Ultracold neutron detectors based on ¹⁰B converters used in the qBounce experiments

Abstract

Gravity experiments with very slow, so-called ultracold neutrons connect quantum mechanics with tests of Newton's inverse square law at short distances. These experiments face a low count rate and hence need highly optimized detector concepts. In the frame of this paper, we present low-background ultracold neutron counters and track detectors with micron resolution based on a ¹⁰B converter. We discuss the optimization of ¹⁰B converter layers, detector design and concepts for read-out electronics focusing on high-efficiency and low-background. We describe modifications of the counters that allow one to detect ultracold neutrons selectively on their spin-orientation. This is required for searches of hypothetical forces with spin–mass couplings.

The mentioned experiments utilize a beam-monitoring concept which accounts for variations in the neutron flux that are typical for nuclear research facilities. The converter can also be used for detectors, which feature high efficiencies paired with high spatial resolution of $1\mathrm{–}2\mathrm{\mu }\mathrm{m}$. They allow one to resolve the quantum mechanical wave function of an ultracold neutron bound in the gravity potential above a neutron mirror.

1. Introduction

In any neutron experiment the detection concept plays a crucial role. The developments are driven by the experimental needs: The next generation of neutron decay experiments [1] as well as high-flux neutron imaging experiments [2,3] require high rate capacities of 100×10⁶ s⁻¹, while gravity tests at short distances need low-background detectors with a spatial resolution of a micron. Time of flight experiments require a temporal resolution of $10\mathrm{\mu }\mathrm{s}$, and experiments at neutron research centers need large area detectors [4,5] of 1–10 m² with about 50,000 pixels per system.

All detection concepts have in common that a neutron is detected by converting it into charged particles by a nuclear reaction. Suited converter materials are ³He, ⁶Li, ¹⁰B, and ¹⁵⁷Gd. They have large absorption cross sections. Detectors for ionizing particles range from gas detectors, scintillation detectors, to solid state detectors. Due to current helium shortage, today's focus is on boron as a neutron converter for large area detectors.

When neutrons become so slow that their corresponding deBroglie-wavelength is bigger than typical interatomic distances of matter, they totally reflect from surfaces under any angle of incidence. Such ultracold neutrons (UCN) are used in various storage experiments, for example to determine the neutrons's lifetime [6] and to search for a permanent neutron's electric dipole moment [7]. Ultracold neutrons form bound quantum states in the Earth's gravitational field [8]. With its tiny non-equidistant eigenenergies in the pico-eV range and its characteristic size of a few tens of microns, gravitationally bound states of UCN offer the fascinating possibility to combine tests of Newton's gravity law at short distances with the high precision resonance spectroscopy methods of quantum mechanics. In the frame of this paper, we present our detector concepts based on a ¹⁰B neutron converter optimized for gravity experiments with ultracold neutrons within the qBounce collaboration:

In the last decade, gravity experiments with ultracold neutrons have made progress. In 2002, the first observation of such bound states succeeded [8]. Also developments started to measure directly the probability densities using track detectors with spatial resolution [9–11]. Since 2009, resonant transitions between different quantum states can be driven [12–14].

In all experiments, the UCN flux is extremely low, which puts serious constraints on the detector design. Therefore, all parts of the detectors and read-out system need to be optimized regarding maximal efficiency and minimal background.

2. ¹⁰B as a neutron converter

A thin layer of ¹⁰B can be used as a neutron converter. In ¹⁰B, neutron capture leads to a prompt decay into $\alpha$ and lithium ions. Two different reaction channels with individual probabilities and kinetic energies of the ions exist:

$$\begin{array}{ccc}\hfill \mathrm{I}:n+{}^{10}\mathrm{B}& \hfill ⟶\hfill & \mathrm{\alpha }+{}^7\mathrm{Li}^{⁎}(94\%)\hfill \\ \hfill & \hfill ⟶\hfill & \mathrm{\alpha }+{}^7\mathrm{Li}^{3+}+\mathrm{\gamma }(0.48\mathrm{MeV})+2.31\mathrm{MeV},\hfill \\ \hfill \mathrm{II}:n+{}^{10}\mathrm{B}& \hfill ⟶\hfill & \mathrm{\alpha }+{}^7\mathrm{Li}^{3+}+2.79\mathrm{MeV}(6\%).\hfill \end{array}$$ (1)

As the kinetic energy of an incident neutron is negligible for thermal or slower neutrons, the momenta of the ions are antipodal and isotropic. The emitted ions can be detected on one or both sides of the thin converter layer. In the following, we discuss the case applicable to the qBounce detectors, where only the charged particles behind the layer in the neutron flight direction are detected. However, other approaches exist [15–17].

