Further development of our work should include application of the methods described above to calculations of excited states of heteronuclear molecules and ions. Here, an additional difficulty is caused by the fact that in calculation of, let us say, threeelectron systems it is necessary to specially take into account the asymmetry in the distribution of the third electron. This, in particular, means that the basis must contain twocenterorbitalswith 8s ~ 0. LITERATURE CITED i. .

V. N. Kirnos, B. F. Samsonov, and E. I. Cheglokov, Izv. Vyssh. Uchebn. Zaved., Fiz., No. n , 103 ( 1984) . v. N. K i r n o s , B. F. Samsonov, and E. I . Cheglokov, I z v . Vyssh. Uchebn. Z a v e d . , F i z . , No.

2, 109 (1985). .

4. ,

6. 7. 8. 9.

B. V. 7, H. B. J. T. J.

F. N. 31 S. K. C. L. C.

Samsonov and V. N. K i r n o s , Paper d e p o s i t e d a t VlNITI, No. 635-85. K i r n o s , B. P. Samsonov, and E. I . Cheglokov, I z v . Vyssh. Uchebn. Z a v e d . , F i z . , No. (1986). Taylor and J. Gerhauser, J. Chem. Phys., 40, 244 (1964). Gupta and F. A. Matsen, J. Chem. Phys., 47, 4860 (1967). Browne, J. Chem. Phys., 45, 2707 (1966). Gilbert and A. C. Wahl, J. Chem. Phys., 47, 3425 (1967). Browne, J. Chem. Phys., 42, 2826 (1965).

METHOD OF DETERMINING THE SODIUM DIFFUSION COEFFICIENT IN INERT GASES UDC 537.525.5:546.33

V. K. Sveshnikov and N. M. Sveshnikova

A method of measuring the sodium diffusion coefficient in neon and argon is presented. It is based on recording the concentration change of sodium in a sodium discharge as it diffuses along a discharge tube. A diagram of the apparatus is given and the technique of measurements is described. It is pointed out that the proposedmethod can also be used for measuring the diffusion coefficient of other alkaline metals in inert gases and their mixtures.

Known methods of determining the diffusion coefficient, which are based on measuring the metal evaporation rate in an inert gas and the metal condensation rate from a vapor-gas mixture, are slow and are not applicable to measurements directly in gas-discharge instruments

[i]. In this work a method of determining the sodium diffusion coefficient in neon and argon is proposed. It is based on recording the concentration change of sodium diffusing along a discharge tube. Sodium mass transfer in the positive column of a low-pressure discharge, at a constant temperature of the discharge tube, takes place under the influence of an alternating electric field and a concentration gradient [2]. When part of the tube is separated from a region saturated with sodium vapor at a concentration N o and at a distance h0, satisfying the inequality

be

h0> 6/ then the sodium concentration N is related to the diffusion coefficient D and the time t by the relation M. E. Evsev'ev Mordovian Pedagogical Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 57-62, August, 1987. Original article submitted April 26, 1985; revision submitted December 9, 1985. 692

0038-5697/87/3008-0692512.509

1988 Plenum Publishing Corporation

(1)

h~ N = A'o crfc 2 I/D-t

(2)

'

where b is the mobility of sodium ions in the discharge, E is the electric field intensity, and f is the alternating field frequency. Formula (2) is the solution of the diffusion equation for the one-dimensional case. Its use presupposes that the concentration of the saturated sodium vapor N O has the steadystate value at the beginning of the measurement. According to [3], the diffusion coefficient can be determined from formula (2) under the conditions of the one-dimensional problem, when the length of the discharge tube h is considerably larger than the diffusion length A, i.e.,

The diffusion length for a cylindrical tube with a radius R can be found [4] from the formula (4) 9

_l a Then, using Eq. (4), inequality (3) takes the form R9

(5)

=9).

h=> 2,-?(1 -

Since for the majority of discharge tube designs R < 5"10 -= m, the value of the right-hand

side of inequality (5) is close to zero, and consequently

h:>>

~2

-

T3)

(6) I,

that is, in the case where

L >> 1,24 R,

(7)

the value of D can be found from formula (2). The concentration of sodium during its diffusion in the discharge can be determined from the luminous flux, which is proportional to the resonance radiation power of the sodium llne. Radiation power can be determined from the formula of Fabrikant [5], which can be reduced to the canonical form

W =hvNn~BF,

(8)

where hv is the energy of the emitted photon,

q s = 2,72q~(Um)

Ne

is the electron

concentration

in the discharge;

a

1

e

1/

8~T.

