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The influence of spatial temperature distribution and measuring configuration on line-reversal temperature
J. Quanr. Spectrosc. Radial. Transfer. Vol. 12, pp. 1673-1683. Perpmon Press 1972. Printedin Great Britain
THE INFLUENCE OF SPATIAL TEMPERATURE DISTRIBUTION AND MEASURING CONFIGURATION ON LINE-REVERSAL TEMPERATURE* KIYOSHI YOSHIKAWA Corpuscular Engineering Laboratory, Institute of Atomic Energy, Kyoto University, Gokasho, Uji, Japan and ITARU MICHIYOSHI Department of Nuclear Engineering, Kyoto University, Kyoto, Japan (Receioed 30 November 1971)
Abstract-The influence of temperature distribution and measuring configuration on line-reversal temperature is investigated under MHD generation conditions at a seed ratio of lo-’ and atmospheric pressure. Under these conditions, dispersion broadening occurs. In order to obtain information about the inner part of the plasma at large optic&l thicknesses, the width of the spectroscope inlet slit and of the spectral region covered by the photomultiplier must be made sufficiently wide.
1. INTRODUCTION
line-reversal method is widely used for plasma temperature measurements in MHD generators. It is a variant of the sodium line-reversal method.“-3) BUNDY and STRONG(~) meashred flame’temperatures with this procedure. Recently, RIEDM~~LLER et a/.@) investigated the conditions which should be satisfied for a potassium line-reversal temperature to be almost identical witli the electron temperature in the plasma and concluded that the typical MHD generator meets these conditions. According to their experiments, the potassium line-reversal temperature agreed well with the electron temperature derived from the mea&red electrical conductivity. In this paper, the authors study the effects of the temperature distribution in an MHD channel, of the width of the inlet slit of the spectroscope and of the spectral region covered by the photomultiplier on the potassium line-reversal temperature.
THE POTASSIUM
2. LINE
BROADENING’“9’
In the case of a potassium-seeded argon plasma, collision broadening and Doppler broadening are important in determining the absorption coefficient k, for the potassium * Orally presented at the Annual Meeting of the Atomic Energy Society of Japan, April 1971. 1673
KIYOSHI YOSHIKAWA and ITARU MICHIYCISHI
1674
resonance line (2, = 7699 A). Half-widths due to these broadenings are shown in Fig. 1, where E and a are seed ratio and a = Av,,/(ln 2)/Av,, respectively. When a is very large, it is sufficient to consider only collision broadening. But when a is of the order of unity, the
a
10
FIG. 1. Half-widths for collision and Doppler broadening and the ratio LIas a function of the temperature.
combined form for collision and Doppler efficients are written as follows :
broadening
ko ‘l” = koR = aJ(n)[l
+(I;/a)‘]
k,,, = k,P, = k, exp( - 5’) k
dy
must be used. Absorption
co-
for collision broadening,
(1)
for Doppler broadening,
(2)
for combined broadening,
(3)
where
v-
M,c2
v.
t=vo-= J
2RT
(v - vo)J(ln 2). AvJ2 ’
(5)
e, R, co, m, and c are electron charge, gas constant, electrical permittivity, electron mass, and speed of light, respectively, and nR , MR and f,,are density of radiating atoms, mole
Influence of spatial temperature distribution and measuring cor@uration
on line-reversal temperature
1675
weight of radiating atoms, and emission oscillator strength for m -+ n transitions, respectively. Dimensionless absorption coefficients P,, based on equations (l)-(3), are shown in Fig. 2; at a temperature of 2000°K and a seed ratio of 10W3,we may use absorption coefficients for pure collision broadening.
FIG. 2. Absorption coefficients for collision and Doppler broadening and combined absorption coefficients at T = 2000°K.
