Energy of the Quasi-free Electron in H2 , D2 and O2 : Probing Intermolecular
Potentials within the Local Wigner-Seitz Model
C. M. Evans,1, a) Kamil Krynski,1 Zachary Streeter,2 and G. L. Findley2, b)
1)
Department of Chemistry and Biochemistry, Queens College – CUNY, Flushing,
NY 11367
2)
School of Sciences, University of Louisiana at Monroe, Monroe,
LA 71209
(Dated: 2 November 2015)
We present for the first time the quasi-free electron energy V0 (ρ) for H2 , D2 and O2
from gas to liquid densities, on noncritical isotherms and on a near critical isotherm
in each fluid. These data illustrate the ability of field enhanced photoemission (FEP)
to determine V0 (ρ) accurately in strongly absorbing fluids (e.g., O2 ) and fluids with
extremely low critical temperatures (e.g., H2 and D2 ). We also show that the isotropic
local Wigner-Seitz model for V0 (ρ) – when coupled with thermodynamic data for the
fluid – can yield optimized parameters for intermolecular potentials, as well as zero
kinetic energy electron scattering lengths.
a)
cherice.evans@qc.cuny.edu
b)
findley@ulm.edu
1
I.
INTRODUCTION
The quasi-free electron energy V0 (ρ) (where ρ is the fluid number density) is important in
understanding electron transport through a fluid,1–4 as well as for modeling electron attachment reactions in fluids.2,5 The recent development of field enhanced photoemission6–8 as a
technique to directly and accurately measure V0 (ρ) has opened the possibility of investigating the density and temperature dependence of the quasi-free electron energy in a variety of
problematic fluids (e.g., fluids with extremely low critical temperatures,7 strongly absorbing
fluids8 (i.e., fluids that absorb light in the ultraviolet or vacuum ultraviolet regions) and
polar fluids9 ). We have shown in previous studies7,10 that the isotropic local Wigner-Seitz
model for V0 (ρ) can give insight into the determination of the intermolecular potential for a
fluid. In this paper, we present the quasi-free electron energy V0 (ρ) in H2 , D2 and O2 . We
use these data, coupled with thermodynamic data for the fluids, to optimize intermolecular
potential parameters. We also show that nonlinear analyses of V0 (ρ) within the isotropic
local Wigner-Seitz model using these intermolecular potentials yield accurate zero kinetic
energy electron scattering lengths for each fluid.
II.
ISOTROPIC LOCAL WIGNER-SEITZ MODEL
Within the local Wigner-Seitz model, V0 (ρ) is given by10,11
3
V0 (ρ) = P− (ρ) + Ek (ρ) + kB T ,
2
(1)
where P− (ρ) is the ensemble average electron/perturber polarization energy, Ek (ρ) is the zero
point kinetic energy of the quasi-free electron, and (3/2)kB T (kB ≡ Boltzmann’s constant)
is the thermal energy of the quasi-free electron.
The density dependent average polarization energy P− (ρ) depends upon the position ri
of each of the N fluid constituents relative to the quasi-free electron at the moment of
electron/constituent interaction. If the interaction potential is isotropic and is written as a
sum of pairwise potentials, then the total electron/fluid interaction potential is4,12–19
N
∑
1
w− (r1 , . . . , rN ) = − α e2
ri−4 f (ri ) ,
2
i
2
(2)
where α is the polarizability of the fluid, e is the charge on the electron, and f (r) is a
screening function that incorporates the repulsive interactions between the induced dipoles
in the fluid. The screening function f (r) is4
∫
∞
f (r) = 1 − απρ
1
ds 2 g(s)
s
0
∫
r+s
dt f (t) θ(r, s, t) ,
(3)
|r−s|
with
θ(r, s, t) =
3 2
(s + t2 − r2 )(s2 − t2 + r2 ) + (r2 + t2 − s2 ) ,
2
2s
(4)
and with g(s) being the fluid radial distribution function. (In eqs. (3) – (4), the integration
variables s and t represent the distance between the constituent of interest and all other
constituents of the fluid, respectively.) Under the assumption of a canonical distribution,
the probability of sampling a particular polarization energy W is given by12–21
∫
P (W ) =
∫
···
δ(W − w− (r1 , . . . , rN )) × e−U (r1 ,...,rN )/(kB T )
/∫
i
∫
···
∏
e
−U (r1 ,...,rN )/(kB T )
∏
dri
(5)
dri ,
i
where U (r1 , . . . , rN ) is the multi-dimensional potential energy prior to the electron/fluid
interaction. A moment analysis of the Fourier transform of eq. (5) yields the average
polarization energy P− (ρ) through the first moment, or12–21
∫
m1 = −4πρ
∞
g(r)w− (r)r2 dr ≡ P− (ρ) .
