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Local thermodynamic equilibrium in an RF argon plasma
J. Qunnr.
Spectrosc.
Rodior.
Transfer.
Vol.
8, pp.
141 l-1418.
Pergamon Press1968. Printed in
Great
Britain
LOCAL THERMODYNAMIC EQUILIBRIUM RF ARGON PLASMA* P. D.
SCHOLZ~
and T. P.
IN AN
ANDERSON$
University of Iowa, Iowa City, Iowa (Received 29 January
1968)
Abstract-The existence of local thermodynamic equilibrium in the center half of an inductively sustained RF argon plasma was investigated using emission spectroscopy as the diagnostic tool. The experiment was conducted for a fixed oscillator frequency of 4.2 MHz, a fixed plate input power of 2.0 kW, a constant average gas velocity of 9.0 cm/set, and for pressure levels of 0.01, 0.1,05, and 1 .O atm. Four emission lines and the continuum centered at three different wavelengths were used in the investigation to show that local thermodynamic equilibrium exists at I.0 and 05 atm and that within the experimental error the bound states 3p, and above are populated with a Boltzmann distribution at all the pressures investigated.
1. INTRODUCTION
A LABORATORY generated plasma is most likely to be in a state of nonequilibrium due to the severe boundary conditions of confinement and to the highly nonequilibrium methods of generating the plasma. Before any thermal-radiation phenomena can be investigated, it is necessary to determine the exact state (i.e. the local values of the macroscopic and microscopic coordinates) of the plasma. If experimental conditions are such that a state of local thermodynamic equilibrium can be identified, the task is greatly simplified and is reduced to determining only the macroscopic equilibrium coordinates (i.e. pressure, temperature, and composition). Once an equilibrium state has been identified, investigations of nonequilibrium effects may be pursued using this equilibrium state as the datum state. The identification of states of local thermodynamic equilibrium as a function of the bulk plasma static pressure is the subject of this publication. From an atomic point of view, the macroscopic thermodynamic equilibrium state for a monatomic ionized gas requires that all of the species have Maxwellian velocity distributions at the same temperature and that the atomic and ionic levels be populated with a Boltzmann distribution at that temperature. In this case the population distribution is determined by the principle of detailed balance involving radiative and collisional processes. Since the diagnostic tool used in this investigation was plasma spectroscopy, a review of some of the techniques is briefly presented in Section 2. Section 3 is devoted to a discussion of the plasma source, the spectroscopic system, and the experimental results. The conclusions of the investigation are given in Section 4. * This work was conducted while both authors were associated western University, Evanston, Illinois. t Assistant Professor of Mechanical Engineering. $ Chairman and Professor of Mechanical Engineering. 1411
with the Gas Dynamics
Laboratory,
North-
1412
P. D. SCHOLZand T. P. ANDERSON 2. SPECTROSCOPIC
CONSIDERATION
The excitation temperature T,, associated with a Boltzmann population distribution among the atomic bound levels, may be determined from the slope of a plot of In nJg, ( = In 4nZ,,/g,A,,hv,,) versus E,, where standard spectroscopic notation has been employed. The parenthetical equality is for line emission neglecting both self absorption and stimulated emission. For a local thermal equilibrium (LTE) plasma the expression for line emission is a function of electron number density and kinetic electron temperature Te. For low degrees of ionization, the LTE model (which is now less restrictive than the conditions specified for local thermodynamic equilibrium) assumes that all bound levels have a Boltzmann distribution, at T,, that the free electrons have a Maxwellian velocity distribution at T,, and that the species number densities are related by the Saha equation at T,.(l) A necessary condition for LTE is given by Griem’s criterion based on a collisional rate which is ten times larger than the radiative transition rate. (2) Using the argon Is,- lp, resonance transition, this criterion becomes n, 2 1.5 x 1015TL/2 [cm-3]
(1)
where T, is in “K. For a plasma with a thermal limit at bound level I, the states below level r are underpopulated, while the upper states are populated with a Boltzmann distribution.@’ The line emission for a transition originating in an upper state, which lies above the thermal limit, is related to the electron number density and the excitation temperature associated with the upper level Boltzmann population distribution. In this case the excitation temperature is equal to the electron kinetic temperature due to the collisional equilibrium between the free electrons and the upper bound levels. Hence,
where the subscript T.L. denotes the use of lines with E, lying above the thermal limit. The electron number density criterion, assuming that a thermal limit exists at E, = 13.46 eV, for an argon plasma is (n,), 2 2.6 x 10’2T~‘2
[cme3]
(3)
where the subscript r denotes that the criterion is based upon level E,. The continuum emission for a neutral, low degree of ionization plasma is given as a function of electron number density and electron temperature
1, = Me9 T,)
(4)
and requires that the free electrons have a Maxwellian velocity distribution.‘4*5’ In complete local thermodynamic equilibrium the thermodynamic composition tables@ may be used (to relate electron number density and temperature) in conjunction with the LTE model to reduce the expression for the absolute line emission to a function only of the equilibrium temperature Teq. The tables may likewise be used with equation (4) to reduce the continuum emission to a function only of equilibrium temperature.