3. BASIC
EQUATION
OF THE
RADIATION
FIELD
The following assumptions are made : (1) the radiation field is in local thermodynamic equilibrium ; (2) continuum radiation is negligible near the resonance line ; (3) atoms and electrons in the plasma are in thermal equilibrium at the temperature 7’; (4) the population distribution of excited levels of radiating atoms is Maxwellian at a temperature T and the transitions of the atomic state are caused mainly by electron collisions ; (5) the temperature has the spatial distribution T = C+(T,-
LJI-($9
(0 S f(x) I I),
(6)
where T, and T’ are temperatures at the plasma center and the wall (surface of the plasma), respectively. The temperature range is restricted between 1500 and 2000°K and the spatial distribution function f(x) is symmetric with respect to the plasma center. Under these assumptions, the equation of the radiation field may be written as
f.%+$ =k,(B,(T)-I,), where I,, B, are specific intensity and Plank function and k, is the absorption coefficient expressed by equation (l), since induced radiation can be neglected. At the steady state, equation (7) can be integrated with the result IL(l) = Z,(O) e-‘icl*o)+
s 0
k,(T)B,(T)
e-rr(l.s) ds,
(8)
KIYOSHI YOWIKAWAand ITARU MICHIY~SI-II
1676
where I,(O) = I, (s = 0). Here the optical thickness of the layer rl(s, s’) is defined as s
TJS, s’) =
4. THE
MEASURING
GEOMETRY
I S’
(9)
k,(T) ds.
AND
EXPERIMENTAL
METHOD
The measuring configuration for the line-reversal method is illustrated in Fig. 3. The plasma is contained in the region between s = 0 and s = 1,where the direction of s denotes the direction of observation. The W-lamp of known brightness temperature is set on the line s (the optical axis) and the light from the W-lamp goes through the plasma into the inlet slit of the spectrometer. The light from the W-lamp is focused at the plasma center
0 Q Slit of I?M.(A)
rE
-
Ts or BA(Ts)
X
FIG. 3. Measuring configuration of the line-reversal method. The specific intensity A@,) corresponds to the equivalent specific intensity expressed byequation (11) and B(&) corresponds to equation (lo) for 1,. The point S gives the,line-reversal temperature.
(s = 1/2) and the inlet slit of the spectroscope. The light emitted from the W-lamp is received by photomultipliers P.M.1 and P.M.2 whose outputs are connected to the X- and Y-axes of an X-Y recorder, respectively. The spectral shape on the focal plane of the spectroscope is sharp in ideal cases but, in practice, because of the finite width of the inlet slit of the spectroscope, distortion occurs. The distorted spectral shape characterized.by a superposed width 6 can be calculated by integrating the spectral shape with no distortion. The superposed width, 6 and the wave length region (A, -A/2,&, + A/2) covered by P.M.2 are shown in Fig. 4.
Influence of spatial temperature distribution and measuring configuration on line-reversal temperature
1677
FIG. 4. A distorted spectral shape (dotted curve) on the focal plane of the spectroscope, where 6 and A are the superposed width and the spectral region covered by a photomultiplier, respectively. Intensities at points a, b, c, d and e are calculated by integrating a non-distorted spectral shape (region of oblique lines).
4.1 s=oA The specific intensity of the light at the wave length 1 emitted from the W-lamp (the brightness temperature Ts) through the plasma is written as I,(K) = B,(~)e-‘a”*o’+
s
’ k,(T)B,(T)e-‘“(‘,“)ds.
(IO)
0
In the absence of the plasma, the specific intensity is given by I,(D) = B*(E).
(II)
When the specific intensities in equations (10) and (11) are equal, the following equation holds : 1
B,( Ts) = B,( Ts) e-rA(l*o) +
I
k,( T)B,( T) e-rz(f,s) ds.
(12)
0
The temperature satisfying equation (12) is defined as the line-reversal temperature (LRT) at the wave length 1. At a constant plasma temperature, equation (12) reduces to B,Vs) = B,(T),
(13)
1678
KIYOSHI YOSHIKAWA
and ITNW
MICHIYOSHI
showing that the LRT is equal to the plasma temperature. over the region (A,,-A/2, &, + A/2) as lo-VA/Z
We define the average LRT( T,)
lo+A/Z
JAdk
s
lo-612
lo-A/2
where Ja =
s
k,(T)B,( T) e-‘A(‘pa)ds.
