Thermodynamic and transport properties of a cesium plasma

Advanced Energy Conversion, Vol. 3, pp. 19-36. Pergamon Press 1963. Printed in Great Britain

T H E R M O D Y N A M I C A N D TRANSPORT PROPERTIES OF A CESIUM PLASMA L. B. ROBINSON*~" Summary--Very little information is available regarding the properties of a cesium plasma. In this report, thermodynamic and transport properties of partially ionized cesium vapor, or plasma, are determined from theory (supplemented by available experimental results). Such information is essential in predicting efficiencies of the cesium plasma diode converter. Emphasis is placed upon the transport properties of electrons and the effects of neutral atoms on the thermodynamic and transport properties. Cesium vapor, in the region of small fractional ionization, is considered as a non-ideal gas obeying the virial equation of state. P 17"-- [1

R7

B(T)]

+ ~J

Once the second virial coefficient B(T) is determined, all of the required thermodynamic properties are known within this framework. The cesium vapor is considered as composed of atoms with pairwise Lennard-Jones (6-12) interactions. For such interactions, the mutual potential energy V(r)is given by

V(r)=4e[(a-r)12--(a)6] where a is the value of r for which V(r)crosses the r-axis and e is the value of V(r) at its minimum. The parameters (a and e) are obtained by giving an extension in the interpretation of some previously published results. From the perturbation effects of noble gases on the hyperfine spectra of cesium, the parameters in the Lennard-Jones potential have been determined for some noble gas interactions with cesium vapor. Since the parameters are known for the noble gases, a very good estimate can be made for the cesium parameters. A knowledge of a and e for cesium can determine B(T), the second virial coefficients, and derivatives of B(T) with respect to temperature. Thermodynamic properties of cesium vapor are readily obtainable from the tabulations of properties of Lennard-Jones gas, given by Hirshfelder, Curtiss, and Bird. The effects of electrons and ions on the vapor properties are also considered. Numerical values are given for (a) entropy, (b) internal energy, (c) heat capacity at constant pressure, and (d) enthalpy. Transport properties of cesium vapor are obtained from solutions of the Boltzmann transport equation involving the familiar collision integrals of Chapman and Cowling and of Hirshfelder, Curtiss and Bird. As in the case of thermodynamic properties, one can obtain the transport properties of a Lennard-Jones gas from the tabulations of Hirshfelder, Curtiss, and Bird, provided both a and e are known. Numerical values are reported for the following transport properties: (a) electron collision frequency (already mentioned), (b) thermal conductivity, (c) diffusion coefficient, and (d) viscosity. The collision frequency between electrons and neutral atoms (in the appropriate energy range) is obtained from quantum mechanical considerations. The problem is taken as the collision between electron plane waves and a scattering potential of limited extent. The potential for the interactions between electrons and neutral cesium atoms is constructed from Slater orbital wave functions. A polarization potential is added to partially compensate for the distortion of the static atom potential and also for electron exchange. The results of the calculation are compared with available experimental evidence. One has to wait for more reliable experimental results before the reliability of the detailed application of the Lennard-Jones (6-12) potential to cesium vapor can be evaluated.

INTRODUCTION A PROGRAM o f o b t a i n i n g r e l i a b l e t h e r m o d y n a m i c a n d t r a n s p o r t p r o p e r t i e s o f p a r t i a l l y i o n i z e d gases ( p l a s m a s ) f r o m t h e o r y is a l o n g r a n g e o n e . B e f o r o o n e c a n d e c i d e o n saris* Permanent Address: University of California, Los Angeles 24, California. t Consultant to Aerospace Corporation, El Segundo, California. 19

20

L . B . ROBINSON

factory theoretical methods, more experimental information of a general nature is required than is available at present. At this time a report will be made on the completion of the first stage of such a theoretical program involving a cesium plasma. The model used for the cesium plasma in this report will be a collection of electrons, cesium ions, and cesium atoms. The cesium atoms are considered to form a vapor whose deviations from ideal behavior can be understood in terms of pairwise Lennard-Jones (6-12) pairwise interactions. The contributions of the charged particles to the properties are determined from the Debye-Hueckel interactions. Three principal results are reported in this article: (a) a new source of data for LennardJones parameters, (b) numerical values for the Lennard-Jones parameters for cesium vapor, and (c) the collision frequency between electrons and neutral cesium atoms. From the basic Lennard-Jones parameters, the following thermodynamic properties of the vapor will be reported: (a) excess entropy function, (b) excess internal energy function, (c) excess heat capacity function, and (d) excess enthalpy function. Thermodynamic excess functions are also reported which result from the Debye-Hueckel interactions. The range of temperatures and pressures in which charged particles play an important role in determining thermodynamic properties will be discussed. In addition to thermodynamic properties, transport properties of the vapor (based on the Lennard-Jones parameters) will be reported. The transport properties tabulated in the report are: (a) diffusion coefficient, (b) viscosity coefficient, and (c) thermal conductivity. From the collision frequency between electrons and neutral atoms and the Spitzer-H~irm expression for the Coulombic conductivity, (d) the electrical conductivity is calculated. Effects of electric charges on the thermal conductivity is also considered. After the properties of the pure constituents of the plasma are determined, the overall plasma properties can be determined by some combining rules. At present, the above procedure seems to be the best that one can do. The long range plan is to determine properties in a more fundamental way by considering all of the collective interactions which occur within the cesium plasma. As mentioned earlier, more detailed experiments will certainly be of great assistance in constructing a more satisfactory theoretical procedure.