Local thermodynamic
equilibrium
in an RF argon
plasma
1413
For a plasma with an electron temperature of 104’K (of the order found in these experiments) inequalities (1) and (3) give n, 2
1.5 x 10”
[cmm3]
(n,), 2 2.6 x 1Ol4 [cmP3],
for E, = 13.46 eV.
The thermodynamic equilibrium electron number density at 104”K and 1-Oatm is l-5 x lOi cmP3 and at 104”K and 0.01 atm it is 2 x 10” crn3. Hence, LTE should not exist at 104’K for an argon plasma operating between 0.01 and 1.0 atm. However, over the same pressure range and temperature, the criterion for a thermal limit at or above 13.46 eV is satisfied (i.e. must use lines with E, > E,). Thus, with measured values of I,, and I, equation (2) may be solved simultaneously with equation (4) to give T, and n, when the plasma at least satisfies the equilibrium requirements for a thermal limit. If T, equals T,, and Teq, then local thermodynamic equilibrium prevails. Departure from equilibrium then can be observed by comparing these three temperatures. It is the comparison of these three temperatures as a function of pressure (i.e. collision frequency) which was used to detect the existence of local thermodynamic equilibrium in this study. The existence of a Boltzmann distribution was verified within 4 per cent above levels of 13.33 eV over an energy spread of 2.2 eV in upper levels at the lowest pressure (i.e. 001 atm) of the range investigated. The plasma was assumed to be optically thin for this investigation. From a rigorous standpoint, this simplication of treating only an optically thin plasma uncouples the radiation field from the thermal field and explicitly implies non-thermodynamic equilibrium.“’ However, if the collisional rates exceed the radiative rates for populating atomic levels, then detailed balancing by collisional processes alone is approximated. Hence, the electrons have a thermal distribution and the radiation field is a result of spontaneous emission from the various line (discrete) and continuum (continuous) levels. In keeping with the transparent assumption, the lines selected for the line emission measurements were thus selected, in part, for a minimal self-absorption.
3. EXPERIMENTAL
INVESTIGATION
(A) Plasma source The argon plasma was contained within a water cooled quartz channel and sustained by the time varying EM field of a solenoid which was driven by an oscillator (i.e. induction heater). The channel pressure of the inductively sustained plasma was easily controlled by the use of a vacuum reservoir system (see Fig. 1). The pressure range for this study was from 0.01 to.1.0 atm. The channel i.d. was 25 mm and the average convective gas velocity was 9.0 cm/set. The induction heater unit was an RFC, variable frequency, oscillator, which, for this study, was operated at 4.2 MHz and with an input power of 2000 W. The power coupling efficiency of the unit at 1.0 atm is given in Ref. 8. The d.c. plate supply of the unit was modified by the addition of a II-section plus an L-section filter with a ripple percentage of 0.025 per cent to eliminate the 120 Hz modulation of the RF tank current. (The effect of the modulated current was to create a 120 Hz pulsating plasma radiation field.)
1414
P. D. SCHOLZ and T. P. ANDERSON SURGE I
TANK7
CHANNEL COOLING WATER IN END COLLAR
w_A3MA
CHANNEL -
TIE
I-c
I I
RODS
TANK COIL
....__..
VACUUM CHANNEL
INLET VALVE
II
II
PUMP
MANOMETER
FLOWMETER
-
PRESSURE TEMPERATURE u
GAS
SUPPLY
FIG. 1. RF plasma
channel
with control
system.
