Electron attachment to atomic hydrogen on the surface of liquid ⁴He
We demonstrate a possibility that helium surface electrons at cryogenic temperatures can be used as a new source of very low energy electrons. Since both electrons (e¯) and hydrogen atoms (H) are bound on liquid helium surface, two-dimensional mixture gas of these two species is available on the...
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| Zitieren: | Electron attachment to atomic hydrogen on the surface of liquid ⁴He / T. Arai, H. Yayama, K. Kono // Физика низких температур. — 2008. — Т. 34, № 4-5. — С. 496–503. — Бібліогр.: 28 назв. — англ. |
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irk-123456789-1169272017-05-19T03:03:09Z Electron attachment to atomic hydrogen on the surface of liquid ⁴He Arai, T. Yayama, H. Kono, K. Электроны над жидким гелием We demonstrate a possibility that helium surface electrons at cryogenic temperatures can be used as a new source of very low energy electrons. Since both electrons (e¯) and hydrogen atoms (H) are bound on liquid helium surface, two-dimensional mixture gas of these two species is available on the surface. We found that low energy collision of e¯ and H drives electron attachment to form a negative hydrogen ion (H¯) in the mixture. From our temperature dependence measurement of the reaction rate, it was found that another H atom participate in the reaction. Namely, the reaction is expressed as H + H + e¯ → H¯ + H. Possible reaction mechanisms are discussed in terms of direct three-body process and dissociative attachment process. Measurements in applied magnetic field (B) show that the reaction rate coefficient is suppressed as ~ B⁻². This implies that electron spin singlet collision is relevant for electron attachment. 2008 Article Electron attachment to atomic hydrogen on the surface of liquid ⁴He / T. Arai, H. Yayama, K. Kono // Физика низких температур. — 2008. — Т. 34, № 4-5. — С. 496–503. — Бібліогр.: 28 назв. — англ. 0132-6414 PACS: 67.63.Gh;67.25.–k;68.03–g http://dspace.nbuv.gov.ua/handle/123456789/116927 en Физика низких температур Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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Электроны над жидким гелием Электроны над жидким гелием |
| spellingShingle |
Электроны над жидким гелием Электроны над жидким гелием Arai, T. Yayama, H. Kono, K. Electron attachment to atomic hydrogen on the surface of liquid ⁴He Физика низких температур |
| description |
We demonstrate a possibility that helium surface electrons at cryogenic temperatures can be used as a
new source of very low energy electrons. Since both electrons (e¯) and hydrogen atoms (H) are bound on liquid
helium surface, two-dimensional mixture gas of these two species is available on the surface. We found
that low energy collision of e¯ and H drives electron attachment to form a negative hydrogen ion (H¯) in the
mixture. From our temperature dependence measurement of the reaction rate, it was found that another H
atom participate in the reaction. Namely, the reaction is expressed as H + H + e¯ → H¯ + H. Possible reaction
mechanisms are discussed in terms of direct three-body process and dissociative attachment process. Measurements
in applied magnetic field (B) show that the reaction rate coefficient is suppressed as ~ B⁻². This
implies that electron spin singlet collision is relevant for electron attachment. |
| format |
Article |
| author |
Arai, T. Yayama, H. Kono, K. |
| author_facet |
Arai, T. Yayama, H. Kono, K. |
| author_sort |
Arai, T. |
| title |
Electron attachment to atomic hydrogen on the surface of liquid ⁴He |
| title_short |
Electron attachment to atomic hydrogen on the surface of liquid ⁴He |
| title_full |
Electron attachment to atomic hydrogen on the surface of liquid ⁴He |
| title_fullStr |
Electron attachment to atomic hydrogen on the surface of liquid ⁴He |
| title_full_unstemmed |
Electron attachment to atomic hydrogen on the surface of liquid ⁴He |
| title_sort |
electron attachment to atomic hydrogen on the surface of liquid ⁴he |
| publisher |
Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
| publishDate |
2008 |
| topic_facet |
Электроны над жидким гелием |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/116927 |
| citation_txt |
Electron attachment to atomic hydrogen on the surface
of liquid ⁴He / T. Arai, H. Yayama, K. Kono // Физика низких температур. — 2008. — Т. 34, № 4-5. — С. 496–503. — Бібліогр.: 28 назв. — англ. |
| series |
Физика низких температур |
| work_keys_str_mv |
AT arait electronattachmenttoatomichydrogenonthesurfaceofliquid4he AT yayamah electronattachmenttoatomichydrogenonthesurfaceofliquid4he AT konok electronattachmenttoatomichydrogenonthesurfaceofliquid4he |