LENNARD-JONES PARAMETERS Very little information exists regarding the form of an intermolecular potential which exists between alkali atoms. The Lennard-Jones (6-12) potential is reasonably satisfactory for many gases. However, in the interpretation of some phenomena, including alkalis and large rare gas atoms [1], the author has found this potential unsatisfactory in a quantitative way. No conclusive evidence has been found (by the author) of the usefulness of the Lennard-Jones potential in alkali-alkali interactions. In this paper, it will be accepted as a satisfactory approximation for describing such interactions. On the basis of the LennardJones parameters, one can determine the thermodynamic and transport properties. Corresponding values of the properties of rubidium will be given as a basis for comparison when it seems appropriate. Some published data regarding the effects of foreign gases on the spectral lines of rubidium and cesium can be reinterpreted to obtain the constants in the Lennard-Jones potential for these alkalis [1]. The published results can now be looked upon in a slightly different fashion to obtain the Lennard-Jones parameters for rubidium-rubidium interactions and cesium-cesium interactions, as explained below.

Thermodynamic and Transport Properties of a Cesium Plasma

21

The Lennard-Jones potential for the interaction of two types of molecular species, i and j, separated a distance r, is given by

V,,=4e,,[(7)zz--(7) 6]

(1)

where all symbols have their usual significance. The attractive (negative) term is the London dispersion term in the van der Waals energy. This dispersion energy term can be written more compactly as V~ (attraction) --

Ci~ r6

(2)

where Cij = 4Eocro6. The C~j in equation (2) can be calculated with a high degree o f reliability for the alkalis (here i = j). The E and cr cannot be calculated separately, however. The various approximate forms which have been used for the C~j give rise to the inequality [2] that C~ ~ C~iCjj (3) or equivalently ,,jtr~. ~ (,,lejj) 1/2 tr3,trf.] (4) The subscripts ii and jj refer to interactions of like pairs of molecules. One would expect the equality in equations (3) and (4) to be approached more closely as the molecules become more and more alike. Alkali and noble gas atoms are as unlike as atoms can be, hence the equality in equations (3) and (4) is not to be expected. The inequality (4) (or the equality) can be used as a basis for assigning parameters for unlike interacting parameters when the parameters for the individual species are known separately. There are many possibilities which occur when one tries to make a calculation on the basis of an inequality. There seems to be no theoretical foundation for the usual assignments that ,~j = (,,qj)l/.~

(5)

and that ~r~j : ½ ( ~ q- ~rjj)

(6)

However, such assignments as in equations (5) and (6) have been quite useful especially in the cases of similar molecules. One could just as well use cr2j = gl~jj or

Equations (6) and (7) give results which are quite similar when m, is not too different from %7. In this report a new method will be used for the assignment of the parameters, say ~j.j and q~ when a , , aej, ~ , and E,zjare known. The consistency of this new method with other known properties of the alkalis will be pointed out. The method o f this article consists of the following three steps: (a) equation (7) for the calculation of ~jj, (b) equation (2), with i = j, for the calculation of qj, once ~jj is known, and (c) the inequality (4) as a check for consistency. This writer has already published values of ¢~j and ~¢~ for Rb-He, Rb-Ne, Cs-He, and Cs-Ne interactions [1]. The Lennard-Jones parameters for He and Ne are well known. The basic parameters are given in Table 1. From the previously published results, equation (7) gives aej = 4.55 x 10-8 cm for cesium. It is interesting to note that one obtains by the same method and source ~jj = 4.17 x 10-s cm for rubidium. These results are shown in Table 1.

22

L.B. ROBINSON TABLE1. PARAMETERSIN THELENNARD-JONES(6-12) POTENTIAL Interaction He-He Ne-Ne Rb-He Rb-Ne Rb-Rb Rb-Rb Rb-Rb Cs-He Cs-Ne Cs--Cs Cs-Cs Cs~s

( × 10as cm)

Reference

2" 56 2" 78 3"24 3' 43 4" 10 4.23 4" 17 ± 0.07 3.39 3' 58 4"49 4.61 4.55 ± 0.06

Footnote 4 Footnote 5 Footnote 1 Footnote 1 From Rb-He by equation (7) From Rb-Ne by equation (7) Average Footnote 1 Footnote 1 From Cs-He by equation (7) From Cs-Ne by equation (7) Average

There are varying values of the above parameters listed in different sources. A cr of 2-56 angstroms is taken for helium because this value also agrees with results of quantum considerations. Different choices of these parameters for helium and neon will, of course, give different final results. Calculations on the basis of equation (8) give Rb, e = 0.466 eV and for Cs, e = 0.387 eV. In proceeding to the last step, one must calculate the parameter Ci1 in the dispersion energy. P. R. Fontana [3] has made very reliable calculations of this dispersion parameter for many substances, including the alkalis. The specific adaptation of equation (2) in order to determine ~jj is

= 4~

(8)

Fontana obtains Cjj = 1570 × 10-60 ergs cm 6 for R b - R b interactions and Cjj = 2200 × 10-6o ergs cm B Cs-Cs interactions. The above values yield ~jj = 0.466 eV for Rb and 0.387 eV for Cs. Numerical checks will show that the inequality in (4) is maintained. The reasonableness o f the parameters reported here can be established in an interesting fashion. First, it is of interest to compare the parameter ~ (the mutual potential energy of two alkali atoms at the potential minimum) with the dissociation energy of the diatomic molecules. After a careful evaluation of all o f the data available, Herzberg [6] lists the dissociation energy for Rbz as 0.49 eV and that of Csz as 0.45 eV. The dissociation energies compare favorably with the results given here for the parameter E. If equation (5) is used to calculate the o's, the values are about an order of magnitude smaller than the dissociation energies of the diatomic molecules. The reasonableness of the values o f a can be established as follows. One can make use of the properties o f the diatomic molecules as in the case o f E's. Evans, Jacobson, Munson, and Wagman [7] estimate (from band spectra) the equilibrium internuclear separation in Rbz to be 4- 13 x 10-s cm and that for Cs2 to be 4.46 × 10-s cm. These results compare very favorably with the calculated values o f cr. Regardless of the detailed model of the potential o f the diatomic molecule, similar values should result for the equilibrium internuclear distance and the potential minimum. THERMODYNAMIC PROPERTIES In calculating the thermodynamic properties of a cesium plasma, the following contributions are taken into account: (a) translation and electronic energy states o f the atoms,