The efficiency of neutron conversion is crucial for the detection performance. Thicker layers increase the total absorption efficiency of neutrons, but decrease the probability of ion transmission through the remaining layer. Therefore, the layer thickness needs to be adapted to the neutron's velocity. The optimization of ¹⁰B layers is discussed in detail in Ref. [15] and is summarized and updated in the following. The velocity (v) and depth (x)-dependent absorption probability p_abs for neutrons in a ¹⁰B layer is given by

$$p_{\mathit{abs}}(x,v)=\alpha (v)e^{-\alpha (v)x}\mathrm{with}\alpha (v)=\frac{v_0{\sigma }_0{\rho }_{10B}{N}_A}{\mathit{Av}}.$$

here $v_0=2200\mathrm{m}/\mathrm{s}$ is the velocity of thermal neutrons, ${\sigma }_0=3840$ barn the absorption cross-section for thermal neutrons in ¹⁰B [18], ${\rho }_{10B}=2.17\mathrm{g}/{\mathrm{cm}}^3$ the density of pure ¹⁰B,² N_A the Avogadro number, A=10,013 u the atomic mass of ¹⁰B [19] and v the neutron velocity in m/s.

Natural boron contains approx. 20% of ¹⁰B with small negative scattering length and 80% of ¹¹B, which has a low neutron absorption cross-section and a positive scattering length. Therefore, the ratio between the two isotopes can be matched such that neutron reflection from the converter layer is negligible. This occurs for a ¹⁰B enrichment of approximately 97 wt%.

The depth-dependent transmission probability of the ions through the layer is computed from their maximum range in ¹⁰B. Table 1 shows the values of the ion ranges and energies.

Table 1.

Energies and maximum range for the ions from neutron induced fission of ¹⁰B. The ranges are calculated with SRIM-2010 [24]. The roman numerals represent the decay channel according to Eq. (1).

Property	$\alpha$, I	7Li, I	$\alpha$, II	7Li, II
Ekin (MeV)	1.47	0.84	1.78	1.01
Rmax$(\mathrm{\mu }\mathrm{m})$	3.55	1.84	4.39	2.06

Given a total layer thickness d and a neutron penetration depth x, the fraction of transmitted ions in the neutron direction per decay channel is given by

$$P_{\mathit{trans}}(x)=\{\begin{array}{cc}{\displaystyle \frac{1}{2}}\left(1-{\displaystyle \frac{d-x}{R_{\mathit{max}}}}\right),\hfill & (d-x)\le R_{\mathit{max}}\hfill \\ 0,\hfill & (d-x)>R_{\mathit{max}},\hfill \end{array}$$

where R_max is the maximum range for a certain ion species and energy.

For a given neutron velocity v, the yield of transmitted ions per neutron that are transmitted through the layer can be calculated as follows:

$${\epsilon }_{\mathit{species},\mathit{channel}}(d,v)={\int }_0^d{P}_{\mathit{trans}}(x)p_{\mathit{abs}}(x,v)\mathit{dx}=\frac{1}{2\alpha (v)R_{\mathit{max}}}\{\begin{array}{cc}\left(\begin{array}{c}1+\alpha (v)·(R_{\mathit{max}}-d)-\hfill \\ (1+\alpha (v)R_{\mathit{max}})·e^{-\alpha (v)d}\hfill \end{array}\right),\hfill & d\le R_{\mathit{max}}\hfill \\ \left(\begin{array}{c}{e}^{-\alpha (v)·(d-R_{\mathit{max}})}-\hfill \\ (1+\alpha (v)R_{\mathit{max}})·e^{-\alpha (v)d}\hfill \end{array}\right),\hfill & d>R_{\mathit{max}}\hfill \end{array}$$