| first_indexed |
2025-07-08T11:19:50Z |
| last_indexed |
2025-07-08T11:19:50Z |
| _version_ |
1837077461154660352 |
| fulltext |
Fizika Nizkikh Temperatur, 2008, v. 34, Nos. 4/5, p. 496–503
Electron attachment to atomic hydrogen on the surface
of liquid 4He
Toshikazu Arai
Research Center for Low Temperature and Materials Sciences, Kyoto University, Kitashirakawa-Oiwake-cho,
Sakyo-ku, Kyoto, 606-8502, Japan
E-mail: toshikaz@scphys.kyoto-u.ac.jp
Hideki Yayama
Department of Physics, Kyushu University, 4-2-1 Ropponmatsu, Chuo-ku, Fukuoka, 810-8560, Japan
Kimitoshi Kono
RIKEN, Low Temperature Physics Laboratory, Hirosawa 2-1, Wako-shi, Saitama, 351-0198, Japan
Received November 6, 2007
We demonstrate a possibility that helium surface electrons at cryogenic temperatures can be used as a
new source of very low energy electrons. Since both electrons (e�) and hydrogen atoms (H) are bound on liq-
uid helium surface, two-dimensional mixture gas of these two species is available on the surface. We found
that low energy collision of e� and H drives electron attachment to form a negative hydrogen ion (H�) in the
mixture. From our temperature dependence measurement of the reaction rate, it was found that another H
atom participate in the reaction. Namely, the reaction is expressed as H + H + e��H� + H. Possible reaction
mechanisms are discussed in terms of direct three-body process and dissociative attachment process. Mea-
surements in applied magnetic field (B) show that the reaction rate coefficient is suppressed as � �B 2. This
implies that electron spin singlet collision is relevant for electron attachment.
PACS: 67.63.Gh Atomic hydrogen and isotopes;
67.25.–k
4
He;
68.03–g Gas-liquid and vacuum-liquid interfaces.
Keywords: surface state electrons, atomic hydrogen, electron attachment.
Introduction
Slow electron impact on atoms and molecules is a gen-
eral phenomenon that plays an important role in interstel-
lar chemical reactions, ionosphere processes, discharge,
and so forth [1]. Also, it has been reported that slow elec-
tron collisions may induce substantial damages in DNA
strands and other biomolecules [2]. Slow electron scatter-
ing phenomena have been studied mainly using electron
beams. In spite of the recent progress of experimental
techniques, spectroscopy at electron energies much less
than 100 meV with resolutions better than 5 meV is still
difficult [1]. Therefore, phenomena occurring in dark
nebulae, for example, where the temperature is approxi-
mately 10 K ( � 1 meV), are experimentally inaccessible.
In order to study such phenomena, much slower incident
electrons are required.
When electrons (e�) are subjected to cryogenic envi-
ronment and they are in equilibrium with surrounding
temperature, we may acquire very low energy electron
source. By using a modern dilution refrigerator, tempera-
ture range as low as 10 mK is an easy access. Thermal ki-
netic energy of electrons at 10 mK is less than 0.001 meV.
By varying temperature, the electron energy is tunable.
Electrons outside but very close to the surface of liquid
helium at cryogenic temperature is known to form bound
states in the attractive image potential [3]. The motion of
those electrons perpendicular to the surface is restricted,
whereas their motion along the surface is nearly free. Col-
lection of helium surface state electrons (SSE) are then
interpreted as a two-dimensional (2D) gas and it is ther-
© Toshikazu Ara, Hideki Yayama, and Kimitoshi Kono, 2008
mally in equilibrium with the underlying liquid helium.
The properties of SSE are well understood [4]. We dem-
onstrate here that SSE can be used as a source of very low
energy electrons.
As a collision target, we employ atomic hydrogen (H)
in this work. Properties of H gas at low temperature has
been extensively studied, motivated by the interest that
cold H gas behaves as a quantum gas [5,6]. Since H atom
is a Boson, cold H gas exhibits quantum phase transitions
such as Bose–Einstein condensation in three dimensions
[7] and Kosterlitz-Thouless transition in two dimensions
[8]. For the stabilization of H gas at low temperatures, it is
crucial to minimize recombination rate of the atoms into
H 2 molecules. In order to avoid rapid recombination of
adsorbed H atoms on the sample cell walls, wall coating
by superfluid helium film is necessary. Binding energy of
a H atom on superfluid 4He surface is 1.0 K and it is or-
ders of magnitude smaller than metal or other solid sur-
faces. Superfluid wall coating was originally developed
by Silvera and Walraven [9], and it is now a standard tech-
nique for cryogenic H experiments.