Thermodynamic and Transport Properties of a Cesium Plasma

23

(b) energy states resulting from atomic pairwise interactions, and (c) energy states resulting from the multiple Coulombic interactions among the electrons and positive ions. The translational and electronic contributions to the partition function are calculated under ideal gas conditions and the other contributions represent deviations from such behavior. Table 2 contains a very detailed list of the most important thermodynamic functions TABLE 2. THERMODYNAMICPROPERTIESOF CESIUM

"r -(~-h°)/v it--nO 10 20 30 40 50 60 70 80 90

100 110 120 130 140 150 160 170 180 190 200 225 250 275 298.15 300 325 350 375 400 425 450 475 500 550 600 650 700 750 800 850 900 950 1000 1050 1100 1150 1200 1250 1300

20"109 23"553 25.567 26"996 28.105 29.011 29.777 30.440 31"025 31'549 32'022 32.454 32"852 33'220 33'563 33"884 34-185 34'469 34-737 34-992 35.577 36"101 36-574 36-976 37.007 37.404 37-772 38.115 38"436 38"737 39"021 39"290 39'544 40.018 40"450 40'848 41.216 41"559 41"879 42"181 42"465 42"733 42.988 43"230 43"462 43-682 43.894 44'097 44-292

49'681 99"362 149-044 198.725 248-406 298.087 347"769 397"450 447.131 496.812 546.494 596-175 645.856 695'537 745'219 794'900 844.581 894.262 943"944 993.625 1117"828 1242"031 1366"234 1481-246 1490"437 1614'641 1738"844 1863-047 1987.250 2111.453 2235"656 2359"859 2484"062 2732-469 2980.875 3229.281 3477.687 3726.094 3974.500 4222.907 4471.313 4719'721 4968'132 5216-546 5464"967 5713.401 5961.856 6210.341 6458"873

~

~e

25.077 28.521 30"535 31.964 33.073 33.979 34'745 35.408 35.993 36.517 36.990 37.422 37"820 38.188 38.531 38"852 39"153 39.437 39-705 39.960 40"545 41"069 41"542 41.944 41"975 42.372 42.741 43-083 43-404 43.705 43"989 44.258 44'513 44-986 45'418 45.816 46.184 46.527 46.848 47.149 47.433 47.701 47.956 48-199 48.430 48.651 48'862 49.065 49"260

4'968 4'968 4"968 4"968 4-968 4.968 4.968 4.968 4'968 4.968 4.968 4.968 4.968 4"968 4"968 4'968 4'968 4.968 4.968 4'968 4'968 4'968 4'968 4'968 4'968 4.968 4"968 4.968 4.968 4.968 4"968 4.968 4"968 4'968 4"968 4.968 4-968 4-968 4.968 4.968 4'968 4'968 4-968 4.968 4.969 4.969 4.969 4"970 4.971

r -(e-h°)/r h - h g 1350 1400 1450 1500 1550 1600 1650 1700 1750 1800 1850 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000 3200 3400 3600 3800 4000 4200 4400 4600 4800 5000 5200 5400 5600 5800 6000 6200 6400 6600 6800 7000 7200 7400 7600 7800 8000 10000

44"479 44"660 44-834 45-003 45"166 45.323 45-476 45'625 45"769 45-909 46-045 46"178 46'433 46'677 46"909 47'132 47-345 47'551 47"749 47-940 48"125 48"305 48'480 48.816 49-138 49-448 49"748 50.041 50-329 50"614 50'898 51"182 51'466 51'753 52"042 52-334 52'629 52.927 53.226 53.526 53.827 54.128 54.427 54.725 55"021 55.313 55-601 55"886 58.448

6707"470 6956'158 7204'969 7453"942 7703"124 7952"568 8202"338 8452.508 8703"156 8954"375 9206'263 9458-929 9967"063 10479-806 10998'300 11523.795 12057'632 12601'241 13156"131 13723-901 14306-238 14904"937 15521-914 16819"059 18216"003 19733'811 21396'126 23228"349 25256"068 27502"822 29987'415 32721'219 35705'879 38931-919 42378.514 46014-566 49800.921 53693"341 57645"755 61613.347 65555'074 69435'441 73225.451 76902.899 80452.032 83862.929 87130'608 90254.091 114719.255

~

~e

49"448 49.628 49"803 49.972 50"135 50.294 50-447 50"597 50.742 50-884 51.022 51.156 51.417 51'667 51"908 52"142 52-369 52"591 52'809 53-023 53-235 53"445 53.654 54.072 54"496 54-929 55.378 55-848 56-343 56"865 57"417 57'999 58.608 59"240 59"890 60.551 61.216 61"875 62.523 63-153 63"760 64.339 64.888 65.406 65.893 66'347 66.772 67'167 69'920

4.973 4.975 4'978 4'981 4.986 4.992 4.999 5.008 5.018 5"031 5.045 5.062 5-103 5.154 5.218 5.294 5.385 5.490 5.611 5.748 5.902 6.075 6.268 6.719 7"268 7.930 8.715 9.629 10'668 11'815 13.040 14.299 15.539 16.703 17.736 18.591 19.235 19.650 19.836 19.805 19.582 19.197 18.684 18.077 17.406 16-699 15.977 15.259 9.817