(B) Spectroscopic system The spectroscopic system consisted of a Perkin-Elmer grating monochromator, an optical system, and a photoelectric detection system. The optical system included achromatic collimating and condensing lenses, optical filters to exclude the undesired overlapping orders from the grating, and a traversing mechanism for spatially scanning the plasma. The monochromatic output was detected by an RCA lP21 photomultiplier and was recorded by a strip chart recorder as a function of lateral plasma position, thus giving a lateral intensity profile. The strip chart deflections, corresponding to relative intensities, were calibrated by the use of a calibrated photometric tungsten ribbon lamp. The absolute lateral intensity profiles were inverted to give absolute radial profiles by an Abel integral inversion technique. Refraction effects, due to the curved composite channel wall, were found to have negligible effect on the Abel inversion over the center half of the plasma. Wall reflections from the inside wall of the quartz channel tube, which resulted in non-zero plasma intensities at
Local thermodynamic
equilibrium
in an RF argon
plasma
1415
the wall, were also found to be negligible in the inversion when only the center half of each inverted profile was retained for consideration. (C) Results The results of the spatially resolved absolute line and absolute continuum intensities were obtained for the following experimental conditions: a fixed oscillator frequency of 4.2 MHz ; a fixed average argon gas velocity of 9 cm/set ; a fixed power input of 2 kW ; and a fixed tank coil-channel geometry. Spectroscopic measurements were taken at static channel pressure levels of 1.0, 05, 0.1, and 0.01 atm. The argon atom lines included the 4259A1, 43OOA1, 4345A1, and the 5572AI. The continuum was determined at 4259A, 4300 A, and 6965 A by averaging the continuum measurements taken at equal wavelength increments above and below the lines at 4259, 4300, and 6965 A, respectively. The lateral line intensity profile for each line was determined in the normal manner by subtracting the folded continuum lateral profile from the folded line plus continuum profile. The absolute intensities of the three blue lines were checked in the first and second orders. The exit slit bandpass for the first order measurements varied from 6-OA to 12.2 A, and for the second order, the range was from 4.3 to 4.4 A. A Kodak Wratten blue filter was used to isolate these lines from the overlapping orders. The first and second order results agreed to within 4 per cent to indicate that the full line width was passed by the monochromator exit slit. The entrance slit was set at 10 p for all the measurements. The spectral traverses were all taken at an axial position mid-way between the top two turns of the five turn solenoid. The centerline lateral intensities were reproducible to within 4 per cent for traverses with the same experimental conditions. Figure 2 shows a plot of the half radial temperature distribution at 1.0 atm. Shown are the temperatures determined from the slope of In nu/g, versus E, (i.e. T,,) using the four lines and the temperatures determined from the simultaneous solution of equations (2) and (4) using the measured line and continuum intensities at 4259 A (i.e. T,). Also shown are the ranges of Teq, which are based on the a priori assumption of local thermodynamic equilibrium and were determined from the absolute line intensities of the four lines and from the three absolute continuum intensities. The transition probabilities of the lines were calculated at 1.0 atm using the radial line intensities and the average equilibrium temperature determined from the three different continuum intensities at each of the radial zones. The transition probabilities were averaged over the first half of the total number of radial zones. In essence, the transition probabilities were adjusted to force the equilibrium temperatures determined from the line intensities to agree with those from the continuum intensities at 1-Oatm. This is a valid process as long as the plasma at 1.0 atm is in a state of local thermodynamic equilibrium. (This was verified by showing that T_ determined from the absolute continuum intensities was equal to T,, and T, at 1.0 atm.) The values of the transition probabilities are given in Table 1. These values are listed for completeness and are representative of the values listed in the literature.‘9-‘5’ The solid curve of Fig. 2 is based on the absolute continuum intensity at 4259 A and the equilibrium assumption. The error bars of the slope temperature are due to the uncertainty in determining the best fit line through the data points of the four lines. The maximum error for the simultaneously determined temperatures is f3.3 per cent for a possible + 4 per cent error in the measured line and measured continuum intensities. The simultaneous temperatures are about 3 per cent high due possibly to the neglect of the ionization
1416
P. D. SCHOLZ and T. P. ANDEMON
r
-
I
I
r
FROM 5 ,N?ENSlTY
AN. CON?. AT 4259 8
RANGE OF T, FROM CONT. AT 4 51, 4300, AND 6965 % RANGE OF TaR FROM 4259Al, 43OOAI, 434!5Al, AND 5572Al ADS. LINE INTENSITIES FROM
SLOPE
\ \ \
OF
\ q
SYULT. TEMP. (T 1 FROM 42SSAl AND COk AT 4259 8
.2 *4 NONMMENSIONAL
G -8 PLASMA RADIUS FIG. 2. Radial temperature distribution at 1.0 atm. TABLE
Line
I. EXPERIMENTALLY DETERMINED TRANSITION PROBABILITIES
Averaged A,, and percent variation over one-half of radial zones at 1.O atm
A,, x IO-’ [set-‘1 4259Al 4300AI 4345Al 5572Al
I.0
0,339 0.033 1 0.0273 0.0647
[%I + 0.73 +2.2 +2.8 ,2.5
potential lowering effect in equation (2). Radial temperature distributions at 0.5, 0.1, and 0.01 atm were also determined using the same techniques as were used in calculating the profiles shown in Fig. 2. The results of these distributions at the plasma non-dimensional radius of 0.17 are given in Fig. 3.
1417
Local thermodynamic equilibrium in an RF argon plasma
Figure 3 is a plot of temperature versus channel static pressure at a non-dimensional radius of 0.17 (i.e. the fourth radial zone in the inversion scheme). The nomenclature is the same as used for Fig. 2, where the solid curve is Teqand is based on the absolute continuum intensity at 4259 A.
