Collision probability for momentum transfer of low energy electrons in nitroge

Holt, E. H.

Noon, J. H.

Quine, J. P.

English

NASA Technical Reports Server

https://ntrs.nasa.gov/search.jsp?R=19650011862 2020-03-16T23:47:36+00:00Z
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Rensselaer Polytechnic Institute 
Troy, New York 
MICROWAVE MEASUREMENT OF THE 
PROBABILITY OF COLLISION OF IOU 
ENERGY ELEXTRONS IN NITROGEN 
PIASMA RESEARCH LABORATORY 
TECHNICAL REpoT(T NO. 17, November 1964 
This work was partially supported by the National Aeronautics and Space 
Administration under grant no. NsG 48-60. 
This report is substantially the same as paper G5 of the lTth Annual 
Gaseous Electronics Conference, Atlantic City, October 1964. 
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Introduction 
The probability of coll ision for momentum transfer  (P ) of low energy 
electrons (4  1 e V )  i n  nitrogen has been determined as a function of energy 
by a number of workers. 1-7 However, results of different  groups based on 
microwave conductivity measurements are not i n  agreement, although i n  one case 
(Phelps e t  a l )  values of 
mobility and diffusion coefficients i n  D.C. f i e ldse5  
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6 
Pc obtained are consistent with those derived from 
I n  the present work 
both microwave conductivity and electron temperature have been measured 
d i rec t ly  and values of P calculated as a function of electron velocity. If  
a Maxwellian velocity dis t r ibut ion i s  assumed fo r  the electrons, the results 
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fo r  Pc agree with those of Phelps e t  al. However, electron-temperature 
measurements as a function of time i n  a decaying nitrogen p l a sm indicate that 
a Maxwellian dis t r ibut ion does not exist ear ly  i n  the afterglow. It i s  believed 
t h a t  t h i s  f a c t  i s  the  explanation of the discrepancy between previous results 
obtained by other workers using microwave techniques without d i rec t  measure- 
ment of the  electron temperature. 
6 Phelps e t  a1 used a resonant cavity technique i n  which the s h i f t  i n  
resonant frequency and the change i n  conductance of the cavity were measured 
during the decay of a nitrogen discharge. A 10 cm pulsed magnetron w a s  used 
t o  break the gas down i n  the cavity periodically and microwave measurements 
were carried out late i n  the afterglow at  times ranging from 200 p seconds t o  
1. 
2. 
3-  
4. 
5- 
6. 
7. 
J. S. Townsend, V. A. Bailey, P h i l .  Mag. 42, 873, (1921). 
Bruche, Ann. Physik 82, Ser. 4, 912, (1927). 
C.  E. Normand, Phys. Rev. 35, 1217, (1930). 
C. Ramsauer, R. Kollath, Ann. Physik 4, 91, (1930). 
R. W. Crompton, D. J. Sutton, Proc. Roy. SOC. (London) Ml5, 467, (1952). 
A. V. Phelps, 0. T. Fundingsland, and S. C .  Brown, Phys. Rev., 84, 
J. M. Anderson, L. Goldstein, Phys. Rev. 100, 1037, (1955). 
u 
559, (1957). 
Abstract 
The probability of coll ision for  momentum transfer  (P,) of low energy 
electrons ( 4  1 e V )  i n  nitrogen has been experimentally determined using 
microwave transmission techniques. A bakeable c e l l  constructed of X-band 
waveguide contained the plasma. A nitrogen discharge w a s  produced by a pulsed 
D.C. supply and the phase sh i f t  and attenuation of the microwave signal 
measured as a function of t i m e  i n  the  afterglow plasma. The change i n  elec- 
t ron  temperature with t i m e  w a s  monitored using a gated microwave radiometer. 