(6)
0
The zero-point kinetic energy Ek (ρ) is obtained from solving the Schrödinger equation10,11,22
∇2 ψ +
2me
[Ek − Vloc (r)]ψ = 0
~2
(7)
for the quasi-free electron in a dense fluid. In eq. (7), Vloc (r) is a short-ranged potential that
accounts for local dynamic polarization of a fluid constituent by the quasi-free electron, me
is the mass of the electron and ~ is the reduced Planck constant. For isotropic fluids, the
local potential Vloc (r) satisfies an average translational symmetry10,11,22
Vloc (r) = Vloc (r + 2rb ) ,
3
(8)
where rb is the interaction range. At any density, the minimum distance between a quasifree electron with low kinetic energy and a single fluid constituent is given by the absolute
value of the scattering length A. Under the assumption that interactions in the first solvent
shell dominate the dynamics of the problem, the maximum distance rℓ for a short-ranged
interaction is one-half of the spacing between two fluid constituents in this shell. This
distance, otherwise known as the local Wigner-Seitz radius, is10,11
√
rℓ ≡
3
3
,
4πgmax ρ
(9)
where gmax is the maximum of the fluid radial distribution function g(r). Thus, the interaction range for the local short-ranged potential is rb = rℓ − |A|. By applying these
boundary conditions to the asymptotic solutions of eq. (7), the zero-point kinetic energy of
the quasi-free electron is10,11,22
Ek (ρ) =
~2 η02
.
2me (rℓ − |A|)2
(10)
In eq. (10), η0 is a phase shift induced by the short-ranged potential, and the density
dependence arises from rℓ . The phase shift η0 represents the only empirical parameter in
the local Wigner-Seitz model, if the zero kinetic energy electron scattering length A has
been determined independently from other experimental techniques (see, for example, cross
section measurements23–25 or measurements of the shift of high-n dopant Rydberg state
energies26,27 for the fluids in this study) and if the polarization potential P− (ρ) is known.
(The theoretical treatment of the short range potential leading to η0 is a question of active
interest that will be the subject of a future publication by us.)
III.
COMPUTATIONAL METHODOLOGY
Since the calculation of P− (ρ) involves g(r), while rℓ is ascertained from the maximum
of g(r), accurately determining the radial distribution function for each fluid over a large
density and temperature range is extremely important. We have shown that a direct integration of the Ornstein-Zernike relation with a Percus-Yevick closure reduced numerical
instabilities in the determination of g(r) at high densities and on near critical isotherms.10,26
4
This direct integration is accomplished through the homogenous Percus-Yevick model28
g(r) = r−1 e−U (r)/kB T Y (r) ,
where
∫
r
Y (r) =
dt
0
(11)
dY (t)
dt
(12)
with
∫ ∞
[
]
dY (r)
= 1 + 2πρ
dt e−U (r)/kB T − 1 Y (t)
dr
0
[
]
r − t −U (|r−t|)/kB T
−U (r+t)/kB T
× e
Y (r + t) −
e
Y (|r − t|) − 2t .
|r − t|
(13)
Stable numerical integration of eqs. (11) – (13) was achieved using an optimization routine
developed by Ng et al.,29 which employs a weighted linear combination of m ≥ 3 previous
iterations to generate the input for the (m + 1)th iteration.
Since one of the goals of this study was to test the use of the isotropic local WignerSeitz model for V0 (ρ) as a means of selecting an adequate intermolecular potential, we
chose to calculate radial distribution functions via eqs. (11) – (13) using the two most
common intermolecular potentials, namely the Lennard-Jones 6-12 potential (LJ6-12) and
the exponential-6 (Exp-6) potential. The functional forms of these potentials are given in
Table I. Since recent studies30–35 have shown that different cut-off radii for the OrnsteinZernike approach to determining radial distribution functions can yield different potential
parameters, we chose to validate the radial distribution functions obtained from the chosen
intermolecular potentials by comparison with thermodynamic data via36
2
p(ρ, T ) = kB T ρ − πρ2
3
∫
rc
dr r3 g(r)
0
dU (r)
,
dr
(14)
where rc is a cut-off radius. In this study, we found that rc ≈ 8σ ensured numerical stability
in the near critical point calculations. Moreover, increasing rc beyond 8σ did not lead to
appreciable change in the intermolecular potential parameters optimized to thermodynamic
data at noncritical temperatures (both above and below the critical temperature).
5
IV.
EXPERIMENTAL PROCEDURES
The direct determination of the quasi-free electron energy V0 (ρ) using field enhanced
photoemission (FEP)6–8 requires the acquisition of photoemission spectra under the influence
of a minimum of four different electric fields at each fluid density and for each temperature.
The difference ∆i in the near threshold photoemission from a metal electrode (which was
Pt for the experiments reported here) in the presence of a fluid and under the influence of
two different electric fields is6–8
(√
√ )
∆i
= [hν − ϕ(ρ)]
ΛH − ΛL
B(T, ρ)
1
+ (ΛH − ΛL )
2
(15)
where
e2 Ej
εr (T, ρ)
Λj =
(16)
with j = L, H for the low and high applied electric field Ej , respectively. In eqs. (15) and
(16), B(T, ρ) is an empirical constant that depends on the fraction of electrons that can
absorb a photon and escape, hν is the photon energy, ϕ(ρ) is the metal work function in the
presence of the fluid, and εr ≡ ε(T, ρ)/ε0 is the temperature and density dependent fluid relative permittivity, with ε0 being the permittivity of the vacuum. Additional rearrangement
of eq. (15) gives the primary FEP equation:6–8
hν = b∆iE + a(ρ) ,
(17)
with a slope of b = 1/B(T, ρ) and an intercept of
a(ρ) = ϕ(ρ) −
√ )
1 (√
ΛH + ΛL ,
2
(18)
where ∆iE is defined as
∆iE ≡ √
∆i
√
.