(15)
0
Since A/n, CC1, the left-hand side of equation (14) can be rewritten as &,+A/2
&+A/2
B,(TA)x(l-e-‘~(‘*o) s lo-A/2
)dJ=B,,(T,)
j+
(l-e-‘“(l*o))drl.
(16)
Lo-A/2
For large r,(l, 0), equation (12) reduces to = B,(Tw).
g,(Ts) = B,(T)
(17)
s=l
For small z,(l, 0), on the other hand, equation (12) becomes
Note that the LRTs satisfying equation (18) correspond to the asymptotic IA-lo/ -+ co and are independent of the interaction length 1.
LRT(T,,)
at
4.2 S # 0 A Because of the finite width of the slit, the distorted spectral shape appears as shown in Fig. 4 (dotted curve). In this case, the LRT at the wave length 1 is defined as 2+a/2
1 s s
B,( 7%)x (1 - e -n(l*o)) drl = Z,(n),
(19)
1-812
where I+aj2 W
=
f
J,
1
dl.
(20)
1 --a/2
The average LRT(T,) over the region (A, - A/2,1, + A/2) is similarly defined as lo +A/Z lo + A/2 s 10-A/2
4,UJ
dA =
s do-A/2
I,@)
dl,
(21)
Influence of spatial temperature distribution and measuring configuration on line-reversal temperature
1679
where
4,(T,)
= f
BI(TA) x
J
(1 -e-‘2”Vo’) dl.
(22)
a-a/t
5. RESULTS
OF CALCULATIONS
All calculations were made for the potassium resonance line (no = 7699 A) at E = 10m3 and atmospheric pressure. Under these conditions, the specific intensities I, and the LRTs may be regarded as being almost symmetric with respect to the resonance line center. 5.1 Constant temperature At a constant temperature (T = 2OOO”K),spectral intensities emitted from the plasma alone are shown for I = 1 and 10 cm in Fig. 5 (solid curves).
01 -12
I
’
I
-8
I
I
-4
I
I
0 A-&
I 4
FIG.5. Intensity profiles at constant temperature for l
I
I 8
I
I
(A) '2
= 1 cm and 10 cm.
Since the intensity emitted from the plasma is proportional to [l -exp(- k,o], the spectral shape shows a sharp peak around the line center for small 1. With increasing 6, the shape becomes flattened and the intensity also decreases. The dotted curves show intensities of the light emitted from the W-lamp at Ts = T, and 6 = 0 A after passing through the plasma. 5.2 Variable temperatures Three spatial distribution
functions of the temperature f&4 = II-
2x/4,
f&x) = (l-11 f&x)
=
-2x/9)“‘,
sin(xx/l).
are chosen as follows: (23) (24)
(25)
1680
Kt~wn
YOSHIKAWA and ITARU Mtc~rvasm
The spatial distribution function fi(x), often encountered in an MHD experiment, changes markedly at the boundary of the plasma; f3(x) is chosen for comparison with JJx); fi(x) is chosen for the special case where the temperature at the boundary is highest. Spectral intensity profiles corresponding to jr(x) and Jz(x) for 1 = 1 and 10 cm are shown in Figs. 6 and 7, respectively. The solid and dotted curves refer to the same conditions as in Section 5.1. It is obvious that the intensity near the line center is independent of the intensity emitted from the W-lamp because of the large optical thickness of the plasma. It may also be seen from Fig. 7 that self-reversal occurs for distributions having a maximum temperature at the plasma center, and, with increasing 6, the concave shapes that are characteristic of self-reversal diminish. The LRTs for each distribution function are shown in Figs. 8 and 9. For the function j-i(x), the LRT has a maximum at the line center for small 6 and is equal to the plasma temperature Tw for 6 = OA, as predicted by equation (17). Similarly, the LRT for fi(x) and&(x) has a minimum at the line center. These figures show 4 .... .. ..__ ._ A..