-
I I
f 0
Te, FROM ABS. CONT. IliiENSlTY AT 4259 8 FROM RANGE OF Tlo CONT. AT 4259, 4300, AN0 6965 % FROM RANGE OF TIP 4259AI, 43OOAI, 4345Al, AN0 5572Al ABS. LINE INTENSITIES T
FROM
I:‘$/Q,
VI
SLOPE
OF
E,
SIMULT. TEMP. (T 1 FROM 425SAI AND COJkT. AT 425s %
I
I
o-01
0.1 CHANNEL
STATIC
I
O-5 PRESSURE
FIG. 3. Temperature vs. channel pressure at a non-dimensional
I
I.0
(otm) plasma radius of 0.17.
(D) Discussion of results The off-axis temperature peak observed in Fig. 2 disappears as the pressure decreases. The radial temperature gradients in the center half of the plasma are relatively small (i.e. IVTl < 530”K/cm). (It was also observed that the axial gradients are even smaller in the region of these observations at all the pressures.) In the center half the excitation temperatures agree within 3 per cent with the kinetic electron temperature and with the equilibrium line and continuum temperatures (i.e. T,, = T, N T,,).
1418
P. D. SCHOLZ and T. P. ANDERSON
The results at a non-dimensional radius of 0.17, shown in Fig. 3, are typical of the temperature-pressure results obtained at each of the radial stations in the center half. At 0.5 and 0.1 atm the temperatures agree within 4 per cent. However, at 0.1 atm it was observed that the excitation and electron temperatures began to diverge from the equilibrium temperatures near the non-dimensional half radius. This divergence increased with radial distance. At 0.01 atm the excitation temperature agrees within 4.5 per cent with the electron temperature. However, the so-called line and continuum equilibrium temperatures fall outside the error bars and are 17 per cent low. The electron number densities obtained from equations (2) and (4) agreed very well (i.e. within 1.5 per cent) with the equilibrium values at all the pressures except at 0.01 atm where they differed by an order of magnitude. 4.
CONCLUSIONS
The central core of the RF argon plasma, operating as specified above, is in local thermodynamic equilibrium at 1-O and at O-5atm, and very nearly so at 0.1 atm. This state of equilibrium is due in part to the existence of low (i.e. /VT1 I 530”K/cm) axial and radial temperature gradients. At these pressures the plasma is in an apparent state of LTE and the thermal limit extends down to the ground state. The LTE electron number density criterion given by equation (1) appears to be conservative for this plasma. Significant departure from local thermodynamic equilibrium begins to occur at about 0.1 atm and increases as the pressure decreases. At O-01atm the plasma is no longer near a state of local thermodynamic equilibrium. However, the upper bound levels are Boltzmann populated (within the experimental error), and the thermal limit lies between the ground state and the 3p, (4300AI) level. This result is in agreement with the thermal limit criterion given by equation (3). REFERENCES 1. R. W. P. MCWHIRTER, Plasma Diagnosiic Techniques (Edited by R. H. HUDDLESTONEand S. L. LEONARD), pp. 201-206. Academic Press, New York (1957). 2. H. R. GRIEM, Phys. Rev. 131, 1170 (1963). 3. R. WILSON. JQSRT2,477 (1962). 4. J. POMERANTZ. The Influence of the Absorption of Radiation in Shock Tube Phenomena, Navord Report 6136, U.S. Naval Ordnance Laboratory (1958). 5. C. J. CHEN, Valid Conditions for the Kramers-Unsiild Continuum Theory in a Non-Equilibrium Plasma, JPL Technical Report No. 32-707 (1964). 6. K. C. DRELLISHAK, C. F. KNOPP and A. B. CAMBEL, Partition Functions and Thermodynamic Properties of Argon, Gas Dynamics Report A-3-62, Northwestern University (1963). 7. R. N. THOMAS, Some Aspects of Non-Equilibrium Thermo&namics in the Presence of a Radiation Field. University of Colorado Press (1965). 8. P. D. SCHOLZ and T. P. ANDERSON, AIAA J. 5, 1022 (1967). 9. H. N. OLSEN, JQSRT 3, 305 (1963). 10. H. W. DRAWIN. Z. Physik 146,295 (1956). 11. B. D. ADCOCK and W. E. G. PLUMTREE,JQSRT 4,29 (1964). 12. R. H. GARSTANG and J. VAN BLERKOM, J. Opt. Sot. Am. 55, 1054 (1965). 13. W. E. GERICKE, Z. Astrophys. 53, 68 (1961). 14. C. H. POPENOEand J. B. SHUMAKER,JR., J. Res. natn. Bur. Stand. 69A, 495 (1965). 15. P. B. COATES and A. G. GAYDON, Proc. R. Sot. A293,452 (1966).