H atoms in a wall-coated sample cell are in equilibrium
between bulk gas phase and surface adsorbed phase. Ad-
sorbed H atoms behave like 2D gas as well as SSE. Since
either electrons and H atoms are bound on liquid helium
surface, we are able to mix them and study the phenomena
driven by low energy e�–H collisions.
Our preliminary measurements showed that a chemi-
cal reaction of e� with H takes place on the surface and
gives rise to a reduction of SSE density at temperature-de-
pendent rates [10–12].
In this paper, we show that the reaction involves two H
atoms and one electron, namely, H H e H H� � � �� � .
On the basis of this result, we discuss two possible reac-
tion mechanisms: direct three-body process and
dissociative attachment process.
Apparatus
In this section, we show our apparatus for the reaction
rate measurement of e� and H in 2D mixture gas on liquid
helium surface.
We used a dilution refrigerator to obtain low tempe-
ratures. Our apparatus consists of a sample cell on the
mixing chamber, a H 2 dissociator and a 0–12 T supercon-
ducting magnet for the magnetic field dependence mea-
surement. The H 2 dissociator is thermally anchored on
the still in the zero-field measurement and on the mixing
chamber in the magnetic field dependence measurement.
SSE is prepared in the sample cell and H atoms are cre-
ated in the dissociator. H atoms are guided to the sample
cell through a capillary. H atoms in the sample cell are
partly adsorbed on the liquid helium covered surface and
mixed with SSE.
The sample cell contains a filament as an electron
source, an assembly of electrodes for SSE confinement
and a vibration capacitor electrode (VCE) for surface
charge measurement. The sample cell is 44 mm inner di-
ameter and 30 mm depth copper cylinder. Electrodes are
arranged as illustrated in Fig. 1. The upper and lower
electrodes are parallel disks made from copper plated
glass-epoxy printed circuit board, 25 mm in diameter and
3 mm appart. The cell is partly filled with liquid 4He. The
liquid level, which is precisely known from the capaci-
tance, is set 1 mm above the lower electrode.
A tungsten filament at a tiny hole of the upper elec-
trode is briefly heated to emit thermoelectrons. It is oper-
ated at 1.5 K where good amount of helium vapor exists to
slow down the hot electrons by collisions before they
reach the surface. It is required to heat the filament while
H atoms are absent in the cell, otherwise the heat burns off
the superfluid film on the filament and all H atoms rapidly
recombine on the bare surface.
A positive DC potential, VDC , is applied to the lower
electrode while the potentials of the upper electrode and
the guard ring are ground level. The electric field between
upper and lower electrodes presses SSE to the surface.
The guard ring produces a radial confinement potential
for SSE.
Total amount of surface charge is measured by em-
ploying VCE [11]. Our VCE consists of the upper elec-
trode which is mechanically connected to, but electrically
isolated from, the piezo actuator and the positively biased
lower electrode. SSE layer screens the electric field be-
tween the electrodes in proportion to the SSE density � e .
The piezo actuator drives an vibration of the upper elec-
trode. The vibration in the electric field induces an AC
current to the upper electrode. The current amplitude I is
given by
Electron attachment to atomic hydrogen on the surface of liquid 4He
Fizika Nizkikh Temperatur, 2008, v. 34, Nos. 4/5 497
Oscillator
To Amplifier
a
b
g c
h
d
e
VDC
f
Fig. 1. Arrangement of the electrodes in the experimental cell:
piezo actuator (a), upper electrode (b), guard ring (c), SSE (d),
lower electrode (e), mechanical connection (f) of (a) and (b),
tungsten filament (g),liquid
4
He (h).
I
A a
d
b e Ve
e DC� � �
�
�
2 0( ). (1)
Here, Ae is the area of an electrode disk, � is the angular
frequency, d is the separation of the electrodes, b is the
distance of the SSE layer from the lower electrode, �e is
the electronic charge and 0 is permittivity of vacuum.
The relative dielectric constant of liquid helium is ap-
proximated to unity. From Eq. (1), we see that VCE is sen-
sitive at large a and �. We operate our VCE at a and � as
large as possible on conditions that its linearity is main-
tained and heating of the cell is negligible. An aquadag
film bolometer [13] in the cell allows us to sense the heat-
ing. In this experiment VDC � 90 V. VCE is operated at
�
/ 2 880� Hz and the driving voltage of the piezo actua-
tor 5.0 V peak-to-peak. We estimate that a � 76 nm from
measured I for � e � 0. Since I � 0, � e is necessarily
smaller than saturation density � e DCV besat ( / )� 0 . � e
sat is
the maximum SSE density and it is 50 10 8. � electrons/cm2
for our conditions. The VCE resolution is 2 10 5� elec-
trons/cm2 which is 0.04% of � e
sat .