Summing up both channels and ion species leads to the total conversion efficiency

$${\epsilon }_{\mathit{tot}}(d,v)=0.94·({\epsilon }_{\alpha ,I}(d,v)+{\epsilon }_{{}^7\mathit{Li},I}(d,v))+0.06·({\epsilon }_{\alpha ,\mathit{II}}(d,v)+{\epsilon }_{{}^7\mathit{Li},\mathit{II}}(d,v)).$$ (2)

Fig. 1 shows the optimal conversion efficiency regarding neutron absorption and ion transmission for different neutron velocities used in the qBounce experiments. Table 2 gives accordingly the maximal efficiencies and layer thicknesses.

Fig. 1.

¹⁰B Conversion efficiency vs. layer thickness: the blue (red) curve shows the calculated efficiency for a measured UCN spectrum with a $v_{\mathit{mean}}=6.5(9.1)\mathrm{m}/\mathrm{s}$. The black curve, calculated for thermal neutrons with $v=2200\mathrm{m}/\mathrm{s}$, explains the low background sensitivity of ¹⁰B converter layers optimized for UCN (adapted from: [22, Fig. 3.1, p. 32]). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Table 2.

Optimized thickness and conversion efficiency for a ¹⁰B layer are used to convert neutrons to charged particles emitted in the neutron flight direction. The velocities marked with ⁎ are mean values for measured UCN spectra used in the qBounce experiment and the appropriate values are calculated for a part of this spectrum.

v (m/s)	Optimal layer thickness	Conversion efficiency (%)
6.5⁎	221 nm	90.9
9.1⁎	280 nm	87.1
2200	$3.2\mathrm{\mu }\mathrm{m}$	6.1

An analytical description of the theoretical energy spectrum can be found in Ref. [15]. Fig. 2 shows a measured energy spectrum of our qBounce UCN detector together with its theoretical prediction.

Fig. 2.

Measured energy spectrum of the qBounce UCN detector. The channels are proportional to the energy of the decay products. Data points within (outside of) the region of interest (ROI) are shown in blue (green). The red line indicates the theoretical expectation (adapted from: [22, Fig. 3.16, p. 54]). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Neutron converter layers made of boron are produced by electron beam evaporation methods. The boron layer is coated on thin aluminium foils that act as entrance windows for proportional counters (see Section 3), and on plastic substrates that are used as track detectors (see Section 4). Thickness variations from the production process between center and border of the layer can be minimized by rotating the coated sample and increasing the distance between the boron source and the substrate. In our case, thickness variations on the 5%-level are achieved for a layer diameter of 120 mm. They are uncritical for the efficiency of UCN detection.

3. Low-background UCN counters for the qBounce experiments

3.1. Counter design and characteristics

Simple proportional counters, with an optimized layer of ¹⁰B attached on the inside of the detector entrance window as a neutron converter, combine the advantages of solid state detectors and gas counters. The UCN counters used in the qBounce experiment follow this principle. They are highly optimized regarding detector efficiency and background, and can be operated in vacuum. The detector is read out using custom low-background electronics for pulse-height discrimination. For maximum flexibility, the counters can be also read out with standard nuclear electronic modules. A picture of the detector is shown in Fig. 3. Fig. 4 sketches its basic design: The ¹⁰B neutron converter layer is coated on the inner side of a $60\mathrm{\mu }\mathrm{m}$ thick entrance window made of AlMg₃. This very stable aluminum alloy shows very good UCN transmission [21]. The boron converter layer (thickness 220 nm, total efficiency 90.9%) is optimized for an UCN velocity of 6.5 m/s and the active area is reduced to the minimum needed to ensure covering the path of all relevant UCN from the experiment (3×110 mm²). This also reduces the sensitivity to fast and thermal neutrons usually contributing significantly to background at a nuclear research facility.

Fig. 3.

Dismounted detector made of brass and coated with copper. The upper half shows the active boron converter, the anode and the HV feed-through (adapted from: [22, Fig. 3.10, p. 47]).

Fig. 4.

Side view of the described detector design (adapted from: [22, Fig. 3.9b, p. 45]).