Spectrosc.
Rodior.
Transfer.
Vol.
8, pp.
141 l-1418.
Pergamon Press1968. Printed in
Great
Britain
LOCAL THERMODYNAMIC EQUILIBRIUM RF ARGON PLASMA* P. D.
SCHOLZ~
and T. P.
IN AN
ANDERSON$
University of Iowa, Iowa City, Iowa (Received 29 January
1968)
Abstract-The existence of local thermodynamic equilibrium in the center half of an inductively sustained RF argon plasma was investigated using emission spectroscopy as the diagnostic tool. The experiment was conducted for a fixed oscillator frequency of 4.2 MHz, a fixed plate input power of 2.0 kW, a constant average gas velocity of 9.0 cm/set, and for pressure levels of 0.01, 0.1,05, and 1 .O atm. Four emission lines and the continuum centered at three different wavelengths were used in the investigation to show that local thermodynamic equilibrium exists at I.0 and 05 atm and that within the experimental error the bound states 3p, and above are populated with a Boltzmann distribution at all the pressures investigated.
1. INTRODUCTION
A LABORATORY generated plasma is most likely to be in a state of nonequilibrium due to the severe boundary conditions of confinement and to the highly nonequilibrium methods of generating the plasma. Before any thermal-radiation phenomena can be investigated, it is necessary to determine the exact state (i.e. the local values of the macroscopic and microscopic coordinates) of the plasma. If experimental conditions are such that a state of local thermodynamic equilibrium can be identified, the task is greatly simplified and is reduced to determining only the macroscopic equilibrium coordinates (i.e. pressure, temperature, and composition). Once an equilibrium state has been identified, investigations of nonequilibrium effects may be pursued using this equilibrium state as the datum state. The identification of states of local thermodynamic equilibrium as a function of the bulk plasma static pressure is the subject of this publication. From an atomic point of view, the macroscopic thermodynamic equilibrium state for a monatomic ionized gas requires that all of the species have Maxwellian velocity distributions at the same temperature and that the atomic and ionic levels be populated with a Boltzmann distribution at that temperature. In this case the population distribution is determined by the principle of detailed balance involving radiative and collisional processes. Since the diagnostic tool used in this investigation was plasma spectroscopy, a review of some of the techniques is briefly presented in Section 2. Section 3 is devoted to a discussion of the plasma source, the spectroscopic system, and the experimental results. The conclusions of the investigation are given in Section 4. * This work was conducted while both authors were associated western University, Evanston, Illinois. t Assistant Professor of Mechanical Engineering. $ Chairman and Professor of Mechanical Engineering. 1411
with the Gas Dynamics
Laboratory,
North-
1412
P. D. SCHOLZand T. P. ANDERSON 2. SPECTROSCOPIC
CONSIDERATION
The excitation temperature T,, associated with a Boltzmann population distribution among the atomic bound levels, may be determined from the slope of a plot of In nJg, ( = In 4nZ,,/g,A,,hv,,) versus E,, where standard spectroscopic notation has been employed. The parenthetical equality is for line emission neglecting both self absorption and stimulated emission. For a local thermal equilibrium (LTE) plasma the expression for line emission is a function of electron number density and kinetic electron temperature Te. For low degrees of ionization, the LTE model (which is now less restrictive than the conditions specified for local thermodynamic equilibrium) assumes that all bound levels have a Boltzmann distribution, at T,, that the free electrons have a Maxwellian velocity distribution at T,, and that the species number densities are related by the Saha equation at T,.(l) A necessary condition for LTE is given by Griem’s criterion based on a collisional rate which is ten times larger than the radiative transition rate. (2) Using the argon Is,- lp, resonance transition, this criterion becomes n, 2 1.5 x 1015TL/2 [cm-3]
(1)
where T, is in “K. For a plasma with a thermal limit at bound level I, the states below level r are underpopulated, while the upper states are populated with a Boltzmann distribution.@’ The line emission for a transition originating in an upper state, which lies above the thermal limit, is related to the electron number density and the excitation temperature associated with the upper level Boltzmann population distribution. In this case the excitation temperature is equal to the electron kinetic temperature due to the collisional equilibrium between the free electrons and the upper bound levels. Hence,
where the subscript T.L. denotes the use of lines with E, lying above the thermal limit. The electron number density criterion, assuming that a thermal limit exists at E, = 13.46 eV, for an argon plasma is (n,), 2 2.6 x 10’2T~‘2
[cme3]
(3)
where the subscript r denotes that the criterion is based upon level E,. The continuum emission for a neutral, low degree of ionization plasma is given as a function of electron number density and electron temperature
1, = Me9 T,)
(4)
and requires that the free electrons have a Maxwellian velocity distribution.‘4*5’ In complete local thermodynamic equilibrium the thermodynamic composition tables@ may be used (to relate electron number density and temperature) in conjunction with the LTE model to reduce the expression for the absolute line emission to a function only of the equilibrium temperature Teq. The tables may likewise be used with equation (4) to reduce the continuum emission to a function only of equilibrium temperature.
