RADIO SCIENCE Journal of Research NBS/USNC- URSI Vo!' 69D, No.2, February 1965

Momentum Transfer Collisions in Oxygen for Thermal Electrons

Michael H. Mentzoni

Contribution From the Applied Research Laboratory, Sylvania Ele ctronic Systems, a Division of Sylvania Electric Products, Inc ., Waltham, Mass. 02154

(R eceived August 10, 1964 ; revised Sep tember 25, 1964)

T he electron collision frequency for momentu m transfer V m has been determ ined by m icrowa ve methods in an oxygen plasma for te mperatures bet\~een 300 and 900 oK. It w~s found . t hat the measured average electron-neutral collis ion freque ncy was consist ent WIth V ,m bemg proportlOnal to the square of the electron veloc ity when a Maxwelli a n velocity di~tribut i on is assumed for the electrons; e.g., v ,~.(u) = 4.4 X lO-sX N (0 2) x 1, (sec- I), where U IS electron ene.rgy. In electr on volt s an.d N (0.2). IS the nymber density of oxygen molecu les . The res ult s also ll1 dlCatecl that electro n-I on CollISIon s arc Important at the lowe r t emperntures with the values of v" in agreement wit h t he on es rece nt ly report ed by Che ll.

1. Introduction

The electron collision frequency for momentum transfer, V m, and its energy dependence in atmos­pheric gases is of special t echnical interest.

In atmospheric environment the t otal electron collision frequency for momentum transfer is deter­mined by adding the collision frequencies for the various gas constituents in proportion to their number densities. Thus by laboratory determina­t ions of V m as a function of electron energy for gases such as N 2, O2, NO, N, 0 , one is able to compute the composite value of v m. This value will vary with altitude according to the density profiles of the species and the electron temperature, the latter covering the temperature range of the present investi­gation between alt itudes from about 110 to 150 km [Nawrocki and Papa, 1961 ].

The collisional absorption and its effects on earth­space r adio propagation has been recently reviewed by Lawrence et al. [1964] who, in this respect, points out the importance of D layer, the physical processes of which have been surveyed by R eid [1964]. Oalculations of EM-wave propagation based upon the Appleton-Hart ree magneto-ionic theory incor­porate an effective velocity independent collision frequency thus simplifying the analysis. I t has been shown by Phelps [1960], that such a simplifica­tion m ay lead to significant errors.

During reent ry, signal absorption due to the ionized shock layer and wake constitutes a serious problem the solution of which necessitates a knowl­edge of V m in shock heated air [Bachynski et al. , 1960].

T emperatures in a wake are in general less than 3,000 °K for reentry speeds and the Vm measured here would pertain to the far wake which is of importance for radar detectability . A number of labora tory s tuclies have been repor ted In r ecen t years pertaining to ni trogen. For oxygen and dry air the ayailable inform ation is still rather limi ted [Nielsen and Bradb ury, 1937 ; H ealey and R eed, 194 1; Van Lint et al. , 1959; Phelp , 1960 ; Shkarofsky et al. , 196 1 ; Oarruthers, 1962; Gilardini, 1963]. F or thermal and near thermal energies, the dependence of Vm on the electron energy u has been difficul t to determine experimentally. This is so because the generation of monoenerge tic electron beams, used so successfully in scattering experi­ments above 1 e V becomes increasingly difficult at lower energies. H owever, Pack and Phelps [1 959] have improved the design of the mobili ty tube allowing measurements at thermal energies. It seems that the reanalysis by Shkarofsky et al. [1961] of resul ts from the early drift tube experiments of Nielsen and Bradbury [1937] wi th o"-j'gen pertaining to u higher than the gas temperature, Tgaa, is con­sis ten t with a collision frequency given by the formula

VemN( 0 2)-1=4 X 1O-9ul/2+5X lO - 8u C1113 sec- I

where N(0 2) is the molecular number denisty of O"-j'gen and V ern refers to elastic collisions bet:v~en electrons and molecul es. M easuremen ts pertaInmg to Tgas = 300 oK , assuming thermalized electrons, give resul ts that are in agreement with the VemN (0 2)-1 = 7X 1O-8u cm3 sec- I (Phelps, 1960) ; it seems

213

desirable to perform direct measurements at some oth er gas ternperature.s within the suggested range of applicability of this equation and thus ob tain a more complete verification.