Assuming a Maxwellian velocity distribution for  the electrons, Pc w a s  calculated 
as a functicn of electron energy and values obtained agree with those of 
Phelps e t  a1 but disagree with those of Anderson and Goldstein. However the 
radiometer measurements indicate that the electron energy dis t r ibut ion i s  not 
Maxwellian i n  the ear ly  afterglow. This i s  important i n  explaining the dis-  
agreement between values of Pc 
standing the heating e f fec t  of nitrogen metastables i n  the afterglow. 
calculated by earlier workers and i n  under- 
2 
several milliseconds. 
values of the e l ec t r i c  f i e l d  i n  the cavity used i n  determining electron t e m -  
perature. Thermal heating of t he  gas i tself  w a s  a l so  employed t o  change the 
electron temperature- Calculated values of Pc versus electron velocity 
are shown i n  Fig. 1. 
The power incident on the cavity was monitored and 
Anderson and Goldstein 7 used microwave transmission at  9.4 Gc/s through 
a nitrogen discharge i n  a glass vessel located i n  a section of waveguide t o  
measure the phase shif t  and attenuation of a 0.5 m i l l i w a t t  "sensing" wave. 
The electron-molecule col l is ion frequency 1) 
w i t h  nitrogen molecules w a s  calculated, assuming tha t  50 p seconds after a 
2 p second D.C. voltage pulse the plasm had become isothermal a t  /v 300°K. 
I n  addition, at  different times i n  the afterglow an 8.6 Gc/s square-pulsed 
"disturbing" wave w i t h  a power level & 0.5 watts w a s  propagated simultaneously 
w i t h  the  sensing wave. From these microwave conductivity measurements, together 
w i t h  the assumption t h a t  the electron temperature 
re la ted  t o  the amplitude of the incident disturbing wave, values of L' versus 
Te were calculated. From these figures Pc as a function of electron velocity 
v w a s  derived (see Fig. l), using the re la t ion  
fo r  room temperature electrons m 
Te i n  the plasma could be 
m 
where po i s  the gas pressure, normalized t o  O°C. 
8 Formato and G i l a r d i n i  using a plasma microwave conductivity technique 
similar t o  that of Anderson and Goldstein, but measuring electron temperature 
as w e l l ,  showed that electron temperatures are much higher than room temper- 
ature as late as 200 microseconds in the  afterglow. They suggested t h a t  values 
8.  D. Formato, A. Gilardini, Fourth Intern. Conf. on Ioniz. Phen. i n  Gases, 
Uppsala, 0959)  
3 
f o r  Pc obtained a t  post-discharge times of order 50 microseconds where 
electrons were assumed t o  be a t  room temperature would be unreliable. 
ever the  temperature range covered by Formato and Gilardini w a s  limited and 
they had insuff ic ient  data t o  calculate values of P as a function of 
e l e  ctron energy. 
How- 
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Work has also been carried on i n  t h i s  laboratory which i s  relevant t o  
9 the  problem. 
fo r  electron temperature measurements i n  nitrogen afterglow plasmas and Ajello 
has reported calculations of 
conductivity and electron temperature. 
here involved varying the discharge conditions used t o  excite the nitrogen 
plasma -- notably the discharge current and pulsing t i m e  of the voltage applied 
t o  the  discharge electrodes. 
conflicting r e su l t s  on electron temperature relaxation times i n  nitrogen after- 
glow plasmas obtained by Formato and Gilardini 
Stotz has used the gated radiometer developed i n  t h i s  laboratory 
10 
Pc based on combined measurements of microwave 
The extension of t h i s  work reported 
This i s  of par t icular  in te res t  i n  view of the 
a 11 and by Mentzoni and Row. 
Experimental Details 
The apparatus used i n  the  present work i s  shown i n  Fig. 2. A nitrogen 
discharge was in i t i a t ed  i n  a bakeable constructed of X-band waveguide 
by a voltage pulse which could be varied i n  length from 5 t o  50 microseconds 
and repeated a t  100 cps rate. A 9.15 Gc/s s ignal  at  a low power l eve l  
( 5 5 m W )  w a s  propagated through the  discharge and a microwave bridge tech- 
nique used f o r  attenuation and phase s h i f t  measurement. I n  the late afterglow 
9. K. C. Stotz, Rensselaer Polytechnic Ins t i tu te ,  Plasma Research Lab. 
TR9, (1963). 