3
"E 5 52 A -: l-
! !
\
P=lOcm
i
i
0 -8
-12
-4
FIG. 6. Intensity
0 A-L
profiles for fi(x) and
3
0
A\
_..-
..
I’
P=lcm
*
(A)
'2
(A)
l2
I = 1cm or 10 cm.
I I i
I
/
1
/
;
-.
\
P=lOcm
I
0 -12
4
-8
FIG. 7. Intensity
-4
r-Ox
4
8
profiles for h(x) and 1 = 1 cm or 10 cm.
Influence of spatial temperature distribution and measuring configuration on line-reversal temperature
T4778.5 -12
i -8
-4
0 X-L
4
1681
I 8
(A)
l2
FIG.8. Line-reversal temperatures for ii(x) (A = 0 A) and I = 1 cm or 10 cm.
that the LRT depends strongly on both the spatial-distribution function and on the superposed width. The LRT obtainedt5’ experimentally has a behavior that is similar to that shown in Fig. 9 at moderate 6. The temperatures T,,including effects of both 6 and A, are shown in Figs. 10-12. With increasing A, every curve approaches the curve with 6 = 0 A, and TAwill be almost independent of 6 if A is taken to be at least 20 A. Comparing these average LRTs at large A, TAfor fi(x) is closest to the temperature at the plasma center because it maintains higher temperatures in the vicinity of the plasma boundary. In an MHD experiment, it is not desirable that the observed LRTs depend strongly on 6 or A. This conclusion follows because it is impossible to explain an observed LRT on the basis of the coupled functions 6 or A with an unknown spatial distribution of the plasma temperature. Furthermore, it is desired to measure the LRT close to the plasma center, from which a theoretical electrical conductivity can be determined. Note that the superposed width S and the spectral region covered by the photomultiplier A are interchangeable when either of them is small.
FIG.9. Line-reversal temperatures for j*(x) (A = 0 A) and I = 1 cm or 10 cm.
1682
KIYOSHIYOSHIKAWAand ITARU MICHIYCSHI
16OOl
I
I
L
6
I 12 A
16
20
,
J
24 (8, 26
FIG. 10. Average line-reversal temperatures for fI(x) and
1= 1 cm or 10 cm.
1600
6
12
A
16
20
FIG. 11. Average line-reversal temperatures forf,(x)
2& (A, 26
and 1 = 1 cm or 10 cm.
Influence of spatial temperature distribution and measuring configuration on line-reversal temperature
1683
A
u
4
u
12 A
16
20
24 (A, 28
FIG. 12. Average line-reversal temperatures for fa(x) and I = 1 cm.
6. CONCLUSION
The absorption coefficient is well approximated by collision broadening. For 6 = 0 A, the intensity profile is very sharp ; with increasing 6, its variation with wavelength becomes small. Self-reversal cannot be identified for large 6. The line-reversal temperatures (73) become Tw at the resonance line center for 6 = 0 A in an optically thick plasma ; with an increase of 6, T, vs. A-I, curves become flattened, thereby leading to a constant temperature at 6 = 20 A near the resonance-line center and they approach Tsco. Including effects of 6 and A, the average LRT should be obs,erved by widening 6 or A in order to get as much information as possible from the inner region of the plasma. It is, however, very difficult to guess the temperature in the inner region of the plasma from these observed LRTs since they are very sensitive to the spatial distribution of the plasma temperature and to the measuring configuration. Acknowledgement-The authors are much indebted to Dr. M. Numano, Department of Nuclear Engineering, Kyoto University, for suggesting this study.
