1.4929455

Computational study on Kerr constants of neutral and ionized gases
M. Sato, A. Kumada, and K. Hidaka
Citation: Appl. Phys. Lett. 107, 084102 (2015);
View online: https://doi.org/10.1063/1.4929455
View Table of Contents: http://aip.scitation.org/toc/apl/107/8
Published by the American Institute of Physics
APPLIED PHYSICS LETTERS 107, 084102 (2015)
Computational study on Kerr constants of neutral and ionized gases
M. Sato,a) A. Kumada, and K. Hidaka
Department of Electrical Engineering and Information Systems, The University of Tokyo, 7-3-1 Hongo
Bunkyo-ku, Tokyo 113-8656, Japan
(Received 1 June 2015; accepted 12 August 2015; published online 25 August 2015)
In order to quantitatively examine the measurement capability of Poisson’s field using electro-optic
Kerr-effect (EOKE), Kerr constants of neutral molecules and ions are examined by means of first
principle calculations. We have systematically computed Kerr constants of neutral molecules and
ions of several molecular symmetry groups, with consistent theory level and basis sets. Computed
Kerr constants of neutral molecules (N2, CO2, SF6, and CF3I) ranging across two orders of magnitudes are within 50% error of the experimental values, which are comparable to the scattering
between experimental values itself. The results show that SF6 has smaller Kerr constant due to its
high molecular symmetry compared to those of N2 and CO2. In contrast, CF3I has large Kerr constant
due to its permanent dipole. Computed Kerr constants for anions are larger by two orders of magnitude than those of neutral molecules, probably due to the shielding effect. For cations, the opposite
holds true; however, due to anisotropic polarizability, computed Kerr constants for some cations are
comparable to neutral molecules, while others show smaller values. The ratio of Kerr constants of
ions to those of neutral molecules are at most 102; EOKE is valid for measuring electric field in
C 2015 AIP Publishing LLC.
weakly ionized gas whose ionization degree is smaller than 103. V
[http://dx.doi.org/10.1063/1.4929455]
One of the most fundamental parameters required to
understand discharges in gas is the electric field profile during discharges (Poisson’s field), which is different from the
static electric field (Laplace field) due to the existence of
electrical charges. Therefore, measurement of Poisson’s field
has been desired. It is well known that electric field in liquid
has been measured by electro-optic Kerr effect (EOKE).1
However, it was assumed that EOKE cannot be applied to
measure electric field in gas, since the number density of gas
molecules is smaller by three orders of magnitude than that
of liquid. Despite the fact, recently, we have developed a
high voltage measuring apparatus based on EOKE in gas and
measured the Laplace field.2,3 Since electric field measurement by EOKE has multiple advantages such as high temporal and spatial resolution, and low disturbance by
measurements, it is desirable to apply this technique to measure the Poisson’s field.
In this research, in order to quantitatively examine the
measurement capability of Poisson’s field using EOKE, Kerr
constants of molecules and ions were computed by means of
quantum chemical calculation.
P
Under the external electric field
a;n Ea ðxn Þ ðV=mÞ
whose angular frequency is xn and the direction is a, the
dipole moment of a molecule can be written as4
l0a dðxr
¼ 0Þ þ aab ðxr ; x1 ÞEb ðx1 Þ
X
1
b ðxr ; x1 ; x2 ÞEb ðx1 ÞEc ðx2 Þ
þ
2! x1 ;x2 abc
1 X
þ
c ðxr ; x1 ; x2 ; x3 Þ
3! x1 ;x2 ;x3 abcd
l a ðx r Þ ¼
E b ðx 1 ÞE c ðx 2 ÞE d ðx 3 Þ þ ;
a)
Electronic mail: sato@hvg.t.u-tokyo.ac.jp
0003-6951/2015/107(8)/084102/5/$30.00
(1)
where l0 (Cm), aab (Cm2/V or C2m2/J), babc (Cm3/V2 or
C3m3/J2), and cabcd (Cm4/V3 or C4m4/J3) are permanent
dipole, polarizability, first hyperpolarizability and second
hyperpolarizability tensors, respectively, and xr is the angular frequency of polarization. It should be noted that there
are different definitions for polarizability and hyperpolarizabilities5 such as
1~
1
0
la ¼ la þ e0 ~a ab Eb þ b abc Eb Ec þ ~c abcd Eb Ec Ed þ ;
2!
3!
