# Electron thermalization length in solid para-hydrogen at low-temperature

A. F. Borghesani<EMAIL_ADDRESS>CNISM unit, Department
of Physics & Astronomy, Università degli Studi di Padova
and Istituto Nazionale Fisica Nucleare, sez. Padova, Padua, Italy  G. Carugno
Istituto Nazionale Fisica Nucleare, sez. Padova, Padua, Italy  G. Messineo
Istituto Nazionale Fisica Nucleare, sez. Ferrara, Ferrara, Italy  J. Pazzini
Department of Physics & Astronomy, Università degli Studi di Padova and
Istituto Nazionale Fisica Nucleare, sez. Padova, Padua, Italy

###### Abstract

We report the first ever measurements of the thermalization length of low-
energy electrons injected into solid para-hydrogen at a temperature $T\approx
2.8\,$K. The use of the pulsed Townsend photoinjection technique has allowed
us to investigate the behavior of quasi-free electrons rather than of massive,
slow negative charges as reported in all previous literature. We have found an
average thermalization length $\langle z_{0}\rangle\approx 260\,$Å which is 3
to 5 times longer than that in liquid helium at the same temperature.

##  I Introduction

An active field of research is the measure of the permanent electric dipole
moment of the electron to look for CP violation phenomena beyond those
included in the standard model of particle physics Vutha  _et al._ (2010);
Aggarwal  _et al._ (2018); Andreev  _et al._ (2018). Barium monofluoride (BaF)
is a promising polar molecule to be used for this goal because its outermost
electron in the HOMO is acted upon by an intrinsic electric field whose
strength is otherwise unattainable in laboratory Sandars (1965).

The PHYDES collaboration aims at embedding BaF molecules in a solid matrix of
para-hydrogen (p-H2) at cryogenic temperatures or in other cryogenic noble gas
solid environments. This matrix isolation technique limits the host-guest
interaction, enhances the density of the embedded molecules, and reduces their
diffusion prowess Guarise  _et al._ (2017, 2019).

Solid para-hydrogen at low temperature is a matrix of great interest because
of its remarkable properties due to its quantum nature. The antiparallelism of
the nuclear spins leads to a configuration of lowest energy. The solid
crystallizes in hexagonal closed-packed structure (hcp ), the average nearest-
neighbor distance is $\approx 3.8\,$Å with a molecular mean-squared
displacement of $\approx 0.48\,$Å Silvera (1980); Driessen and Silvera (1987);
Meyer (1998); Prisk  _et al._ (2023). Its number density at $T=4.2\,$K is
fairly high, $N\approx 2.6\times 10^{28}\,$m-3, and the solid can be
considered to consist of an assembly of molecules all translationally
localized at lattice sites but freely rotating even in the zero-temperature
limit.

Embedding neutral BaF molecules in the solid p-H2 matrix is yet an unsolved
technical problem. Actually, solid BaF2 is evaporated in a glow discharge
chamber. The BaF2 molecules are dissociated and ionized. BaF+ cations are then
extracted by means of electrostatic lenses and mass-selected by a Wien filter.
Eventually, the cations are electrostatically slowed down to a final kinetic
energy of $\approx 5\,$eV and are admitted into the low-temperature chamber
where they mix with the p-H2 gas that has to be condensed to form the solid.
Details on this technique will appear in a future publication.

We plan to investigate the possibility of neutralizing the cations embedded in
the solid by injecting into it electrons from a photocathode irradiated with a
short UV laser pulse. To achieve this goal the knowledge of the energetics and
dynamics of low-energy electrons in the solid p-H2 matrix is of paramount
importance. However, to the best of our knowledge, there are only a couple of
papers about excess negative charge carriers in solid p-H2 Levchenko and
Mezhov-Deglin (1990); Levchenko, Mezhov-Deglin, and Shtinov (1991) and a few
more in solid Le Comber, Wilson, and Loveland (1976); Levchenko and Mezhov-
Deglin (1988) and liquid H2 Levchenko and Mezhov-Deglin (1992, 1994); Lerner
and Sokolov (1994); Berezhnov  _et al._ (2003).