Finally, el ectron-ion collisions, Vei, which are important at high electron-ion densities and lower temperatures are considered and their collision cross section evaluated. For a review of Veln and Vel per­taining to the atmosphere the reader is referred to publications by Nicolet [1959], Nawrocki and P ap a [1961], and Dalgarno [1961].

2. Experimental Method and Procedure

The experimental arrangement h as been described previously [:Mentzoni, 1963; Mentzoni and R ow, 1963] and only the salient points will be discussed here.

A short duration, pulsating d-c voltage, periodi­cally created an oxygen plasma inside a cylindrical quartz tube which also functioned as a waveguide operating in the " quasi" TEll mode due to a gold coating on the ou ter surface. A low power X-band microwave signal propagating along the 80 cm long plasma column acted as the probing wave. An interferometer technique together with accurate measurements of plasma insertion losses permitted determin ation of the ratio.

(1)

here ~A, ~r/> , ~a , and ~{3 represents changes in the total attenuation (nepers), phase shifts (radians) , attenuation and phase constant respecti,-ely due to the plasma, L is length of the interaction region, E is electric field in tensity, R is inner radius of the quartz cylinder and rTr and rT i are the real and imagi­nary components of the rf conductivity

(2)

Here N ., e, m , and v are el ectron number density, charge, mass, and velocity , respectively, and 10 is the spherically symmetrical part of the electron velocity distribution function normalized such that

( ( m )3/2 _ m~' . . ) 10= 2rrkTe e 2k T ' m the Maxwelhan case .

Equation (1) is valid provided the first order pertur­bation analysis is suffi ciently accurate [Golant and Zhilinskii, 1960] which means that

with wj2rr = 8.97X 103n~~2 and,

_ .{RNeE2rdr /27r l RNerdr

A F- l RE21'dl' I rrR2

'

AF goes from unity to 1.3 for the TEll mode as N e goes from. a uniform to a J o(2.405rj R ) distribution. Actually the quartz walls cause a further (and in this case insignificant) perturbation of the TEll mode as shown by Unger [1957].

E"en though the probing field is nonuniform, it is selected to be ,-ery weak, satisfying the criterion

2G(u)m ( 2+ 2) 2 W Vm e

with (il) and G the average electron energy and energy loss factor (G- factor) , respectively, so that one has that the average electron energy is much larger than the energy perturbation caused by the probing field , i.e.,

and thus can be regarded as sp atially uniform. When a larger field intensity is used, such as in the cross modulation experiment, the distribution func­tion can still be regarded as independent of position prmrided the thermal conductiyity of the electrons is very large.

In the present arrangement the gas is heated to any desired ambient temperature up to about 1000 oK by surrounding the waveguide-discharge con­figuration by an el ectric o\'en. A microwave radiom­eter utilizing the noise sampling technique is used to monitor the electron radiation temperature, one can thus find the t ime after which isothermal condi­tions prevail during the afterglow [Mentzoni, 1964] which ranged typically from about 30 to 50 fJ,sec. J n comparison, the total measurabl e time decay for electrons and ions, char acterized by N e> 5X 108 cm- 3,

lasted always more than 1 msec. In general one has vm= V.,n+ vet+ v,., where the

new right-side terms indicate the frequencies for electron-ion and electron-electron collisions. In the following Vee will not be considered. Whereas Vern is independent of spatial coordinates, this is, in general , not so with Vei which depends on the ion density , N t according to the relation [Shkarofsky , 1961].

- 0 N j 1 [0 T 3/2N- l /2] Vel- 1 -3 n 2 e e V

where 0 1 and O2 are constants. Ignoring the logarithmic term one can write,

since N i=N .,

214

(, I

where F(r) describes the electron density profile defined by N e(r) = NoF(r) , with N o being the electron density at the center.

When isothermal conditions prevail (Te= Tgas) the electrons obey a Maxwellian distribution so

that (u) =~kTgas, this is also a reasonable assump­

tion when Te is slightly in excess of Tgas due to an rf heating pulse (cross modulation). In this case and for vm< < w (low pressure approxi­mation) one ha; from eqs (1) and (2)

~A 47r [ r oo 3 0 fo d ~¢ = -3w Jo Ve"'V 2)V V

(3)

It is seen that for a nearly uniform electron dis tribu­t ion, F(r) ~ lone has

~A _ _ 47rr oo 30fod· ~¢ - 3wJ o v",v ov v. (3a)

