Nonuniform electron distributions in a solenoidal ioniser

Solenoidal ionisers are a new class of highly efficient helium detectors that are increasingly important for high resolution atom scattering, molecular scattering and scanning helium microscopy. They operate via electron ionisation, where the electrons are trapped by the magnetic field of a solenoid and additional electrostatic potentials. Their ionisation efficiency scales with the electron population they contain, motivating large devices with high emission currents. However, these detectors typically become unstable at high electron densities, constraining their performance improvement. Through imaging the electron population at the exit of the ioniser, we demonstrate that these instabilities arise from non-uniformities in the electron distribution. Considering the ioniser as a non-neutral plasma leads to the proposal of the formation of a virtual cathode and a plasma instability as the origins of the non-uniformity.

The scattering of atoms and molecules from surfaces has played a critical role throughout the history of experimental physics discovery, with one of the earliest and most famous examples being Estermann and Stern's demonstration of the wave nature of matter by the diffraction of helium atoms from LiF [1]. More recent applications include measuring the electron-phonon coupling constant of various samples including 2D materials with helium scattering [2][3][4][5] and determining the quantum rotational state to state transition probabilities of a hydrogen molecule [6]. As experiments become more ambitious and complex they typically involve measuring an ever smaller flux of atoms or molecules. Such requirements demand improved detectors; a difficult task for inert species such as helium. * Author to whom any correspondence should be addressed.
Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Solenoidal ionisers are a class of recently developed electron ionisation devices designed specifically for the highly efficient detection of atomic and molecular beams [7][8][9]. While many devices utilise electron ionisation [10,11], it is techniques such as scanning helium microscopy [12][13][14], helium spin echo [15] and molecular scattering [6,16] that have adopted and driven the development of solenoidal ionisers due to the demand for high efficiency detectors tailored to low mass species. These instruments are named for their use of a solenoidal magnetic field to radially contain a population of electrons along the axis of the device. Radial confinement of electrons allows for detectors with long axial lengths, providing a large ionisation volume and thereby maximising the detection efficiency. Reflection of the ions at one end of the ioniser can also lead to increased electron densities.
The ionised species are themselves radially confined by the electronic space charge and also experience an axial potential, that is either flat or has a gradient to extract ions more rapidly. In instruments that use a flat axial potential [9], it is believed that the ions perform randomly orientated ballistic motion along the ioniser till they reach the exit of the ioniser, Figure 1. (a) Cross-sectional schematic of the PMSI and probe apparatus. Blue-Stepped hollow cylindrical NdFeB permanent magnet with iron yoke collars. Red-Floated filament, 5A maximum drive current, floatable to 5 kV. Grey-Shaped cylindrical liner electrode and 1 kV DC power supply. Purple-Coaxial probe mounted on xyz-translation stages. The core and shield of the probe are isolated and each floated by a 1 kV DC power supply. An ammeter monitors the current on the core of the probe. Yellow-Electron pathway from emission at the filament to extraction at the exit. The mesh grid prevents most of the electrons from returning to the filament, reducing the effect of electron extraction on the emission rate of the filament. (b) Currents collected by the coaxial probe, Ip (multimeter current), outer probe shield, I sh , filament emission current, Ie and total current, Itot, during a scan of the probe. The currents all sum to a constant value, indicating that the total current seen leaving the filament is accounted for. A well defined distribution is observed on the probe scan, while halos appear in the shield when the current strikes the side of the probe. Additional speckle noise in the outer shield current is thought to be where the electron beam strikes some exposed insulating material. The total emission current variation is always below ∼2% indicating that the measurement is not having a significant effect on the emission. however there is little control of the process. By contrast, active ion extraction is typically achieved by imposing an electric potential gradient to ensure that ions move towards the exit of the ioniser. The potential gradient is obtained by varying the depth of the electron space charge well by changing the distance between the electron beam and the electrode wall with either a tapered electrode [7] or a nonuniform magnetic field that compresses the electron beam [17].
Previous demonstrations of solenoidal ionisers typically exhibit either unstable [9] or hysteretic [18] behaviour presenting a serious issue for measurements. It has been suggested that the changes in behaviour are due to sudden alterations of the electron population inside the ioniser [9,18], but to date no direct evidence has been observed.
In the current work, we report the first experimental map of the electron distribution inside a solenoidal ioniser using a coaxial probe on a motorised translation stage. We observe a non-uniform electron distribution at high emission currents that explains the instabilities observed in other similar solenoidal ionisers. The paper concludes with a proposal for potential mechanisms for the instability and suggestions for how such mechanisms could be mitigated in future instruments.