Almost all the experimental papers found in literature are dealing with
negative charges of unspecified nature. In those experiments ionization is
obtained by either using an energetic electron beam of $\approx 45\,$keV Le
Comber, Wilson, and Loveland (1976) or $\beta$-rays from Tritium sources whose
average energy is in excess of $5\,$keV Levchenko and Mezhov-Deglin (1988);
Levchenko, Mezhov-Deglin, and Shtinov (1991); Levchenko and Mezhov-Deglin
(1990, 1992, 1994) whereas the dissociation energy of H2 is $\approx 4.5\,$eV.
These energy values are sufficient for both ionization and repeated
dissociation events of the hydrogen molecules that may lead to the production
of stable H- ions or even (H2)${}_{n}^{-}$ cluster ions.

The analysis of the phenomenology of the drift mobility measurements of the
negative excess charges in both liquid and solid hydrogen and in solid p-H2
convincingly demonstrates that different ionic species were detected rather
than quasi-free electrons.

In liquid H2 in the range $12\,\mbox{K}<T<34\,\mbox{K}\,$, the value of the
drift mobility $\mu$ of negative species is comprised in the range $1.5\times
10^{-2}\,\mbox{cm${}^{2}$/Vs}<\mu<3\times 10^{-2}\,\mbox{cm${}^{2}$/Vs}$,
comparable to the value in liquid helium Borghesani (2007), and shows evidence
of the presence of two different negatively charged species Sakai, Bőttcher,
and Schmidt (1983); Levchenko and Mezhov-Deglin (1992). This behavior is
rationalized by assuming that, owing to the very large and positive ground-
state energy of an electron in the conduction band of the liquid $V_{0}\approx
1.9\,$eV Lerner and Sokolov (1994) and because of the strong, repulsive,
short-range exchange forces both electron bubbles (i.e., electrons self-
localized in empty cavities) and ionic bubbles Khrapak, Schmidt, and Volykhin
(1995) (i.e., H- ions or (H2)${}_{n}^{-}$ cluster ions surrounded by a void
shell) both coexist in the liquid Berezhnov  _et al._ (2003).

In solid hydrogen the mobility of the negative charges is very low, roughly
ranging from $4\times 10^{-7}\,$cm2/Vs to $8\times 10^{-6}\,$cm2/Vs in the
range $12\,\mbox{K}<T<14\,\mbox{K}$ Le Comber, Wilson, and Loveland (1976);
Levchenko and Mezhov-Deglin (1988), also depending on the applied drift field,
and strongly decreases with decreasing temperature. In order to rationalize
these results several hypotheses have been suggested. The first one is that
electrons deform the not-so-rigid solid (its surface tension is estimated to
be $\sigma\approx 4.9\times 10^{-3}\,$J/m2 at $T=2.8\,$K Corruccini (1965))
and self-localize in empty bubbles giving origin to electron bubbles as was
suggested for the case of solid helium Cohen and Jortner (1969), in which the
mobility values are of comparable magnitude Keshishev, Mezhov-Deglin, and
Shal’nikov (1971); Sai-Halasz and Dahm (1972); Dionne, Young, and Tomizuka
(1972). The second one is that the charge carriers are localized by the
formation of small polarons and their transport occurs via phonon-assisted- or
hopping processes Le Comber, Wilson, and Loveland (1976); Shikin (1977).
Whatever the explanation might be, it is clear from the experiments that the
charge carriers cannot be quasi-free electrons.