Figure 2 shows the diagram of electronics for VCE
measurement. Dry batteries supply VDC . The signal cur-
rent is measured by a current-to-voltage preamplifier
(Stanford Research Systems, Model 570). The upper elec-
trode is virtually grounded in this preamplifier. Unwanted
signals from direct capacitive coupling of the upper elec-
trode with the piezo actuator and with the lower electrode
are canceled by employing a phase shifter, an attenuator
and a differential amplifier (NF corporation, 5305). VCE
is calibrated as follows. First, we take the signal at � e � 0
and VDC � 0. This signal corresponds to � �e e� sat at
given VDC . Next, we apply VDC and measure the signal at
� e � 0. The signal at finite � e at applied VDC moves be-
tween these two points as a linear function of � e .
Hydrogen atoms are created in a cryogenic H 2
dissociator. A helical resonator [14] in the dissociator is
excited by rf pulses to ignite glow discharge of helium va-
por. It is supposed that the charged particles hit and disso-
ciate the frozen H 2 molecules at the wall. This technique
was originally developed by Hardy et al. [15] and we re-
ferred to van Roijen et al. [16] to design our dissociator.
Figure 3 is a diagram of rf discharge electronics. The
pulse generator generates 100
s pulses and it activates
the rf synthesizer (Anritsu MG3642A) output during the
pulse. We tuned the helical resonator to 430 MHz so that
we are able to use less expensive amateur radio power am-
plifiers. The power amplifiers are purchased from Tokyo
Hy-Power. The first one (PRA-15-430, custom made) am-
plifies 40 dB and the second one (HL-250 DX) 14 dB.
Maximum available power is 54 dBm (250 W) at 0.0 dBm
synthesizer output. High power pulses are only required
for the first ignition after cooling. Once the discharge oc-
curs, much smaller power is enough to start discharge
from the second ignition. We suppose that is because
charged particles live long in the dissociator. For the mea-
surements described below, the synthesizer output is re-
duced to �9 0. dBm. The circulator (Tokyo Hy-Power
A8713, custom made) is necessary to protect the ampli-
fier from reflected waves. The reflected waves are termi-
nated by a 50 � terminator. Discharge ignition can be rec-
ognized from sudden change of the reflected wave
amplitude as it is shown on the screen of the oscilloscope
in Fig. 3.
H atoms thus created are guided to the sample cell
through a capillary tube. Discharge conditions, such as rf
power, pulse width, number of pulses, pulse repetition
rate and temperature of the dissociator, are carefully
maintained so as to regulate the amount of H atoms fed to
the cell. Discharge is operated after SSE containing ex-
perimental cell is cooled from 1.5 K down to working
temperature.
Zero magnetic field measurements
We describe our measurements in the absence of ap-
plied magnetic field in this section.
As mentioned in the previous section, it is necessary to
fill H atoms in the sample cell after SSE is prepared. We
recorded VCE output before and after H atoms were
filled. As a fixed amount of H atoms are fed into the cell,
the VCE output starts to vary, signaling the loss of SSE.
The influence of the H atoms is to diminish SSE from the
surface. We confirmed that without H 2 stored in the
dissociator, the discharge had no influence on the SSE
498 Fizika Nizkikh Temperatur, 2008, v. 34, Nos. 4/5
Toshikazu Arai, Hideki Yayama, and Kimitoshi Kono
piezo actuator
phase shifter attenuator
differential
amplifier
bi-phase
lock-in amplifier
VDC current-to-voltage
preamplifier
Fig. 2. Diagram of the detection electronics.
Pulse
Pulse
modulation
Generator
< 0 dBm
+ 40 dB + 14 dB
oscilloscope
circulator
cryogenic
part�50
Fig. 3. Electronics for cryogenic pulsed rf discharge hydrogen
dissociation.
density. This means that atomic hydrogen does influence
the SSE. Typical traces of the VCE output are shown in
Fig. 4,a. Here, the vertical axis is the number of SSE (N e)
which is converted from the VCE output. At t � 0, H at-
oms are fed into the cell. Compared to the characteristic
time ( �10 3 s) of SSE loss curve, a few seconds filling
time of H atoms is short and we neglect it in the analysis.
After H atoms are filled, the dissociator is switched off.
The shown data were measured at T = 0.30 K and VDC =
= 90 V. Between the two curves only the initial number of
SSE, N e ( )0 is different.
We attribute the SSE loss to the attachment of elec-
trons to H atoms to form negative hydrogen (H �) ions.
The reaction product H � must leave the surface since we
measured the loss of surface charge by VCE. It is less
probable for charges to escape from the surface into the
free volume, because there is a potential barrier compara-
ble to eVDC that needs to be overcome. Then, the ions
penetrate into the underlying liquid helium. The electro-
static energy gain may dominate over the loss due to the
zero-point motion of the ion inside liquid He, which is
essential for the electron bubble to form.
