J. Plasma Fusion Res. SERIES, Vol.7 (2006) 314-318

Electron Collision Cross Sections of Mercury
SUZUKI Susumu, KUZUMA Kiyotaka1 and ITOH Haruo1
Department of Physics, Chiba Institute of Technology, Shibazono, Narashino 275-0023, Japan
1 Department of Electrical, Electronics and Computer Engineering, Chiba Institute of Technology, Tsudanuma, Narashino 275-0016, Japan（Received: 6 October 2004 / Accepted: 5 October 2005）

Abstract
In this paper, we propose a new collision cross section set for mercury which revises the original set
summarized by Hayashi in 1989. Hanne reported three excitation collision cross sections (6 3 P0 , 63 P1 , 63 P2 )determined from an electron beam experiment in 1988. As a matter for regret, no attentive consideration wasgiven to combining these three excitation cross sections with the cross section set of Hayashi. Therefore we
propose a new set where these three excitation cross sections are included. In this study, other two excitation
cross sections (61 P1 , 63 D3 ) except for the three excitation collision cross sections (6 3 P0 , 63 P1 , 63 P2 ) are takenfrom the original set of Hayashi. The momentum transfer cross section and the ionization collision crosssection are also taken from Hayashi. A Monte Carlo Simulation (MCS) technique is applied for evaluating
our new cross section set. The present results of the electron drift velocity and the ionization coeﬃcientare compared to experimental values. Agreement is secured in relation to the electron drift velocity for
1.5 Td < E/N < 7 Td and to the ionization coeﬃcient for 400 Td < E/N < 3000 Td, where E/N (V · cm 2 ) isthe reduced electric field, E (V/cm) is the electric field, N (1/cm3 ) is the number density of mercury atoms at0 ℃, 1 Torr, E/N is also equal to 2.828 × 10−17 E/p0 from the relation of the ideal gas equation, p0 (Torr) isgas pressure at 0 ℃, 1 Torr = 1.33322 × 10−2 N/cm2 and 10−17 V/cm2 is called 1 Td. Thus it is ensured that
our new cross section set is reasonable enough to be used up to 100 eV when considering with the electron
drift velocity and the ionization coeﬃcient.
Keywords:mercury, electron collision cross section, excitation collision cross section, Monte Carlo simulation
1．Introduction
Mercury is the only metal kept in a state of liquid
at ordinary temperatures, and has widely been utilized
in electrical equipment such as light sources, switches,
vacuum pumps, thermometers, etc. Mercury, cadmium,arsenic, and lead are regarded as environmental pollutants recently, and mercury is, above all, said to be thegravest pollutant. For this reason, its use is prohibited inindustry. Despite the above, mercury, with which noneof adequate substitutes for a strong emission source ofultraviolet ray can hardly be found, is still a prominent
source of fluorescent light since no adequate substitutefor a strong emission source of ultraviolet ray can befound.
The new collision cross section set is based on theset summarized by Hayashi [1], except with three excitation collision cross sections (63 P0 , 63 P1 , 63 P2 ) basedon the experimental data of Hanne [2]. Remarkable difference is noticed in a portion of the built-up 63 P1 crosssection. Evaluation of the proposed collision cross section set is made by means of the MCS. Concurrently

with this, comparison is made with the experimental
values of the electron drift velocity and ionization coeﬃcient by others [3-9].
A rapid rise is noticed in the momentum transfer
cross section of the mercury in the range from 0.05 eV
to 0.2 eV. Then a monotonoic decrease is made after thecross section reaches a peak value of 2 × 10−14 cm2 [13, 7-9]. The momentum transfer cross section is completely diﬀerent in its shape with respect to Ramsauer
gases such as Ar, Kr and Xe. However in this studydetailed results brought about from the problems of thecross section are not considered as a matter of course,
because the high region of electron energy is to be exclusively dealt with in this paper.
The electron collision cross section set of mercury
has been reported by Rockwood [10], Nakamura [4],Liu [11], Hayashi [1], Sakai [12] and England [9]. In
this paper, the validity of the proposed electron crosssection set is shown by comparing the present MCS calculation with the experimental swarm data.
C 2006 by The Japan Society of Plasma

Science and Nuclear Fusion Research

Corresponding author’s e-mail: susumu.suzuki@it-chiba.ac.jp314

Electron Collision Cross Sections of Mercury

cross section Q ( 10-16cm2)