Finally, in solid p-H2 the drift mobility of excess negative charge carriers
turns out to be extremely low. In the temperature range
$13.6\,\mbox{K}>T>5.8\,\mbox{K}$ $\mu$ decreases from a value $\mu\approx
6\times 10^{-6}\,$cm2/Vs down to $\mu\approx 2\times 10^{-11}\,$cm2/Vs
Levchenko and Mezhov-Deglin (1990); Levchenko, Mezhov-Deglin, and Shtinov
(1991). Above $T=10\,$K the diffusion coefficient $D$ of negative charges,
obtained from the mobility by exploiting the Nernst-Debye-Einstein
relationship $D/\mu=k_{\mathrm{B}}T/e,$ in which $k_{\mathrm{B}}$ is the
Boltzmann constant and $e$ is the electron charge, closely resembles the
diffusion coefficient of ortho-hydrogen molecules and is therefore
rationalized by assuming a mechanism of thermally activated diffusion of
charged point defects interacting with vacancies Andreev and Savishchev
(1989). Below $T=8\,$K also a mechanism of quantum tunneling of vacancies has
to be introduced. In this way the activation energy and the free energy for
vacancy formation are determined and are in good agreement with the
theoretical estimate Ebner and Sung (1972) and with their determination in NMR
experiments Zhou  _et al._ (1990). Moreover, the authors warn that some
differences in the experimental outcome are to be attributed to the
anisotropic crystal structure Levchenko and Mezhov-Deglin (1990).

The nature of the negative charge carriers has been identified with localized
point charge defects that interact with the vacancies in the crystal Levchenko
and Mezhov-Deglin (1990). It is very plausible that such entities are massive
H- anions or the even more massive (H2)${}_{n}^{-}$ cluster ions as possible
products of the technique used to inject charges in the solid. Moreover, the
presence of H${}_{2}^{-}$ anions has been observed in $\gamma$-ray irradiated,
pure solid p-H2 in high-resolution ESR experiments Kumada  _et al._ (1996),
whereas electron bubbles are produced in place of H${}_{2}^{-}$ anions only if
the solid p-H2 contains a small amount of deuterium Miyazaki  _et al._ (2002).
Actually, similar mechanisms of interaction between charge carriers and
vacancies have been observed in solid deuterium, in which negative charges are
assumed to be electron bubbles Trusov, Mezhov-Deglin, and Levchenko (1996).

The whole wealth of experimental results and their interpretation strongly
indicates that until now the behaviour of quasi-free electrons in fluid and
solid hydrogen, including solid p-H2, has not been investigated yet.
Therefore, we have carried out an experiment in which bunches of low-energy
electrons are photoinjected into the solid by means of short pulses of a UV
laser. We report in this paper the results of the measurements of the
thermalization length of such quasi-free electrons in solid p-H2 at $T\approx
2.8\,$K.

The paper is organized as follows. In Sect. II details on the experimental
setup and procedure are given. In sect. III we present and discuss the
experimental results. Finally, some conclusions are drawn in Sect. IV.

##  II Experimental details

A thorough description of the whole apparatus will appear in a subsequent
paper. Here we only give a brief and schematic description of the experimental
setup and technique.

The measuring cell is located in a cryostat and is cooled down to $T\approx
2.8\,$K using a commercially available cryocooler. The cell can be optically
accessed through suitable viewports in the cryostat and is evacuated down to a
residual pressure $P_{\mathrm{res}}\approx 10^{-6}\,$Pa.

Normal hydrogen gas is enriched in the para-hydrogen fraction by exploiting
standard techniques Tom  _et al._ (2009); Sundararajan  _et al._ (2016);
Bhandari  _et al._ (2021). The gas is precooled to $T\approx 77\,$K by slowly
flowing it in a capillary immersed in a liquid N2 bath. Subsequently, the gas
is flown through a copper tube, kept at $T\approx 22\,$K by means of an
ancillary cryocooler. The tube is packed with Fe2O3 fine powder that acts as a
catalyst for the ortho$\rightarrow$para conversion. The gas flow rate is
controlled by a leakage valve and measured by a flow meter. The gas enriched
in the para fraction is then admitted into the measuring cell at low pressure.
During the growth of the solid p-H2 film the gas pressure is about $P\approx
7\times 10^{-5}\,$Pa.