The permanent magnet solenoidal ioniser (PMSI) is represented schematically in figure 1(a) and has a similar principle of operation to previous instruments that use an electromagnet [8,9]. An axially magnetised, stepped, cylindrical permanent magnet (details provided in the appendix) is mounted around the outside of a vacuum chamber to provide an increasing axial magnetic field down the length of the device. A shaped cylindrical electrode (liner) is placed inside the vacuum chamber to control the electric potential inside the ioniser.
Thermionic emission from a tungsten filament provides the source of electrons at the entrance to the ioniser. The potential difference between the filament and the liner accelerates the electrons into the body of the ioniser, while the axial magnetic field ensures the electrons are trapped radially during transit. In addition, under normal operation, electrons are reflected at the exit of the device by the drop in potential from the liner voltage to that of the grounded chamber, increasing the probability of ionisation events and thus ioniser sensitivity.
To view the electron distribution, additional instrumentation was added to the system. In particular, a coaxial probe was mounted on a motorised xyz-translation stage (z-axis: VAb linear manipulator, xy-axes: Varian 2 axis manipulator, actuators: Thorlabs ZFS25B with KST101 controllers) at the exit of the ioniser. The central conductor of the probe (∼1 mm diameter) is connected to an ammeter, which is floated to the liner potential. The outer conducting sheath of the probe, also floated close to the liner potential, is insulated from the central conductor. This configuration ensures that electrons striking the side of the probe are collected by the sheath, while the central conductor only collects electrons impinging directly on the tip.
A mesh grid was installed near the liner exit to: (a) shield the ioniser from external electric fields and (b) minimise the number of passes the electrons make along the ioniser. The installed grid has a transmission probability of 11%, meaning that 99% of the measured current originates from electrons performing a single pass through the ioniser. Additionally, only ∼1% of the current is expected to pass back to the filament, therefore any perturbation of the emission or electron distribution due to the measurement probe is minimised. . At low filament currents the population imaged at the exit of the device was contained on the axis in a single region. As the filament current increased the electron distribution underwent a complex evolution into a multi-region structure. (b) Emission current and total flux of current measured onto the probe as the filament current increases. As the space charge effects begin to be significant around I f = 4.1 A, we also see that the total flux exiting the ioniser reaches a maximum.
The coaxial probe can be rastered in both the horizontal and vertical directions to image the electron population at the exit of the device. While the probe core is about 1 mm in diameter, it is typically moved in steps of 0.1 mm, meaning the distributions are significantly over-sampled. Scans were taken at a range of filament drive currents, thereby probing the electron distribution as a function of electron density in the ioniser.
To test the operation of the coaxial probe for mapping the electron population, it was first confirmed that all currents in the device were conserved and the manipulator position has minimal influence on the filament emission. Figure 1(b) shows the currents measured on various components throughout a scan of the probe. The current measured onto the central core of the probe represents the electrons exiting the ioniser at that (x, y) position and can be seen to vary with the probe position. Electrons that do not collide with the central part of the probe can then strike either the probe sheath or the liner (through either direct collision with the wall or through hitting the mesh at the exit). The total current emitted from the filament varies less than ∼2% with probe position, indicating that very few electrons reflect from the probe and contribute to the space charge near the filament, as predicted. Figure 2(a) shows an example set of images of the current measured onto the central core of the coaxial probe about 2 mm behind the mesh grid. The axial magnetic field is at its peak strength around the mesh and thus the electron population is expected to be maximally compressed (see magnetic field illustration in figure 1). At low emission currents (I f ⩽ 3.9 A), the electron distribution appears as a relatively uniform continuous region. However, as the emission current is increased, the distribution becomes less homogeneous and comprises a distinct repeating motif appearing across several images. At high emission currents (I f ⩽ 4.1 A), the electron distribution consists of discrete maxima and no longer appears as a continuous uniform distribution, a divergence from the expected behaviour. Figure 2(b) demonstrates how the emission current and current onto the probe integrated across the entire image vary with filament current. Above a filament current of I f ≈ 4.1 A, the emission current no longer increases exponentially, following the typical behaviour for space charge limited thermionic emission [19,20]. At approximately the same filament current, the total current onto the probe also stops increasing and thus despite an increasing emission current no additional current is passing down the full length of the liner.
Given the magnetic compression and dimensions of the filament, the electron beam is expected to be ∼3 mm wide 1 , but the probe is ∼1 mm wide. Therefore, the electron population images shown in figure 2 arise from the convolution of the probe shape and the actual electron distribution. As such, the observation of a distinictly repeating motif in several of the images (e.g. at I f = 4.3 A) indicates that the electron distribution comprises several sharp, dense regions. Figure 3 illustrates how the effect of the probe shape can be removed from the data through deconvolution. In general, performing a deconvolution can be difficult, however the problem is easier to solve when the true signal is sparse and a repeating motif is visible [24,25]. Figure 3 demonstrates the result of performing a sparse blind deconvolution using an alternating descent method 2 to recover the point spread function from the experiment performed at I f = 4.3 A. The top row of figure 4 shows how the recovered point spread function is then used to deconvolve the remaining experimental data using a Lucy-Richardson deconvolution procedure implemented in MATLAB. Again, at low emission we see that a narrow line structure is obtained, but as the emission current is increased the pattern breaks apart and in some cases can form small dense electron regions.