Um._U~

~T e

l--

=m

is the average velocity of the plasma electrons, q B ( ~ ) is the maximum value of the effective cross section of atom excitation, U B is the excitation potential of the resonance level, is the potential for which the effective cross section takes on the maximum value, and T e is the electron temperature. For small values of the discharge current density, j < 40 A'm -2, and at an inert gas pressure of the order of 2.0 kPa, electron temperatures in the inert gas or in a mixture of gases are similar. Therefore, at constant discharge current density (9)

F Fo

W II ~,

,\' Nn

693

P2

i

....T

Fig. 1 Here, F and F 0 are luminous fluxes corresponding respectively to sodium concentrations N and No in the discharge. Thus, with allowance for Eq. (9), Eq. (2) takes the form

F_.= eric ho

~o

(10)

2Vb~"

Thus, when conditions (i) and (7) are fulfilled and at the steady-state value of the pressure of saturated sodium vapor in the evaporator, the diffusion coefficient in the alternating current discharge can be determined from the time t and the ratio of luminous fluxes from the region saturated with sodium vapor and from the part located at a distance h 0 away. Determination of the diffusion coefficient was carried out by means of the apparatus shown schematically in Fig. i. The apparatus consists of a thermostat equipped with chambers 4 and 5. The discharge channel of tube 1 in chamber 4 is heated to the operating temperature. The evaporator 2 is an ampul of sodium. It is located in chamber 5 of the thermostat. To shorten the heating time of sodium to the tube temperature, it is heated in two stages. A preheating to the sodium melting temperature takes place in chamber 5 of the thermostat, and further temperature increase to the operating temperature is accomplished stepwise using heater 3. The evaporator temperature is precalibrated against the power demand of heater 3. The settling time 9 of the sodium temperature in the evaporator is 9 see. The temperature in the thermostat chambers is controlled by thermometers 9 and 10. The discharge tube is powered by an alternating voltage taken from the output of amplifier 12. The amplifier input is connected to the reference generator 13. The stability of the discharge current in the system is sustained by the electronic stabilizer ii. The luminous flux of the discharge radiation is measured in relative units by silicon photocells 6 and 7. Linear dimensions of the photocells used were(10 • 6).10 -3 m. A light filter was used for selective separation of sodium lines from the spectrum emitted by the discharge. Photocell 6 was designed to measure sodium radiation in the area saturated with sodium vapor. From photocell 7, place d at a fixed distance h 0 from photocell 6, a signal is taken whose magnitude is determined by the sodium concentration in the tube at a given instant of time. The electric signals from photocells 6 and 7 are applied to the input of the HZZ8-2 two-channel automatic recorder 8. Signal recording on chart paper is done by switching on heater 3 with switch PI, which is mechanically coupled to switch P2. To measure the sodium diffusion coefficient, the discharge tube is placed in the thermostat. At a frequency of 300-400 Hz the discharge in the tube is ignited and the discharge current, whose density does not exceed 40 A-m -2, is set up, after which heaters in chambers 4 and 5 are switched on. Chamber 4 is heated to the operating temperature of 493-553~ * Chamber 5 is heated to a temperature of 370~ which is close to the melting temperature of sodium. Upon establishing steady-state temperature conditions in the chambers, the heater and the automatic recorder, which registers signals from photocells 6 and 7, are switched on with switch PI. To calculate D from formula (10), one records the time t of sodium diffusion in the tube from the instant of establishment of a steady-state value for the concentrated sodium vapor pressure in the evaporator, and hence for the photocurrent from photocell 6, which is registered by the automatic recorder. In this case, the settling t i m e o f the sodium temperature in the evaporator should be significantly smaller than the diffusion time of sodium in the tube, counted from the moment of heater switching, i.e., *Noticeable radiation of sodfum discharge begins at a temperature above 450~

694

,

m29sec"I

'

.

t

lJ ~ - - - - ~

x Experiment r

2-----~

5/3

495

Fig.