A set of windows and electrodes is located in the cell to allow us to monitor
the growth and purity of the p-H2 film and to inject electrons in the solid
film and collect them. A conceptual, simplified, schematics of the electrodes
and windows assembly is shown in Figure 1.

Figure 1: Conceptual scheme of the setup for measuring both the thickness of
the p-H2 solid film during its growth and the amount of photoinjected charges
after the film growth is stopped. Symbol meaning: W = optical substrate. K =
$100\,$Å thin gold cathode, p-H2 = para-hydrogen film of thickness $h$. RA =
retractable anode. $d=$ drift distance. V = d.c. power supply. C = integrating
capacitor. $R=$ high-impedance resistor. CA = charge amplifier. FTIR =
Fourier-Transform InfraRed interferometer.

A sapphire window, coated with a $\approx 100\,$Å thick Au film, is mounted in
the cell so as to achieve good thermal contact with it. The thin Au film is
grounded and acts as the photocathode in a diode configuration of the
electrodes.

At regular time intervals the thickness of the growing p-H2 film is monitored
by recording its near-infrared (NIR) absorption spectrum, as described in
literature Fajardo (2011, 2019). The radiation of the near-infrared source
(Thorlabs, model SLS202L/M) impinges on the back of the window, passes through
the p-H2 film and is detected by a MCT detector (Hamamatsu, mod. P7163) whose
output is amplified and fed to the FTIR interferometer (Bruker, mod. Equinox
55), thereby yielding the absorption spectrum $I(\tilde{\nu})$ as a function
of the wavenumber $\tilde{\nu}$ in cm-1.

According to literature the film thickness $h$ and the ortho-H2 fraction $f$
can be computed from the optical density
$\mbox{OD}=-\log\left[I(\tilde{\nu})/I_{0}(\tilde{\nu})\right]$, in which
$I_{0}$ is the spectrum recorded in absence of p-H2, respectively as Fajardo
(2011, 2019)

$h(\mbox{mm})=0.048\times\int\limits_{4495}^{4525}\mbox{OD}(\tilde{\nu})\,\mathrm{d}\tilde{\nu},$
(1)

$f=\frac{0.124}{h}\times\int\limits_{4151}^{4154}\mbox{OD}(\tilde{\nu})\,\mathrm{d}\tilde{\nu},\quad\mbox{or
}f=\frac{0.0787}{h}\times\int\limits_{4732}^{4742}\mbox{OD}(\tilde{\nu})\,\mathrm{d}\tilde{\nu}$
(2)

A typical OD spectrum recorded after $\approx 3\,$hrs condensation is reported
in Figure 2. In all experimental runs we obtained a constant growth rate of
$\approx 100\,\mu$m/hr and a constant ortho fraction $2\,\%<f<3\,\%$.

Figure 2: Optical density spectrum of the solid p-H2 film after $3\,$ hrs
condensation. The ro-vibrational transitions used for computing the film
thickness are enclosed in the dashed ellipse. The lateral bands are analyzed
to obtain the ortho-H2 concentration. The meaning of the notation of the ro-
vibrational transitions can be found in literature Bhandari  _et al._ (2021).

After the solid film thickness has reached a value between $500$ and
$600\,\mu$m the gas inlet is closed and the film growth is almost stopped, its
growth rate dropping down to $\approx 0.5\,\mu$m/hr, whereas the ortho-
hydrogen content remains constant for a longer time interval after the
photoinjection experiment has been completed.

Once the solid film has reached the desired thickness, the retractable anode
is set in position in front of the cathode by means of a suitable manipulator.
The NIR source is replaced by a 266-nm pulsed laser. The laser, whose spot has
a diameter $\phi\approx 8\,$mm, impinges on the back of the Au cathode and
photoextracts electrons. The laser pulse has a duration $\Delta t\approx
10\,$ns and an energy per pulse that can be varied in the range
$50\,\mu\mbox{J}<W<150\,\mu\,$J, but is kept constant during the measurement.
The laser pulse repetition rate is set at $10\,$Hz.