Here we would like to determine the reaction rate
equation from the experiment. In general, one electron
can react with n H atoms to form negative hydrogen,
for which the chemical reaction formula is written as
e H H H� �� � � �n n( )1 . This reaction occurs only in
area Ae of lower electrode, where SSE are confined. An-
other relevant reaction that should be taken into account
is the recombination of H atoms to form H 2 molecule:
H H� � H 2. H atoms exist on the entire surface area A of
the cell with surface density �H as well as in the free vol-
ume V with volume density nH [6]. The recombination
occurs both on the surface and in the volume.
The rate equations for the total number of SSE,
N Ae e e� � , and of hydrogen atoms, N A VnH H H� �� ,
may be written as
dN
dt
K N Ne
e e
n� � ( )� H , (2)
dN
dt
K N N K Ne e
n
s
H
H
eff
H� � �( ) ( )� � 2, (3)
� �� � � �[ ( / ) ]
/
A V
E k Ta B
th e 1. (4)
Here, K e and K s
eff are the rate coefficients for elect-
ron attachment and H–H recombination, respectively.
We assumed the adsorption isotherm of H, � �H H th� �n
� exp ( / )E k Ta B , with the surface binding energy Ea
a n d t h e t h e r m a l d e B r o g l i e w a v e l e n g t h � th �
� ( / ) /2 2 1 2
� m k TBH , where mH is the atomic hydrogen
mass. We used the most reliable experimental value of
Ea , 1.0 k B [17], in the analysis. The coefficient � relates
�H and N H. Substitution of nH in the N H expression by
the adsorption isotherm leads � �H H� N . The effective
recombination rate coefficient K s
eff is expressed in terms
of surface and volume recombination rate coefficients,
K s and K
v
, as K AK VK E k Ts s a B
eff
th� � ��
v
� 2 2exp ( / ) [6].
One of the most prominent properties of the observed
SSE loss curves N te ( ) is that when the loss curves are
normalized by the initial number of SSE N e ( )0 , all
the curves with different N e ( )0 fall on a single curve
(Fig. 4,b). This property implies that d N dte(log ) / must
be independent of N e , and accordingly the evolution of
N tH( ) should be independent of N e . This condition is sat-
isfied if the first term in Eq. (3) is negligible in compari-
son with the second term, that is, the loss rate of H atoms
is dominated by recombination rather than electron at-
tachment.
The characteristic measure of the SSE loss rate is
given by
v0
0
1
0
� �
�
�
�N
dN
dte
e
t( )
. (5)
This is the initial slope of the N t Ne e( ) / ( )0 curve, indi-
cated in Fig. 4,b as the dotted line. We refer to v0 as the
initial rate hereafter. The measured v0 values at various
temperatures are plotted in Fig. 5 as a function of 1/ T .
The difference between the circles and the squares is
the amount of H atoms fed at t � 0. The circles and the
squares correspond to 50 and 30 discharge pulses, respec-
tively. In the temperature range,1/ T � 3 K �1, v0 is the ex-
Electron attachment to atomic hydrogen on the surface of liquid 4He
Fizika Nizkikh Temperatur, 2008, v. 34, Nos. 4/5 499
0 5 10
t, 10 s
3
2.0
1.5
1.0
0.5
1.0
0.8
0.6
N
(t
)/
N
(0
)
e
e
N
(t
),
1
0
e
9
a
b
Fig. 4. Density decay curves of SSE (T � 0.30 K, VDC � 90 V).
Hydrogen atoms are fed at t � 0. These curves are measured un-
der the same conditions except that the initial SSE densities
are different (a). Two curves in (a) are normalized by the value
at t � 0. Perfect scaling behavior is observed. (There are actu-
ally two lines overlapping each other.) The dotted line is the
tangent at t � 0, which defines v0 (b).
ponential function of 1/ T . The dashed lines in Fig. 5 are
proportional to exp ( / )2E k Ta B and these lines fit the data
well in 1 5 1 3. /K K–1 –1� �T .
From Eq. (2), v0 is expressed as
v0 0� �K Ne
n( ( ))� H . (6)
Since the temperature dependence of K e is weak (this will
be verified later), the temperature dependence of v0
should be dominated by � n . At high temperatures where
the second term in the bracket of Eq. (4) dominates,
� �� ( / ) /th exp ( )V E k Ta B . I g n o r i n g t h e T �1 2/ d e-
pendence o f � th , v0 shou ld b e p ropo r t iona l to
exp ( )nE k Ta B/ . Therefore, we conclude that n � 2. This
means that two H atoms participate in the electron attach-
ment to form negative hydrogen, that is, the reaction
H H e H H� � � �� � is occurring in our system.