ΛH − ΛL
(19)
Thus, the FEP signal is a plot of photon energy as a function of ∆iE via eq. (17). Fitting
the linear region of this signal by least squares analysis yields the work function ϕ(ρ) from
the intercept a(ρ) in the limit of zero electric field. The density dependent quasi-free electron
6
energy V0 (ρ) is then directly determined from
V0 (ρ) = ϕ(ρ) − ϕ0 ,
(20)
where ϕ0 is the reference work function for the specific metal electrode. Because ϕ(ρ)
is temperature dependent, and in order to minimize errors introduced by fluid/electrode
interactions, a series of photoemission spectra are obtained at very low fluid density and at
various voltages for each isotherm to ascertain the reference work function ϕ0 .
All photoemission data for these studies were measured with monochromatic vacuum
ultraviolet synchrotron radiation using the University of Wisconsin Synchrotron Radiation Center stainless steel Seya-Namioka beamline equipped with a high energy (5–35 eV)
grating.26,37 The radiation, which had a resolution of ∼ 10 meV in the energy region of
interest, entered a copper sample cell fitted with a 3 mm thick MgF2 window capable of
withstanding up to 100 bar of pressure. The photoemitter was a 10 nm thick strip of Pt
sputter deposited along the window diameter. A stainless steel electrode was attached to
the window perpendicular to this strip of Pt, with the spacing between the two electrodes
being 0.1 cm. The electric field was applied to the stainless steel electrode, while the photocurrent was detected at the Pt electrode using a Keithley 6514 electrometer. (To increase
the signal-to-noise ratio for these low signal measurements, the high voltage power supply
and the electrometer drew power through an AC line conditioner to remove electrical line
noise.) The flux from the Seya beamline was monitored for consistency using a nickel 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 beam current in
the Aladdin storage ring, since the lowest photoemission threshold of the nickel mesh occurred in the spectral region of interest. The cell temperature was adjusted, and maintained
to within ±0.05 K, using an Advanced Research Systems DE-204SB 4K closed cycle helium
cryostat system.
Oxygen (Matheson Trigas, 99.998%), hydrogen (Matheson Trigas, 99.9999%) and deuterium (AirGas, 99.999%, 99.8% isotopic enrichment) were used without further purification.
Prior to the introduction of any of these gases to the sample cell, the gas handling system
and sample cell were baked to a base pressure of at least low 10−8 Torr. The number densities of O2 ,38,39 H2 40 and D2 41 were calculated using the NIST Standard Reference Database
7
23 Version 9.1.42
Significant loss of photoemission current was encountered in the measurement of photoemission spectra of Pt in all of these fluids at very high densities (i.e., ρ > 1.8 × 1022 cm−3
or ρr ≡ ρ/ρc > 1.7) and, therefore, only a few data points were obtained in this regime.
Since the minimum field strength used in these studies was 1.0 kV/cm (corresponding to a
kinetic energy of 100 eV for a free electron in a vacuum) while the maximum field strength
was in excess of 40 kV/cm, electron localization and electron attachment reactions to fluid
impurities should be minimized, even when one accounts for the significant reduction in the
kinetic energy of the electron due to electron/fluid interactions. Therefore, we attribute
the almost complete loss of photoemission signal at very high densities to fluid stabilized
dissociative electron attachment.43,44
Each FEP spectrum depends on the relative permittivity of the fluid and, therefore,
this latter parameter was obtained by a least squares analysis of the total charge on the
electrodes, in the absence of photons, as a function of voltage across the electrodes at all
fluid densities, temperatures and applied electric fields. The Clausius-Mossotti functions45,46
for the experimentally determined relative permittivities for all fluids in this study are shown
in Fig. 1 plotted versus the reduced fluid number density ρr . These data indicate that the
relative permittivity exhibits a slight critical point effect, which was not observed in our
earlier measurements of He7 and N2 ,8 and which – to the best of our knowledge – has
not been reported previously. Modeling of these results, along with relative permittivity
measurements from simple polar fluids, is currently in progress.47
V.
RESULTS AND DISCUSSION
Because of the large number of photoemission spectra required for this study, we have
chosen for brevity to show in Fig. 2 representative spectra of Pt in H2 , D2 and O2 obtained
for various electric fields near the critical point of each fluid. A distinct enhancement of
the photocurrent as a function of electric field is observed in all systems. Similar series
of photoemission spectra were obtained at various fluid number densities on noncritical
isotherms and on an isotherm near the critical temperature. The near critical isotherm was
selected to be +0.5 K above the critical temperature so as to maximize temperature and
pressure stability near the critical point and to minimize the possibility of forming two phases
8
(i.e., gas and liquid) in the sample cell. Electric fields were chosen to maximize the field effect
and to minimize the probability for electron localization or electron attachment reactions,
while not creating dielectric breakdown in the cell. Because ϕ(ρ) is temperature dependent,
a set of four low density field dependent photoemission spectra were measured at every
temperature, thereby providing a ‘zero’ density signal that corrects for these temperature
variations.