A d.c. power supply energizes the drift capacitor up to $1\,$kV and sets the
desired value of the electric field in the drift region of length $d=1\,$cm.
The current induced in the circuit by the drifting photoextracted electrons is
integrated by a charge amplifier (i.e., an active integrator) whose time
constant is $\tau\approx 400\,\mu$s and whose output amounts to $\approx
0.25\,$mV/fC. The charge amplifier output is connected to a digital scope. For
each setting of the drift field, 2000 signals are averaged together in order
to improve the signal-to-noise ratio. The averaged signals are then fetched by
a PC for offline analysis.

##  III Experimental results and discussion

In this Section we report the experimental results and their rationalization.

###  III.1 Characterization of photoextraction in vacuo

First of all, the behavior of the system has to be characterized in vacuo. We
report in Figure 3 the charge photoextracted under vacuum conditions.

Figure 3: Electric field dependence of the charge collected at the anode in
vacuo. The solid is only a guide for the eye.

The amount of photoextracted charge increases with increasing field, as
expected, and saturates for an electric field strength as weak as $E\approx
15\,$V/cm, when practically the whole amount of electrons produced by the
laser pulse is collected.

The energy distribution of the photoelectrons is not known, as is the work
function of the deposited Au film. We directly measured the highest energy of
the electrons by reversing the electric field polarity in order to nullify the
collected charge, thereby obtaining $\epsilon_{\mathrm{max}}=(0.56\pm
0.05)\,$eV. The amount of the collected charge proportionally varies with the
laser pulse energy, its electric field dependence remaining the same.

###  III.2 Measurements with solid p-H2 in place

The measurements of photoextraction have been carried out after the thickness
$h$ of the solid p-H2 film reached a value between $500\,\mu$m and
$600\,\mu$m. If the applied potential difference between anode and cathode is
$V$, the electric field strength $E$ in the dielectric film is given by

$E\,(\mbox{V/cm})=\frac{V}{K(d-h)+h}\approx 0.792V,$ (3)

in which $V$ is measured in Volt, $d=1\,$cm is the drift distance and
$K\approx 1.28$ is the relative dielectric constant of solid p-H2 Younglove
(1968); Constable, Clark, and Gaines (1975). An accurate determination of $h$
is not extremely important as a $20\,\%$ uncertainty in $h$ brings about a
negligible $0.2\,\%$ uncertainty in $E.$ The electric field strength in the
p-H2 vapor is $E_{v}\,(\mbox{V/cm})=KE\approx 1.01V\,\mbox{(V)}.$ We have
investigated the range $10\,\mbox{V/cm}\lesssim E\lesssim 600\,\mbox{V/cm}$.

####  III.2.1 Signal waveform analysis

Some preliminary pieces of information can be gathered by inspecting the
acquired signal waveforms, such as the one shown in Figure 4.

Figure 4: Time evolution of the collected charge in presence of the solid p-H2
film recorded with $E=280\,$V/cm.

First of all, we note that the amount of collected charge is a factor more
than 2000 smaller than that collected in vacuo with a much smaller electric
field strength. We will discuss this reduction later in greater detail. For
now, we want to stress the fact that the signal rise time $\Delta t\simeq
0.7\,\mu\mbox{s}$ is independent of the applied electric field and is solely
due to the response of the electronics. If electrons injected into the solid
were prone to dissociate H2 to form H- anions or to form electron bubbles,
their drift time through the solid would be no shorter than several tens of
seconds. However, we do not observe any charge pile-up in the solid in spite
of the quite high laser pulse repetition rate.