In general Vm is a function of electron energy 10( = 7~mv2) and it can be shown [Phelps et al. , 1951] that one can deduce from the measured average quantities the energy dependence of Vern under certain condi tions. In this study the condition v! < < w2 is satisfied and the right side of (3) can be written, assuming vern> > Vei'

(4)

Based on previously reported results it is reasonable to assume a power law vem= NCvh and one obtains from (4)

(vem)=~ 7r- 1 /2 (2;T ) -~Ncreth) (4a)

where C and h are constants which may be found empirically from a best fit procedure, and N is gas density in molecules per cubic centimeter. If indeed there exists such a simple power law for Vern and measurements are performed with the same N, but various temperatures, T e= Tgns, one obtains h from the simple relation

(Vem) Tek= (t.A !~¢)Tek=(Tek) ~. (5) (Vemhel (~A!t.¢)Tel Tel

If h can be determined in this manner, 0 can be determined from (3) and (4).

I U

16

14

~ 12

00

o

A 10 E

> v

10

-- -----

0 3 . 0 TORR

• 1.5 TORR

6 0 .6 TORR

• •

66

o o

•

20 30 40

MICROSECONDS

o

50

•

FlGUUE l. - T ypical i'ilne variations of <Vm > as computed j1-om plasma insert'ion-loss measmements dllring the ajte1'­glow.

The scatter of the experimental pOints represents diITercnccs heiween sC\7eral I uns of 1I1cas llrcmenis. 'rhe broken lines indicaie wbere iho s tead y-sLaie va lue of <Vm> should have been, based 011 the one IOllncl at p=3.0 tO I'I' .

3. Results a.nd Discussion

In the preceding section it was shown how it is possible to obtain vem(u) for the low pressure approxi­mation under the assumption that Vem depends on v according to a simple power law. Figure 1 shows a plot of (~ A l t.¢)w versus t ime at Tgas =894 Ole It is seen that isothermal conditions are not present during the very early afterglow, thus perhaps invalidating (4). This is so because the assumption fo being 11axwellian may not be true even though the inequality vm< < w ( = 271" 1010 cycles sec - I) is well satisfied. At later times constancy of ( ~ AI ~¢)w is obtained indicating T e= Tgas in agreement with electron radiation temperature measurements. The results for this case are given in table 1 and it is seen that h= 2 yields good agreement with experi­mental resLuts. The relationship thus found reads Vern(u) = 4.4 X 1O- 8X u N(02) sec- I with u in electron vol ts or < vem>=3.0 X 105 pT. sec- I, with p (torr) referred to 273 OK. At temperatures below 569 OK steady state values of < vm> as fou~d for t!J.e .higher temperatures were not as concluslve pOllltmg to the possible noticeable contribution from < vet> ·

The spatial distrib.ution of N~ and t~ereby v.et must in this connectIOn be cons1dered sm ce (3) 1S the basis for the interpretation of the measurements. As has been pointed out [Mentzoni, 1965] the plasma seems to be axially and tangentially unifo~m based on luminosity measurements. The rad1al variation of N e(r) depends on the initial dis~ri­bution and the rate of ambipolar diffusion durmg

215

the afterglow. At the pressures pertaining to the present experimen t, i t is reasonable to assume a radially uniform initial distribution. The ratio

l RE(r)2F(r)2 r driRE(r) 2F(r) rdrappearing in (3) is

the spatial average F(r) of the electron-ion pro£le weighted by the quantity E(r) 2 F(r) with E(r) vary-

ing approximately as Jo(lci), thus, F(r) is most

sensi tiye to F(r) in the center region.

TABLE I.- Ratios of <vern > for various temperatures under isothermal condi tions compared wi th the case of V <m = NC v b (h = 2) and <V.m> being averaged over a Maxw ellian velocity d istribulion

T .k T . , < Vem> Tek /<Vem> Ttl (T , k/ T . ,)'12 h=2

894 734 1. 27 1. 218 734 569 1. 29 1. 290 894 569 1.64 1. 571 569 450 1.4 1.264

TABLE 2. Electron-ion density profiles f01' various times dW'ing the aftel'glow, T = SOO 0[(, p = O. S torr and D aP = 110 cm 2

sec- 1 torr 1

Time N . (t,r)/N,= F (t ,r )

r/R =O r/R =0.2 r/R=0 .4 r/R =0 .6 r/R =0.8 - -- - --------

¢iCC 10 0.980 0.980 0.980 0.980 0.961 20 .980 .980 .980 .979 .880 30 . 980 .980 .980 . 973 .796 40 .980 .980 .980 .959 . 725 50 .980 . 980 .978 .959 .667 60 .980 .980 .975 .915 .618

