the standard model, at a level where Monte-Carlo simulations often meet difficulties. In a recent publication [22] we had sketched the derivation of the PSF and its singularities in the context of neutron -decay, and listed over a dozen neutron decay experiments that use magnetic guiding of charged reaction products for high precision measurements. The paper is organized as follows: Section 2 sketches the conventional approach to the problem, which leads to a smooth hyperbolic PSF, used already some 30 years ago [7], and still in use up to these days [8]. Section 3 gives analytical proof that this conventional approach fails for any finite number n of gyrations: At certain values Rn of the particle's displacement R on the detector (Fig. 1), strong resonances appear in the PSF, in spite of the wide angular range of emission angles θ. Section 4 shows that this singular behaviour of the PSF makes the local response of a detector very sensitive to minute changes of instrumental parameters like field amplitude B, particle energy E, detector position z0, or its angular adjustment. Section 5 extends these algebraic calculations to anisotropic sources and to nonuniform guide fields. Section 6 sketches a recent experiment [23] done at LANL, in which the resonances predicted in [22] were observed, and discuss some possible uses of the true magnetic PSF.

The conventional magnetic point spread function (PSF)
Without loss of generality, let the charged particles be monoenergetic electrons. Their radius of gyration r depends on their polar angle of emission θ as and on their kinetic energy E via the maximum radius of gyration as 2 0 for a field of amplitude B, with electron charge e, mass m, and relativistic momentum p. We assume a uniform magnetic field, for the non-uniform case see Sect. 5. The pitch of the helix is also indicated in Fig. 1. These formulas are found in most textbooks on electromagnetism.
Upon arrival of an electron on the detector, its total number of gyration orbits is where the slash reminds us that n' needs not be an integer. The total phase angle of gyration is hence related to the angle of electron emission θ as The smallest occurring phase angle, reached for electron emission under θ = 0, is is the corresponding minimum number of orbits in the limit θ → 0 (where in fact a gyration is no longer visible). On the detector, the electron's point of impact (circular dot in Fig. 1) is displaced from its projected starting point (grey dot), reached for θ = 0, by the distance where ( modulo 2 )      is the phase angle seen on the surface of the detector, with 02    .
The conventional approach to the problem is to assume that all phase angles occur with the same probability, given by dp0/d = 1/, see [23] for a straightforward derivation of the magnetic PSF under this assumption. In this approach, the probability for finding an electron Inserting, for a given r, the derivative (dR/d) −1 as a function of R from Eq. (7) leads to For an isotropic source, integration of dp0/dR over angle of emission θ then gives a probability distribution that no longer depends on R, with R = (x 2 +y 2 ) 1/2 . The singularity at R = 0 reflects the fact that all orbits cross the origin, for arbitrary values of emission angles θ and φ, and of energy E. Figure 2 of [24] shows a rough measurement of such a 1/R response. Note that in [22] we inadvertently called the function g(R) the PSF, and not the function f(R). However, this conventional result cannot be the full truth, because, in its derivation from Eq. (7), the phase angle  and the pitch angle θ are treated as independent variables. In other words: for a given emission angle θ, the electron on the detector is erroneously assumed to run on a circle through all values of .

Derivation of the true magnetic PSF
In reality both angles  and θ are uniquely linked to each other by Eqs. (4) and (5) as After emission under θ, the electron hence arrives at one fixed and predetermined position on the detector, given by the electron displacement 22 1 2 00 from Eq. (7). To increase the size of the phase angle  on the detector, one has to increase the emission angle θ at the source, and with it the gyration radius r from Eq. (1), such that the trace on the detector is no longer a circle but some sort of a spiral.  We now come to the calculation of the true PSF. Often the polar angular distribution of the particles emitted from a source is developed in Legendre polynomials as functions of cosθ.
Therefore the PSF is best written as We first treat the isotropic case d /d cos 1 P   , for anisotropic sources see Sect. 5 However, we need the PSF not as a function of , but as a function of R. Inversion of the multi-valued function R() of Fig. 2  , as seen in the example of Fig. 2. Within a given cycle on the detector, numbered by the integer n, the slow envelope can therefore be piecewise approximated by a constant value Rn, which is best chosen to be the maximum of R() in the n th interval as indicated by the dashed horizontal lines in Fig. 2. The true function R() from Eq. (13) then is piecewise replaced by the approximate functions The dashed curves in Fig. 2 show these invertible functions Eq. (17), indicating the high quality of the approximation. For each orbit, the approximate R() can be resolved for , where + holds for the rising branches of R() in Fig. 2. This approximation holds for the rising branches of R() in Fig. 2, and  for the falling branches.
These approximate 's then must be inserted in Eq. (15). In this way one obtains for every cycle a partial PSF that we call fn. These partial PSFs must then be summed up to obtain the magnetic PSF, For integer n0, one finds in the denominator of Eq. (15) with the plus sign for the rising branches, the minus sign for the falling branches. After Eqs. (20) Usually n0 is not an integer, in which case special attention must be given to the lowest orbit, then numbered by the next-lower integer nf = floor(n0). The width of the lowest interval in Fig. 2 and Eq. (17)    The fluctuating R() and its smooth envelope |sinθ()| coincide near each maximum Rn. This means that  and θ in Eq. (7) are strongly correlated there, contrary to the conventional assumption of independence of  and θ. Therefore the deviations of the true PSF from the conventional PSF are strongest at R  Rn. They are singular because the derivative d/dR in Eq. (14) diverges whenever R() reaches a maximum Rn near a half-integer number of revolutions, where R() becomes stationary. If R is increased above a particular Rn, the correlation between  and θ in the corresponding orbit is suddenly lost, cf. Fig. 2, and only the rather uncorrelated terms from the higher orbits contribute to Eq. (19). Therefore the true PSF falls steeply back to the conventional PSF whenever R rises beyond one of the Rn.