On the other hand, the transit time of electrons emerging into the vapor is
estimated to be several orders of magnitude smaller than the rise time of our
electronics because of the extremely low density of the vapor at $T\approx
3\,$K. Unfortunately, there are no measurements of electron drift velocity
$v_{D}$ in p-H2 gas for so high an electric field strength and at so low a
temperature except those at $T=76.8\,$K, at a lowest pressure $P\sim 500\,$Pa,
and at the highest reduced field $E/N\sim 2.6\,$Td
($1\,\mbox{Td}=10^{-21}\,\mbox{V\,m}^{2}$), yielding a drift velocity
$v_{D}\approx 3.3\times 10^{6}\,$cm/s Crompton and Robertson (1971); Robertson
(1971). Just to carry out a back-of-the-envelope estimate, we assume that the
pressure in our experiment is so low that the vapor can be treated as an ideal
gas because the equation of state of p-H2 is only known for $T>13.8\,K$
Leachman  _et al._ (2009). We, thus, get a gas number density $N\approx
2\times 10^{18}\,$m-3, thereby yielding $E/N\approx 5\times 10^{5}\,$ Td for
$E=10\,$V/cm. As there are no hints at how fast electrons travel at such huge
a reduced field strength, we just assume that they move in vacuo and obtain a
maximum final speed $v\sim 2\times 10^{8}\,$cm/s, thereby leading to a transit
time $\tau\sim 5\,$ns, much shorter than the rise time of the electronics.

This analysis of the waveform leads to the conclusions that: i) electrons are
actually injected into the solid in which they move at high speed, and ii)
neither low mobility electron bubbles nor anions form.

####  III.2.2 Electric field dependence of the photoextracted charge

The typical outcome of the experiment is shown in Figure 5, in which the
amount of the charge collected at the anode is plotted as a function of the
electric field $E$ in the p-H2 film.

Figure 5: Collected charge $Q$ vs the electric field $E$ acting on the p-H2
film.

As previously noted, the electrons photoinjected into the solid p-H2 film are
epithermal. Upon injection electrons will lose their excess energy by
collisions in the solid until they get thermalized and then propagate in the
conduction band of the crystal. The model used to describe the injection
process has been thoroughly described in literature Onn and Silver (1969,
1971); Smejtek  _et al._ (1973). We briefly recall here its main features.

The electrons injected in the solid film close to the Au/p-H2 interface are
under the influence of their image force, which acts so as to push them back
to the cathode, and of the externally applied electric field $E$, which pulls
them towards the collector. The potential energy (in eV) of the electrons is
given by

$\varphi(z)=V_{0}-{A}{z}^{-1}-Ez\quad\mbox{for }z\geq 0.$ (4)

Here, $z$ is a coordinate orthogonal to the Au/p-H2 interface where the
coordinate origin is located, and $V_{0}$ is the electron ground state energy
at the minimum of the conduction band in the solid. The coefficient $A$ takes
into account the dielectric properties of the solid film and is given by
Griffiths (2017)

$A=\frac{e}{16\pi\epsilon_{0}K}\left(\frac{K-1}{K+1}\right)\approx
0.345\,\mbox{eV\,\AA}.$ (5)

$\varphi(z)$ has a maximum at $z_{m}=\sqrt{A/E}$ where the Schottky barrier
lowering (in eV) amounts to $\Delta\varphi=-2Ez_{m}$. In our experimental
conditions $200\,\mbox{\AA\ }\lesssim z_{m}\lesssim 2000\,\mbox{\AA}$, whereas
$4\times 10^{-5}\,\mbox{eV}\lesssim|\Delta\varphi|\lesssim 3\times
10^{-4}\,\mbox{eV}$ is negligible.

We note that the distance of the potential maximum from the Au/p-H2 interface
is always much shorter than the solid film thickness, $z_{m}\ll h$. However,
we cannot be sure that $z_{m}$ is greater or comparable to the electron mean
free path (mfp) $\ell$ in the solid. We can reasonably assume that this is the
case. Actually, on one hand, if we could make the hypothesis that the
molecules in the solid were a collection of independent scattering centers, we
would get $\ell\approx 4\,\mbox{\AA}$ by using the recommended value for the
e-H2 scattering cross section at low energy ($\sigma_{s}\approx 9.2\,$Å2) Yoon
_et al._ (2008). This mfp value, surprisingly close to the nearest-neighbor
distance in the solid, is far too short to be reasonable. Actually, at low
energies where the electron wavelength is comparable or even larger than the
typical intermolecular distance, we expect deep modifications of the gas-phase
behavior. Coherent scattering of the electron wave function within the solid
may even deeply lower the gas-phase cross section Sanche (1988). On the other
hand, a mfp values of a few tens of times larger than the nearest-neighbor
distance might be more realistic. In any case, we assume that
$z_{m}\gtrsim\ell$.