24

L . B . ROBINSON

(for condition (a) in the preceding paragraph of cesium as a function of temperature. The temperatures range from 10°K to 10,000°K. In the table, ~ is the molar Gibbs free energy, /~ is the molar enthalpy, ~ is the molar entropy and ~p is the molar heat capacity. The zero superscripts and subscripts refer to standard states. The electronic contributions to the partition function, from which the thermoydnamic properties are calculated, are obtained by summing over the listed energy states of cesium.t Deviations from some of these ideal thermodynamic properties, as a result of different types of imperfections, are given in Tables 2 and 3. One can determine the ranges of temperatures and pressures where these contributions become important. It is also possible to compare the relative magnitudes of these effects. The equation of state for cesium vapor used in this report is the virial equation. The range of temperatures and pressures of interest is restricted to the range in which the second virial coefficient B(T) is sufficient to represent deviations of the vapor from ideal behavior• The equation of state is

where ~" is the molar volume and the other symbols have their usual meanings. The second virial coefficient can be calculated once the pairwise interactions are known. B(T) is given by cO

f [1 -

B(T) = 27rN0

]

e-V(r)/~r] r 2 dr

(10)

0

where, in the present case V(r) is the Lennard-Jones (6-12) potential and No is Avogadro's number. One may write the integral in equation (10) as cO

B*(T*) = 3 f x 2 { 1 - - e x p [ ;

(12

16)]} dx

(11)

0

where B* ~ B(T)][]TrNoe a] and T* = kT/E. Hirshfelder, Curtiss, and Bird [81 have tabulated B* as a function of T* such that the range of physically significant temperatures is covered. Hence, once (r and E are known for a given substance, B(T) is readily determined. Having determined the second virial coefficient, one can obtain all of the required thermodynamic properties within the framework of this model of the vapor. Hirschfelder, Curtiss, and Bird [9] provide the equations and tables for the thermodynamic properties per mole given below: ~ (entropy), j~ (internal energy), ~p (heat capacity at constant pressure), and /-7 (enthalpy). The appropriate equations are

(TS -- S°)/P ÷ (R]P) ln (P/Po) = -- (~--~)

(12) (13)

(14)

(15) "~ See FOWLERand GUGGENI-IEIM, Statistical Thermodynamics pp. 256-258 for discussion of this method of establishing partition functions.

Thermodynamic and Transport Properties of a Cesium Plasma

25

As shown above, the pertinent thermodynamic properties of such a gas are expressed in terms of B(T) and the first and second derivatives of this coefficient with respect to temperature. From equation (10) dB(T)

_

2~'NOkT 2 ;

e_V(r)/k T V(r) r 2 dr

(16)

[[V(r)kT 2

(17)

dT

o 2~NOkT z f

dgB(T)dT 2 _

T2] e-V (r)/kT V(r) r 2 dr

o Since Hirshfelder, Curtiss, and Bird [9] have calculated B*(T*), T*[dB*(T*)]/dT* and (T*) 2 [d2B*(T*)]/dT *~, it is a relatively simple matter to determine the required thermodynamic properties. It is clear that (2~ No(r~ T* dB* _ T dB / 1,3 dT* dT

(18)

so that the left hand sides of equations (12), (13), (14), and (15) are readily calculated. These thermodynamic excess functions are given in Table 3. TABLE 3.

THERMODYNAMIC PROPERTIES OF ALKALI VAPORS

T*

B

--T ~-T

(cm2)

(cma)

(cm3)

Rubidium 1"0 1 "5

2-0 2"5 3"0 3-5 4.0 5.0

--232 --109"8 --57"4 --28"6 --10"54 1'735 10'54 22- 3

--405 --221 --149"1 --111.3 --87'8 --72"0 --52'7 --45.1

--637 --331 --206 --139"9 -98-3 -70.3 --42'2 -22-8

Cesium I "0

1-5 2.0 2.5 3"0 3.5 4-0 5"0

--302 --142"7 --74"6 --37-2 --13"70 2.25 13-70 28'9

--526 --287 --193.8 --144-6 --114'1 --93.5 --78-7 --58.6

--828 --430 --268 --181"8 --127.8 --91-3 --65"0 --29-7

Myers, Buss, and Benson [10] have considered the applicability of the Debye-Hueckel theory of charged particle interactions, to the problem of determining thermodynamic excess functions of a plasma. The excess function in this case is the measure of the contri-

26

L . B . ROBINSON

bution of the charged particle interactions to the plasma properties. They conclude that the Debye-Hueckel theory is adequate to provide the desired information. According to their formulation, the effects of the Coulombic interactions can be expressed completely in terms of the quantity k T l n ~,~, where y~ is the activity coefficient of the ith species. For each of the ith species in the plasma (i.e., electrons and cesium ions) In ~,~ =

Z2_q2 KD 2DkT

(20)

where '
(21)

In the above two equations qe is the charge on the electron, n~ is number of ions of type i per unit volume, and Z~ is the number of charges (positive or negative) on each ion, and D is the dielectric constant. The other symbols have their usual meanings. When the concentration is given in terms of moles per liter, rather than ions per cm 3, equation (20) becomes In ~,~ --

Z~q 3 (~No~ $1/2 (kT)a/2 \ 5 0 0 ]

(22)

where No is Avogadro's number and s, the ionic strength is given by

s = ½Zt c~Z2

(23)

In equation (23), c~ ---- 1000 ni~No. In terms of the activity coefficient, the partial molar Helmholtz free energyexcess function ~i~.l is given by .~l = R T In 7~ (24) The partial molar entropy excess function ~ 1 is ~ 1 = ½R In y~

(25)