(2)
or
^ Eb Ec þ ^c Eb Ec Ed þ Þ; (3)
la ¼ l0a þ e0 ð^a ab Eb þ b
abc
abcd
where e0 (C/(Vm)) is the vacuum permittivity. The angular
frequencies are omitted for simplicity. The relations among
different definitions are as follows:
aab ¼ e0 ~a ab ¼ e0 ^a ab ;
~ ¼ 2! e0 b
^ ;
babc ¼ e0 b
abc
abc
(4)
cabcd ¼ e0~c abcd ¼ 3! e0^c abcd :
Relations between different conventions for hyperpolarizabilities are discussed in detail in Refs. 6 and 7. We will adopt the
convention defined in Eq. (1), which is often used to describe
polarizabilities and hyperpolarizabilities of molecules.4,8,9 For
DC-Kerr effect, the terms that one should consider is
permanent dipole, frequency dependent polarizability
aeab aab ðxe ; xe Þ, static polarizability aab aab ð0; 0Þ, and
frequency dependent hyperpolarizabilities, babc ðxe ; xe ; 0Þ
and dabcd ðxe ; xe ; 0; 0Þ, where xe denotes the angular
frequency of the laser. Making use of quantum chemical
calculation, these values can be computed either by
107, 084102-1
C 2015 AIP Publishing LLC
V
084102-2
Sato, Kumada, and Hidaka
Appl. Phys. Lett. 107, 084102 (2015)
perturbative approach10 or by Finite Field method11 at
Hartree-Fock (H-F) level, second order Møller-Plesset (MP2)
level12 and density functional method (DFT). It is known that
DFT methods can reproduce second hyperpolarizability computed with accurate electron correlation techniques for various
small molecules with much less computational cost.9,13 The
theoretical expression of molar Kerr constant in terms of the
above mentioned values is derived by Buckingham in his
seminal work.4 The expression in SI units can be found in
Ref. 14. Molar Kerr constant mK (m5/(V2 mol)) is defined as
mK
ðn2
6nðnk n? ÞM
;
2
þ 2Þ ðer þ 2Þ2 dE2
(5)
where n and er are the isotropic refractive index and relative
permittivity of the gas, respectively, (nk n?) is the refractive index difference for light polarized parallel and perpendicular to the electric field, and d (kg/mol) and M (kg/m2 are
density and molar weight of the gas, respectively.15 Note
that n and er goes to unity in dilute gas. On the other hand,
Kerr constant B (m/V2), which is used in measurement of
electric field in liquid1 and gas2,3 is defined as
B
ðnk n? Þ
;
k0 E2
(6)
where k0 (m) is the vacuum wavelength of laser. Thus, if B
is given, the electric field can be evaluated by measuring the
phase shift between the light field polarized parallel and perpendicular to the applied electric field. Taking into consideration Eqs. (5) and (6), for axisymmetric molecules, the
expression of molar Kerr constant derived by Buckingham
can be simplified as
"
2 #
N
2l0 b aeaniso aaniso aeaniso l0
þ
B¼
þ
cþ
2
5kT
6e0 k0
3kT
5ðkT Þ
N
½K1 þ K2 þ K3 þ K4 ;
6e0 k0
(7)
where N (1/m3), k (J/K), and T (K) are the number density of
gas molecules, Boltzmann constant, and temperature,
respectively, and1 the following relations hold: aaniso ½ð3aab
aab aaa abb Þ=22 ;
b ðbk b? Þ=3;
bk ðbzaa þ baza
þ baaz Þ=5; b? ðbzaa 3baza þ 2baaz Þ=5; c 3ðck c? Þ=2,
ck ðcaabb þcabba þ cabab Þ=15 and c? ð2cabba caabb Þ=15,
where Einstein summation convention is applied. The z axis
corresponds to the axis of molecular symmetry. Since the
expression of molar Kerr constant is derived by classical
statistical mechanics, averaging over all orientation of a
molecule with a Boltzmann-type weighting factor, Kerr
constants depend on temperature, except for spherically symmetric molecules.
Prior to calculating Kerr constants of neutral molecules
and ions, we have estimated the charged species produced by
discharges. Rate coefficients were obtained with Bolsigþ.16
The electron collision cross section database in Refs. 17–21,
22, 23, and 24 were used for N2, CO2, SF6, and CF3I, respectively. The rate coefficients for inelastic collisions of N2 and
SF6 are shown in Fig. 1. It is seen in Fig. 1 that there are
excitation and ionization for N2, whereas electron attachment
does not appear in the figure. Thus, the majority of charged
molecules produced in N2 will be Nþ
2 . Likewise, we decided
þ
to consider COþ
2 for CO2, SF5 ; SF6 ; SF5 , and F for SF6
þ
and CF3I and I for CF3I as primary ions. Though most
types of collisions lead to excitation, de-excitation rate is
usually sufficient especially in atmospheric pressure.25 Thus,
effects of electronically excited species are neglected. In
addition to reactions between electrons and molecules, there
are molecule-molecule, radical-molecule, ion-molecule reactions, and so on.26 Although these reactions may be essential
for discharges in gas mixtures,27 most ions produced by
secondary reactions are neglected and ruled out from our
computation of Kerr constants.