A first attenuation of the amount of charge injected in the solid with respect
to the vacuum case is due to the quantum reflection of the electron
wavefunction at the barrier $V_{0}$. This phenomenon strongly reduces the
amount of charge that can be collected at the anode, as experimentally
observed.

Once successfully injected into the solid, an electron loses its excess energy
by scattering off molecules, or phonons, or structural defects. Thus,
electrons undergo a random walk through the medium with the possibility that
somewhere thermalization occurs due to some inelastic process (as, for
instance, by inducing rotational excitation in the H2 molecules). Several
possible paths for electrons can be singled out. The first possibility is that
the electron is backscattered towards the cathode at its first collision Young
and Bradbury (1933) or may backdiffuse towards the cathode before
thermalization occurs. These two processes do not depend on the position of
the potential maximum. A second possibility is that the electron thermalizes
for $z<z_{m}$ and backdiffuses towards the cathode. All these processes lead
to a reduction of the amount of charge that can be collected at the anode. The
third and final possibility is that thermalization occurs for $z>z_{m}$ and
the electron is collected at the anode because of the pull exerted by the
external field. In some sense, the potential maximum acts on the electrons as
a gate at a distance $z_{m}$ from the cathode.

By taking into account all the aforementioned effects, it has been shown that,
at low temperature and high density, the amount of charge injected into the
solid and collected at the anode is accurately described by the following
equation Onn and Silver (1969); Silver  _et al._ (1970); Onn and Silver (1971)

$\frac{Q}{Q_{0}}=\frac{B}{D}\mathrm{e}^{-z_{m}/z_{0}},$ (6)

in which $Q_{0}$ is the amount of charge that would be collected in vacuo, and
$z_{0}$ is the thermalization length. $B$ is a coefficient that describes the
effect of the barrier and could be computed if the energy distribution
function of the photoelectrons were known Broomall, Johnson, and Onn (1976);
Gushchin, Kruglov, and Obodovskii (1982). In our case, $B$ is constant because
the energy distribution function of the photoextracted electrons does not
change from run to run. $D$ is a term that describes the backdiffusion
process. It can be written as $D=1+Cz_{0}/\ell$, where $C$ is a numerical
constant, and does not depend on $z_{m}$. Moreover, as all the measurements
with the solid film in place have been carried out for $E\gtrsim\,15\,$V/cm,
$Q_{0}$ can also be considered as a constant.

In Figure 6

Figure 6: The charge collected at the anode $Q$ as a function of the
coordinate $z_{m}$ of the potential energy maximum measured in the three
experimental runs. The charge in each run has been normalized to the laser
energy. The solid lines are the best fit curves based on Eq. (6).

we report the experimental results for the collected charge as a function of
the coordinate $z_{m}$ of the potential energy maximum obtained in the three
different runs we have carried out at roughly one month interval from each
other. The charge amount in each run has been normalized to the laser energy.

The experimental data satisfactorily follow the exponential law Eq. 6. We
obtain three independent determinations of the thermalization length that are
reported in Table 1.

run |  $z_{0}$ (Å) |  uncertainty (Å)
---|---|---
1 | 256 | 20
2 | 291 | 28
3 | 257 | 12
Table 1: Thermalization length determinations in the three experimental runs
with the relative uncertainties.

The three different determinations of the thermalization length are in good
agreement with each other, thereby yielding a weighted average

$\langle z_{0}\rangle=(261\pm 10)\,\mbox{\AA}.$ (7)

We note that the ratio of the thermalization length to the film thickness is
$d/z_{0}\sim 2\times 10^{4}$. This high value of the ratio ensures that
boundary effects are negligible, thereby also lending credibility to the
statistical approach leading to Eq. 6.