T h e partial enthalpy excess function H~t is

~ l = 2 R T In ~

(26)

The partial molar Gibbs free energy excess function G~ is

~ t = _3R T In ~,, 2

(27)

TABLE 4. EXCESS PARTIAL MOLAR ENTHALPY OF A CESIUM PLASMA AS A RESULT OF COULOMBIC INTERACTIONS

Hi (cal mob -i) T (°K) 1000 1500 2000 2500 3000

P = 10-1mmHg P = InunHg --5"0 --1.2 --3-9 --8'4 --1-2

× × × × x

10-4 10-a 10-a 10-a 10-2

--1"7 --1.6 --7.1 --1"6 --2-3

× x × × x

10-a 10-a 10-a 10-2 10-2

P=10-immHg

P= lmmHg

--1.0 --3'6 --14-6 --42 --72

--3.4 --4"8 --28 --80 --138

Thermodynamic and Transport Properties of a Cesium Plasma

27

Values of the partial molar enthalpy excess function are given in Table 4 as a function of T for a pressure of 10-~ mm of mercury and also 1 mm Hg.

TRANSPORT

PROPERTIES

The Boltzmann transport equation [11] is the starting point from which the transport properties of a plasma are calculated. For each of the ith constituents, a distribution function J~ obeys

a-i+ v~. vgq +

a~. vvt~ = ~ t ! ~

(28)

For collisions of a binary nature, the collision term on the right hand side of equation (20) may be written as

;;;

St

= Ej j dvj j j j

(f~'fj-js~) t'J(v,,,X)sinx dedv,

(29)

As is customary, the primes refer to the situation following the collision. The symbol L,~j means the relative speed between the ith a n d j t h particles. I(t,~j, x) represents the differential scattering cross section, dQ/d~2, such that

dQ _ 1(vii, x)

(30)

dO

where dO = sin x dx de. The form of I(vtj, X) depends upon the types of particles involved in the collision. Some specific forms of the differential cross section will now be given. Collisions between electrons and neutral atoms yield the following differential cross section [12]:

I(vij, X) --

I 4~z I Z' (2l q- 1) ( e ~ l -- 1) el(cos x) 12

(31)

l=0

where K = mvljh -1, 8t is the Ith order phase shift and Pt(cos x) is the lth order Legendre polynomial. For collisions between neutral atoms [13],

I(v,~, x) sin x dx = b db

(32)

and

~

dr/r z (33)

In equation (33), b is the impact parameter, rm is the distance of closest approach, V(r) is the mutual potential energy between the two molecules, and F is the reduced mass. The parameters of equations (32) and (33) are illustrated in Fig. I. Various types of collision cross sections, Q~g], are required in dealing with transport properties, as we shall see shortly. The r~<,> zcm/, are defined as Q ~ = 2~r f (1 -- m cos n X) I@~j, ~) sinx dx 0

(34)

28

L.B. ROBINSON

for the electron-molecule collision and

Q~") (m) = 2~r ? [1 -- rn Cos'* X(v~, b)] b db

(35)

0

for the molecule-molecule collision. In both equations (34) and (35) m = 0 or I and n = 1, 2, 3 . . . . . When m = 1, it is usual to omit the subscript and write Q(n). It is clear that Qlgl is the cross section referred to in equation (30). Another cross section which occurs frequently is QI~I or a(1) given by Qo) = 4= ~ (l + 1) sin 2 (8l+1 - 81)

(36)

K2 l=0

An alternate form for Q~]~ is [14] ~ (2l ÷ 2) sin 8~ sin 8t+1 cos (St -- ~l+1) Q(1) = 4~,~2trio ~ (2l q- 1) sin 2 8t - 4rr ~ l=0 Particle trajectory

FXG. 1. Parameters involved in the scattering of a particle by a center of force located at O. A two body scattering problem can be represented as shown above.

These forms for Q(1) occur when the I(vtj, X) of equation (31) is used in equation (34). Q(1) is called the diffusion cross section or momentum transfer cross section. In neutron physics, it is called the transport cross section. In addition to the various cross sections given in equations (26) and (27) calculated at one energy or velocity, cross sections averaged with various powers of the velocity are required in obtaining transport properties. These averaged cross sections are commonly called collision integrals or omega integrals, and are given by oo

[ kZ ]1/2

f

),2 ),~t+3

(38)

O where the symbol m is usually I and generally omitted. In equation (38),/x~jis the reduced

mass of the colliding partners of types i and j, and =

v,j

is the reduced relative speed of the colliding with actual relative speed vo.

Thermodynamic and Transport Properties of a Cesium Plasma

29

Solutions of the Boltzmann transport equation yield relatively simple expressions for transport parameters. The equations for the diffusion coefficient ~, the viscosity coefficient 7, and the thermal conductivity h are given below for a pure substance [15] _

(3 2rrkZTa/lz)l/2 16 P~ra2 Q(1,1)*

(39)

In equation (39) t~ is the reduced mass, P is the pressure and a is defined in equations (1) and (7). The symbol m is omitted from the omega integrals.