Properties of the magnetic PSF
In terms of emission angle θ, the ring-shaped singularities in Fig. 3

Anisotropic sources and non-uniform guide fields
with integers m ≤ l. An example is particle emission from atoms or nuclei that carry a vector or tensor polarization, see for instance chapters 19.3 and 20.5 in [25].
We begin with the conventional PSFs for anisotropic sources.
For l = 0 this coincides with the result (10) for isotropic sources. For l = 1 this last equation gives the PSFs for angular correlation functions, like the parity violating  asymmetry, see [22] for details. Another example is particle emission from surfaces obeying Lambert's law. For l = 1, m = 1 we have 1 2 2 , with the complete elliptic integral E(x), and so forth. Figure 5 shows Our second extension treats the case that the magnetic field, while still axially symmetric, is not uniform. In many experiments the field decreases continuously from B at the source to B' at the detector, which avoids glancing incidence on the detector for particles emitted near θ = /2. In the experiments [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] cited in the introduction, non-adiabatic transitions are strictly suppressed because their angle dependent energy losses would corrupt the measurements. In the adiabatic approximation, for BB   the inverse magnetic mirror effect makes the gyration radius widen from r at the source to r' on the detector, while the helix angle decreases from  to ', with We expect that the usual adiabatic invariants, on which the above equations are based, guarantee that the particle distribution on the detector remains unchanged, stretched, however, by a factor 1/b. While this conjecture sounds reasonable, we better check it analytically. To this end we set both quantities being related to each other as 22 ( The electron angular distribution then changes on the way from the source to the detector from d / d Pu (taken, e.g., from Eq. (27) , for a decreasing field b < 1 this angular distribution narrows to arcsin c b    , and diverges when the critical angle θ'c is approached from below. and the phase angle ' remains the same. First we note that the size of R' is determined between the particles' last focus and the detector, separated from each other by the short distance δz. This focus exists also for gyration in a non-uniform field, according to Busch's theorem [1]. The distance δz has at most the length of one pitch, δzd   , which latter is of the order of the local gyration radius dr  , cf. Eq. (3). On the other hand, the adiabatic condition requires that variations of the field B(z) are negligible over distances of one gyration radius r', hence also over δzr  . Therefore the field is uniform between the last focus and the detector, and Eq. (7), and with it Eq. (9), are valid also for R' and '.
From this we conclude that the integral Eq. (10) can also be used for transport in non-

Experimental verification, and possible applications of the new PSF
A recent experiment [23], done at the Los Alamos National Laboratory on the ultracold neutron decay spectrometer UCNA [26], in [23], where the conventional PSF would predict a very smooth response. The agreement with simulated expectations is excellent. The Monte Carlo result in their Fig. 4 can be compared to our result in Fig. 2b, calculated with identical parameters, namely, n0 = 1.6. Note that our resonances are narrower and steeper than the Monte Carlo result from [23]. Hence care must be taken when applying these calculations to possibly inherently broadened experimental data.
What are the possible benefits of having a new PSF? First, knowledge on the ring-shaped singularities in the PSF may promote understanding of experimental data and avoid surprises, for instance in reaction microscopy and similar experiments. Second, even when these rings remain unresolved, one must investigate their effect in high precision experiments, as was done for neutron decay in [22,23]. Third, these rings may serve as an analytical tool to assess the proper working of magnetic guiding systems.
As an example for this last point we take the retardation spectrometer of the neutrino mass experiment KATRIN [13]. Its length from the effective 3 H source to the detector is properly shaped. The Bi-source should be installed somewhat upstream of the initial field maximum to limit θmax, and the detector should be moved further downstream of the pitch field region to adapt gyration radii to detector size. In this way the entire spectrometer volume could be probed for possible discrepancies. These thoughts serve merely to remind the reader that new insights may generate new opportunities. (In the meantime I learned from C. Weinheimer that at 976 keV, adiabatic transport is no longer guaranteed in the KATRIN spectrometer.) Similar studies could be done on the neutron decay spectrometers PERC [28] with n0 = 200, Nab [29] with n0 = 170, or Perkeo-III [30] with n0 = 15. Note added: A recent preprint [31] combines our initial approach with a special numerical method and finds results that coincide precisely with our result in Fig. 3b. The discrepancies mentioned in this note 16 refer to the first version arXiv:1501.05131v2 [physics.ins-det] of the present preprint, where a less precise approximation had been used.

Conclusions
We calculated the point spread functions for charged particles in magnetic guide fields, which differ significantly from previously used results, as seen in Fig. 3. Algebraic results are derived for isotropic and anisotropic point sources, for uniform and non-uniform guide fields, valid also for sources rather close to the detector. The singularities found move rapidly across the PSF when the number of gyration orbits is changed by as little as a fraction of one orbit, see Fig. 4. A recent experiment done at LANL corroborates these results.