R×B drift momentum spectrometer with high resolution and large phase space acceptance.

We propose a new type of momentum spectrometer, which uses the R×B drift effect to disperse the charged particles in a uniformly curved magnetic field, and measures the particles with large phase space acceptance and high resolution. This kind of R×B spectrometer is designed for the momentum analyses of the decay electrons and protons in the PERC (Proton and Electron Radiation Channel) beam station, which provides a strong magnetic field to guide the charged particles in the instrument. Instead of eliminating the guiding field, the R×B spectrometer evolves the field gradually to the analysing field, and the charged particles can be adiabatically transported during the dispersion and detection. The drifts of the particles have similar properties as their dispersion in the normal magnetic spectrometer. Besides, the R×B spectrometer is especially ideal for the measurements of particles with low momenta and large incident angles. We present a design of the R×B spectrometer, which can be used in PERC. For the particles with solid angle smaller than 88 msr, the maximum aberration is below 10-4. The resolution of the momentum spectra can reach 14.4 keV/c, if the particle position measurements have a resolution of 1 mm.

Introduction

The beam station PERC (Proton and Electron Radiation Channel) [1] for the experiments of free neutron decay is under development [2], as the successor of PERKEO [3], PERKEOII [4-8], and PERKEOIII [9], which measured the angular correlation coefficients A, B, and C in the free neutron decay.

The motivation of PERC is to supply an intense beam of well-defined electrons and protons (e−/p+) from free neutron decay. With the general-purpose e−/p+-beam, various quantities related to the physics in and beyond the Standard Model can be measured [10-14].

PERC provides in the instrument a neutron decay volume, where the free neutrons are imported and decay into charged electrons, protons, and neutral electron anti-neutrinos. With a series of coils, a specified static magnetic field is applied in the instrument. The charged decay e−/p+ spiral along the magnetic field lines, and are guided from the decay to the detection area. Fig. 1 sketches the principle of PERC.

Fig. 1

The principle of the PERC beam station. The charged decay e−/p+ are guided by a magnetic field from the decay volume to the detection area.

The neutron decay volume has a magnetic field of B0=1.5 T. After the decay volume, a magnetic field barrier B1=6 T is applied. The e−/p+ particles propagate adiabatically in PERC, hence the pitch angles of them, i.e., the angles between the particle momentum p and the magnetic field B, fulfil

Therefore, only the e−/p+ with pitch angles at B0 smaller than the critical angle can pass through the B1 field barrier

After the B1 barrier, the guiding field is gradually decreased to B2=0.5 T at the detection area, where the e−/p+ particles can be processed and measured.

As for the measurements of the particles, the loss free energy spectroscopy for electrons has been demonstrated in Refs. [3,15]. While in the experiments with PERC, a spectrometer for the e−/p+ momentum measurements besides the energy sensitive detector is desired. The principle of a normal magnetic spectrometer after PERC is sketched in Fig. 2.

Fig. 2

The principle of a magnetic spectrometer, that is attached to the exit of PERC.

At the end of PERC, the magnetic spectrometer must firstly shield the guiding field B2, and apply a vertical analysing magnetic field B3. The incident e−/p+ from PERC pass through a small aperture, then disperse in the B3 field. The position sensitive detectors for electrons and protons are placed on both sides of the incident window. The dispersion distance of a particle in the spectrometer is thenwhere p and q are the momentum and charge of the particle, is its incident angle according to the normal of the detector plane.

Compared with the energy resolving detectors, the magnetic spectrometer is versatile, that it can realize various measurements in PERC [1], also detect the electrons and protons at the same time. Technically, the position sensitive detectors highly suppress the backscattering problem of e−/p+ and the background. The momentum spectrum of the decay electrons has higher resolution in the low energy scale, thus is especially needed for the estimation of the Fierz interference term b in the neutron decay [10].

However, because of the strong magnetic guiding field of PERC, we found difficulties in the design of the magnetic spectrometer. The spectrometer has to drastically decrease the guiding field B2 to zero at the incident window, whereas the magnetic lines do not vanish, but spread in vertical directions. When the e−/p+ pass the vertical field, the pitch angles of them are highly distorted either in adiabatic or in non-adiabatic transports, as shown in Fig. 3.

Fig. 3

Sketch of e−/p+ motions in the drastically decreased B2 field. In adiabatic transports, the particles follow the magnetic field lines to vertical directions. While in non-adiabatic transports, the e−/p+ are bended in the vertical field. In both cases the pitch angles of the particles are highly distorted.

The distortion of the e−/p+ pitch angles strongly depends on the field distribution and the e−/p+ momenta, thus are neither predictable nor controllable [16]. Therefore, the distribution of the particles in the spectrometer can hardly represent their momenta.