5 (,rmkT) 1/2

(40)

'7 -- 16 rm2 Q(2,2)* In equation (40) m is the mass. 2 t - 25 (rrmkT) 1/2 Sv 32 rra 2 ~2(2'2)* m

(41)

In equation (41), sv is the specific heat of the v a p o r at constant volume. In this paper the viscosity coefficients of both rubidium vapor and cesium vapor have been calculated and compared with one other known calculation for cesium. In a paper called Some Selected Properties of Cesium, Irving Granet calculated transport properties of cesium v a p o r in a straightforward fashion [16]. He used the mean free path theory from the simple kinetic theory of gases. The diameter of cesium atoms was estimated from the density of solid cesium. Granet assumed that cesium v a p o r is essentially monatomic, evidently a very reasonable assumption. Granet found for cesium ~ ----0.98 × 10 -5 ~ / T poise. The present calculations give ~ ---- 1-45 × 10 -5 (~/T/T2(2,2)*) poise for a = 4.55 ~.~ 10-s cm. The collision integral 12(2, 2). is of the order of unity, varying from about 0 . 5 - 2 . 8 . Granet's simple calculation gives results not too different from the present calculation especially since 1-5 is a rough average value of ~(2,2). in the temperature range of interest. Table 5 lists the transport properties given in equations (39), (40) and (41) for various values of the reduced temperature, T*. To convert T* to degrees Kelvin, one must multiply T* by 5400 for rubidium and by 4490 for cesium. A simplifying assumption, Sv ~ 3R,'(2M), has been made in equation (41); R is the molar gas constant and M is the atomic mass number (atomic weight) in grams. TABLE 5. TRANSPORT PROPERTIES OF ALKALI VAPORS

T*

"~ × 104(poise)

~ × 10~(calcm lsec ldeg-lK)

.~P (cm~sec-latm)

Rubidium 1.0 1-5 2'0 2"5

6"54 9"65 12"45 15"05

5.69 8"42 10"9 13"2

4.50 9.90 35.40 53.10

Cesium 1.0 1.5 2'0 2.5

6.25 9"22 11 '9 14.3

3'51 5.18 6"67 8.03

2.28 5.06 8.67 13.00

30

L . B . ROBINSON

ELECTRICAL RESISTIVITY The electrical resistivity is a transport property of the plasma deserving special attention. Contributions to the electrical resistivity, and hence electrical conductivity, are of two types: (a) the charged particle interactions and (b) the collisions between electrons and neutral atoms. These two contributions shall be considered separately and later combined in order to obtain the overall electrical resistivity, in a manner similar to that used by Lewis and Reitz [17]. The Coulombic contribution to the electrical resistivity of the cesium plasma is obtained on the basis of a modified model of a Lorentz gas. A Lorentz gas is a fully ionized gas in which the negative changes (electrons) do not interact with each other and the positive charges (cesium ions) are at rest. The electrical resistivity of pc of such a gas is [18]

~3/2 ml12 ZqZe c 2 In A pC :

2(2kT)a/2

(42)

me is the mass of the electrons, Z is the charge on the ions, c is the speed of light in a vacuum, and, 3 (k3T3] 1/2 (43)

A=

2ZZlq~

\ ~rn~/

The specific modification of the Lorentz gas to be used here is that given by Spitzer and Harm [19]. In the Spitzer-H~irm model o f conductivity, the resistivity p~ is given by =

Pdw

(44)

where yE depends on Z. Spitzer and H~irm obtained the value yE = 0. 582 for Z = 1. Using numerical values of the parameters given in equation (42), one obtains

pc = 3.80 × 108 Z(ln A) T-3/~ Y2 cm and p~ = 6.53 × 10z ZT-3/2 In A

(45)

p~ depends on ne weakly through In A. Numerical values of p~ are given in Table 6. TABLE 6.

ELECTRICAL RESISTIVITY OF A CESIUM PLASMA FROM COULOMBIC INTERACTIONS

pl (t2 - cm) T(°K)

P=10-2mmHg

P=10-1rnmHg

P = lmmHg

1500 2000 2400 2600 3000

0.87 0.44 0.29 0" 25 0-20

0.81 0.41 0" 27 0- 22 0" 16

0.76 0.37 0' 24 0" 20 0- 14

The contribution o f the collisions between electrons and neutral cesium atoms can be examined from the point of view of the Boltzmann transport equation. For a homogeneous plasma, equation (28) can be written as (with the external field X in the x-direction only) a3~(vt, t)

at

X(t) af(v~, t) _ [af(vt, t)~ q- me Ovlx k at ]c

(46)

Thermodynamic and Transport Properties of a Cesium Plasma

31

When elastic collisions between electrons and neutral atoms are the only type of importance, the collision term becomes a special form of equation (29), namely

Ot] = f f VrI(Vr,X)d£2[f~f~- tiff]

dvj

(47)

The symbol i means electrons and j means neutral particles, and Vr is v#. One form of the Maxwell transfer equation [20] is obtained when both sides of equation (46) are multiplied by viz and integration is performed with respect to dr,. The result is --

Ot

v~x - - - - -

me

Oviz

dv~ =

v~z

\ Ot ] c

dv~

(48)

It is evident that the above equation may be written as 0 {t'~z)-

v~z

Ot where

(v~> = (f~(vt, t) V~x dr,.

\ Ot ] c

dv~-

viz -

me

OVix

-

dvi,

(49)

Some simplification can be made in the term on the right

side of equation (49) as follows

f fir ,,,z~;,;Of~dv~z]dviydv,z=--I

(50)

Equation (50) is obtained by integrating the expression in brackets by parts with the use of

f udv = u v - f vdu.

the well-known formula

Here we use u =

viz,

dv = 0ff-'0e i.v.

du = dv~x, and

v = ft. We also assume that fi = 0 at v~z = -4- ~ . Because of the above relationships, we may rewrite equation (49) as

Ot

J Vix \Ot]e

dr/--

me

(51)

If (v~z) depends only on t, then the partial derivative is replaced by the total derivative. The collision term (the integral on the left-hand side of the above equation) can be simplified by making use o f the following transformation:

f f fv,xf'ifjl(vr, X) vrdg2dv, dvj= f f fvlz~fi.I(vr, X) vrdt2dv, dv]

(52)

It follows that

f ?iq When the magnitude of v~ remains essentially constant following a collision, v;~ -

v~

=

v~

(cos

x -

1).

