Quasi-free electron energy in near critical
point helium
Yevgeniy Lushtak a,b , Samantha B. Dannenberg a ,
C. M. Evans a,b,∗ , and G. L. Findley a,c
a Department
of Chemistry and Biochemistry, Queens College – CUNY, Flushing,
NY 11367; Tel: (718) 997-4248; Fax: (718) 997-4216
b Department
c Department
of Chemistry, Graduate Center – CUNY, New York, NY 10016
of Chemistry, University of Louisiana at Monroe, Monroe, LA 71209
Abstract
We present for the first time the quasi-free electron energy V0 (ρ) in helium from
low density to the density of the triple point liquid (gaseous helium/liquid helium
I/liquid helium II triple point), on a noncritical isotherm and on an isotherm near the
critical temperature. These data were obtained using field enhanced photoemission,
a new experimental technique for the characterization of dense fluids. A novel critical
point effect in helium is presented and is accurately fitted to the local Wigner-Seitz
model. This model, which was developed for attractive fluids, is thereby shown to
be applicable to repulsive fluids as well.
Key words: field enhanced photoemission, local Wigner-Seitz model, repulsive
fluids, radial distribution functions, interatomic potential, quasi-free electron
energy
PACS: 79.60.-i, 34.80.-i, 82.80.Pv
∗ Corresponding author.
Email addresses: cherice.evans@qc.cuny.edu (C. M. Evans),
findley@ulm.edu (G. L. Findley).
Preprint submitted to Elsevier
11 April 2012
1
Introduction
The interaction of quasi-free electrons with helium has been investigated extensively [1–32] because of the unusually low electron mobility, the highly
positive quasi-free electron energy, and the localization of excess electrons in
dense helium. The more recent of these studies focus on the structure, energetics and spectroscopy of the localized electron in helium [18–26], and on the
behavior of excess electrons near the surface of the gas/liquid helium interface [27–32]. However, the modeling of electron localization in helium requires
a thorough understanding of the quasi-free electron energy V0 (ρ), since the
energetic stability of the localized electron is directly related to V0 (ρ) [18–21].
Similarly, modeling the behavior of an excess electron near the gas/liquid helium phase boundary using a classical image potential is parameterized by
V0 (ρ) [27, 29, 30].
No experimental studies of the quasi-free electron energy V0 (ρ) in dense helium exist across the entire helium density range, however, because of the
difficulties inherent in obtaining accurate, low noise V0 (ρ) data at low temperatures [6, 33, 34]. (In fact, the only direct studies of V0 (ρ) are for liquid
helium [1–5] or in low density helium [6], while the other experimental measurements over a broader helium density range extracted V0 (ρ) from electron
surface barrier studies [7] or from dopant field ionization data [33].) We recently developed a new technique, field enhanced photoemission [34], to measure V0 (ρ) directly in fluids at very low temperatures, or in strongly absorbing
fluids. In this Letter, we determine for the first time V0 (ρ) in helium from a
low density of ≤ 1.0 × 1021 cm−3 to a high density of 2.2 × 1022 cm−3 (the density of the gaseous helium/liquid helium I/liquid helium II triple point [35]),
2
on a noncritical isotherm and on an isotherm near the critical temperature,
using field enhanced photoemission. These data are then employed to extend
the local Wigner-Seitz model [37,38], which we originally developed for understanding V0 (ρ) in attractive fluids, to strongly repulsive fluids (where repulsive
fluids are defined as those with positive zero kinetic energy electron scattering
lengths).