10- J. Ajello, NASA CR 104, (1964). 
11. 
12. 
M. H. Mentzoni, R. V. Row, Phys. Rev. 130, 2312, (1963). 
K. C. Stotz, NASA TN D-226, (1963); E. H. Holt, K. C. Stotz, Rev. Sci. 
Inst .  34, 1285, (1963). 
4 
where the plasma attenuation i s  very low, a special  technique i s  required fo r  
measuring small changes i n  a large s ignal  as a function of t i m e .  As shown i n  
Fig. 2, a r e s i s to r  network and a pair of crystals, one of reversed polarity, 
were used i n  a balancing arrangement t o  bring the signal l eve l  down t o  a suit- 
able value fo r  observation on the oscilloscope when the plasma attenuation 
was small. 
t h i s  means. 
Plasma attenuations as low as 0.01 db were eas i ly  detectable by 
The electron temperature was measured separately a t  post-discharge times 
i n  the afterglow up t o  300 microseconds using a gated radiometer which has been 
described p r e ~ i o u s 1 y . l ~  Typical resul ts  are shown i n  Figs. 3, 6 and 7. 
Probability of Collision 
The electron-molecule coll ision frequency L) can be determined from 
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the  r a t i o  
of the plasma. A knowledge of the electron density ne i s  not required, but 
the electron velocity distribution must be known. Assuming a Maxwe1lia.n dis- 
t r ibu t ion  and following the general theory of Margenau14, w e  may w r i t e  
CT / E  r i  of the  real and imaginary parts of the microwave conductivity 
where 
2 u = m v / 2  
m = the  electron mass 
w = k T e  
w = the angular frequency = 2x x 
6 As done by Phelps e t  a1 the low pressure 
9-15 Gc/s 
approximation may be used, since 
13. W. C. Taf t ,  K. C. Stotz, E. H. Holt, IEEE Trans. IM-12, 90, (1963). 
14. H. Margenau, Phys. Rev. 69, 508, (1946). 
f 
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. Y /co 4 0.01. It i s  known that th i s  in tegra l  can be represented approximately m 
by a polynomial of the first kind.14 Using a re la t ion  of the form 
the experimental data may be used t o  evaluate the coefficients al, a2 and a3. 
Provided that a suff ic ient  range of electron energies has been measured 
experimentally, these coefficients can then be used t o  obtain values fo r  
a m ( u )  and P,(u) from the relations 
and 
Fig. 4 shows P (u) versus the electron velocity (v) calculated f o r  a 
number of different  gas pressures from experimentally measured values of 
( rr/r. ) over a range of measured electron temperatures Te 
35OO0K t o  5OO0K. It can be seen tha t  the r e su l t s  a re  more i n  agreement with 
those of Phelps e t  a1 than those of Anderson and GoldsteinO7 However the 
va l id i ty  of the main assumption used i n  calculating 
measurements, namely that a Maxwellian d is t r ibu t ion  of electron ve loc i t ies  
ex i s t s  throughout the afterglow i s  highly questionahle. 
shown by a consideration of the electron temperature measurements. 
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from approximately 
1 
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Pc from the microwave 
This point i s  c lear ly  
Electron Temperature Decay 
If one assumes that the time required f o r  electrons t o  achi-eve thermal 
equilibrium w i t h  the heavier gas constituents depends primarily on col l is ions 