Several publications discussing similar instruments, either for studying non-neutral plsamas [26][27][28][29][30] or as ion sources themselves [31,32], note that the electrons are susceptible to various instabilities that lead to non-uniformities forming. Thus, solenoidal ionisers can be considered in terms of nonneutral plasmas [33], with a performance that is dominated by the non-ideal behaviour that occurs with increasing electron density. Specifically as the emission current is increased we see two features that deviate from ideal operation: (1) the appearance of non-uniformities in the electron distribution and (2) the saturation of the electron current exiting the ioniser. We propose two mechanisms that can explain the observation of these features, a plasma instability such as the diocotron instability and the creation of a virtual cathode.
A mechanism for the electron beam non-uniformity at higher emissions is the diocotron instability often observed in non-neutral plasmas [30,33]. This instability grows at a rate dependent on several factors such as: electron density, distance from the conductor to the electrons, magnetic field strength and hollowness of the beam. Thus, there is a characteristic time over which small perturbations in an otherwise uniform beam can grow and create vortices of electrons. For example, consider an annular electron layer inside a conducting tube. For low plasma densities, the electron layer will rotate due to the ⃗ E × ⃗ B drift at the diocotron frequency given by, where n is the density of the plasma, e is the electronic charge, ϵ 0 is the permittivity of free space and B is the magnetic flux density [33]. It can then be shown that the characteristic time for the diocotron instability to develop is given by, where b l , c l are defined in the appendix.
To contextualise the growth time of the instability, we compare it to the expected transit time of an electron down the ioniser. The speed of the electrons within the ioniser volume, v, is given by the electrostatic acceleration of the electrons, v = √ 2qU/m, where U is the potential difference between the filament and the liner. The distribution of the velocity components between the directions parallel, v ∥ , and perpendicular, v ⊥ , to the magnetic field is determined by the precise electric and magnetic field shape close to the filament. Let us define a parameter α = v ∥ /v as the proportion of the velocity in the direction parallel to the axis of the ioniser. Given that the transit time is ∆t = L/(αv), the ratio of the transit time to the characteristic growth time of the diocotron instability when the beam is an annular layer is given by, where r 1 , r 2 are the radii of the edges of our electron layer. We would expect that the instabilities will be important when ∆t/τ > 1, which for our current setup (assuming r 1 = 1.45 mm, r 2 = 1.55 mm and l = 3 mode) corresponds to α crit < 1.5. Given that 0 < α < 1, we determine that the diocotron instability will form in our ioniser and needs to be considered in more detail. The behaviour of the diocotron instability was accurately modelled using the two dimensional particle-in-cell (PIC) package, xpdc2 [21][22][23]. The 3D problem was converted to a 2D simulation by assuming a uniform magnetic field and considering a frame of reference moving down the ioniser at the same parallel velocity as a single electron in the beam, v ∥ . Given the aspect ratio of the ioniser, we can then assume an infinitely long ionisation volume. The 2D simulation then evolves for a time sufficient for a test electron to propagate down the full length of the ioniser, ∆t = L/v ∥ . While the assumption of a 2D system is valid given that the change in electron distribution is sufficiently slow, the effect of the spatially varying magnetic field is difficult to quantify given that steps in the B-field are known to excite plasma rotations and instabilities [34], as well as magnetically compressing the beam. As such, the simulations provide a semi-quantitative analysis of the experiment.
If we assume perfect transport of electrons down the ioniser, it can be shown (details in appendix) that the total number of electrons enclosed inside the ioniser is given by, where I e is the emission current into the liner. The value of α, as well as the thickness and shape of the initial electron distribution were chosen to obtain a good match to the experimental data. A value of α = 0.8 was assumed for the parallel velocity component, which lead to a transmission time of ∆t = 76 ns for electrons with U = 70 V of energy and N = 4.73 × 10 8 electrons inside the liner for each 1 mA of electron emission.
Assuming an average beam radius of 3 mm down the ioniser leads to an estimated average electron density of 6 × 10 13 m −3 . The bottom row of figure 4 shows the result of the simulations with an average magnetic field strength of 0.1 T and demonstrates that as the emission current is increased, electron vortices begin to form, breaking the continuous distribution apart and correlating well with the experimental data shown in the top row of figure 4. The precise location and size of the inhomogeneities is not of interest here, instead we highlight that the onset of the instability is consistent with the experiment.