2
Experimental
As described below, field enhanced photoemission [34] involves photoemission
spectra acquired at several different applied electric fields from a metal electrode immersed in a fluid. For the studies presented here, photoemission from
a Pt electrode was measured with monochromatic synchrotron radiation [39]
using the University of Wisconsin Synchrotron Radiation Center stainless steel
Seya-Namioka beamline equipped with a high energy (5−35 eV) grating. The
radiation, which had a resolution of ∼ 12 meV in the energy region of interest,
entered a copper sample cell [34] capable of withstanding pressures of up to
100 bar. This cell is equipped with a LiF window possessing a 10 nm thick
strip of sputter deposited Pt along its diameter to serve as the photoemitting electrode. The other electrode, placed in a parallel configuration at a
distance of 0.1 cm, was stainless steel. The positive voltage was applied to the
stainless steel electrode, while the photocurrent was measured from the Pt
electrode with a Keithley 617 electrometer. (The electrometer was also used
to obtain the capacitance at each temperature, density and electric field, in
order to determine the permittivity of helium as a function of helium density
and temperature.)
3
The cell temperature was maintained to within ±0.05 K using an Advanced
Research Systems DE-204SB 4K closed cycle helium cryostat. To achieve the
near critical temperature of 6.5 K, the sample cell was allowed to cool for 24
hours and was thermally shielded using a thin cylinder of oxygen free copper
containing holes punched as necessary for cell alignment, wire leads and the
incident radiation. Commercial helium (Matheson Gas Products 99.9999%)
was used without further purification. Prior to any measurements, the system
was baked to a base pressure of low 10−8 Torr at room temperature. The flux
from the Seya beamline was monitored for consistency using a Ni mesh intersecting the beam prior to the sample cell. However, the photoemission spectra
were normalized to monochromator flux using a GaAsP diode adjusted for
the beamcurrent in the Aladdin storage ring, since the lowest photoemission
threshold of the Ni mesh occurs in the spectral region of interest. To minimize error induced by helium/electrode interactions, a series of photoemission
spectra were obtained with minimal helium pressures at various voltages for
each temperature. To minimize any error possibly produced by any residual
impurity oxide layer formation on the electrode, a series of field induced photoemission spectra were obtained with minimal helium pressures both before
and after the addition of dense helium. There was no change in work function
as a result of the addition of dense helium. Since it requires ∼ 3.8 × 1015 helium atoms to coat the surface of the Pt electrode (under the assumption of a
Lennard Jones fluid [36]), these low density spectra (ρ ≃ 2 × 1020 cm−3 ) were
used to establish the “zero” density work function ϕ0 for each isotherm.
4
3
Results and Discussion
In the presence of an electric field and at photon energies above threshold,
Fowler’s equation for photoemission from a clean electrode in a vacuum is
[34, 40–43]
[
2
i = C kB T
+
2
1 2
π
6
(
1
2kB 2 T 2
)2 ]
√
3
hν − ϕ0 + e E
,
(1)
where C is an empirical constant that depends on the fraction of electrons
that can absorb a photon and escape; kB is Boltzmann’s constant; T is the
temperature of the system; hν is the photon energy; ϕ0 is the electrode work
function in a vacuum; e is the electron charge; and E is the applied electric
field. When the metal electrode (Pt for this study) is immersed in helium
as opposed to a vacuum, the helium interacts with the electrode, thereby
changing the potential step, and the photoemitted electron undergoes multiple
helium scattering events, thus changing the fraction of electrons that reach
the detector. In this instance, C → B(T, ρ). Moreover, changes in the helium
permittivity ε(T, ρ) as a function of density and temperature must also be
considered in accounting for the effects of the applied electric field. Under
these conditions, eq. (1) becomes [34]
[
2
i = B(T, ρ) kB T
2
v
u
u
hν − ϕ(ρ) + t

+
1
2kB 2 T 2
1 2
π
6
e3 E
εr (T, ρ)
2 
 
 ,
(2)
where ϕ(ρ) is the Pt work function in the presence of helium, and εr ≡
ε(T, ρ)/ε0 is the temperature and density dependent helium relative permittivity, with ε0 being the permittivity of the vacuum. Following considerable
rearrangement, the difference ∆i in the photoemission from the electrode in
5
the presence of helium and under the influence of two different electric fields
is, therefore [34],
(√
√ )
∆i
= [hν − ϕ(ρ)]
ΛH − ΛL
B(T, ρ)
1
+ (ΛH − ΛL ) .