If the distribution function for the neutral cesium atoms remains essentially the equilibrium distribution then

v~ \ Ot ] c dvt = -- Ng

f f~v~xrQM(V)

dv

(54)

32

L . B . ROBINSON

where QM(V) is the momentum transfer cross-section. The subscript r has been dropped from Vr since we assume that [Vr [ = [v, [. The symbol Ng is the particle density of neutral gas atoms. In equation (54)

QM(V) = f (1 -- cos X) I(v, X) dO

(55)

We may now write equation (51) as d(v~x)dt + Ng f fiv~x vQM(V) dv -- X(t)me

(56)

If QM(V) oc 1/v, as is the case approximately for many situations of interest, then equation (56) becomes (regardless of the form of the distribution function) d(v~z~ _}_ No vQM(v) (vix) -- X(t)

dt

me

(57)

Writing (v~) = (d(x~)/dt), one finds the very simple equation

_ _ _ _ + v d(x¢) d2(xi) --- X(t) dt 2

dt

(58)

me

where v = f Ng VQM(v)fdv is the collision frequency. The symbol (x~) means average value of x~. When vQM(v) is not constant, equation (56) becomes

d(v~z~ + No(vQM(V ) v~z) -- X(t) dt me

(59)

On the basis of the following two equations for the electric current density J = X/(pnqe) and J:neqe(Vx), a simple expression for On (the plasma resistivity resulting from collisions between electrons and neutral cesium atoms) can be found. (v~) is obtained for present purposes by solving equation (57). Even though vQ(v) is not constant for cesium atoms, it is nearly so in part of the temperature and pressure region of interest. This present report contains results for only the equilibrium case. The more general case is being investigated by one of this writer's graduate students, Robert C. Beaty. He is using the methods developed by Burgers and Pipkin for calculations for a cesium plasma. Using X = qeEx ei~e, the solution of equation (57) is

(vz)-

qe Ez eI'°t me v + i~

(60)

When one uses 1/pn = neq~ (vz)/X, from the above paragraph, the result for the low frequency case is meNu'if: (61) pn

=

2 neq e

The symbol ~1 represents the average collision frequency per neutral atom and is given by f(v) vI(v, X) (1 -- cos X) sin xdx dv

51 = 2~r f o

o

In the present casef(v) is the Maxwell-Boltzmann distribution.

(62)

Thermodynamic and Transport Properties of a Cesium Plasma

33

The details of the manner in which the intensity distributions I(v, X) of equation (31), and (62) and the cross sections Q(Z) of equation (34) have been calculated are reported elsewhere. A summary of the calculations is given in Fig. 2. Both the averaged collision Ire-

15

~ %,

~

15

~

f

~

.

~

o_

5

06

5

t

z

3

4

5 6 r(i000'K)

7

a

9

to

@

FIG. 2. A v e r a g e d cross section and collision frequency as a function of t e mpe ra t ure .

quency and the averaged momentum transfer cross section are presented. The units for the cross section are atomic units, where 1 atomic unit of area equals 0. 280 x 10-z6 cmL The collision frequency per atom has units of cm 3 see-1. Equation (61) can be written as gn = 3.57 x 103 ~z ( ~ s f s )

(63)

where fs ---- ~e/(Ng q- ne) is the fraction of ionized gas. This fraction fs can be obtained from the famous Saha equation, which for cesium becomes f~ 1 - - f ] --

2.4x

10-4T5/s (44~00) P exp --

(64)

In the Saha equation, P is in mm Hg. A graph of f versus T, with P as a parameter is given in Fig. 3. Table 7 contains values of pn as a function of T for various values of P. It is evident TABLE 7. RESISTIVITY FROM ELECTRON AND NEUTRAL CESIUM ATOM INTERACTIONS p~ ( o h m -- cm) T(°K)

Q(10-ZScm ~)

1500 2000 2500 3000

36.4 34.2 32'2 30'8

P = 10-2ram H g P = 10-Zmm H g 6"42 0"118 6"36 x 10 -3 3-39 x 10 -4

19"4 0"338 2 ' 5 4 x lO -~ 2"86 x lO -s

P = 1 mmHg 64.2 1"05 8"4 x 10 -2 1"37 x 10 -~

34

L.B. ROBINSON

from Table 5, which contains values of p~, that there is a large range of temperatures and pressures in which p~ and pn are comparable. |

I0°

:mmHg) 10-2

f

10-3

$

p-I#-

10-4

I0"

I

I

I

I000

ZOO0

3000

T

T *K

FIG. 3. Fractional ionization in cesium vapor. The contributions o f the Coulomb interactions to the thermal conductivity can be determined within the framework of irreversible thermodynamics once the electrical conductivity is known. Assuming that a liner relationship exists between the "flows" or currents d~ ( i = 1, 2, . . . n) and the "affinities" or forces X~, one precedes as follows: J~ ---- ~ L~e Xe

(65)

k=l

A proper choice of currents and forces for the electrical current and thermal currents established the following relationships among the phenomenological coefficients L~e: /-4k = Lk~ (i, k = 1, 2 . . . . .

n)

(66)

Equation (66) represents what is commonly known as the Onsager theorem. On the basis o f such considerations, the following expression is obtained for the thermal conductivity AL o f a Lorentz gas: •~L = 20

~_2~3/2

br/

(kT)5/2 k 1012 T 5/2 = 4" 67 × cal sec -1 deg -1 cm -1 q4e Z In A Z In A

me 1/2

(67)