Obtaining FEP signals from the photoemission data and then fitting these signals by
linear least squares analyses of eq. (17) yields intercepts a(ρ) for each density at each
√
√
temperature. These intercepts are shown in Fig. 3 plotted as a function of ΛH + ΛL
for select densities in H2 , D2 and O2 at noncritical temperatures. Fig. 4 shows similar data
for the near critical isotherms in H2 , D2 and O2 . Extrapolating a(ρ) to zero field yields the
work function ϕ(ρ) (i.e., eq. (18)), which can then be used to obtain the quasi-free electron
energy V0 (ρ) from eq. (20). (We should note here that the reference work function was set
to ϕ0 ≈ ϕ(3.0 × 1019 cm−3 ) for each fluid at each temperature in this study.) Fig. 5 presents
V0 (ρ) plotted as a function of the reduced fluid number density ρr ≡ ρ/ρc (with the fluid
critical density42 ρc (in units of 1021 cm−3 ) being 9.3397, 10.435 and 8.209 for H2 , D2 and
O2 , respectively).
To apply the isotropic local Wigner-Seitz model to the data shown in Fig. 5, we began by
determining optimized Lennard-Jones 6-12 and exponential-6 potential parameters for H2 ,
D2 and O2 using eq. (14) with the thermodynamic data of each fluid provided by NIST.42
The optimal parameters were defined as those that minimized the average percent deviation
in the pressure as calculated from eq. (14) in comparison to the experimental thermodynamic
conditions for the fluid when Tr (≡ T /Tc ) > 1, at a near critical temperature (i.e., Tr ≈ 1),
and when Tr < 1, with the critical temperature42 Tc being 33.145 K, 38.34 K and 154.58
K for H2 , D2 and O2 , respectively. (We should note here that we were unable to obtain
V0 (ρ) data at temperatures Tr < 1 for H2 and D2 . However, to ensure consistency over the
broadest density range (i.e., from gas phase up to the density of the triple point liquid),
we optimized potential parameters using additional thermodynamic data42 at Tr < 1 for
all systems in this study.) Fig. 6 shows the optimization results for a Lennard-Jones 6-12
potential for O2 when Tr > 1, when Tr ≈ 1 and when Tr < 1. The solid point in the
individual panels of Fig. 6 represents the area of overlap for each temperature region. Using
this technique allows one to obtain the parameters that best fit the entire fluid phase of O2 ,
9
as well as the error in these parameters. (Contour plots similar to that shown in Fig. 6
were generated for each potential (i.e., LJ6-12 and Exp-6) for each system (i.e., H2 , D2 and
O2 ), but are not shown for brevity.) Table II gives the intermolecular potential parameters
for the exponential-6 and Lennard-Jones 6-12 potentials for H2 , D2 and O2 as determined
from these contour plots in comparison to parameters obtained from other computational
methods.
The radial distribution functions from the optimized potential parameters were used
subsequently to calculate the average electron/fluid polarization energy P− (ρ), from eq. (6),
and the local Wigner-Seitz radius rℓ for use in the determination of the kinetic energy Ek (ρ)
of the quasi-free electron (i.e., eq. (10)). (The polarizability for each fluid was obtained
from S. V. Khristenko et al..48 ) Once an optimized intermolecular potential was found, the
only adjustable parameters in the isotropic local Wigner-Seitz model are in the calculation
of the kinetic energy Ek (ρ) of the quasi-free electron, namely the zero kinetic energy electron
scattering length A and the phase shift η0 . Thus, optimization of the isotropic local WignerSeitz model involves the adjustment of one or both of these parameters. However, for the
fluids in this study, the zero kinetic energy electron scattering length A had been previously
determined.27 Nevertheless, we decided to ascertain if one could obtain the zero kinetic
energy electron scattering length directly from the quasi-free electron energy data through
the local Wigner-Seitz model.
Thus, employing a technique similar to the one used in the optimization of the intermolecular potential parameters, we measured the average percent deviation between the
calculated V0 (ρ) in comparison to experimental data as a function of both A and η0 . As
an example, Fig. 7 shows the contour plots for the optimization of V0 (ρ) for D2 at noncritical temperatures and near the critical temperature using the exponential-6 potential for
all calculations. The solid point in the individual panels of Fig. 7 represents the area of
overlap between the noncritical and critical isotherm data for those contours that deviate
from experiment by less than the experimental error. (Similar contour plots were generated
for the other fluids but are not shown here.) The results of these optimizations are presented
in Table III along with those obtained from the calculation of V0 (ρ) (i.e., η0 ) with fixed A.