Geometry of ions and neutral molecules was optimized
with Gaussian 09 package28 at ground state electron configuration. For open-shell systems, unrestricted methods were
used. Becke 3-Parameter, Lee, Yang, and Parr (B3LYP)
DFT level and MP2 ¼ FULL level of theory was selected. In
general, it is desirable to use large basis set to calculate high
order polarizabilities. For a prominent example, t-aug-ccpV5Z basis are used to calculate the fourth-order hyperpolarizability of Argon.8 However, due to limitation of
FIG. 1. Rate constants of inelastic collisions between electrons and N2 and
SF6.
084102-3
Sato, Kumada, and Hidaka
Appl. Phys. Lett. 107, 084102 (2015)
computational resources, doubly augmented correlation on
all atoms (daug-cc-pVTZ)29 was chosen as a basis set. In
order to enable comparison of the computed values, diffuse
functions were added even when computing neutral molecules and cations. Preliminary calculation showed that basis
set errors of N20 s second-order polarizability was less than
2% when daug-cc-pVTZ was used. Since iodine has large
atomic number, so as to reduce the computational cost and to
account for relativistic effect, we used aug-cc-pVTZ basis
with small-core relativistic Pseudo-Potential (aug-cc-pVTZPP),30 as an exception. Computed bond lengths and angles of
each molecule were in good agreement with experimental
values and Couple Cluster calculations;31–36 most errors
were within 0.02 Å for bond length and 1 for angles, respectively. Static and frequency dependent polarizability,
frequency dependent first and second hyperpolarizability of
neutral molecules and ions were calculated with Gaussian 09
package at the same level or theory as geometry optimization. The optimized geometry obtained at B3LYP level was
chosen for the input of atomic coordination. Since all molecules have Cn axis of molecular symmetry, where n is larger
than 2, Kerr constants were calculated by Eq. (7). The computed values are shown in Table I.
As shown in Table I, static polarizabilities of neutral
molecules, anions, and cations calculated at B3LYP level are
in reasonable agreement with those calculated at
MP2 ¼ FULL level and other post H-F methods, except for
Nþ
2 ; for most molecules and ions, the difference between
them are within 5%. For neutral molecules whose experimental static polarizabilities are available, excellent
TABLE I. Computed static and frequency dependent polarizability, hyperpolarizabilities, and Kerr constants of molecules. Kerr constants were calculated at
T ¼ 300 K, k0 ¼ 632.8 nm: wavelength of He-Ne laser. The single digit number “0” is determined from the molecule’s point group.
Molecule
Point
group
1040aisoa
(Cm2/V)
1040 aeiso a
(Cm2/V)
1080 aaniso aeaniso
(C2m4/V2)
1050b
(Cm3/V2)
1061c( ¼ K1)
(Cm4/V3)
1061K2
(Cm4/V3)
1061K3
(Cm4/V3)
1061K4
(Cm4/V3)
1019B
(m/V2)
N2 (calc.)
N2 (expt.)
D1h
D1h
1.97, 1.93b
1.94e
2.00, 1.94c
1.97e
0.604
0.599e
0.00
0
0.00
0
2.92
2.90
0.00
0
Nþ
2 (calc.)
Nþ
2 (expt.)
CO2 (calc.)
CO2 (expt.)
COþ
2 (calc.)
SF6 (calc.)
SF6 (expt.)
D1h
D1h
D1h
D1h
D1h
Oh
Oh
2.39, 0.00b
2.65 6 0.06i
2.86, 2.93b
2.93e, 2.89j
2.46, 2.17b
5.27, 5.02b
5.06l
1.97
1.55
0.00
0.756, 0.686c, 0.701d
0.9e, 0.792f1
0.642 6 0.0075 f2
3.00, 6.71h1, 7.94h2
0.00
7.48
0.00
2.67
2.55e
5.2g
3.27
2.90, 2.91c
3.26e
2.58
5.32
4.66
6.15e
7.35
0.00
0
0.00
0
0.00
0.00
0
0.00
0
0.00
0.00
0
22.5
29.7e
3.55
0.00
0
0.00
0
0.00
0.00
0
17.1
23.8e
2.6
0.928
1.2m
SFþ
5 (calc.)