2
(3)
In eq. (3), Λi = e3 Ei /(εr (T, ρ)), with i = L, H for the low and high applied
electric field, respectively. Additional rearrangement of eq. (3) gives [34]
hν = b ∆iE + a ,
(4)
where the slope is b = 1/B(T, ρ),
∆iE = √
∆i
√
,
ΛH − ΛL
(5)
and the intercept a(ρ) is
a(ρ) = ϕ(ρ) −
1
2
(√
ΛH +
√
ΛL
)
.
(6)
Thus, the field enhanced photoemission signal, which is a plot of photon energy
as a function of ∆iE , should possess a linear region with an intercept that
yields the work function ϕ(ρ) in the zero field limit.
Fig. 1 presents representative photoemission spectra of Pt in helium measured
both at 14.7 K and at 6.5 K at various electric field strengths. Both at low
density (i.e., Fig. 1a, for example) and at high density (i.e., Fig. 1b, for example), a distinct enhancement of the photocurrent as a function of electric field
is observed. (The significant decrease in photocurrent between the low density
and high density spectra is presumably caused by multiple scattering effects
changing B(T, ρ).) Similar series of photoemission spectra were obtained at
various helium number densities on a noncritical isotherm (14.7 K) and on
an isotherm (6.5 K) near the helium critical temperature of 5.2 K. (The near
6
critical point temperature is selected to maximize temperature and pressure
stability near the critical point and to minimize the possibility of forming a
liquid in the sample cell.) Electric fields were chosen to maximize the field
effect but to minimize the probability for electron localization, while not creating dielectric breakdown in the cell. Since the possibility of bubble formation
might increase near the He critical point, the electric fields chosen to measure
FEP spectra on the critical isotherm were significantly higher than those used
to obtain FEP spectra on the noncritical isotherm. (Schoepe and Rayfield [5]
investigated field emission of electrons from helium bubbles (i.e., localized
electron states) on the helium surface and concluded that the electric field
required to extract a localized electron from a bubble in liquid helium is 150
V/cm at the gas/liquid interface. Thus, the electric fields for measuring field
enhanced photoemission were selected to be significantly above this cutoff field
threshold.) Because ϕ(ρ) is temperature dependent, a set of four low density
field dependent photoemission spectra were measured at both temperatures,
thereby providing a “zero” density signal that corrects for these temperature
variations.
Since ∆iE depends on the relative permittivity in the fluid, this latter parameter was obtained by a linear least squares analysis of the total charge
(measured using a Keithley 617 electrometer) on the electrodes in helium, in
the absence of photons, as a function of voltage across the plates for all helium densities, temperatures and applied electric fields. The Clausius-Mossotti
function [44,45] for the experimentally determined relative permittivity in helium is shown in Fig. 2, plotted as a function of helium number density. These
data indicate that the relative permittivity is independent of temperature for
our system and is nonlinearly dependent on the helium number density.
7
Fig. 3 presents the photon energy as a function of ∆iE for the six possible
pairwise differences that can be taken between the spectra shown in Fig. 1.
Fitting these data with a linear least squares analysis to eq. (4) yields an
√
√
intercept that is shown in Fig. 4, plotted as a function of ΛH + ΛL for
various densities from low density to the density of the gaseous helium/liquid
helium I/liquid helium II triple point for the noncritical isotherm of 14.7 K
(cf. Fig. 4a) and for an isotherm (6.5 K) near the critical temperature (cf. Fig.
4b). Extrapolating these data sets to zero field yields the work function ϕ(ρ).
The quasi-free electron energy V0 (ρ) is then obtained from
V0 (ρ) = ϕ(ρ) − ϕ0 ,
(7)
where ϕ0 is the intercept from the analysis of the lowest number density data
set (i.e., ϕ0 = ϕ(2.0 × 1020 cm−3 ), as discussed below in the Experimental
Methods section).