Local thermodynamic
equilibrium
in an RF argon
plasma
1413
For a plasma with an electron temperature of 104’K (of the order found in these experiments) inequalities (1) and (3) give n, 2
1.5 x 10”
[cmm3]
(n,), 2 2.6 x 1Ol4 [cmP3],
for E, = 13.46 eV.
The thermodynamic equilibrium electron number density at 104”K and 1-Oatm is l-5 x lOi cmP3 and at 104”K and 0.01 atm it is 2 x 10” crn3. Hence, LTE should not exist at 104’K for an argon plasma operating between 0.01 and 1.0 atm. However, over the same pressure range and temperature, the criterion for a thermal limit at or above 13.46 eV is satisfied (i.e. must use lines with E, > E,). Thus, with measured values of I,, and I, equation (2) may be solved simultaneously with equation (4) to give T, and n, when the plasma at least satisfies the equilibrium requirements for a thermal limit. If T, equals T,, and Teq, then local thermodynamic equilibrium prevails. Departure from equilibrium then can be observed by comparing these three temperatures. It is the comparison of these three temperatures as a function of pressure (i.e. collision frequency) which was used to detect the existence of local thermodynamic equilibrium in this study. The existence of a Boltzmann distribution was verified within 4 per cent above levels of 13.33 eV over an energy spread of 2.2 eV in upper levels at the lowest pressure (i.e. 001 atm) of the range investigated. The plasma was assumed to be optically thin for this investigation. From a rigorous standpoint, this simplication of treating only an optically thin plasma uncouples the radiation field from the thermal field and explicitly implies non-thermodynamic equilibrium.“’ However, if the collisional rates exceed the radiative rates for populating atomic levels, then detailed balancing by collisional processes alone is approximated. Hence, the electrons have a thermal distribution and the radiation field is a result of spontaneous emission from the various line (discrete) and continuum (continuous) levels. In keeping with the transparent assumption, the lines selected for the line emission measurements were thus selected, in part, for a minimal self-absorption.
3. EXPERIMENTAL
INVESTIGATION
(A) Plasma source The argon plasma was contained within a water cooled quartz channel and sustained by the time varying EM field of a solenoid which was driven by an oscillator (i.e. induction heater). The channel pressure of the inductively sustained plasma was easily controlled by the use of a vacuum reservoir system (see Fig. 1). The pressure range for this study was from 0.01 to.1.0 atm. The channel i.d. was 25 mm and the average convective gas velocity was 9.0 cm/set. The induction heater unit was an RFC, variable frequency, oscillator, which, for this study, was operated at 4.2 MHz and with an input power of 2000 W. The power coupling efficiency of the unit at 1.0 atm is given in Ref. 8. The d.c. plate supply of the unit was modified by the addition of a II-section plus an L-section filter with a ripple percentage of 0.025 per cent to eliminate the 120 Hz modulation of the RF tank current. (The effect of the modulated current was to create a 120 Hz pulsating plasma radiation field.)
1414
P. D. SCHOLZ and T. P. ANDERSON SURGE I
TANK7
CHANNEL COOLING WATER IN END COLLAR
w_A3MA
CHANNEL -
TIE
I-c
I I
RODS
TANK COIL
....__..
VACUUM CHANNEL
INLET VALVE
II
II
PUMP
MANOMETER
FLOWMETER
-
PRESSURE TEMPERATURE u
GAS
SUPPLY
FIG. 1. RF plasma
channel
with control
system.