Below 0.33 K (1 3/ T � K–1), v0 starts to saturate or
even decreases as temperature decreases. This behavior is
qualitatively consistent with the temperature dependence
of �. As can be seen from Eq. (4), � approaches the con-
stant value A �1 at low temperatures. For our cell geome-
try, however, A �116 cm 2 and V � 25 cm 3, and the cross-
over from the exponential temperature dependence to the
constant value is expected to be at approximately 0.06 K
( /1 15T � K �1), which is five times lower than 0.33 K. Un-
less we assume either an unrealistic surface area of the
cell walls or a different temperature on the surface from
the bulk liquid He, no plausible explanation has been de-
termined yet. One of the possibilities that we cannot ex-
clude is heating of the surface due to H recombination en-
ergy. It is said that weakness of ripplon-phonon coupling
in 4He is the bottleneck of cooling mechanism [18]. In
this respect, the experiment on 3He is worthwhile because
3He can carry the heat from the surface more efficiently
than 4He.
In the following discussion, we focus our attention on
the temperature range of 1.5 K � � �1 1/ T 3.0 K �1, where
v0 2� exp ( / )E k Ta B . Putting n � 2 in Eq. (2) and neglect-
ing the first term of Eq. (3), we can solve Eqs. ( 2)–(4) an-
alytically to obtain
N t
N
K N t
K N t
e
e
e
s
( )
( )
exp
( )
( )0
0
0 1
2 2
2
� �
�
�
�
�
�
�
�
�
�
�
�
H
eff
H
. (7)
By fitting the measured decay curves to Eq. (7), we obtain
two fitting parameters, N H( )0 and K e . We obtained the
value of K s
eff from Arai et al. [19]. As shown in Fig. 6,
Eq. (7) fits the data well.
The parameters thus obtained are N H( ) ( )0 8 1 1012� � �
atoms and ( . . )35 0 5 1012� � atoms for the circles and the
squares in Fig. 5, respectively, and K e � � � �( )5 3 10 16 cm4/s
for both circles and squares. No significant temperature
dependence is found for N H( )0 and K e within our resolu-
tion.
Measurements in applied magnetic filed
When the electron spin of H atoms is polarized in
strong magnetic field, H-H recombination rate is known
to be suppressed [6]. It is because electronic state of
bound H 2 molecule is spin singlet and thus spin triplet
pair of H atoms does not proceed to H 2 unless spin flip
takes place during the collision.
500 Fizika Nizkikh Temperatur, 2008, v. 34, Nos. 4/5
Toshikazu Arai, Hideki Yayama, and Kimitoshi Kono
10
–2
10
–3
10
–4
10
–5
10
–6
v
0
,
s–
1
0 2 4 6 8
1/T, K
–1
Fig. 5. Temperature dependence of the initial rate v0. Black
and gray circles denote the data for 50 and 30 discharge
pulses, respectively. The dashed lines are proportional to
exp ( / )2E k Ta B .
1.0
0.8
0.6
0.4
0.2
N
(t
)/
N
(0
)
e
e
0 5 10
t, 10 s
3
Fig. 6. Measured SSE density decay at T � 033. K (circles) and
its fit to the solution Eq. (7) of the rate equation (solid line).
Likewise one can expect that electron attachment to H
atom is suppressed by magnetic field since electronic spin
state of H � is singlet. In this section we show our results
of electron attachment reaction rate measurements in ap-
plied magnetic fields.
Magnetic field is applied by a superconducting magnet
up to 12 T. The liquid level in the sample cell is adjusted
so that it coincides with the magnet center. As in the
zero-field experiments, H atoms are introduced into the
sample cell where SSE is prepared in advance. Since we
found that the SSE loss rate is strongly suppressed in
magnetic field, we kept filling H atoms at constant flux �
(atoms/s) in order to make the SSE loss rate faster by in-
creasing H density in the sample cell and collide with SSE
more frequently. In this case the � term is added to the rate
equation for N H as
dN
dt
K N N K Ne e s
H
H
eff
H� � � �( ) ( )� �2 2. (8)
Like the zero-field data, we recognized the same scaling
behavior of the SSE loss curves at all magnetic fields. Ac-
cordingly, we may ignore the electron attachment term in
Eq. (8).
The solution of the coupled rate equations including �
is given by
N t
q
qtH tanh( ) ( )�
�
(9)
for N H and
N t
N
p
q
qt ate
e
( )
( )
exp ( )
0
� �
�
�
�
�
�
�tanh , (10)
for N e , where p K Ke s� � / eff and q K s� �� eff .