Thermodynamic and Transport Properties of a Cesium Plasma

35

The Spitzer-H~irm correction is ~ion = 3TAL (68) where for cesium, 3T ---- 0" 225. Some numerical values are given for ;~ionin Table 8, so that comparison can be made with the results obtained for the neutral particle interactions. This article concludes with a brief report on the method which was used by this author to obtain the momentum transfer cross section and collision frequency for collisions of electrons with cesium atoms. So far as this author knows, this analysis is the first such detailed analysis for cesium. TABLE 8. THERMAL CONDUCTIVITYOF A CESIUM PLASMA FROM COULOMBIC INTERACTIONS

2ton (watt T(K)

1300 1500 1700 2000 2200 2400

p=10

-2mmHg

c m - 1 deg-1) ~: 10 5 p=

1-26 2.06 3'15 5' 41 7" 36 9-78

10 - l m m H g

1 '35 2.21 3.42 5"89 8' 19 10"9

p=lmmHg

1 "44 2'37 3"72 6"46 8"89 12. l

The collision frequency between electrons and neutral atoms in low energy range is obtained from quantum mechanical considerations. As is customary, electron scattering by neutral atoms is considered as a collision between an electron plane wave and scatterer of limited extent. Hartree-Fock wave functions and the Thomas-Fermi function have served as bases for construction of scattering potentials. In the case of cesium, the Hartree-Fock functions are evidently not available and the Thomas-Fermi function does not represent the alkali atoms too well. This author used the Slater orbital wave functions in the construction of the potential for the scattering of electrons by cesium [21 ]. A polarization potential was included to compensate (in an indirect way) for the neglect of the distortion of the static atom potential by the incident electrons as well as for exchange between this electron and the atomic electrons. Collision properties are calculated from the Schroedinger radial equation

dZy~_~.[(Jgao)Z.+U(x) dx 2

/ ( / x2 + l) + VI(X)] yt ~ 0

(69)

In equation ( 6 9 ) ~ is the wave number [2mEh-2] 1/~, ao is the Bohr radius (0.529 × 10-s cm), U(x) = 2mV(x) h -2 and l is the orbital angular momentum quantum number, E is the initial kinetic energy ½mv,j 2 of equation (31). V(x) is the static atom potential constructed from the Slater orbital wave functions. Vl(X) is the polarization potential which has the form

Vl(x) --

A (B + x2)2

(70)

A and B are parameters and x ~ r/ao. The details of the construction of the potentials and methods of solution are given elsewhere [21]. Solutions of equation (69) provide the phase shifts, and hence the differential scattering cross section I(vtj, x) of equations (31) and (34). In the preparation of this report, this writer has benefited from discussion with G. L. Johnston, who also prepared Fig. 3. R. L. Wilkins kindly provided Table 2.

36

L. ]3. ROBINSON

REFERENCES [1] L. B. ROBINSON,Phys. Rev. 117, 1275 (1960). [2] H. S. TAYLORand S. GLASSTONE,editors, ~4dvanced Treatise on Physical Chemistry, Vol. 2, pp. 351-352, Van Nostrand, New Jersey (1951). [3] P. R. FONTANA,Phys. Rev. 123, 1865 (1961). [4] H. S. TAYLORand S. GLASSTONE,Advanced Treatise on Physical Chemistry, Vol. 2, p. 328, also J. E. KIRKPATRICK,W. E0 KELLER, E. F. HAMMELand N. METROPOLIS,Phys. Rev. 94, 1103 (1954). [5] J. O. HIRSHFELDER,C. F. CURTISSand R. B. BrRt), Molecular Theory of Gases and Liquids, p. 165, Wiley, New York (1954), also L. HOLaO~N and J. OTTO,Z. Phys. 33, 1 (1925). [6] G. HERZBERG, Molecular Spectra and Molecular Structure, I Spectra of Diatomic Molecules (2nd Ed.), van Nostrand, New Jersey (1950). [7] W. H. EVANS, R, JACOBSON,T. R. MUNSONand D. R. WAGMAN,J. Res. Nat. Bur. Stand. 55, 83 (1955). [8] J. O. HIRSHFELDER,F. C. CURTISand R. B. BIRD Molecular Theory of Gases and Liquids, pp. 1114-1115, Wiley, New York (1954). [9] J. O. HIRSHFELDER,F. C. CURTISand R. B. BIRD, Molecular Theory of Gases and Liquids, pp. 230-231, Wiley, New York 0954). [10] H. MYERS, J. H. Buss and S. BENSON, Physical Chemistry in Aerodynamics attd Space Flight, pp. 257-270, Pergamon Press, New York (1961). [11] S. CHAPMANand T. F. COWLING, Mathematical Theory of Non-Uniform Gases, Cambridge University Press, New York (1952). [12] N. F. MoTT and H. S. W. MASSEV, The Theory of Atomic Collisions, 2nd Ed. p. 126, Oxford University Press (1949). [13] J. O. HIRSHFELDER,F. C. CURTISand R. B. BIRD, Molecular Theory of Gases and Liquids, p. 51, Wiley, New York (1954). [14] L. B. ROBINSON,Phys. Rev. 105, 922 (1957). [l 5] J. O. HIRSHFELDER,F. C. CURTISand R. B. BIRD, Molecular Theory of Gases and Liquids, pp. 526-527, WHey, New York (1954). [16] I. GRANET,Amer. Soc. Naval Engrs, 72, 319 (1960). [17] H. W. LEWISand J. R. RErrz, J. Appl. Phys. 30, 1439 (1959). [18] L. SPrrZER, Physics of Fully Ionized Gases, p. 83, Interscience, New York (1956). [19] L. SPITZERand R. HXRM, Phys. Rev. 89, 977 (1953). [20] S. ALTSHULER,P. MOLMUDand M. MOE, unpublished Space Technology Laboratories report. [21] L. B. ROBINSON,Phys. Rev. 127, 2076 (1962).