.,~3

K

2

--~<1. ~- t

(ii)

From the charts obtained in this way, signals proportional to luminous fluxes are determined, as well as the diffusion time calculated from the fixed speed of the chart. Then, from the ratio of luminous fluxes F/F0 = 0.7, the time t, and the distance h0, one determines the diffusion coefficient. Figure 2 shows calculated and experimental temperature dependences of the sodium diffusion coefficient in neon (i) and argon (2). The diffusion coefficient measurement was done in tubes with diameter 1.8"I0 -2 m and length 5.4"10 -I m. The inert gas pressure in the tubes at a temperature of 295~ was 1 kPa. Measurements were carried out at a frequency of 400 Hz with a current of 4"10 -2 A, which satisfies conditions (i) and (7). The distance between photocells was chosen to be equal to 4.0"10 -I m. Inequality (ii) then holds true since the diffusion time of sodium in argon at the discharge tube temperature of 553~ is 72 sec, and in argon it is 148 sec. Therefore condition (ii) is fulfilled for both neon and argon, i.e., 0.i << 1 and 6"10 -2 << I, respectively. Due to the lack of literature data, a calculation of the sodium diffusion coefficient in neon was carried out using the formula derived from kinetic theory by Enskog and Chapman, with an empirical correction by Uilk and Li [6]. Values of the sodium diffusion coefficient in argon were compared with the experimental data taken from [i], which we recalculated for specified temperatues and pressures of argon. 553~

Values of the sodium interdiffusion in an inert gas at a temperature T between 493 and can be approximated by a formula similar to [7]

O = Do where Do = 7"10 -3 and n = 1.4 for neon; the values D O = 2.7-i0 -s and n = 3.5 for argon were obtained from experiment. A comparison of the data obtained with the results of calculations for neon and with the published data of other authors for argon showed that the discrepancy in the determination of the diffusion coefficient does not exceed 10%. Let us consider the sources of possible errors in determining the sodium diffusion coefficient in inert gases using the proposed method. The relative error in determining the diffusion coefficient from formula (i0) is determined by the relative errors 71, 72, 7s, and 74 associated respectively with the settling time of the temperature of the evaporator, the measurement accuracy of the luminous flux, the diffusion time of sodium in the inert gas, and also, as follows from Eq. (12) with the temperature at which the diffusion process is realized. Since the relative errors 71, ~2, 73, and 74 are independent and their values correspond to the same confidence levels, the error 7 in determinng the diffusion coefficient can be calculated as the rms value of the given relative errors -,.... ~

(13)

Let us determine the error YI associated with the settling time of the temperature of the evaporator. Emission of the luminous flux F registered by photocell 6 begins when the sodium temperature in the evaporator T I = 493~ [8]. Therefore, formula (i0) is determined for a

695

TABLE 1 Relative error Inert gas

u Neon Argon

u I u I Y*

3,9 2,7

8,5 8,0

2,8 2,5

0,5 1,2

9,8 8,9

time interval To + t beginning from the moment of the appearance of the luminous flux. Let us find the time interval T O from the onset of the sodium glow to the establishment of the steady-state temperature in the evaporator. Since the establishment of a steady-state temperature in the evaporator occurs in a very short time interval 9 = 9 sec~ one can assume that the temperature in the range from T O ffi 370~ to T 2 ffi 553~ changes linearly. Thus the time T 0 is equal to

T~ -- T, o

-

-

(14)

~T~_T ~

which is equal to 3 sec. Now, using formula (i0), let us calculate the change in the sodium diffusion coefficient in neon and argon when the sodium diffusion time in the tube is changed from t to t + T0, that is, from 72 to 75 sec for neon and from 148 to 151 sec for argon. The change found in the sodium diffusion coefficient in neon and argon is equal respectively to 3.9 and 2.7%. These values are significantly overestimated since we assumed that over the interval T o the evaporator temperature was steady (553~ although in reality it was reached only after that interval. The error in determining the luminous flux is due to nonlinearity of the automatic recorder amplifier H338-2.* For a recording channel width equal to 7"10 -= m, this gives rise to a change of the luminous flux by 2.1% [9], which corresponds to a relative error Y2 of 8.5% in determining the sodium diffusion coefficientinneon and of 8% in argon (Table i). The relative error Ys in determining the sodium diffusion coefficient in neon and argon due to inaccuracy in the time measurement is also determined by the accuracy of the automatic recorder. According to [9], the relative error 6 in the time recording of the automatic recorder can be found from the relation

~___ (__~s 1).lO0/%.72n[o

(15)