2．Calculation method
The MCS [13] is carried out in a uniform electricfield where the gap length is 1cm. An electron is released from a center point of the cathode at zero velocity, and 1.0×104 electrons are calculated on their trajectory until they reach the anode. Positions and velocities
of electron are calculated by Newtonian mechanics atevery short duration ∆t. The collision time is also calculated by comparing the total collision cross section
given by the electron energy of the in-flight electron anda random number at every ∆t. The kind of collision isdetermined by the method of Itoh and Musha [14]. After the collision, the electrons restart with an initial energy that is equal to the electron energy possessed justbefore the collision minus the energy lost by the electron in the collision. The energy loss of the electrons incollisions with gas molecules is given below.
(a) Momentum transfer collision ; εeloss = 2m
M (1 −cos ω)ε eV.
where m and M : masses of electron and mercury,respectively, ω : scattering angle of electron to
the direction of electric field, ε : electron energy
possessed just before the collision.
(b) Excitation collision : εexloss = 4.67 eV (63 P0 ),4.89 eV (63 P1 ), 5.64 eV (63 P2 ), 6.70 eV (61 P1 )and 8.86 eV (63 D3 ).
These values are a constant loss for each excitation threshold.
(c) Ionization collision : εiloss = 10.43 eV.
The value indicates the threshold of ionization.
The data of electron behavior are obtained by thesteady state Townsend (SST) sampling in 200 segmentsof slit section ∆x between the cathode and the anode.
The momentum transfer and ionization collisioncross sections of Hayashi [1] are adopted in this study,
because of the existence of the data covering a wide region ranging from 0.1 eV to 1000 eV. The newest ones
adopted about from Hanne [2] are employed as the three
excitation cross sections. However to interpolate a lackof data for ε > 6 eV for with these cross sections, extrapolation is made to attain a higher energy level tothe extent of 1000 eV. As the method of extrapolation,
to be suitable for the experimental values of the electron drift velocity and the ionization coeﬃcient [3-9],
the correction of these cross sections is repeated up toabout 100 eV that is the maximum value of electron energy distribution in the calculation of MCS to obtain theionization coeﬃcient at E/N = 3000 Td. These crosssections for ε > 100 eV are estimated smoothly. The
two excitation cross sections 61 P1 and 63 D3 are employed from the original data of Hayashi [1]. The setof cross sections adopted in this study is illustrated inQm102

10Qi

61P1

1100

63P110-163D363P263P010-2
10100

101102103

energy ε (eV)Fig. 1 Electron collision cross section of mercury. (to distinguish the cross sections, the line kind is indicated as for 63 P1 (broken line), 63 P2 (dotted line)
and 63 D3 (long broken line).)Fig. 1.
Fig. 2 is the comparison between the excitation crosssections of 63 P0 , 63 P1 and 63 P2 of Hayashi (solid line)and Hanne (broken line). The momentum transfer crosssection Qm is plotted in the same figure. By comparing
the two sets of cross sections, it is clear that the threshold regions of the 63 P0 cross section curve agree wellwith each other. However above the peak in the curve,
the curve of Hanne reaches a much deeper minimum atapproximately 4.8 eV than that of Hayashi. The 63 P1

Qmcross sections Q ( 10-16-2cm )

101



010-1

10

63P010-2
4.5

63P2

63P1

55.5

6electron energy ε (eV)Fig. 2 Excitation collision cross sections (solid line :Hayashi, broken line : Hanne).

315

Electron Collision Cross Sections of Mercury

Table 1 Occurrence number of the excitation.

cross section curve of Hanne begins to rise upward andalmost vertically, starting from 4.9 eV. After reaching
the peak, the curve again makes a rise in a direction of
the second peak, starting from 5.4 eV. Thus the maximum value is obtained as 5.5 eV. On the other hand,the 63 P1 cross section curve of Hayashi makes slower
increase than the other curves. After that, the curve approaches a nearly constant value. The 63 P2 excitation
cross section curves of Hanne are smaller than that ofHayashi in all regions.

Authors
Number of Excitation

3.1 Comparison of excitation collision cross sectionof mercury

Spatial distributions of excitation are calculated by
MCS using each excitation collision cross section of
Hayashi and Hanne. Figure 3 shows an example with
E/p0 = 10 V/cm · Torr, p0 = 1 Torr. In this figure, theposition of the excitation collision occurring in thethree- dimensional space is projected perpendicularly
onto the two-dimensional plane included in the axis
connecting the centers of the cathode and anode. Theexcitation collision starts on a position at a distance of
0.467 cm from the cathode corresponding to the threshold 4.67 eV of 63 P0 and existence of a peak is noticed
at 0.5 cm to be continued up to approximately 0.6 cm.
Building-up of the second group of excitation collisions
occurs beyond the position of 0.9 cm. The first peakis attributed to the peak for 63 P0 , whereas the highest
peak at 0.5 cm complies with the peak for 63 P1 . The

distance from center of electrodey (cm)2185
2002
1894376

3.2 Electron drift velocity

Figure 4 shows the comparison between the present
calculated values of electron drift velocity and the experimental values reported by other researchers [3-5].
Although, the present value diﬀers vastly from most of
the experimental values obtained for E/N < 1.5 Td, it
is necessary to consider the influence of thermal motion of the electrons due to energy transfer from mer-(b) Hanne