For the design of the proportional chamber itself, multiple competing parameters need to be optimized. The proportional chamber consists of a housing that forms the cathode, and a gold-coated tungsten filament as an anode wire, which is supported by an isolated solder contact and a high-voltage (HV) vacuum feed-through that also grounds the cathode. The construction material of the housing is brass, which turns out to have less radioactive impurities (that mainly provide detector background by $\mathrm{\alpha }\mathrm{-}\mathrm{emission}$) than aluminum or steel. An additional reduction of $\mathrm{\alpha }\mathrm{-}\mathrm{emission}$ from the detector housing into the detection volume by radioactive impurities is achieved by coating the inner surface with pure copper. A layer of $50\mathrm{\mu }\mathrm{m}$ stops all $\mathrm{\alpha }\mathrm{-}\mathrm{particles}$ with a kinetic energy up to 10 MeV. Within the housing, a cylindrical hollow builds the detection volume constantly purged with ArCO₂ (90:10) at 1.4 bar. The gold coating on the tungsten wire suppresses oxidation and therefore aging.

High voltage sparks in a proportional chamber can be reduced significantly by decreasing the detector high voltage to its minimal possible value. In order to achieve the same gas amplification for lower voltages, the anode and cathode diameters are reduced [22]. The cathode diameter is limited by the stopping range of decay ions in the counting gas, because all electrons from primary ionization should experience high gas amplification near the anode wire. The most energetic ion ($\alpha$ from channel II) with an energy of 1.78 MeV has a stopping range of 9.26 mm [22]. Therefore, the distance from the entrance window to the anode wire is chosen to be 12 mm and the anode wire is selected with a diameter of $15\mathrm{\mu }\mathrm{m}$. Additionally, the background caused by ionizing particles from outside of the detector is suppressed as the detection volume is shrinked. The gas amplification for different anode and cathode radii are shown in Fig. 5.

Fig. 5.

Gas amplification M vs. detector voltage for different anode radii a and cathode radii b. The gas amplification increases with smaller anode and cathode radii. The practically feasible diameter of the anode wire is mainly determined by the thickness variations (adapted from: [22, Fig. 3.4, p. 36]).

The wire length is chosen to be several centimeters longer than the active area to avoid influences by distortions of the electric field. The detector shows good performance for an HV operating range from 350 to 800 V, see Fig. 6.

Fig. 6.

Pulse height spectra for different operating high voltages: The horizontal black lines mark the zero counts value for each of the spectra. As higher voltage implies higher gas gain, the width of each spectrum is normalized by varying the main amplification to fit all spectra in the same pulse-height range. The red spectra show signatures of space charge effects (adapted from: [22, Fig. 3.12, p. 50]). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

3.2. Read-out electronics

The read-out of the detector system follows the scheme depicted in Fig. 7.

Fig. 7.

Sketch of the detector read-out system. The deposited charge is converted to a voltage signal using a RC-circuit. After pre- and main amplification the signal is digitized. The dashed box indicates the parts operated in vacuum (adapted from: [26, Fig. 2.9, p. 22]).

On the anode wire deposited charges are collected using a RC-circuit and a charge-sensitive preamplifier. Furthermore, they are converted to a voltage signal. In order to suppress electronic noise, this system is mounted into the vacuum as close to the detector as possible. The signal pulse is further amplified to a voltage range between 0 and 10 V. Then, the signal is digitized and recorded.

All parts of the read-out system are well-adapted to the design of the proportional counter and the need to measure signal count rates in the range of 10⁻³ s⁻¹: We use a custom charge-sensitive pre-amplifier developed by Physikalisches Institut/Universität Heidelberg. It is mounted directly on the support of the counter. To minimize electronic noise, all connections are soldered and plugs are avoided where possible. For main amplification, a standard low-noise NIM nuclear electronic module is used.

The digitalization is done using a commercially available four-channel analog-to-digital-converter³ (ADC). Per recorded pulse, it provides a time stamp with an accuracy of $1\mathrm{\mu }\mathrm{s}$, the pulse height in units of up to $2^{16}$ channels and two logic signals (1 or 0).