Employing the fully optimized results, one notices that the scattering length obtained from
the best intermolecular potential (as selected using thermodynamic data and the quasi-free
electron energy data) is within 2% of that determined from dopant field ionization studies.27
10
(Using the zero kinetic energy electron scattering lengths obtained here to determine the
total scattering cross section, via the relationship σT = 4πA2 , gives σT = 3.88 Å2 for H2
and σT = 0.149 Å2 for O2 . The zero kinetic energy total scattering cross section obtained
for hydrogen is comparable to those obtained from other experimental techniques, which
range from 2.5 Å2 to 5.75 Å2 .27,49–51 The difficulties in directly measuring the zero kinetic
energy total electron cross section of O2 have been discussed by various groups.52–56 However, the value presented here is in line with the estimate that σT ≤ 0.4 Å2 .52–55 ) Therefore,
choosing the intermolecular potential that best fits the thermodynamic data allows one to
obtain the zero kinetic energy electron scattering length from the experimentally measured
quasi-free electron energy in the fluid. The calculated quasi-free electron energy V0 (ρ) using
the isotropic local Wigner-Seitz model are shown as lines in Fig. 5. (We should note here
that the lines were calculated to the triple point liquid density (i.e., ρr ≈ 2.5), although we
were unable to obtain experimental data in this region because, as mentioned previously,
the fluids in this study have significant dissociative electron attachment channels.43,44 )
VI.
CONCLUSIONS
We have measured (cf. Fig 5) the quasi-free electron energy in H2 , D2 and O2 from gas to
liquid densities at various temperatures, including an isotherm near the critical temperature.
Moreover, we have shown that these data can be interpreted within the local Wigner-Seitz
model for V0 (ρ). To the best of our knowledge, these results represent the first investigation
of electron energy in these fluids over such a wide density range. Thus, we have further
illustrated8 that field enhanced photoemission can be used to obtain the quasi-free electron
energy in molecular fluids. We have also shown that combining the thermodynamic data
of the fluid with the isotropic local Wigner-Seitz model for the quasi-free electron energy
allows one to select optimized intermolecular potentials and to determine the zero kinetic
energy electron scattering length of the fluid. Since the investigation of the zero kinetic
energy electron scattering length in many complex molecular fluids is difficult, the ability to
extract A accurately from the quasi-free electron energy will allow for future study of more
complex molecular fluids (such as polar fluids). We are currently in the process of extending
the local Wigner-Seitz model to anisotropic fluids.9
11
ACKNOWLEDGMENTS
We thank Dr. Yevgeniy Lushtak (SAES, Inc), Ms. Ollianna Burke (NAVY Officer
Training Command) and Mr. Holden Smith (Louisiana State University) for their assistance
in obtaining the experimental data presented here. All measurements were performed at the
University of Wisconsin Synchrotron Radiation Center, a facility primarily funded by the
University of Wisconsin – Madison with supplimental support from facility Users and the
University of Wisconsin – Milwaukee. This work was supported by a grant from the National
Science Foundation (NSF CHE-0956719).
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J. W. Leachman, R. T. Jacobsen, S. G. Penoncello, and E. W. Lemmon, J. Phys. Chem.
Ref. Data 38, 721 (2009).
41
I. A. Richardson, J. W. Leachman, and E. W. Lemmon, J. Phys. Chem. Ref. Data 43,
013103 (2014).
42
E. W. Lemmon, M. L. Huber, and M. O. McLinden, NIST Standard Reference Database
23: Reference Fluid Thermodynamic and Transport Properties – REFPROP, Version 9.0
(National Institute of Standards and Technology, Gaithersburg, MD, 2010).
43
E. Krishnakumar, S. Denifl, I. Čadež, S. Markelj, and N. J. Mason, Phys. Rev. Lett. 106,
243201 (2011), and references therein.
44
V. Laporta, R. Celiberto, and J. Tennyson, Phys. Rev. A 91, 012701 (2015), and references
therein.
45
P. V. Rysselberghe, J. Phys. Chem. 36, 1152 (1932).
46
J. W. Schmidt and M. R. Moldover, Int. J. Thermophys. 24, 375 (2003).
47
C. M. Evans, K. K. Krynski, K. Yalikun, and G. L. Findley, J. Chem. Phys. (2016),
Charge screening and dielectric permittivity in dense diatomic fluids, in preparation.
48
S. V. Khristenko, A. I. Maslov, and V. P. Shevelko, Molecules and Their Spectroscopic
Properties (Springer-Verlag, New York, 1998).
49
U. Asaf, W. S. Felps, K. Rupnik, S. P. McGlynn, and G. Ascarelli, J. Chem. Phys. 91,
5170 (1989).
50
J. Ferch, W. Raith, and K. Schroeder, J. Phys. B: At. Mol. Phys. 13, 1481 (1980).
51
E. S. Chang, J. Phys. B: At. Mol. Phys. 14, 893 (1981).
52
A. Zecca, G. Karwasz, and R. S. Brusa, Riv. Nuovo Cimento 19, 1 (1996), and references
therein.