In this case, we propose a method of R×B drift momentum spectrometer, which can realize the momentum analyses of the e−/p+ without eliminating the guiding field of PERC.

In Section 2, we introduce the principle of the R×B spectrometer. The systematic corrections of the particle distribution are stated in Section 3. The transfer function of the particles is discussed in Section 4.

Principle of the R×B drift momentum spectrometer

When a charged particle propagates in a curved magnetic field, it has the drift effect perpendicular to the magnetic field B and the field curvature , so called as the R×B drift. The drift velocity of the first order can be expressed as [17]where m is the mass of the particle, and are the velocity components parallel and vertical to the magnetic field line. In a static magnetic field, the velocity components can be expressed with the absolute velocity v and the pitch angle

Suppose that we apply a uniformly curved magnetic field, with the curvature R and the field strength B as constants, then the curved magnetic field lines are distributed parallely and coaxially, as shown in Fig. 4.

Fig. 4

Sketch of the principle of the R×B drift spectrometer, in which a uniformly curved magnetic field B3 is generated. The magnetic field lines are distributed parallely and coaxially, and are bended by an angle of .

According to Eq. (4), the drift velocity is a constant in the uniformly curved magnetic field, and the higher order contributions induced by are zero [18-20]. During a propagating time of T, the drift distance D of a particle iswhere is the bending angle of the route of the particle gyration center during the time Tas marked in Fig. 4. is a factor related to the particle pitch angle

Compare Eq. (6) with Eq. (3), the behaviour of the R×B drift is similar to that of the particle dispersion in the magnetic spectrometer. The drift or the dispersion distances in both cases are proportional to the particle momentum, and inversely proportional to the analysing magnetic field and the particle charge.

With this principle, we propose a design of the R×B drift momentum spectrometer, as shown in Fig. 5.

Fig. 5

The design of the R×B drift spectrometer at the end of PERC, and the simulated trajectories of e−/p+.

At the beginning of the R×B spectrometer, we apply several connected coils to gradually decrease the guiding field of PERC from 0.5 T to 0.15 T, with the field gradient satisfies the adiabatic condition of e−/p+ transports. After that a series of tilted coils generates a 180° bended magnetic field. Along the central line of the tilted coils, the curvature of magnetic field line is R0=40 cm, and the field strength is kept as B3=0.15 T. Fig. 6 plots the magnetic field from the end of PERC to the R×B spectrometer detector.

Fig. 6

The magnetic field strength along the central line of the coils from the end of PERC to the detector of R×B spectrometer. The field is decreased gradually from B2=0.5 T at the end of PERC to B3=0.15 T, and is kept as constant in the tilted coils. The zero position denotes the beginning of the R×B spectrometer.

At the beginning of the tilted coils, we apply an aperture of 1×1 cm2 to define the size of the incident e−/p+-beam. The particles through the aperture follow the curved magnetic lines, and turn 180° then reach the detector on top. During the propagation, they drift along the x-axis according to their charges and momenta. Fig. 7 shows the simulated dispersion of electrons and protons in the R×B spectrometer.

Fig. 7

The front view of the simulated electron and proton trajectories in the R×B spectrometer. The colours of the trajectories denote the particle momenta, that are arranged continuously from 0 to 1.19 MeV/c: (a) electrons and (b) protons.

Hence the R×B drift spectrometer, instead of eliminating the guiding field of PERC, evolves the field smoothly and gradually to the analysing magnetic field. The charged particles can be transported adiabatically during the processes, and their angular information can be kept and measured.

Table 1 lists the parameters of the standard configuration of the R×B spectrometer design.

Table 1

Parameters of the standard configuration of the R×B drift spectrometer.

Parameter	Comment	Value
B3	Analysing field	0.15 T
R0	Field line curvature	40 cm
w	Aperture width	1 cm
h	Aperture height	1 cm
α	Bending angle	π
θ_max	Max. pitch angle	9.1°
rmax	Max. gyration radius	0.42 cm
Dmax	Max. drift	8.29 cm

w and h in Table 1 are the width and height of the aperture along the x- and y-axes. and r are the maximum pitch angle and gyration radius of the decay e−/p+ in the B3 fieldwhere p=1.19 MeV/c is the maximum momentum of e−/p+ from the free neutron decay [21].

With the standard configuration, the maximum drift D for p is 8.29 cm.