15. R. D. Crout, J. App. Math., 19, 34, (1940). 
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with neutral  par t ic les ,  it i s  possible t o  estimate a relaxation time fo r  the 
electrons. If we w r i t e  
X = mean fractional energy loss per col l is ion 
U = average electron energy 
- 
e - -  
1 
(ue-ug) = average excess electron energy above that of gas molecules 
then the average r a t e  of energy loss  by col l is ion 
- -  
d -/dt = A (ue-U,) Urn 
may be taken as 2m/M, 5.5(21n/M),~ or  20(21n/M),~~ according t o  different  
workers, depending on whether account is taken purely of e l a s t i c  col l is ion or  
whether exci ta t ion of low-lying rotat ional  levels  of the nitrogen molecule 
are  considered. However an average value f o r  the energy loss  can be obtained 
from the data of Frost and P h e 1 ~ s . l ~  Taking ( A U , ) / N  -1.5 x 10l1 for  a 
range of electron energies f rom 0.03 t o  0.27 eV, where 
molecules/cc -3.2 x 10 p, and p i s  i n  Torr, the r a t e  of decay of electron 
N = the number of 
16 
temperature may be written, using these approximate numerical values, 
(dTe/dt) = - 3 x lo5 p (Te-T ) = d(Te-Tg)/dt 
Q 
and the electron-temperature relaxation time has the form T - 3 / p  where Z is  
i n  microseconds, p is i n  Torr. This expression f o r  the magnitude of 
agrees quite well with that given by Mentzoni and Row,” and with t h e i r  experi- 
mental r e su l t s  using a mildly driven discharge with a pulsing time of a f e w  
microseconds. 
16. J. C. Bowe, Phys. Rev. 117, 1416, (1960). 
17. L. S .  Frost, A. V. Phelps, P h y S -  Rev. 127, 1621, (1962). 
7 
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Figures 3, 6 and 7 show experimentally measured values i n  t h i s  labor- 
a tory of electron temperature versus t i m e  i n  the afterglow fo r  a nuniber of 
d i f fe ren t  gas pressures. It can be seen that i n  the first few microseconds 
of the post-discharge period the electron temperature does drop rapidly, i n  
agreement with the above relationship f o r  7 ,  and also that 7 increases as 
p decreases. 
a maximum and then f a l l i ng  off slowly. 
However the electron temperature rises again with time reaching 
a The electron temperature measurements reported by Formato and Gilardini, 
who used a 10-microsecond pulsing t i m e  fo r  t h e i r  discharge, were not carried 
out a t  times close t o  the termination of the active discharge and they there- 
fore  only observed a slow electron temperature decay. 
possible generation of energetic electrons after the energy source of the dis- 
charge i s  removed, by the ionizing interact ion of metastable nitrogen molecules 
o r  by superelastic col l is ions between electrons and metastables. Electron- 
ion recombination although removing low energy electrons preferentially,  and 
thus increasing the average electron energy, could not explain the observed 
e f f ec t  quantitatively.  Our results,  showing an i n i t i a l  f a l l  i n  temperature 
followed by a r i s e  and then SLOW decay i n  temperature, give additional support 
for  t h i s  hypothesis of metastables. 
state i s  by no means achieved eazly i n  the  afterglow. 
co l l i s ions  are the  source of energetic electrons our r e s u l t s  indicate that 
an i n i t i a l  increase i n  metastable density i n  the first few microseconds of the 
post-discharge period i s  required. 
the source of the energetic electrons, then the cross-section f o r  t h i s  process 
must be highest a t  low electron velocit ies.  No matter which reaction i s  
considered t o  dominate, an equilibrium s i tua t ion  appears unlikely. 
They suggested a 
Moreover they indicate that an equilibrium 
If metastable-metastable 
I f  electron-metastable interact ions a re  
8 
This point must be considered i n  re la t ion  t o  the assumed Maxwellian 
d is t r ibu t ion  function used both i n  equation (2) and i n  the derivation of 
electron-temperature from the radiometer measurements. Figure 5a shows the 
type of discrepancy which occurs when a Maxwellian dis t r ibut ion function i s  
assuned. 
of electron temperature, a s  measured i n  the afterglow at  different  times. 
The microwave conductivity r a t i o  (u /Ci) i s  plotted as a function r 
Since the sane temperature i s  recorded at  two different  times i n  the  after- 
glow, the two values of (CT /%) obtained a t  these times can be cornpazed, and 
are obviously different,  though equation (2) predicts that they should be 
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the same. A s  shown i n  Fig. 5b the  electron density var ies  monotonically w i t h  
time i n  the afterglow fo r  the same range of times i n  which the microwave ' 
conductivity and electron temperature measurements, shown i n  Fig. 5a, were 
carr ied out. 