53
I. Shimamura, Sci. Pap. Inst. Phys. Chem. Res. (Rikagaku Kenkyusho) 82, 1 (1989).
54
S. A. Lawton and A. V. Phelps, J. Chem. Phys. 69, 1055 (1978).
55
I. D. Reid and R. W. Crompton, Aust. J. Phys. 33, 215 (1980).
56
J. Randell, S. L. Lunt, G. Mrotzek, J. P. Ziesel, and D. Field, J. Phys. B: At. Mol. Opt.
Phys. 27, 2369 (1994).
14
TABLE I. Intermolecular potentials used in this study. For these functions, σ and rm are collision
radii (rm = 21/6 σ for the Lennard-Jones 6-12 potential), ϵ is the potential well-depth, and γ gauges
the steepness of the intermolecular potential.
Name
Functional form
[
( rm )6 ]
6 γ(1−r/rm )
ϵ
U (r) = 1−6/γ γ e
− r
[( )
]
( )6
12
U (r) = 4 ϵ σr
− σr
Exponential-6 (Exp-6)
Lennard Jones 6-12 (LJ6-12)
15
TABLE II. Intermolecular potential parameters for an exponential-6 (Exp-6) and a Lennard Jones
6-12 (LJ6-12) potential for H2 , D2 and O2 obtained from this study, as well as selected parameters
from previous investigations.
Fluid
H2
Potential
Exp-6
ϵ/kB (K)
25.6 ± 0.1
37.3
36.4
rm (Å)
3.291 ± 0.003
3.337
3.43
γ
12.0 ± 0.1
14.0
11.1
Ref
this work
b
c
H2
LJ6-12
28.3 ± 0.1
32.6
37.0
36.7
37.00
36.0
3.243 ± 0.006
3.32
3.29
3.32
3.287
3.29
—
—
—
—
—
—
this work
a
a
a
b
f
D2
Exp-6
28.4 ± 0.1
3.392 ± 0.005
12.0 ± 0.1
this work
D2
LJ6-12
31.6 ± 0.1
34.4
37.0
35.2
3.358 ± 0.004
3.34
3.29
3.31 7
—
—
—
—
this work
a
a
a
O2
Exp-6
112.8 ± 0.1
125.0
125.0
99.495
3.625 ± 0.006
3.86
3.84
3.781
13.5 ± 0.1
13.2
13.0
14.45
This study
d
e
e
O2
LJ6-12
126.0 ± 0.1
106.7
116
3.587 ± 0.004
3.892
3.92
—
—
—
this work
e
f
a
I. H. Hillier, M. S. Islam and J. Walkley, J. Chem. Phys. 43, 3705 (1965). b J. Van de
Ree, J. Los ad A. E. DeVries, Physica 34, 66 (1967). c W. J. Nellis, M. Ross, A. C.
Mitchell, M. van Thiel, D. A. Young, F. H. Ree and R. J. Trainor, Phys. Rev. A 27, 608
(1983). d Q. F. Chen, L. C. Cai, Y. Zhang and Y. J. Gu, J. Chem. Phys. 128, 104512
(2008). e A. Belonoshko and S. K. Saxena, Geochimica et Cosmochimica Acta 55, 3191
(1991). f M. D. McKinley and T. M. Reed III, J. Chem. Phys. 42, 3891 (1965).
16
TABLE III. The zero kinetic energy electron scattering length A, the phase shift η0 , and the
average percent deviation %χnc and %χcr between the calculated quasi-free electron energy V0 (ρ)
and the experimentally measured values for noncritical temperatures and for a near critical point
temperature, respectively. The radial distribution functions used in these calculations are obtained
using either the Lennard-Jones 6-12 potential (LJ6-12) or the exponential-6 potential (Exp-6)
parameters. The zero kinetic energy electron scattering length was obtained either from Evans
et al 27 using dopant field ionization (DFI) or was determined here by optimizing the fit to the
quasi-free electron energy (this work).