Figure 7 shows scaled SSE loss curves at various mag-
netic fields. One can see that strong magnetic field sup-
presses electron attachment reaction rate. The dots are
data points and the lines are fitting results to Eq. (9) with
two adjustable parameters p and q. The SSE loss curves
are very well fitted and hence the rate equations describe
the react ions correct ly. The fi t t ing resul ts give
� � � �( )2 1 1012 atoms/s for B � 0.2 T and K e shown as
dots in Fig. 8. The line in the Fig. 8 is in proportion to
B �2. The data shows that K Be � �2. We will discuss this
point in the next section.
Discussion
We discuss here two possible microscopic mecha-
nisms of the reaction: direct three-body collision process
and dissociative attachment (DA) process. Our result
n � 2 is reasonable because the cross section of radiative
attachment H e H� � �� � h� for slow electron impact is
very small and in this case a third body is required for
energy conservation [20,21].
As for the direct three-body collision H H e� � �, the
rate coefficient, K e , may be estimated roughly as
K ae � 3
vH from a simple consideration of collision fre-
quency [22], where a is the scattering cross length and vH
is the thermal velocity of H atoms. We may consider two
scattering cross lengths, one is the cross length associated
with the interaction between two hydrogen atoms and the
other is associated with the interaction between an elec-
tron and atomic hydrogen. The former cross length might
be on the order of atomic scale (10 8� cm), while the latter
should have a much longer range because of the
Electron attachment to atomic hydrogen on the surface of liquid 4He
Fizika Nizkikh Temperatur, 2008, v. 34, Nos. 4/5 501
1.0
0.8
0.6
0.4
0.2
0
N
(t
)/
N
(0
)
e
e
t, 10 s
4
0 T
2 T
5 T
8 T
11 T
0 1 2 3
Fig. 7. Scaled SSE loss curves at few B (magnetic fields). The
dots are data points and the lines are fitting results. Magnetic
field suppresses the reaction and the lifetime of the two-di-
mensional gas mixture becomes longer.
10
0
10
1
B, T
10
–21
10
–22
10
–23
10
–24
K
,
cm
/s
e
4
Fig. 8. Magnetic field dependence of electron attachment rate
coefficient at T � 03. K. Circles are experimentally determined
values and the line is drawn proportional to B�2.
charge-induced dipole moment of atomic hydrogen,
which may be given by b e k TB� ( / ) /2 2 1 4� H , where
� H � ( / )9 2 3aB is the polarizability of the hydrogen atom,
where aB is the Bohr radius, and e is the elementary
c h a rg e . T h e n u me r i c a l v a l u e o f b a mo u n t s t o
( )3 4 10 7� � � cm. If we assume that a equals to b,
K ae � 3
vH gives a reasonable agreement with the mea-
sured value, otherwise the rate coefficient becomes too
small to explain the experimental observation. It is inter-
esting indeed to have a microscopic calculation for the di-
rect three-body process, which should take into account
the influence of the helium surface.
Another candidate is the following two-step mecha-
nism. (i) H H H� � 2( , )v J , (ii) H e H H2( , )v J � � �� �.
The first step is the surface H recombination which is
known that a rotationally and vibrationally excited mole-
cule H 2 14 4( , ) or H 2 14 3( , ), v and J are vibrational and ro-
tational quantum number, respectively, is produced first
and it relaxes toward ground state by colliding with the
sorroundings [23]. The second step is the DA of an elec-
tron to a H 2 molecule. DA reaction proceeds via the for-
mation of H 2
� molecular anion resonance state [1]. In
Fig. 9, the adiabatic potentials of 1�g
� H 2 with
vibrationally excited levels of J � 0 states and 2� u
� H 2
�
are plotted. From the energy consideration, DA is possi-
ble when the total ( = electronic + nuclear) energy of H 2
�
is above its dissociation limit, i.e., in our case, it is neces-
sary for H 2( , )v J to capture an electron before it relaxes
below the H 2
� dissociation limit since the kinetic energy
of e� on helium surface is too small to excite the molecu-
lar vibrational motion. In other words, DA which we con-
sider now is exothermic. If J � 0, the H 2 10( )v � states
are above H 2
� dissociation limit. Exothermic DA is ex-
pected to exhibit very large cross sections at low energies
[24], however, almost all the previous works are on the
endothermic DA and exothermic DA is not well under-
stood.