Here, s is the distance at which the chart was shifted during the experiments and v n is the nominal speed of the chart. For the H338-2 automatic recorder with v n = 5"10 -3 m/sec, s = 1 m, and t o = 205 sec, the error 6 = 2.5%. This error leads to an error ~3 of 2.8% for the diffusion of sodium in neon and of 2.5% in argon. The temperature measurement in the thermostat chambers was done using mercury thermometers of the U-7 type having a maximum reading of 300~ and an accuracy of • [i0]. Therefore, the error y~ in determining the sodium diffusion coefficient in neon is equal to 0.5% and in argon, 1.2%. In Table 1 the rms relative error y of the sodium diffusion coefficient in neon and argon, calculated from Eq. (13), is also given; it is equal respectively to 9.8 and 8.9%. These values are also smaller than 10%, which further confirms the reliability of the proposed method. As follows from Table i, the most significant contribution to the error in determining the sodium diffusion coefficient in an inert gas is due to the accuracy of the luminous flux measurement. *For the case of identical characteristics of the photocells used in the apparatus.

696

The proposed method can be used to measure the diffusion coefficient of other alkaline metals in inert gases and their mixtures. LITERATURE CITED I. 2. 3. 4. 5. 6. 7. 8. 9. i0.

K. M. Aref'ev, V. M. Borishchanskii, et al., in: Thermal Properties of Gases [in Russian], Nauka, Moscow (1973), p. 49. B. K. Sveshnikov, Izv. Vyssh. Uchebn. Zaved., Fiz., No. 2, 3 (1985). V. V. Karpukhin, I. A. Sokolov, and G. D. Kuznetsov, Physlcochemical Principles of Technology of Semiconductor Materials [in Russian], Metallurgiya, Moscow (1982), p. 53. Yu. P. Raizer, Principles of Modern Physics of Gas-Discharge Processes [in Russian], Nauka, Moscow (1980), p. 85. V. L. Granovskii, Electric Current in a Gas [in Russian], Nauka, Moscow (1971), p. 544. S. Bretsznajder, Properties of Gases and Liquids [Russian translation], Khimiya, Moscow (1966), p. 535. N. B. Vargaftik, Handbook of Thermal Properties of Gases and Liquids [in Russian], Nauka, Moscow (1972). V. I. Gaponov, Electronics [in Russian], Part II, GIFML, Moscow (1960), p. 390. M. G. Berdichevskii, V. A. Ivantsov, B.A. Lapin, and M. G. Yakubov, Electric Recording Instruments [in Russian], ~nergoizdat, Leningrad (1981). E. B. Ametisov, V. A. Grigor'ev, B. G. Emtsev, et al., Heat and Mass Transfer, Thermotechnical Experiment: Handbook [in Russian], E'nergoizdat, Moscow (1982).

ENERGY SPECTRUM OF FAST ATOMS IN A DARK CATHODE SPACE V. V. Kuchinskii and E. G. Sheikin

UDC 537.525

A method is proposed for solving the kinetic equation for fast atoms moving in a proper gas. We find the Green's function which permits computation of the energy spectrum of fast atoms for an arbitrary source function in the volume of a proper gas or an amorphous body having plane boundaries. The energy spectrum of fast atoms is computed in a dark cathode space. The solution obtained is used to compute the contribution of fast atoms to atomization of the cathode surface.

Fast atoms in a dark cathode space exert substantial influence on the gas discharge characteristics. By delivering part of its energy during elastic collisions with thermal atoms of the discharge gas carrier, fast atoms participate in the formation of the gas temperature in the near-cathode area of the discharge. Electron emission from the cathode surface under the influence of fast atoms can alter substantially the existence conditions for a self-consistent discharge. Atomization of the cathode surface by fast atoms, togethe~ with ionic atomization, plays a substantial part in the formation of metal atom concentration in a discharge. The origination of fast atoms in a dark cathode space is due to the charge transfer process proceeding efficiently in this domain. The energy spectrum of fast atoms is formed because of elastic scattering of the charge transfer atoms. The energy spectrum of fast foods was found in [i] under the assumption that the scattering section of the charge transfer atoms by the thermal atoms was zero. By using results presented in [2, 3], it is easy to shows that as the energy of the collidng atoms varies between 100 and 1 eV, the ratio of the elastic scattering section of the colliding atoms a to theresonance charge transfer section varies between 0.i and 0.31 for Ar and 0.15 and 0.38 for Ne. Consequently, the results in [i] can be utilized only for rough estimates. A. A. Zhdanov Leningrad State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 8, pp. 62-67, August, 1987. Original article submitted December 27, 1985.

0038-5697/87/3008-0697512.50

9 1988 Plenum Publishing Corporation

697