SF
6 (calc.)
SF
5 (calc.)
F (calc.)
CF3I (calc.)
CF3I (expt.)q
CF3Iþ (calc.)
I (calc.)
D3h
Oh
C4v
4.19, 4.04o
8.61, 8.25o
7.42, 7.06o
2.99, 2.78p
8.63, 8.23o
7.73r, 9.89s
7.56, 7.06o
10.5, 10.9p
4.23
9.08
7.66
3.47
8.89
0.00
0.00
5.14
0.00
8.45
0.00
0.00
0.04
0.00
0.981
0.992, 0.748c
5.6e, 1.16k, 0.831f2
0.110
1.28
1.49 6 0.1m
0.985 6 0.007, 0.993 6 0.009n
0.523
105
20.1
67.5
10.8
0.00
0.00
33.1
0.00
47.3
0.345
0.00
24.8
0.00
40.8
0.00
0.00
6.75
0.00
106
7.78
11.5
20.6
0.00
0.595
0.00
2.92
45.1
29.7
0.00
99.5
0.00
166
0.00
0.632
76.5
37.8
49.1
149
189t
217
32.8
a
C3v
C3v
C3v
aiso ¼ aaa/3 with Einstein summation convention.
31
þ
National Institute of Standards and Technology (N2: MP2/aug-cc-pVDZ, Nþ
2 :B2PLYP/ aug-cc-pVTZ, CO2: MP2/aug-cc-pVDZ, CO2 : B2PLYP /6-31G).
40
2
For Nþ
,
static
polarizability
was
computed
at
HF/daug-cc-pVTZ
and
MP2/daug-cc-pVTZ
level
additionally;
HF/daug-cc-pVTZ:
0.645
10
(Cm
/V)
and
2
MP2/daugcc-pVTZ: 3.24 1040.
c
Sekino and Bartlett.37 (CCSD(T) with polarization consistent basis set recommended by Sadlej, 644.3 nm, c: EOKE. The relations between atomic units and
SI units are 1 a.u. ¼ 1.649 1041 (Cm2/V) and 1 a.u. ¼ 6.235 1065 (Cm4/V3) for a and c, respectively.)
d
Gubler and Bosshard.5 (Value in the table is converted through Eq. (4). k0 ¼ 1064 nm, c(3x; x, x, x): third harmonic generation (THG).)
e
Buckingham et al.15 (N2: 299 K, CO2: 301 K, k0 ¼ 632.8 nm).
f
Ward and Miller38 (f1: THG, f2: c(2x; x, x, 0): dc-induced second harmonic generation (dc-SHG), k0 ¼ 644.3 nm).
g
Breazeale39 (T ¼ 293 K, k0 ¼ 650 nm).
h
h1: This work, czzzz: 6.71 (k0 ¼ 632.8 nm), h2: Tarazkar et al.13 (EOKE, czzzz, MCSCF/t-aug-cc-pV5Z, k0 ¼ 1101 nm).
i
McCormack et al.40 (Inferred value: see Ref. 40 for details.)
j
Spackman.41 (These values from analysis of several different experiments.)
k
Ward and Elliott.42 (THG, see Ref. 37 for conversion between conventions.)
l
Watson and Ramaswamy.43 See also Ref. 44. (1 (Å3) ¼ 1.113 1040 (Cm2/V)).
m
Buckingham and Dunmur45 (T ¼ 300 K, k0 ¼ 632.8 nm, 1 esu (statvolt2 cm5) ¼ 1.238 1025 (Cm4/V3)).
n
Shelton and Mizrahi.46 (dc-SHG, k0 ¼ 620 and 650 nm, see Ref. 46 for details.)
o
MP2 ¼ full/daug-cc-pVTZ, I: aug-cc-pVTZ-PP (B3LYP/daug-cc-pVTZ, I: aug-cc-pVTZ-PP).
p
Bichoutskaia and Pyper (MP2, see Ref. 47 for basis sets).
q
Dipole moment of CF3I is obtained by Stark effect36 (l: 3.50 1030 (Cm)) and heterodyne beat apparatus48 (l: 3.07 1030), see also Ref. 49 for experimental details. This work: 3.03 1030. (1 a.u. ¼ 8.48 1030 (Cm), 1 D ¼ 1018 esu cm ¼ 3.336 1030 (Cm).)
r
Giacomo and Smith.48 (Obtained from estimated electronic polarization. See Ref. 48 for details.)
s
Marienfeld et al.50 (Estimated by interpolation between different molecules. See Ref. 50 for details.)
t
Kamiya et al.2 (k0 ¼ 632.8 nm).
b
084102-4
Sato, Kumada, and Hidaka
agreement is seen between experimental and computed values. In addition, frequency dependent polarizabilities of N2
and CO2 show good agreement with experimental values.