The quasi-free electron energy V0 (ρ) in helium from eq. (7) is presented in
Fig. 5 as a function of helium number density. Similar to V0 (ρ) extracted
from dopant field ionization studies in neon [48], V0 (ρ) in helium is strictly
positive. Unlike the previous results in neon [48], however, the critical point
effect observed in helium extends across a broader density range. Thus, the
accurate modeling of helium represents a significant test for the applicability
of the local Wigner-Seitz model [37, 38] to repulsive fluids.
Within the local Wigner-Seitz approach [37, 38]
3
V0 (ρ) = P− (ρ) + Ek (ρ) + kB T ,
2
(8)
where P− (ρ) is the ensemble averaged electron/helium polarization energy,
Ek (ρ) is the zero point kinetic energy of the quasi-free electron, and (3/2)kB T
8
is the thermal energy of the quasi-free electron. The ensemble averaged electron/helium polarization energy P− (ρ) is determined from [37, 38, 48]
P− (ρ) = −4πρ
∫
∞
0
g(r) w− (r) r2 dr ,
(9)
where g(r) is the helium/helium radial distribution function, and w− (r) is the
electron/helium interaction potential. The zero point kinetic energy Ek (ρ) of
the quasi-free electron, on the other hand, is calculated from [37, 38]
Ek (ρ) =
~2 η02
.
2 me rb2
(10)
In eq. (10), ~ is the reduced Planck constant, and me is the mass of the
electron. Under the assumption that interactions in the first solvent shell are
dominant, rb = rℓ − |A| is the interaction range for a local short-ranged dynamic electron/helium polarization potential within this shell, while η0 is the
phase shift induced by this short-ranged potential. The maximum distance
rℓ for the short-ranged interaction is defined by rℓ ≡
√
3
3/(4 π gm ρ), with gm
being the maximum of g(r), and therefore represents a local Wigner-Seitz radius. The minimum distance for the interaction is given by the zero kinetic
energy electron scattering length A.
Clearly, the density dependence of both P− (ρ) and Ek (ρ) is dominated by the
local density in the first solvent shell of helium. This local density, in turn,
strongly depends on the helium/helium intermolecular potential. Determining
g(r) by directly integrating the Ornstein-Zernike relation with a Percus-Yevick
closure has been shown to provide the most stable and reliable integral results
over the broadest temperature and density ranges [37]. Unlike the other rare
gases [38,39,48,49], however, a Lennard-Jones 6-12 potential (with parameters
given by Aziz, et al. [36]) did not provide the best fit of the local Wigner-Seitz
9
model to the experimental V0 (ρ) data near the critical temperature (cf. Fig.
5, solid lines), although this potential could be used to fit the experimental
data at noncritical temperatures and densities. The best fit (cf. Fig. 6, solid
lines) of the local Wigner-Seitz approach to the experimental V0 (ρ) data, both
at noncritical temperatures and on an isotherm near the critical isotherm,
required a more complex exponential-6 potential, which was optimized by
Young, et al. [47] by fitting the thermodynamic properties of helium near the
solid/liquid boundary. Therefore, the data in Figs. 5 and 6 not only show that
the local Wigner-Seitz model can be extended to repulsive systems, but also
illustrate that the accurate modeling of the critical point effect in V0 (ρ) within
the local Wigner-Seitz framework can pinpoint better pairwise intermolecular
potentials for dense fluids.
4
Acknowledgement
The experimental measurements reported here were performed at the University of Wisconsin Synchrotron Radiation Center (NSF DMR-0537588). This
work was supported by a grant from the National Science Foundation (NSF
CHE-0956719). We also thank Gabrielle Germain, an NSF Research Experience for Undergraduate (NSF CHE-0956719) student, for her assistance in
acquiring photoemission data, some of which are presented in Fig. 1.