Corrections on R×B drift spectrometer

In the R×B spectrometer, the particle distribution on the detector is influenced by the properties of the particles and the instrument.

e−/p+ pitch angle

Compare Eq. (8) with Eq. (3), both the drift and the dispersion have corrections related to the pitch (incident) angle . However, because of the presence of the component in R×B drift in Eq. (4), the influence of the correction factor in the R×B spectrometer is much smaller than that in the dispersive magnetic spectrometer. In Fig. 8, the magnitudes of both correction factors are plotted.

Fig. 8

The incident angle factors in the dispersive magnetic spectrometer and in the R×B drift spectrometer, versus the incident (pitch) angle from 0° to 45°.

For , i.e., the solid angle less than 88 msr, the in the R×B drift is negligible as less than 10−4 deviated from 1, while the deviation in the magnetic spectrometer is 1.4×10−2. Hence the R×B spectrometer has large acceptance of e−/p+ incident angles, which is a significant advantage.

Gyration radius r and aperture width w

In the R×B spectrometer, the gyration radii of the particles remain during the drifts and the detection. For a given momentum, the maximum gyration radius of the particles is

Through the aperture, the size of the e−/p+-beam on the detector will beas sketched in Fig. 9(a). Therefore, the maximum deviation induced by the gyration radius r relative to the drift D is

Fig. 9

Sketches of the electron pattern on the detector with discrete momenta, influenced by w, r, and H. p denotes the maximum electron momentum in free neutron decay: (a) electron distribution influenced by w and r and (b) electron distribution influenced by H.

Beam height H and curvature R

In the curved magnetic field, the field strength has a gradient along RB3 and R0 are the field strength and the curvature along the central line of the tilted coils. The particles at different positions along R will experience deviated magnetic fields, thus have the driftwhere y is the vertical position of the particle on the detector relative to the beam center. The maximum deviation of the drift is

Hence the e−/p+-beam height relative to the field curvature H/R0 tilts the particle distribution on the detector, as sketched in Fig. 9(b). For a detector only sensitive to the x-position, the measured particle distribution is widened.

Particle distribution on the detector

The dispersion of the particles in the drift spectrometer is mainly influenced by the systematics of w, r, and H. Fig. 9 shows the sketchy distribution of electrons on the detector affected by these factors.

Fig. 10 shows the simulated distributions of electrons on the detector, with discrete momenta from 0 to 1.19 MeV/c.

Fig. 10

Simulated electron distribution on the detector of the R×B spectrometer with discrete momenta. p denotes the maximum electron momentum in free neutron decay as 1.19 MeV/c. The particles with can also be measured in the spectrometer. The deviations caused by r and H are proportional to the drift distance D.

The deviation of the drift caused by w is a constant, while that induced by r and H are proportional to the drift distance. Hence at the low momentum range, the R×B spectrometer has better performance.

Furthermore, in normal magnetic spectrometer, the particles with very small momenta cannot be totally measured if their dispersion distances are smaller than the aperture width . While as shown in Fig. 10, the R×B spectrometer does not have this limit. It can measure the full range of the momentum even when .

Transfer function

The motions of the e−/p+ particles in the R×B spectrometer are clearly defined during the drift processes. The transfer function, i.e., the relation between the momentum spectrum F(p) and the particle distribution on the detector G(x), can be calculated and used in the data analyses. In this section, we discuss the transfer function including the corrections of r, w, and H. The negligible correction is not considered here. And the higher order contributions, e.g., the R deviation induced by v, the r deviation induced by the B3 gradient along R, are not taken into account in the transfer function.

Particle distribution from point source

We assume that the e−/p+ are homogeneously emitted in the decay volume of PERC. In addition, the size of the e−/p+-beam at the end of PERC is much larger than the size of the aperture's open window. Therefore, the particles that pass through the aperture can be treated as emitting from the open window of the aperture.

For the particles emitted from a point source at position x=0 with given momentum and pitch angle, their distribution along the x-axis is given as [22]where K denotes the complete elliptical integral of the first kind.

Angular distribution of particles in B3 field

If we apply unpolarized neutrons in PERC experiments, then the e−/p+ are isotropically emitted in the decay volume [10]. Their angular distribution in the field B0 is

With Eq. (1), when the particles propagate from B0 to B3, their angular distribution is transformed to

If the aperture is sufficiently thin, it does not distort the angular distribution of the particles. We integrate Eq. (16) over , the particle distribution along the x-axis from a point source is then

Fig. 11 plots the distribution L(x,p) of particles in the B3 field with different momenta.

Fig. 11

The distribution L(x,p) of particles from a point source in the B3 field along the x-axis, as given in Eq. (19). Different curves denote the particles with different momenta. p is the maximum momentum of e−/p+ in free neutron decay as 1.19 MeV/c.