Fluid
H2
Potential
Exp-6
A (Å)
0.541 (DFI)
0.556 ± 0.001 (this work)
η0
0.66
0.651 ± 0.002
%χnc
2.1
1.8
%χcr
2.5
1.5
H2
LJ6-12
0.541 (DFI)
0.56 ± 0.01 (this work)
0.71
0.69 ± 0.01
4.5
3.1
10
7.5
D2
Exp-6
0.575 (DFI)
0.589 ± 0.002 (this work)
0.68
0.670 ± 0.001
2.4
1.2
2.1
1.4
D2
LJ6-12
0.575 (DFI)
0.58 ± 0.01 (this work)
0.73
0.70 ± 0.01
5.1
2.1
11
6.8
O2
Exp-6
−0.091 (DFI)
−0.109 ± 0.004 (this work)
0.60
0.592 ± 0.002
3.0
1.4
2.9
1.8
O2
LJ6-12
−0.091 (DFI)
−0.112 ± 0.01 (this work)
0.65
0.67 ± 0.01
7.2
3.5
18
10
17
8
6
e
)2 + r ( / )1 – r ( ×
(a)
4
e
01
2
2
0
0.0
0.5
1.0
1.5
2.0
2.5
1.5
2.0
2.5
1.5
2.0
2.5
rr
8
6
e
)2 + r ( / )1 – r ( ×
(b)
4
e
01
2
2
0
0.0
0.5
1.0
rr
8
(c)
e
)2 + r ( / )1 – r ( ×
6
4
e
01
2
2
0
0.0
0.5
1.0
rr
FIG. 1. The Clausius-Mossotti function [(εr − 1)/(εr + 2)], where εr ≡ ε(T, ρ)/ε0 with ε0 being the
vacuum permittivity, plotted as a function of reduced fluid number density ρr ≡ ρ/ρc for (a) H2 ,
(b) D2 and (c) O2 . The temperatures for H2 were (•) 39.00 ± 0.05 K and (◦) 33.55 ± 0.05 K, which
is near the critical temperature of Tc = 33.15 K. The temperatures for D2 were () 48.09 ± 0.05
K, (N) 43.03 ± 0.05 K and () 39.02 ± 0.05 K, which is near the critical temperature of Tc = 38.34
K. The temperatures for O2 were (H) 158.57 ± 0.05 K, () various noncritical temperatures below
the critical temperature of Tc = 154.58 K, and the near critical isotherm of (△) 155.27 ± 0.05
K. The solid lines represents nonlinear least squares fits of the noncritical temperature data to
[(εr − 1)/(εr + 2)] = c1 ρr + c2 ρ2r + c3 ρ3r . For H2 : ρc = 9.22 × 1021 cm−3 , c1 = 1.2 ± 0.1 × 10−2 ,
c2 = 7.3±0.2×10−3 and c3 = −1.7±0.1×10−3 . For D2 : ρc = 1.32×1022 cm−3 , c1 = 9.8±0.2×10−3 ,
c2 = 8.9±0.3×10−3 and c3 = −2.0±0.1×10−3 . For O2 : ρc = 8.20×1021 cm−3 , c1 = 4.5±0.2×10−2 ,
c2 = −9.1 ± 0.3 × 10−3 and c3 = −1.6 ± 0.3 × 10−3 .
18
8
8
8
(a)
(b)
6
6
2
0
6
)stinu .bra( i
)stinu .bra( i
)stinu .bra( i
4
6
(c)
4
2
0
7
8
9
Energy (eV)
10
6
4
2
0
7
8
9
Energy (eV)
10
6
7
8
9
10
Energy (eV)
FIG. 2. Photocurrent plotted as a function of photon energy for (a) H2 , (b) D2 and (c) O2 . These
data were obtained near the critical point of each fluid with the temperatures and densities being
(a) Tr (≡ T /Tc ) = 1.011 and ρr (≡ ρ/ρc ) = 0.962, (b) Tr = 1.018 and ρr = 1.01 and (c) Tr = 1.004
and ρr = 1.02. (The critical temperature42 Tc is 33.145 K, 38.34 K and 154.58 K; while the critical
density42 ρc (in units of 1021 cm−3 ) is 9.3397, 10.435 and 8.209 for H2 , D2 and O2 , respectively.)
The applied fields (from bottom to top) are (a) 10.0 kV/cm, 14.0 kV/cm, 18.0 kV/cm and 22.0
kV/cm; (b) 24.5 kV/cm, 31.0 kV/cm, 37.0 kV/cm and 44.0 kV/cm; and (c) 7.00 kV/cm, 8.50
kV/cm, 12.0 kV/cm and 14.0 kV/cm.
19
9.0
(a)
8.8
)Ve(
8.6
(
ra
)
8.4
8.2
8.0
7.8
0.00
0.05
0.10
LH
LL
1/2
+
0.15
1/2
0.20
(eV)
9.2
(b)
9.0
8.8
)Ve(
8.6
)
(
ra
8.4
8.2
8.0
7.8
0.00
0.05
LH
1/2
+
0.10
LL
1/2
0.15
(eV)
8.4
(c)
8.2
)Ve(
8.0
(
ra
)
7.8
7.6
7.4
7.2
0.00
0.02
LH
0.04
1/2
+
0.06
LL
1/2
0.08
0.10
(eV)
FIG. 3. The FEP intercept a(ρ) (cf. eq. (18)) of representative field enhanced photoemission plots
for (a) H2 at Tr = 1.182, (b) D2 at Tr = 1.253 and (c) O2 at Tr = 1.03. In all graphs (•) represents
the density of 3.00 × 10−19 cm−3 or a reduced density of ρr ≈ 3.0 × 10−3 . The reduced densities ρr
are (a) (◦) 0.225, () 0.455, () 0.696, (N) 0.942, (△) 1.18, and (H) 1.36; (b) (◦) 0.144, () 0.422,
() 0.719 and (N) 1.01; (c) (◦) 0.242, () 0.479, () 0.966, (N) 1.25 and (△) 1.41. See text for
discussion.