These signals may be used to gate the detectors. In contrast to other ADCs, the detector signal is not physically gated, but the logic signals are stored within the raw data for every recorded signal. This feature is advantageous, because in our case the neutron beam is shared between three different experiments and we may measure the background rate while other experiments are running.

The individual timing information for every pulse is used to implement a software-based burst filter. It may happen that the detection signal registers many unexpected events in a short time, often due to high-voltage sparks from other experiments nearby. These so-called ‘bursts’ are a huge false-effect when measuring very small count rates. In our case, we filter out signals with a relative time difference of less than $100\mathrm{\mu }\mathrm{s}$. For a typical count rate in our experiments of ${10}^{-2}{\mathrm{s}}^{-1}$, the corresponding probability for two true signals in such a time slot is approximately ${10}^{-9}$ and for normal measuring times of several hours negligible.

In order to suppress electronic noise and bursts further, an electromagnetic shielding was applied. The whole detector system was mantled with a conducting layer of 1 mm thickness, see Fig. 8.

Fig. 8.

Electromagnetic shielding. The whole detector system (green) is mantled by a conductive layer of 1 mm thickness (red). The vacuum chamber has no direct connection to conductive layer (adapted from: [26, Fig. 3.1, p. 28]). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

3.3. Beam monitor concept

The incoming neutron flux at a beam position at a nuclear research reactor is subject to fluctuations. This effect is accounted for by normalizing all measured count rates by a measure linearly proportional to the incoming flux. Fig. 9 shows our beam monitor setup. We take advantage of the unavoidable losses occurring at the junction between two neutron beam guides leading to the experiment. These losses are proportional to the incoming flux and are monitored by an additional time-resolving neutron counter. The measured beam monitor count rate is in the order of 100 s⁻¹, which is several orders of magnitude larger than the observed count rates in our experiments. The beam monitor is read out by the same read-out system used for the experiment detector.

Fig. 9.

Beam-monitor concept. At the juncture of two neutron beam guides leading to the experiment, losses occur proportional to the incoming flux. We monitor part of these losses by means of an additional counter (adapted from: [22, Fig. 2.11, p. 23]).

3.4. Spin-dependent UCN detection

In the last years, the search for hypothetical short-ranged spin-dependent forces has received large attention. The qBounce collaboration may contribute to this active area of research [13,14]. For this purpose, a UCN counter which is sensitive to the neutron's spin direction was developed. We use the spin-dependent scattering lengths of iron forming a spin-dependent potential. For UCN below a critical velocity v_crit, this effect is large enough that neutrons behave differently when impinging a polarized soft-iron foil: Neutrons with spin parallel to the iron polarization pass the foil, while neutrons with anti-parallel spin are reflected.

To realize a spin-dependent neutron counter, the existing design described in Section 3.1 is enhanced: The detector is equipped with an entrance foil with a 150 nm thick layer of soft iron on the outside and an active area of ¹⁰B on the inside. The polarization of the foil is achieved using one rectangular coil below and one above the detector. The foil was produced using a sputtering technique. Its hysteresis curve was characterized by means of Kerr magnetometry. The saturation field was found to be 5 mT. The foil requires a holding field of only 1 mT to stay polarized. With this information, the polarization direction can be switched by changing the currents of the detector coils according to the hysteresis curve.

The transmission and polarization of the foil were characterized using UCN in an independent measurement. The experimental setup consists of a polarizer, a spin flipper, and the foil to analyze. The neutron velocity is determined using a standard time-of-flight (TOF) technique. The polarization of the foil is approximately 93%.

3.5. Results

The energy spectrum and the total background behaviour of our current UCN counter is given in Fig. 10. In this plot, all measurements performed in our beam time in 2012 are summed up and normalized to the total measuring times.

Fig. 10.

Total energy spectrum and background behaviour of our UCN counter. All measurements performed in our beam time in 2012 are summed up and divided by the total measuring times. The optimal ROI $[180,920]$ regarding signal-to-noise is indicated in red (black). The exponential increase toward smaller channels originates from electronic noise. A hardware threshold rejects signals with very small pulse heights (adapted from: [26, Fig. 4.9, p. 55]). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

The total counter efficiency is given by the efficiency of the neutron converter, transmission losses in the entrance AlMg₃-foil and the selection of a region of interest (ROI) in the pulse-height analysis.