Applying the DA model, we have to rewrite the rate
equation for N e as
dN
dt
K
A
N Ne DA
e
e� �
*
, (11)
where K DA is the surface DA rate coefficient and N* is
the number of H 2( , )v J in the area Ae whose energies are
above the H 2
� dissociation limit. The rate equation for N
*
is
dN
dt
K A N N
K
A
N Ns e
DA
e
e
*
* *
� � �
1
2
12 2�
H . (12)
The first term is the surface H recombination rate in Ae
and the second term is the relaxation rate of H 2( , )v J with
its lifetime . If we take a steady state approximation
dN dt
*
/ � 0 and assume that N K N N ADA e e* *
/ / �� ,
Eq. ( 11) will be
dN
dt
K K N Ne
s DA e� �
1
2
2 2 � H. (13)
Equation (13) takes the same form as Eq. (2) with n � 2
and
K K Ke s DA�
1
2
. (14)
Therefore our analysis holds without modification. Using
Eq. (14) and the measured K e , we can roughly estimate .
To do this we refer to a calculated maximum value of 3D
endothermic DA cross section �DA � �10 15 cm 2 for ini-
tial state H 2 9 0( , ) [25]. We treat our 2D gas as a squashed
3D gas of thickness d and write K dDA DA e� � v / where
ve B ek T m�
/ 2 is the 2D mean speed of e�. Taking
d � 5�, which corresponds to the thickness of H layer, we
have � �� �10 109 8 s.
The suppression of K e and its B �2 dependence can be
explained as follows. For comparison with our experi-
mental results, a line proportional to B �2 is drawn in
Fig. 8. Since two electrons in the bound state of H � are
spin singlet, large suppression of measured K e by mag-
netic field suggests that contribution from electron spin
triplet collision followed by spin flip is absent or small.
If the reaction proceeds via DA, the rate determining
step would be the H recombination. For our H densities,
the recombination mechanism is well described in terms
of van der Waals recombination [6]. The magnetic field
dependence of van der Waals recombination rate coeffi-
cient is K Bs
eff � �2 and then our measured B �2 depend-
ence can be understood.
502 Fizika Nizkikh Temperatur, 2008, v. 34, Nos. 4/5
Toshikazu Arai, Hideki Yayama, and Kimitoshi Kono
0
–1
–2
v = 14
13
12
11
10
9
8
7
6
5
V
(r
),
eV
0 2 4 6 8 10
r, arb. units
2
1
u
g
+
+
H+H
–
H+H
–
Fig. 9. The adiabatic potential curves of 1�g
� H2 (solid line)
[26] with vibrationally excited levels of J � 0 states [27] and
2�u
� H2
� (dashed line) [28].
If the reaction is via direct three-body collision, we
can understand the B �2 dependence by following the
same discussion as van der Waals recombination. Elec-
tron attachment reaction rate is given by the golden rule
! !! !" � # $ �%2 2
&
�
f T i E E
i f
f i
� ( )
,
, (15)
where i and f denote initial state e� + H + H and final state
H � + H, respectively. If �T operator does not induce spin
flip, singlet character is required for e�–H pair to have
non zero transition amplitude. In strong magnetic fields,
SSE energetically favors spin down state. On the other
hand, there are two high field seeking hyperfine states
of H, in conventional notations, !!!a$ � ��$ � ��$cos sin' '
and !!b$ � ��$, where sin ' is a small coefficient propor-
tional to 1/ B at high field. !� �$ denotes electron spin up
and nuclear spin down, and so on. Therefore, only the
!sin ' ��$ component in !a$ state atom gives non zero tran-
sition matrix and " is proportional to B �2 as well as hy-
drogen recombination rates.
Summary
We studied reaction process in two-dimensional mix-
ture gas of electrons and hydrogen atoms bound on liquid
helium surface at temperatures below 1 K. When these
two species were mixed on the surface, a loss of surface
charge was observed. The loss was attributed to electron
attachment reaction to a H atom. We measured its reaction
rates at various temperatures and magnetic fields. From
the analysis of our data, we concluded that the reaction
includes an electron and two H atoms: H H e� � ��
� ��H H. Two possible mechanisms were proposed: di-
rect three-body collision process and dissociative attach-
ment process.
The measured suppression of the reaction rate coeffi-
cient and its B �2 dependence were well described by elec-
tron singlet collision.
We believe that the cryogenic electron-hydrogen sys-
tem will cast a new light on the study of slow electron im-
pact on atoms and molecules.
Acknowledgments
This work has been supported by President’s Special
Research Grant of RIKEN, Grant-in-Aid for Scientific
Research (B) from JSPS and Grant-in-Aid for Explora-
tory Research from MEXT.
We thank T. Shiino and T. Mitsui for partial collabora-
tion. We are grateful to A. W�rl and P. Leiderer for VCE
installation and valuable discussions. We are grateful to
T. Kumada for informing us the charge-induced dipole in-
teraction. This work was carried out under the Joint Re-
search Program of the Institute for Solid State Physics,
University of Tokyo.
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