Returning to the saturation of the electron current exiting the ioniser, it is well known that virtual cathodes (VCs) form around filaments whose emission is space-charge limited [27,29]. In this emission regime, electrons released from the filament perform complex trajectories as they are repelled from the dense space charge cloud outside the filament. Consequently, the electron distribution is no longer representative of the shape of the filament but instead is characterised by a VC. Moreover, low-energy electron beam current saturation has been observed with increasing emission current, arising from an abrupt change in VC structure (suppression of current near the cathode centre) due to increased space charge depth [27]. Such a mechanism can explain the differences between the experiment and simulations (figure 4) at high emission currents. In particular, the distribution continues to precess in the simulation due to the ⃗ E × ⃗ B drift present, however the experimental data does not since the total current passing down the ioniser reaches a saturation value (figure 2(b)) and therefore the electric field is no longer increasing in strength. Indeed, given sufficient time, the vortices will often recombine to form a continuous distribution as observed in other plasma devices [35].
The direct observation of non-uniform electron distributions propagating down the ioniser with increasing emission currents is a significant result for the operation of solenoidal ionisers. Typically, these instruments rely on the electron distribution to create the space charge necessary for extracting ions [8,9,18]. If the distribution is not uniform, ions cannot be correctly extracted and the performance is hindered. We should note that the addition of the low transmission mesh means the device is essentially operating in 'open' mode [9] while configured for electron imaging, meaning that electrons make a single pass down the instrument. When operating as an ioniser, the device is configured in the 'closed' mode and so we would expect that the electron density would be higher due to the reflection of a significant number of electrons at the exit of the ioniser. However, it is unlikely that operation of the detector in closed mode would smooth out all of the asymmetries caused by instabilities through either multiple reflections or recombination of the vortices [35].
Given that measurements with these class of detectors rely on long term stability, it is important that any instabilities are addressed. The diocotron instability growth rate can be minimised by using an electron beam that extends across the entire liner, rather than using a hollow loop for the filament. The hollow electron beam had previously been selected to allow atomic and molecular beams to pass directly into the ioniser, however the use of an off-axis electron source has been demonstrated and would avoid any obstructions [36].
An alternative to relying on the electron space charge would be to construct devices that use a segmented liner with fixed potentials along the length, such as those found in non-neutral plasma devices [26,28]. Pinning the potential with additional instrumentation would ensure that the extraction potential is correct even if the electron beam suffers from an instability and allow the possibility for dynamic correction of the fields.
In summary, directly imaging the electron beam exiting a solenoidal ioniser reveals the presence of instabilites in the spatial electron distribution as the emission current is increased. Non-uniform space charge poses a serious issue for the operation of solenoidal ionisers since ion extraction relies on the potential created by the electron beam. PIC simulations show that the observed behaviour of the charge distribution is consistent with the presence of a diocotron instability in the non-neutral plasma. At high emission currents the electron current down the ioniser reaches a maximum value, most likely driven by changes in the behaviour of the space charge limited virtual cathode. These results are expected to guide future development in solenoidal ioniser technology aimed at more stable detectors.

Data availability statement
The data that support the findings of this study are openly available at the following URL/DOI: https://doi.org/10.5281/ zenodo.7582247.

Acknowledgments
The work was performed in part at the Materials and NSW node of the Australian National Fabrication Facility, a company established under the National Collaborative Research Infrastructure Strategy to provide nano and microfabrication facilities for Australia's researchers. The plasma simulations were performed using XPDC2 provided by PTSG, while the deconvolution was performed using code developed by Wright et al https://github.com/deconvlab/sas-deconv.

Appendix A. Characteristic diocotron instability growth time
Following the treatment of Davidson [33], in the limit of low electron density and the guiding centre approximation, let us consider a perturbation of the form, (A.1) The dispersion relation for a perturbation on an annular electron layer, with radii r 1 , r 2 , inside a conducting tube of radius R, is given by, where if there is no central conductor, . (A.4) By solving the quadratic equation, we see that the perturbation is unstable if b 2 l > 4c l and the characteristic growth time will be given by,

Appendix C. Permanent magnet characteristics
A complete description of the PMSI is provided by Martens [37], however a brief summary is included here. The magnetic field is generated from a set of hollow NdFeB cylinders (total length ∼300 mm long, 35 mm diameter central bore) depicted in blue in figure C1(a), the grey sections are soft iron components used to engineer the magnetic field. Figure C1(b) shows the resultant axial magnetic flux density from the assembly, with the field increasing along the length of the assembly to a maximum of ∼0.18 T. The vacuum chamber is then assembled inside the magnet assembly, with a typical operating pressure during measurements of ∼10 −9 mbar.