1 D, is ambipolar diffusion coeffi cient pertaining to the ions o~ .

10-IOXN. p A= IO-' X <v.m> ' B = IO- ' X < v, ,> b A+B lO-sX < vm> '

cm-3 TorT sec-1 sec-1 sec-1 sec-1

3.325 3.0 2.70 1. 53 4.23 4. 35±. 85 6.650 3.0 2. 70 2.89 5.59 6.07±1.17 9.975 3.0 2. 70 4.20 6. 90 8.5

6.650 1. 5 1. 35 2.89 4.25 3. 65±. 05 9.975 1. 5 1. 35 4. 20 5.55 5. 0

• Based on present resnlts, < v. m> =3.0XPXT.X10' (8e0-1).

N· (20X10'T " ' ) b Based on Chen 's 0xperimen tal results, < v.,>=3.6 T .;/2 In . N .'/2 '

(sec-1).

'Present direct measnrements.

Considering diffusion losses under isothermal con­ditions only [Mentzoni, 1964] F(r) can be computed as illustrated in table 2. At room temperature, Tgas= 300 OK, some of the results obtained are illus­trated in table 3. Due to the lack of isothermal conditions in the time range of the high electron-ion

densities at the lowest pressures the experimental points tend to lie below those to be expected from theory and previous experiments (see fig. 2) assuming Te= Tgas= 300 OK which in general follow the l aw

where Ct, C 2 are constants. Several theoretical and two experimental evaluations of 01 and O2 have been made previously yielding values of 01 which varies between the limits 2.25 and 4.84 X 103, and O2

between 3.32 X 103 and 20 X 103. It should be noted that Ginzburg and Gurevich

[1960] and Nicolet [1959] quote expressions for <Vei> 'which are somewhat modified in form as compared to (6) being <vel>~ 5.5NITe-3/2 In (2 X 102T e/N//3) and < ve;> = [59.1 + 4.18 10gIO (TN N e)] N ,fT e3/ 2 10- 6, respectively, in the latter equation the electron density is given per meter.3 At the pres­sures p = 3.0 mm Hg and p = 1.5 mm Hg i t is seen from table 2 that the agreement with Chen's [1964] results is fair considering the experimental error of the measurements.

From figure 2 it is seen that the experimental points for the two different pressures converge for the lower electron densities and electron temperature indicating that < Vel> is dominating. At the higher densi ties on the other h and, the increasing divergence of the experimental points for the two pressures is to be expected in the case < vem> is dominating. The solid curve in figure 2a indicate that even under nonisothermal conditions the present result for < vem> coupled with < Vel> as found by Chen is reasonably consistent with experimental observation of < I'm> and electron radiation temperature meas­urements. As the electron energy increases the

1 1

4000 \ 1

3500

>< 'L 3000

::J

~ 2500

~ 2000 Z

~ 1500

~ 1000

500

10

1

\ \ 1 1 1 1 \ \

o P <0< 0 .30 TORR o p " 0.15 TORR

v e i (CHEN)

T • T

12

\1

'0

9 =-\

. \: / , 8 ;;

7 .!'! \ \ \

~ (b)

6 ..... -<.-

5 ' ro

o

:xl 30 40 50 411 511 6" 7n 811 9~ Ian II" )211 MI CROSECONDS DIFFERENT1A l PHASE SHIFT , A <JI cc n

FIGURE 2.- (b) < v,.> vel'SUS the diffel'ential phase-shift 6¢ for T gas = SOO 0 [{ and pressures O. SO and 0.15 t01'1-.

The measured values are indicated by paints, the dash-dotted curve repre­sents <Vei> under isotbennal conditions according to Chen , and tbe solid curve is computed [rom < Vm> = < V.m>+<v. ,> =3.0 P T .XlG'+3.6

N , (2.0XlO' T ,312 ) X T,3/2 In N .lf2

with p=0.30 torr (referred to 273 OK ) and T , following tbe time decay cnrve shown in (a). Tbe solid portion of this cnrve (time2:20 I'sec) has beeu measured.

216

inelastic elec tron collisions involving rotational a~d vibrational excitation of the OA'Ygen molecules w~l also become increasingly important, however, theIr contribution to <vm> is expected to be small at the tempera tures considered hore.