106

0.0

0.0

-0.5

-0.5

300

300

200

200

100

100

0.2
0.4
0.80.6
distance from cathode x (cm)1.000.0

drift velocity vd (cm/s)

number of collisional excitation

9503305
12143770.5(a) Hayashi

00.0Hayashi

diﬀerence between the two cross sections appears distinctly in the spatial distribution. The peak of Hanne is
1.5 times higher than that of Hayashi.
The total number of excitations using the cross sections of Hanne is 4377, whereas that using the data ofHayashi is almost of the same value, 4376. However
the numbers of excitations with the Hanne data are of
the value of 950, 3305 and 121 for the 63 P0 , 63 P1 and63 P2 excitations, respectively, whereas those from theHayashi data are 2185, 2002 and 189. These numbers
are listed in Table 1. No generation of other excitation
or ionization can be seen under such conditions. The energy loss caused by these three excitations is 1.04 times
greater using the data of Hanne than that using the dataof Hayashi.

3．Results and discussion

0.5

63 P063 P1
63 P2
total

Hanne

0.2
0.4
0.8
0.6
distance from cathode x (cm)1.0105104 -1
10Fig. 3 Spatial distributions of excitation obtained by using
cross sections shown in Fig. 2 (E / p0 = 10 V/cm・Torr, p0 = 1 Torr).

Elford(573.2K)
Nakamura(520K, 68.7Torr)
Nakamura(608K, 502Torr)
McCutchen(350Torr)
Present

100
E / N (Td)101

Fig. 4 Comparison between present result and experimental values of the drift velocity reported by otherresearchers.
316

Electron Collision Cross Sections of Mercury

101
100
10-18
10107

Overton and Davies(1968):Exp.
Davies and Smith (1965):Exp.
Badareu and Bratescu (1944):Exp.
Present values :MCS
αi/N : ionization coefficient
αex/N : excitation coefficient100η= ٨٤100٤20100105104 -1
10101-10106
10

α i /N-10500

1000

1500

2000

E/ N (Td)100

101
E / N (Td)

102

2500

-20
3000

persentage of error η %102

αex /N102

Sakai(BE A )Nakamura(BE A ,520K)
Nakamura(BE A ,608K)
R ockwood (BE A )L iu(MCS)
Present

α i / N, αex / N (10-17 cm2 )drift velocity vd (cm/s)

mean energy <ε> (e V )103

Fig. 6 Obtained ionization and excitation coefﬁcients.

103

than the present values. Sakai et al. [12] obtained results similar to those of the present study, and it is interpreted that the discrepancy is due to Penning ionization
by metastable atoms of mercury in the experiment. Thisis a significant eﬀect at such low E/N. Penning ionization is not included in our MCS.
Furthermore, the calculated excitation coeﬃcient
αex /N is shown for reference in Fig. 6.

Fig. 5 Comparison between present results and othercalculation values of mean energy and drift velocity.

cury atoms. Furthermore Nakamura et al. [15] mademention of the existence of the pressure dependence inthe drift velocity of the mercury. Thus increase was noticed with the value of the density ratio of the mercury
molecules to the mercury atoms with respect to the gas
temperature [4], that is, they described the gas pressure varies depending on the gas temperature. In thepresent calculation, no consideration is taken for thisthermal motion in the collision process. Meanwhile inE/N  1.5 Td, the calculations are closely consistent
with the values obtained from the experiments.
Figure 5 shows the comparison between the present
results and other calculations of the electron mean energy and drift velocity [10 – 12,15]. Although the meanenergy of the present results is lower than the values
previously obtained in the region E/N < 3 Td, it is almost identical with the other calculations for E/N 
3 Td. This diﬀerence originates from the cross sections
used. However, the drift velocity is in agreement withthe other results for 0.1 Td < E/N < 2000 Td.

4．Conclusion
A new electron collision cross section set for mercury was based on the cross section sets of Hayashi andHanne is investigated. The evaluation of the new collision cross section set is made by MCS, and comparisonof the electron drift velocity with the ionization coeﬃcient is also made. The results are in agreement with thepreviously reported experimental results within a few
percent. Thus it is ensured that the new cross section
set is reasonable enough to be used with the energies up
to 100 eV, when considering the electron drift velocity
and the ionization coeﬃcient.
Acknowledgements
The authors of this paper wish to express their sincere gratitude to late Dr. M. Hayashi of Gaseous Electronics Institute, Professor Emeritus of Nagoya Instituteof Technology who was kind enough to continuously
encourage the authors.

3.3 Ionization coefficient

Figure 6 shows the ionization coeﬃcient αi /N in the
reduced electric field ranging from 200 Td to 3000 Td
calculated by MCS for comparison with experimentalresults. The present data are consistent with those ofOverton et al. [6] covering a wide region, with diﬀerences less than ±5 % for E/N > 400 Td. However adistinct diﬀerence arises for E/N < 400 Td, where thevalues of Overton are approximately 1.4 times higher[1][2][3][4]317

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Electron Collision Cross Sections of Mercury

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