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13
Fig. 1. [Color online] Photocurrent plotted as a function of photon energy for helium
number densities of (a, a′ ) 2.0×1020 cm−3 and (b, b′ ) 1.8×1022 cm−3 at temperatures
of (a, b) 14.7 K and (a′ , b′ ) 6.5 K, which is near the critical temperature of 5.2 K. In
(a, a′ ) the applied fields are (—) 1.0 kV/cm, (· · · ) 2.0 kV/cm, (− −) 3.0 kV/cm and
(− · −) 4.0 kV/cm. In (b) the fields are (—) 4.0 kV/cm, (· · · ) 7.5 kV/cm, (− −)
12.0 kV/cm and (− · −) 18.0 kV/cm. In (b′ ) the fields are (—) 6.0 kV/cm, (· · · )
10.0 kV/cm, (− −) 15.0 kV/cm and (− · −) 20.0 kV/cm. (Note: The increased noise
in (b) was caused by transient noise on the transmission line at SRC, which has
since been reduced with a line filter. These spectra were not remeasured, however,
since the noise level was correctable.)
14
Fig. 2. [Color online] The Clausisu-Mossotti function [(εr − 1)/(εr + 2)], where
εr ≡ ε(T, ρ)/ε0 , with ε0 being the vacuum permittivity, plotted as a function of
helium number density ρ. The solid markers represent data taken at the noncritical
temperature of 14.7 K, while the open markers represent data taken at 6.5 K, which
is near the critical temperature of 5.2 K. The solid line is a nonlinear least squares
analysis to (εr − 1)/(εr + 2) = a1 ρ + a2 ρ2 + a3 ρ3 , with a1 = 1.01 ± 0.03 × 10−3
cm3 , a2 = −3.44 ± 0.8 × 10−5 cm6 and a3 = 8.5 ± 0.4 × 10−7 cm9 .
15
Fig. 3. [Color online] Photon energy plotted as a function of ∆iE (ρ) for helium number densities of (a, a′ ) 2.0 × 1020 cm−3 and (b, b′ ) 1.8 × 1022 cm−3 at temperatures
of (a, b) 14.7 K and (a′ , b′ ) 6.5 K, which is near the critical temperature of 5.2
K. The lines represent the differences in the spectra presented in Fig. 1, with the
lowest set of fields given by (—) and the highest by (- -). Before taking the differences, each photocurrent spectrum was smoothed using a Gaussian algorithm [46],
as implemented in Wavemetrics IgorPro, to reduce noise in the differences.
16
Fig. 4. [Color online] The intercept a(ρ) of representative field enhanced photoemission plots (cf. Fig. 3, for example), measured at (a) 14.7 K and at √
(b) 6.5 K
√ (which is
near the critical temperature of 5.2 K), plotted as a function of ( ΛH + ΛL ). The
number densities range from (•) 2.0 × 1020 cm−3 to () 2.4 × 1022 cm−3 in (a), and
from (•) 2.0 × 1020 cm−3 to () 1.8 × 1022 cm−3 in (b). Each line represents a linear
least squares analysis with an intercept that yields ϕ(ρ). See text for discussion.
17
Fig. 5. [Color online] The quasi-free electron energy V0 (ρ) in helium – determined
from eq. (7) using the data in Fig. 4 (as well as data at other densities that are not
shown in Fig. 4 for brevity) – plotted as a function of helium number density ρ. The
solid markers represent data obtained at 14.7 K, while the open markers represent
data obtained at 6.5 K, which is near the critical temperature of 5.2 K. The lines are
a local Wigner-Seitz calculation with the helium/helium interaction modeled using
a Lennard-Jones 6-12 potential [36]. In the modeling of V0 (ρ), A = 0.63 Å [50] and
η0 = 0.73 (as determined here). See text for discussion.
18
Fig. 6. [Color online] The quasi-free electron energy V0 (ρ) in helium plotted as a
function of helium number density ρ. See Fig. 5 for details about the experimental
data. The lines are a local Wigner-Seitz calculation with the helium/helium interaction modeled using an exponential-6 potential [47]. In the modeling of V0 (ρ) ,
A = 0.63 Å [50] and η0 = 0.78 (as determined here). See text for discussion.
19