(B) Spectroscopic system The spectroscopic system consisted of a Perkin-Elmer grating monochromator, an optical system, and a photoelectric detection system. The optical system included achromatic collimating and condensing lenses, optical filters to exclude the undesired overlapping orders from the grating, and a traversing mechanism for spatially scanning the plasma. The monochromatic output was detected by an RCA lP21 photomultiplier and was recorded by a strip chart recorder as a function of lateral plasma position, thus giving a lateral intensity profile. The strip chart deflections, corresponding to relative intensities, were calibrated by the use of a calibrated photometric tungsten ribbon lamp. The absolute lateral intensity profiles were inverted to give absolute radial profiles by an Abel integral inversion technique. Refraction effects, due to the curved composite channel wall, were found to have negligible effect on the Abel inversion over the center half of the plasma. Wall reflections from the inside wall of the quartz channel tube, which resulted in non-zero plasma intensities at
Local thermodynamic
equilibrium
in an RF argon
plasma
1415
the wall, were also found to be negligible in the inversion when only the center half of each inverted profile was retained for consideration. (C) Results The results of the spatially resolved absolute line and absolute continuum intensities were obtained for the following experimental conditions: a fixed oscillator frequency of 4.2 MHz ; a fixed average argon gas velocity of 9 cm/set ; a fixed power input of 2 kW ; and a fixed tank coil-channel geometry. Spectroscopic measurements were taken at static channel pressure levels of 1.0, 05, 0.1, and 0.01 atm. The argon atom lines included the 4259A1, 43OOA1, 4345A1, and the 5572AI. The continuum was determined at 4259A, 4300 A, and 6965 A by averaging the continuum measurements taken at equal wavelength increments above and below the lines at 4259, 4300, and 6965 A, respectively. The lateral line intensity profile for each line was determined in the normal manner by subtracting the folded continuum lateral profile from the folded line plus continuum profile. The absolute intensities of the three blue lines were checked in the first and second orders. The exit slit bandpass for the first order measurements varied from 6-OA to 12.2 A, and for the second order, the range was from 4.3 to 4.4 A. A Kodak Wratten blue filter was used to isolate these lines from the overlapping orders. The first and second order results agreed to within 4 per cent to indicate that the full line width was passed by the monochromator exit slit. The entrance slit was set at 10 p for all the measurements. The spectral traverses were all taken at an axial position mid-way between the top two turns of the five turn solenoid. The centerline lateral intensities were reproducible to within 4 per cent for traverses with the same experimental conditions. Figure 2 shows a plot of the half radial temperature distribution at 1.0 atm. Shown are the temperatures determined from the slope of In nu/g, versus E, (i.e. T,,) using the four lines and the temperatures determined from the simultaneous solution of equations (2) and (4) using the measured line and continuum intensities at 4259 A (i.e. T,). Also shown are the ranges of Teq, which are based on the a priori assumption of local thermodynamic equilibrium and were determined from the absolute line intensities of the four lines and from the three absolute continuum intensities. The transition probabilities of the lines were calculated at 1.0 atm using the radial line intensities and the average equilibrium temperature determined from the three different continuum intensities at each of the radial zones. The transition probabilities were averaged over the first half of the total number of radial zones. In essence, the transition probabilities were adjusted to force the equilibrium temperatures determined from the line intensities to agree with those from the continuum intensities at 1-Oatm. This is a valid process as long as the plasma at 1.0 atm is in a state of local thermodynamic equilibrium. (This was verified by showing that T_ determined from the absolute continuum intensities was equal to T,, and T, at 1.0 atm.) The values of the transition probabilities are given in Table 1. These values are listed for completeness and are representative of the values listed in the literature.‘9-‘5’ The solid curve of Fig. 2 is based on the absolute continuum intensity at 4259 A and the equilibrium assumption. The error bars of the slope temperature are due to the uncertainty in determining the best fit line through the data points of the four lines. The maximum error for the simultaneously determined temperatures is f3.3 per cent for a possible + 4 per cent error in the measured line and measured continuum intensities. The simultaneous temperatures are about 3 per cent high due possibly to the neglect of the ionization
1416
P. D. SCHOLZ and T. P. ANDEMON
r
-
I
I
r
FROM 5 ,N?ENSlTY
AN. CON?. AT 4259 8
RANGE OF T, FROM CONT. AT 4 51, 4300, AND 6965 % RANGE OF TaR FROM 4259Al, 43OOAI, 434!5Al, AND 5572Al ADS. LINE INTENSITIES FROM
SLOPE
\ \ \
OF
\ q
SYULT. TEMP. (T 1 FROM 42SSAl AND COk AT 4259 8
.2 *4 NONMMENSIONAL
G -8 PLASMA RADIUS FIG. 2. Radial temperature distribution at 1.0 atm. TABLE
Line
I. EXPERIMENTALLY DETERMINED TRANSITION PROBABILITIES
Averaged A,, and percent variation over one-half of radial zones at 1.O atm
A,, x IO-’ [set-‘1 4259Al 4300AI 4345Al 5572Al
I.0
0,339 0.033 1 0.0273 0.0647
[%I + 0.73 +2.2 +2.8 ,2.5
potential lowering effect in equation (2). Radial temperature distributions at 0.5, 0.1, and 0.01 atm were also determined using the same techniques as were used in calculating the profiles shown in Fig. 2. The results of these distributions at the plasma non-dimensional radius of 0.17 are given in Fig. 3.
1417
Local thermodynamic equilibrium in an RF argon plasma
Figure 3 is a plot of temperature versus channel static pressure at a non-dimensional radius of 0.17 (i.e. the fourth radial zone in the inversion scheme). The nomenclature is the same as used for Fig. 2, where the solid curve is Teqand is based on the absolute continuum intensity at 4259 A.