The conversion efficiency is given by 90.9%, and the transmission of the entrance foil is 95.1%. From a detailed background analysis, we find an optimal ROI for channels [180,920]. This corresponds to an energy interval between [478,1861] keV and contains 89% of the full spectrum. This results in a total efficiency of the UCN counter of $ϵ=77\%$.

The total detector background can be extracted from Fig. 10 to $(0.65\pm 0.02)\times {10}^{-3}{\mathrm{s}}^{-1}$. This tiny value is quite remarkable considering the experimental conditions very close to a research reactor core. The low background rate allows to measure signal rates in the range of 10⁻³ s⁻¹.

The background stability during approx. 100 days is shown in Fig. 11.

Fig. 11.

Background stability. Individual measurements of the background taken over approx. 100 days show the long-term stability of the read-out system of our detector (adapted from: [26, Fig. 4.14, p. 60]).

4. Spatial resolution detector

Above a horizontal mirror, the gravity potential leads to discrete energy levels of a bouncing massive particle. The corresponding quantum mechanical motion of a massive particle in the gravitational field was named the quantum bouncer. The discrete energy levels occur due to the combined confinement of the matter waves by the mirror and the gravitational field.

Fig. 12 shows on the left side the Schrödinger-wave function ${\Psi }_n$ for the first five states n, when a neutron traverses a slit between two mirrors in the gravity potential of the earth. $\rho =\sum |C_n{|}^2\times {\Psi }_n^{⁎}{\Psi }_n$ can be interpreted as the probability of detecting a neutron above the mirror with state populations $|C_n{|}^2$ in an incoherent superposition. The right side of Fig. 12 shows the corresponding experimental setup. Spatial resolution detectors visualize ultracold neutron density distributions behind the mirrors. Fig. 13 displays a measurement of the incoherent superposition of gravitationally bound quantum states. 70% of the neutrons are found in state one and 30% are found in state two.

Fig. 12.

UCN form gravitationally bound quantum states above a horizontal mirror. On the left side, the first five eigen-states and -energies of neutrons in a horizontal slit between two mirrors are shown. These mirrors form infinitely high potential walls, drawn in black. The figure on the right side shows a corresponding experimental setup to measure incoherent superpositions of these states (right, adapted from: [32, Fig. 5.1, p. 61]).

Fig. 13.

Incoherent superpositions of gravitationally bound quantum states of UCN. The height histogram is obtained from a track-detector measurement using the setup depicted in Fig. 12, right. Totally 70% of the neutrons are found in state one (grey) and 30% are found in state two (green); (adapted from: [14, Fig. 4]). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Such measurements require UCN detectors with a spatial resolution of $1\mathrm{–}2\mathrm{\mu }\mathrm{m}$. Spatial-resolution detectors have been developed within INTAS 991-705, with special consideration of ²³⁵UF₄ as converter material on CR39. In this case nuclear fission converts a neutron into a detectable track on CR39. A measurement of the neutron density using a high-resolution plastic nuclear-track detector with uranium coating is presented in [9,23]. The development of CR39 with ¹⁰B converters has been closely related to Cascade detector developments at University of Heidelberg [15,16].

For the qBounce measurements, we use the ¹⁰B converter described in Section 2 on CR39 plastics as track detectors. As the boron layer needs to be removed after exposure, we insert a few mono-layers of copper between boron and CR39. CR39 plastic results from dietilenglycol bis allylcarbonate by polymerization, which is hard and insoluble in most solvents. CR39 is well known in radiation detection applications, when ion recoils cause tracks, which are enlarged by an etching process in a caustic solution. Here, neutrons are captured in the coated ¹⁰B layer in a Li-$\mathrm{\alpha }\mathrm{-}\mathrm{reaction}$. The decay products are emitted back-to-back. Therefore, one ion per nuclear reaction leaves defects in the chemical structure on its way through the CR39 plastic, before it is stopped. The mean track lengths are dependent on the reaction channel (see Section 2). They are about $4.5(4.8)\mathrm{\mu }\mathrm{m}$ for Li-ions and $8.1(9.9)\mathrm{\mu }\mathrm{m}$ for $\mathrm{\alpha }\mathrm{-}\mathrm{particles}$ for reaction channel I(II). After exposure with UCN, the ¹⁰B-coating is removed chemically using sulphuric acid and hydrogen peroxide in an ultrasonic bath. An etching technique using sodium hydroxide makes these tracks visible under an optical microscope. Depending on etching time and temperature, the traces are enlarged with respect to the trade-off between spatial resolution and visibility, see Fig. 14. Typically 500 nm of CR39 are removed in this process, as shown in Fig. 15.