The author thanks R. V. Row for his interest and stimulatino' discussions, and J. Donohoe for his able technical a~sistance during this research, This work was partly supported under contract with Air Force Cambridge Research Laboratories.

4. References

Bachynski, M. P., T. W. Johnston, and L P. Shkarofsky (1960), Electromagnetic properties of hIgh-temperature air, Proc. IRE <l8, 347.

Carruthers, J . A. (1962), Conductivi ty of electrons in gases weakly ioni zed by X-rays, Can. J. Phys. <l8, 1528.

Chen , C. I. . (1964), Electro n collisions in neon plas ma, Phys. R ev. 135, A627 .

Dalgarno, A. (l9G1 ), Charged parLicles in t he upper atmos­phere, Ann . Geophys. 17, 16.

Gilardini, A. (1963), pri vate co mmuni cation . . Ginzburg, L. , and A. V. Gurevich (1960), No~llD ear phe­

nomena in a plas ma located In an a lterna t Il1 g electro­magnetic field, Usp . F iz. Na uk. 70, 201.

Golant, V. E., and A. P . Zhilinskii (1960),. Propagation.of electromagnetic waves t hrough wavegLildes filled wIth plasma, Sov. Phys. Tech. Ph ys. 5, 12.

H ealey, R. H., J. W. Reed (1941),The behavior. of slow elec­t rons in gases (Amalgamated Wll'eless Lelt., SIdn ey).

Lawrence, R. S., C. G. Little, and II . J . A. Chi vers (1964), A survey of ionospheric effects upon earth-space l'adlO propagation , Proe. IEEE 52, 4. . . .

M entzoni, M. II. (1963), Effective electron reCOmbll1 atIO n Jll

h ea ted nitrogen, J. Geophs. Res. 68, 4181. M entzoni, lVI. n . (1964), Ambipolar diffusion in an oxygen

plasma at elevated te mpera t ures, Phys. Rev. 13<l, A80.

.i\Ientzoni, M . I-I. (1965), E lectron r emoval durin g t he early oxygen afterglow, J . Appl. Phys. . ..

Mentzon i, M . II ., and R. V. Row (19G3), Ro ta tIonal eXCItatIon a nd electron r elaxation in nitrogen, Phys . R ev. 130, 2312.

Nawrocki, P. J., and R. Papa (1961), Atmospheri c processes, Geophys ics Corp. of Ame~'i ca, R eport No. 61 - 37- A . .

Nicolet, M. (1959), The collISIOn frequency of electro ns 1Il the terrestrial atmosphere, Phys. Flui ds 2, 95.

Nielsen, R. A., and N . E. Bradbury (1937), :Electron a nd n egative ion mobili t ies in oxygen, air, nitrous ox ides and ammon ia, Phys. R ev. 51, 69.

Pack, J . L ., and A. V. Phelps (1959), Drif t veloc iLi cs of slow electron s in ni t rogen, argon and n eon, Bull . Am. Phys. Soc. II, <l, 317.

Phelps, A. V. (1960), Propagation constants for electro­magnetic waves in weakly ionized dry air, J. Appl. Phys. 31, 1723.

Phelps, A. V., O. T. Fundingsland, and S .. C. Brown ,<1951), Microwave determination of the probabIlIty of collISIOn of slow electron s in gases, Phys. R ev. 8<l, No.3, 559.

Reid, G. C. (1964), Physical processes in t he D-reglon of the ion osphere, R ev. Geophys. 2, 311. .

Shkarofsky, J . P. (1961), Values of t he t ranspor t coe ffiCIents in a plas ma for any degree of ionization based on a Maxwellia n d istribution , Can. J. Phys. 39, 1619.

Shkarofsky, J . P ., M . P. Bachynski, and .1'. W. J ohnston (Jun e 1961) Collision frequency assocIated WIth hI gh temperat m e' a ir and scatterin~ cross-sections of the co nstituents, Plan etary Space SCI. 6, 24- 46 . .. .

Un ger, n . G. (1957), Circula r electri c wave trans mIsS Ion In a dielectr ic coated " 'a veguido, Bell System Tech, J. 36, 1253.

Va n Lint V. A. J., E. G. Wi cker, and D . L . Trueblood (1959), Recomhination and attachment in ioni zed gases, R eport No. TR- 59- 43, General Atomic Di vis ion of General Dy namiCS Corporation, San Diego, Calif.

(Paper 69D2-455)

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