-
I I
f 0
Te, FROM ABS. CONT. IliiENSlTY AT 4259 8 FROM RANGE OF Tlo CONT. AT 4259, 4300, AN0 6965 % FROM RANGE OF TIP 4259AI, 43OOAI, 4345Al, AN0 5572Al ABS. LINE INTENSITIES T
FROM
I:‘$/Q,
VI
SLOPE
OF
E,
SIMULT. TEMP. (T 1 FROM 425SAI AND COJkT. AT 425s %
I
I
o-01
0.1 CHANNEL
STATIC
I
O-5 PRESSURE
FIG. 3. Temperature vs. channel pressure at a non-dimensional
I
I.0
(otm) plasma radius of 0.17.
(D) Discussion of results The off-axis temperature peak observed in Fig. 2 disappears as the pressure decreases. The radial temperature gradients in the center half of the plasma are relatively small (i.e. IVTl < 530”K/cm). (It was also observed that the axial gradients are even smaller in the region of these observations at all the pressures.) In the center half the excitation temperatures agree within 3 per cent with the kinetic electron temperature and with the equilibrium line and continuum temperatures (i.e. T,, = T, N T,,).
1418
P. D. SCHOLZ and T. P. ANDERSON
The results at a non-dimensional radius of 0.17, shown in Fig. 3, are typical of the temperature-pressure results obtained at each of the radial stations in the center half. At 0.5 and 0.1 atm the temperatures agree within 4 per cent. However, at 0.1 atm it was observed that the excitation and electron temperatures began to diverge from the equilibrium temperatures near the non-dimensional half radius. This divergence increased with radial distance. At 0.01 atm the excitation temperature agrees within 4.5 per cent with the electron temperature. However, the so-called line and continuum equilibrium temperatures fall outside the error bars and are 17 per cent low. The electron number densities obtained from equations (2) and (4) agreed very well (i.e. within 1.5 per cent) with the equilibrium values at all the pressures except at 0.01 atm where they differed by an order of magnitude. 4.
CONCLUSIONS
The central core of the RF argon plasma, operating as specified above, is in local thermodynamic equilibrium at 1-O and at O-5atm, and very nearly so at 0.1 atm. This state of equilibrium is due in part to the existence of low (i.e. /VT1 I 530”K/cm) axial and radial temperature gradients. At these pressures the plasma is in an apparent state of LTE and the thermal limit extends down to the ground state. The LTE electron number density criterion given by equation (1) appears to be conservative for this plasma. Significant departure from local thermodynamic equilibrium begins to occur at about 0.1 atm and increases as the pressure decreases. At O-01atm the plasma is no longer near a state of local thermodynamic equilibrium. However, the upper bound levels are Boltzmann populated (within the experimental error), and the thermal limit lies between the ground state and the 3p, (4300AI) level. This result is in agreement with the thermal limit criterion given by equation (3). REFERENCES 1. R. W. P. MCWHIRTER, Plasma Diagnosiic Techniques (Edited by R. H. HUDDLESTONEand S. L. LEONARD), pp. 201-206. Academic Press, New York (1957). 2. H. R. GRIEM, Phys. Rev. 131, 1170 (1963). 3. R. WILSON. JQSRT2,477 (1962). 4. J. POMERANTZ. The Influence of the Absorption of Radiation in Shock Tube Phenomena, Navord Report 6136, U.S. Naval Ordnance Laboratory (1958). 5. C. J. CHEN, Valid Conditions for the Kramers-Unsiild Continuum Theory in a Non-Equilibrium Plasma, JPL Technical Report No. 32-707 (1964). 6. K. C. DRELLISHAK, C. F. KNOPP and A. B. CAMBEL, Partition Functions and Thermodynamic Properties of Argon, Gas Dynamics Report A-3-62, Northwestern University (1963). 7. R. N. THOMAS, Some Aspects of Non-Equilibrium Thermo&namics in the Presence of a Radiation Field. University of Colorado Press (1965). 8. P. D. SCHOLZ and T. P. ANDERSON, AIAA J. 5, 1022 (1967). 9. H. N. OLSEN, JQSRT 3, 305 (1963). 10. H. W. DRAWIN. Z. Physik 146,295 (1956). 11. B. D. ADCOCK and W. E. G. PLUMTREE,JQSRT 4,29 (1964). 12. R. H. GARSTANG and J. VAN BLERKOM, J. Opt. Sot. Am. 55, 1054 (1965). 13. W. E. GERICKE, Z. Astrophys. 53, 68 (1961). 14. C. H. POPENOEand J. B. SHUMAKER,JR., J. Res. natn. Bur. Stand. 69A, 495 (1965). 15. P. B. COATES and A. G. GAYDON, Proc. R. Sot. A293,452 (1966).