The temperature estimated from the radiometer reading i n  which the 
assumptions of a Maxwellian dis t r ibut ion and black-body radiation by the plasma 
are  made, m y  well be unreliable early i n  the afterglow where it i s  obvious 
that a non-equilibrium s i tua t ion  exists. 
Discussion 
Our electron temperature measurements confirm the  slow temperature decay 
i n  the nitrogen afterglow reported by Formato and Gilardini8 and indicate that 
temperatures of order 1000% can exist  a t  times of order 50 microseconds i n  
the afterglow. 
lead t o  an overestimate f o r  Pc(u) as can be seen from equation (1). I n  
addition, our temperature measurements indicate t h a t  the basic a s s b p t i o n  of 
a Maxwellian d is t r ibu t ion  throughout the afterglow i s  invalid. 
An underestimate of the temperature of the decaying plasma could 
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If one assumes that the heating which occurs i n  the afterglow is  due 
t o  metastables then the r i s e  i n  temperature a f te r  the initial f a l l  requires 
that the  cross-section for  electron-metastable interact ion i s  highest fo r  
low electron velocit ies.  
s h i f t s  t o  l a t e r  times f o r  lower pressure discharges indicates that the rate 
The fact  that the temperature peak i n  the afterglow 
of 
by 
metastable interact ion i s  influenced by the neutral  gas background. 
The conflicting values of electron-temperature relaxation times obtained 
8 Formato and Gilardini and Menteoni and  ROW,^ emphasize the importance of 
the influence of the discharge pulsing t i m e  on electron heating mechanisms. 
By varying parameters of the dischaxge it i s  possible t o  study t h e i r  e f fec t  
on the formation and r a t e  of decay of nitrogen metastables. Pulsing fo r  
20 microseconds gives a higher temperature i n  the active discharge than pulsing 
f o r  5 microseconds, but it has been found t h a t  increasing the pulsing t i m e  
beyond th i s  has l i t t l e  further effect  i n  increasing the temperature of the 
act ive discharge. Whether the metastable population has had suf f ic ien t  time 
$0 achieve equilibrium, however, i s  a question which has s t i l l  t o  be answered. 
It i s  noticeable (see Fig. 7) that a high current fo r  a short time does not 
cause as great a heating e f fec t  i n  the afterglow plasma as  a smaller current 
for a longer t i m e .  
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ELECTRON ENERGY- Ev 
Fig. 1 Probability of Collision f o r  Momentum 
Transfer of Low-Energy Electrons. 
. 
KLYSTRON 
PHASE 
SHIFTER 
A T T E N U A T O R  
Fig. 2 Schematic D i a g r a m  of the Microwave Bridge Circui t  For 
Measuring Attenuation and Phase Shi f t  of the Plasma. 
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0 5 0  I00 150 
TIME-M ICROSE CONDS 
Fig. 3 Electron Temperature versus Post-Discharge Time I n  
the Afterglow for Several Pressures. 
(CM-TO 
Fig. 4 Collision Probability Obtained Experimentally -- Plotted 
Versus Electron Velocity, Together with Results of Other Workers. 
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j= 0.1. 
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'ELECTRON TEMPERAT-U-RE~K); 
.- 
Fig. 5a Measured Values of the  Conductivity Ratio as a 
Function of Electron Temperature During a Portion 
of the  Post-Discharge Tim. 
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Fig. 5b Calculated Electron Densities During the Same 
Portion of the Post-Discharge Time. 
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T I ME-MIC ROS ECONDS 
_ -  
Fig. 6 Electron Temperature Variation i n  the Early Afterglow. 
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Ti M E-MI CRO S ECON DS 
Fig. 7 Electron Temperature in the E a r l y  Afterglow 
for Vaxious Discharge Conditions. 


Microwave measurement of the probability of collision of low energy electrons in nitrogen technical report no. 17

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