Although polarizability of CF3I is not measured directly,
computed dipole moment of CF3I shows good agreement
with experimental values, as shown in the footnote (q) in
Table I.
In order to account for the discrepancy between static
polarizabilities of Nþ
2 computed at different theory levels,
we have additionally computed the static polarizability of
Nþ
2 at H-F level and MP2 level for comparison. As denoted
in the footnote (b) of Table I, MP2 value shows large deviation from other values; moreover, the sign is incorrect. These
results show that electron correlation has a great effect on
polarizability of Nþ
2 as discussed in Ref. 13. This is a wellknown fact for open-shell systems. In addition, it is known
that MP2 level of theory works well for computing polarizability and hyperpolarizabilities of closed-shell system; however, it often fails to compute those of open-shell
system.13,51 In contrast to MP2 values, B3LYP/daug-ccpVTZ level shows reasonable agreement with experimental
values. Unlike Nþ
2 , it is easier to account for electron correla
tion for SFþ
;
SF
5 ; F , and I , whose ground state is a sin5
glet state. This is consistent with the fact that post H-F
results and B3LYP results agrees well for these ions.
Computed values of frequency dependent second hyperpolarizabilities (hereinafter c) of N2 and SF6 are in good
agreement with experimental values and other computational
values, and the errors are within 20%. It is notable that c of
CO2 obtained by Buckingham is larger compared with other
values. This is presumably due to the drawbacks of the determination of second hyperpolarizability by EOKE. Since
molar Kerr constant contains all the contribution from l, a,
b, and c, determination of hyperpolarizabilities by EOKE
requires fitting experimental molar Kerr constants to a function of reciprocal of the absolute temperature, in order to
separate each contribution. However, the contribution of c is
usually small, and thus, the accuracy is greatly reduced.14
This is consistent with the fact that K1 of CO2 is an order of
magnitude smaller than K3, i.e., the contribution of c to
molar Kerr constant is small, while K1 and K3 of N2 are comparable. Regarding SF6, since c is the sole contributor to
molar constant, EOKE measurement should be valid.
Although there are few studies on c of ions, especially for
13
polyatomic ions, c of Nþ
2 is studied in detail by Tarazkar.
þ0
Counterintuitive negative value of N2 s c is discussed in
detail in Ref. 13.
Computed Kerr constants of neutral molecules are in
good agreement with experimental results. CF3I has the largest Kerr constants followed by CO2, N2, and SF6. Large Kerr
constant of CF3I is due to its permanent dipole and small
Kerr constant of SF6 is a result of its high molecular symmetry; K2, K3, and K4 goes to zero. The term K3, which is due
to anisotropy of the polarizability, is dominant for CO2 and
N2. This is probably the result of their anisotropic molecular
geometry.
It can be seen that I shows large polarizabilities. This
is a well-known fact. Iodine has large shielding effect due to
its large atomic radius. Shielding effect is enhanced when iodine is negatively charged, resulting in large polarizability.
Appl. Phys. Lett. 107, 084102 (2015)
Likewise, polarizabilities of anions tend to be larger than
neutral molecules. Thus, anions have larger Kerr constant
than neutral molecules. For cations, the opposite holds true;
it is known that polarizabilities of atomic cations are smaller
by more than an order of magnitude than those of neutral
atoms.52 However, this does not always hold true for molecular cations. In addition, values such as anisotropic polarizabilities or hyperpolarizabilities are not necessarily smaller
compared to those of neutral molecules. As a result, the calculated values of Kerr constants of cations tend to be smaller
or comparable to those of neutral molecules.
These results show that at least with respect to N2, CO2,
SF6, and CF3I, Kerr constants of anions and cations are at
most 102 times larger than those of neutral molecules. When
birefringence induced by ions is negligible compared to that
induced by neutral molecules, EOKE measurement using
Kerr constants of neutral molecules is valid. Taking into
account that Kerr constants are proportional to number density of molecules for dilute gas, EOKE measurement of
Poisson’s field seems to be valid for measuring electric field
in weakly ionized gas whose ionization degree is smaller
than 103.
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