Fig. 14.

Optimization of the etching process. Depending on etching time and temperature, the traces are enlarged performing a trade-off between spatial resolution and visibility. For a temperature of 42 °C, the bulk etching rate is approx. 100 nm/h. The traces are etched approx. three times faster. The lower figure shows the track diameter D in dependence of the etching time (adapted from: [25, Fig. 3.10, p. 39]).

Fig. 15.

Vertex reconstruction process. Depending on the angle of the impinging ion, the aspect ratios $2r/l$ of the tracks differ. Using image and pattern recognition methods, the true vertex (green dot) can be reconstructed. The red dot indicates the center of gravity (adapted from: [13, Fig. 2.9, p. 32]). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

The position of the tracks is retrieved optically by an automatic scanning procedure. The microscope is equipped with a micrometer scanning table and a CCD-camera. A drawback of this procedure is that it is very time-consuming and does not allow for fast feed-back. An electronic alternative, which allows an online-access to the data, are modern CCD [27] or CMOS chips [28].

The efficiency of such a detector is limited by a minimal detection angle. Traces with smaller angles to the plane cannot be etched because the plastic bulk is removed at a faster rate v_b than the normal component of the trace etch velocity v_t. The minimal angle is given by $\theta =\arcsin v_b/v_t$ with $v_t/v_b\simeq 3$. The angle of acceptance for the detection traces in CR39 is thus 20°. The efficiency is $ϵ=1-\sin \theta \simeq 67\%$ [29]. The total efficiency of the detector is slightly reduced to 61% due to a ¹⁰B detection efficiency of 90.9%, see Section 2.

The angular distribution of the decay products from the nuclear reaction is isotropic. The minimal angle $\theta =20°$ results in a theoretical spatial resolution of about $1.8\mathrm{\mu }\mathrm{m}$ for a trace diameter of $1.6\mathrm{\mu }\mathrm{m}$.

The spatial resolution – depending on the needs – can be further improved using automated image- and pattern recognition to reconstruct the vertex of the nuclear reaction. The method is illustrated in Fig. 15, top. Using the aspect ratio, the orientation, and the weighted and unweighted average of each trace, the position of the nuclear reaction can be reconstructed. Fig. 15, bottom, shows two examples of traces, where no or a small correction is applied. This method can improve the spatial resolution by $\simeq 0.7\mathrm{\mu }\mathrm{m}$.

5. Summary and outlook

Detectors based on ¹⁰B-converters have an efficiency of 77% for ultracold neutrons and the background rate can be as low as $(0.65\pm 0.02)\times {10}^{-3}{\mathrm{s}}^{-1}$. The spatial resolution is, depending on the effort, on the micron scale. The purpose of our detector developments is to probe the relation of gravitation and quantum theory by investigations of low energy bound states in the gravitational field. Most sensitive tests are based on quantum interference. So far the potential is barely exploited. As the quantum bouncing ball is concerned, quantum interference is found in the spatial and in the time domain. Due to interference, the quantum carpet of the quantum bouncing ball has a spatial modulation of about $6\mathrm{\mu }\mathrm{m}$, and is influenced by hypothetical gravity-like interactions. These interactions may or may not couple to the spin of the neutron. A careful analysis of this modulation is within the reach of our detectors. The time domain can be exploited by Gravity Resonance Spectroscopy [12]. Due to the low count rate, ultralow background rates are highly desirable. Probing the neutron's electric neutrality [30] with Ramsey spectroscopy of gravitational quantum states of ultracold neutrons [31] has the need for robust detectors as well.