Transmission function of aperture

Since the distance from the aperture to the detector is much longer than the helical pitches of the particles, we assume that the particle distribution on the detector from any point in the aperture follows Eq. (19).

Define the transmission function of the aperture along the x-axis as

We treat the particles as emitted from the aperture's open window. On the detector, the particles then have a distribution ofwhich is the convolution of the aperture function T(x) and the distribution L(x,p) of a point source.

Correction of beam height H

For the correction of beam height H, we consider the particle distribution along the y-axis. For simplicity, we use the condition w=h, so the distribution along the y-axis is the same as Eq. (21), i.e., P(y,p). Since the factor H/R0 tilts the particle distribution on detector, P(y,p) is projected on the x-axis. According to Eq. (14), the projection can be written as

Transfer function

All together, we take the corrections related to r, w, H, as well as the drift D(p) into account. The total transfer function is then the convolution of the distributions in Eqs. (21) and (22)

For a given momentum spectrum F(p), the particle distribution on the detector can be derived

Fig. 12 shows examples of given momentum spectra F(p) and the resulted G(x) from Eq. (24).

Fig. 12

Examples of given F(p) and resulted G(x) according to Eq. (24). (a) and (b) denote discrete momentum and position distribution same as shown in Fig. 10. (c) and (d) denote the theoretical momentum spectrum of electrons in free neutron decay and their position distribution.

In experiments, G(x) can be measured by position sensitive detectors. To evaluate F(p) from Eq. (24), which is the Fredholm integral equation of first type, a numerical method is used to convert the integral into the quadrature calculation, the F(p) and G(x) into arrays, thus convert the kernel function M(x,p) into a square matrix [23,24]. The transfer function then can be written asand the momentum spectrum F can be evaluated

In this case, the G and F have the same number of rows, thus the same resolution. In the standard configuration, if G has a resolution of 1 mm, the momentum spectrum F can reach a resolution of 14.4 keV/c.

Conclusion

The R×B drift spectrometer offers the opportunity of momentum measurements of charged particles in the instrument with guiding fields, in which case the normal magnetic spectrometers cannot work well. In this proposed drift spectrometer, the guiding field is not eliminated, but gradually evolved to the analysing field. The drifts of the particles in the uniformly curved magnetic field have similar behaviours as in the normal dispersive magnetic spectrometer.

As a conclusion, the R×B spectrometer has the advantages:

Adiabatic transports of particles. As shown in Fig. 6, from the guiding field to the detector of the R×B spectrometer, the charged particles can be adiabatically transported. The angular distribution of the particles can be kept and measured.

Low momentum measurements. As shown in Fig. 10, the particles with very small momentum can be measured in the R×B spectrometer, while they cannot be totally detected in normal magnetic spectrometer if their dispersion .

Large acceptance of incident angle. As shown in Fig. 8, when , corresponding to 88 msr solid angle, the direct aberration induced by the e−/p+ incident angle is very small as less than 10−4.

As a conceptual design of R×B spectrometer, the particle drifts are considerably influenced by the systematics related to both the instrument and the particle properties. Table 2 lists the maximum sizes of the corrections in the standard configuration.

Table 2

Estimated maximum correction sizes on the e−/p+ particles dispersion in the R×B spectrometer.

Correction	Comment	Max. Size
4rc/D	Gyration radius	2.0×10−1
w/Dmax	Aperture width	1.2×10−1
H/R0	Aperture height	6.7×10−2
ΔB_3	Field homogeneity	8×10−3
f(θ_max)	Incident angle	8×10−5

However, the motions of the e−/p+ particles are clearly defined during the drifts, thus the transfer function of the particles can be well known. Experimentally, one is able to fit the momentum spectrum F(p) to the measured particle distribution G(x), or numerically evaluate F(p) from G(x). In the standard configuration, if the position sensitive detector has a resolution of 1 mm, the momentum spectra can reach a resolution of 14.4 keV/c. Additionally, by performing detector calibration with defined particle sources, the systematic errors can be controlled at low level.

Besides, there is also room for improvement of the R×B spectrometer design. For further development, we can decrease the corrections, e.g., by optimizing the bending angle , the curvature R, and the analysing magnetic field B3. The transfer function in this paper only considers the motions of the e−/p+ particles to the first order. The higher order contributions, e.g., the acceleration , the deviation of curvature R induced by the drift D, are small but complicated [19,20]. More accurate transfer function can be calculated, and results in two-dimensional particle distribution on detector. In the area of large acceptance magnetic spectrometer, we notice that the differential algebraic techniques are applied in the calculation of non-linear aberrations [25,26]. In future investigation, the transfer function can also be determined numerically with the desired precision, expected to be less than 10−2 aberrated.