The present value of the thermalization length $\langle z_{0}\rangle$ turns
out to be 3 to 5 times longer than in liquid helium at approximately the same
temperature Onn and Silver (1969).

A lack of knowledge of the electron drift mobility and/or scattering cross
section in the solid p-H2 at the low temperature of our experiment does not
allow us to gather some deeper pieces of information regarding the electron
thermalization time that may be useful for the design of the experiment aimed
at the neutralization of cations embedded in the solid.

Nevertheless, we conclude that thermalization of quasi-free electrons occurs
well within the bulk. Upon thermalization, the electron thermal speed at
$T=2.8\,$K is $v_{\mathrm{th}}\approx 11\times 10^{3}\,\mbox{m/s}$ to be
compared to the average value of the sound speed $v_{S}\approx 2\times
10^{3}\,$m/s Kuroda  _et al._ (2001); Bezuglyi, Plakhotin, and Tarasenko
(1971); Wanner and Meyer (1972, 1973). Thus, it is reasonable to assume that
polarization of the medium via phonons is too slow to respond to the electron
motion, which could otherwise hinder the Coulomb interaction of the electrons
with the embedded BaF+ cations.

##  IV Conclusions

For the first time we have carried out measurements of the thermalization
length of quasi-free electrons in solid para-hydrogen at low temperature. The
goal is to investigate their energetics and dynamics in order to ascertain the
possibility to neutralize cations embedded in a solid matrix of this material.

The outcome of the present experiment has several facets. First of all,
electrons of low energy are injected into the conduction band of the solid by
exploiting the photoelectric effect of a gold cathode irradiated with a short
pulse of a UV laser whose photon energy is just close to the dissociation
energy of H2. By so doing, the production of very slow H- anions is very
unlikely. An analysis of the time evolution of the waveforms we have recorded
indicates that only fast negative charges are present.

Secondly, we have been able to determine the thermalization length $z_{0}$ of
quasi-free electrons. It turns out that $z_{0}$ is quite large though well
much shorter than the solid film thickness. This is a useful piece of
information regarding the possibility of neutralizing the cations that have to
be embedded in the solid p-H2 matrix in order to carry out high-precision
measurements of the electric dipole moment of the electron.

However, the thermalization length of electrons in a matrix is an important
quantity in several more physics fields. It is well known that there might be
a relationship between electron thermalization and mobility Sano and Mozumder
(1977). Moreover, among others, electron thermalization plays a role in the
physics of geminate ion recombination or of free ion yield which is important
in the field of radiation detectors Aprile  _et al._ (2006); Ferradini and
Jay-Gerin (1991), just to mention some applications.

In this first experiment we have investigated the electron thermalization at
one single temperature and density. We had to assume that there is a barrier
to the injection of electrons into the solid without actually measuring it
because its determination requires a detailed knowledge of the density
dependence of the thermalization process. Moreover, it was impossible, in the
present experimental configuration, to measure the electron drift time.
Therefore, we are planning for the near future to extend our measurement
campaign to fill in this gap.

## Acknowledgments

We wish to thank dr. D. Sali, Prof. A. C. Vutha, Prof. T. Momose, Prof. J. D.
Weinstein, and Prof. C. Caesar for helpful suggestions. A. F. Borghesani
gratefully acknowledges useful discussions with Prof. A. G. Khrapak, Prof. L.
P. Mezhov-Deglin, and Prof. J.-P. Jay-Gerin. We also express our thanks to
Fulvio Calaon for technical support. G. Messineo acknowledges financial
support from the European Union’s Horizon2020 research and innovation
programme under the Marie Skłodowska-Curie grant no. 754496.

## Author declarations

### Conflict of Interest

The authors have no conflicts to disclose.

### Authors contributions

All authors equally contributed to the experiment.

### Data availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

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