20
9.0
(a)
8.8
8.6
)Ve(
8.4
(
ra
)
8.2
8.0
7.8
7.6
0.00
0.05
0.10
LH
LL
1/2
+
0.15
1/2
0.20
(eV)
9.4
(b)
9.2
9.0
)
8.4
8.6
(
ra
)Ve(
8.8
8.2
8.0
7.8
7.6
0.00
0.05
0.10
LH
LL
1/2
+
0.15
1/2
0.20
(eV)
8.5
(c)
8.0
)Ve(
(
ra
)
7.5
7.0
6.5
0.00
0.05
LH
1/2
+
0.10
LL
1/2
0.15
(eV)
FIG. 4. The FEP intercept a(ρ) of representative field enhanced photoemission plots for (a) H2
at Tr = 1.011, (b) D2 at Tr = 1.018 and (c) O2 at Tr = 1.004. In all graphs (•) represents the
density of 3.00 × 10−19 cm−3 or a reduced density of ρr ≈ 3.0 × 10−3 . The densities ρ are (a)
(◦) 0.311, () 0.455, () 0.696, (N) 0.964, and (△) 1.80; (b) (◦) 0.173, () 0.400, () 0.614,
(N) 0.918, (△) 1.39, (H) 1.76, and (▽) 1.87; (c) (◦) 0.295, () 0.731, () 0.931, (N) 1.34, (△) 1.47,
and (H) 1.65. See text for discussion.
21
1.2
(a)
1.0
0.8
)Ve(
0.6
0
V
0.4
0.2
0.0
-0.2
0.0
0.5
1.0
1.5
2.0
2.5
3.0
2.0
2.5
3.0
rr
1.2
(b)
1.0
0.8
)Ve(
0.6
0
V
0.4
0.2
0.0
-0.2
0.0
0.5
1.0
1.5
rr
(c)
FIG. 5. The quasi-free electron energy V0 (ρ) in (a) H2 , (b) D2 and (c) O2 – determined from
eq. (20) using data in Figs. 3 and 4 (as well as data at other densities and temperatures that
are not shown for brevity) – plotted as a function of reduced number density ρr . In all cases, (•)
represents data obtained at noncritical temperatures above the critical temperature (i.e., Tr > 1),
(, N, H) represents data obtained at noncritical temperatures where Tr < 1, and (◦) represent
data obtained near the critical temperature (i.e. Tr ≈ 1.02). The lines are the optimal isotropic
local Wigner-Seitz calculation. See text for discussion.
22
(a)
(b)
0.7
5
0.
1
0.1
1
0.2
140
140
100
80
90
40
140
140
120
100
90
70
128
90
70
50
30
3.20
20
30
40
3.15
60
70
60
0.
11
80
0.
0.
127
180
90
41
.0
0
1
0.0
5
.5
5
.9
0
1
31
.4
0
0.6
0.5
0.4
0.5
.6
5.5
5
5.4
5.
8
0.
.8
0
1
.0
0
0
126
Bk / e
0.8
125
200
200
50
3.10
124
120
)K(
0.7
5.5
3.15
3.25
6.0
0.9
1
.1
220
.0
160
0.8
1
0.11
.2
0
8.0
0.9
0.4
.0
0.5
5
3.10
123
3.30
1
1
3.20
5.3
0
0.0
Bk / e
5.5
5.4
0.3
Bk / e
3.15
9.0
3.25
)K(
)K(
3.20
0.4
3.25
(c)
3.30
7
0.
3.30
123
3.10
124
125
s (Å)
126
s (Å)
127
128
123
124
100
0
10
90
125
126
127
128
s (Å)
FIG. 6. Contour plots showing the percent average deviation of the pressure calculated from eq.
(14) in comparison to experimental thermodynamic pressures for O2 at low densities up to the
triple point liquid density at temperatures (a) above the critical temperature, (b) near the critical
temperature, and (c) below the critical temperature as a function of the Lennard-Jones potential
parameters ϵ/kB and σ. The solid point represents the area of overlap among the three panels.
See text for discussion.
23
(a)
(b)
0.69
0.69
8.5
9.5
5.5
6.5
0.68
4
7.5
4.5
5.5
8.5
9.5
4.5
5
6.5
9.5
8
h
4.5
3.5
3
5
8 7.5
4.5
3.5
6
6.5
8
7
5
6.5
9 8.5
6.5
5.5
0.57
0.58
5
0.59
7.5
8.5
4.5
0.66
0.56
4.5
0.67
5.5
7 6.5
9
6
0
h
0
0.67
8.5
7
7
5.5
13
11
10
4
0.68
5
12
10.5
6
7.5
8
6
9
11
10
9
7
13
12
10.5
0.66
0.60
10
0.56
A| (Å)
9
6
5.5
7
5
4
8
0.57
0.58
0.59
0.60
A| (Å)
|
|
FIG. 7. Contour plots showing the percent average deviation of the quasi-free electron energy V0 (ρ)
calculated from eq. (1) in comparison to V0 (ρ) in D2 obtained from field enhanced photoemission
(cf. Fig. 5) at (a) a noncritical temperature (i.e., Tr = 1.253 and (b) near the critical temperature
(i.e., Tr = 1.018) as a function of the zero kinetic energy electron scattering length A and the
phase shift η0 . The solid point represents the area of overlap among the two panels. See text for
discussion.
24