REFERENCES 1. Ch. F~?RY, Compf. rend. 137,909 (1903). 2. H. KOHN,Ann. Physik 44,749 (1914). 3. E. GRIFFITH and J. H. AWBERY,Proc. R. Sot. (Land.) 123,401 (1929). 4. H. M. STRONG and F. P. BUNDY, J. Appl. Phys. 25,1521 (1954). 5. W. RIEDM~LLEN, G. Banonmow and M. SALVAT, Z. Nafurforsch 23a, 731 (1968). 6. A. C. G. MITCHELL and M. W. ZBMANSKY, Resonance Radiation and Excited Atoms. Cambridge University Press, New York (1961). 7. H. R. GRIEF,Plasma Spectroscopy. McGraw-Hill, New York (1964). 8. T. HIRAMOTO, J. Phys. Sot. Japan 26,785 (1969). 9. S. S. PEN-, Quantitative Molecular Spectroscopy and Gas Emissivities. Pergamon Press, London (1959).
THE INFLUENCE OF SPATIAL TEMPERATURE DISTRIBUTION AND MEASURING CONFIGURATION ON LINE-REVERSAL TEMPERATURE* KIYOSHI YOSHIKAWA Corpuscular Engineering Laboratory, Institute of Atomic Energy, Kyoto University, Gokasho, Uji, Japan and ITARU MICHIYOSHI Department of Nuclear Engineering, Kyoto University, Kyoto, Japan (Receioed 30 November 1971)
Abstract-The influence of temperature distribution and measuring configuration on line-reversal temperature is investigated under MHD generation conditions at a seed ratio of lo-’ and atmospheric pressure. Under these conditions, dispersion broadening occurs. In order to obtain information about the inner part of the plasma at large optic&l thicknesses, the width of the spectroscope inlet slit and of the spectral region covered by the photomultiplier must be made sufficiently wide.
1. INTRODUCTION
line-reversal method is widely used for plasma temperature measurements in MHD generators. It is a variant of the sodium line-reversal method.“-3) BUNDY and STRONG(~) meashred flame’temperatures with this procedure. Recently, RIEDM~~LLER et a/.@) investigated the conditions which should be satisfied for a potassium line-reversal temperature to be almost identical witli the electron temperature in the plasma and concluded that the typical MHD generator meets these conditions. According to their experiments, the potassium line-reversal temperature agreed well with the electron temperature derived from the mea&red electrical conductivity. In this paper, the authors study the effects of the temperature distribution in an MHD channel, of the width of the inlet slit of the spectroscope and of the spectral region covered by the photomultiplier on the potassium line-reversal temperature.
THE POTASSIUM
2. LINE
BROADENING’“9’
In the case of a potassium-seeded argon plasma, collision broadening and Doppler broadening are important in determining the absorption coefficient k, for the potassium * Orally presented at the Annual Meeting of the Atomic Energy Society of Japan, April 1971. 1673
KIYOSHI YOSHIKAWA and ITARU MICHIYCISHI
1674
resonance line (2, = 7699 A). Half-widths due to these broadenings are shown in Fig. 1, where E and a are seed ratio and a = Av,,/(ln 2)/Av,, respectively. When a is very large, it is sufficient to consider only collision broadening. But when a is of the order of unity, the
a
10
FIG. 1. Half-widths for collision and Doppler broadening and the ratio LIas a function of the temperature.
combined form for collision and Doppler efficients are written as follows :
broadening
ko ‘l” = koR = aJ(n)[l
+(I;/a)‘]
k,,, = k,P, = k, exp( - 5’) k
dy
must be used. Absorption
co-
for collision broadening,
(1)
for Doppler broadening,
(2)
for combined broadening,
(3)
where
v-
M,c2
v.
t=vo-= J
2RT
(v - vo)J(ln 2). AvJ2 ’
(5)
e, R, co, m, and c are electron charge, gas constant, electrical permittivity, electron mass, and speed of light, respectively, and nR , MR and f,,are density of radiating atoms, mole
Influence of spatial temperature distribution and measuring cor@uration
on line-reversal temperature
1675
weight of radiating atoms, and emission oscillator strength for m -+ n transitions, respectively. Dimensionless absorption coefficients P,, based on equations (l)-(3), are shown in Fig. 2; at a temperature of 2000°K and a seed ratio of 10W3,we may use absorption coefficients for pure collision broadening.
FIG. 2. Absorption coefficients for collision and Doppler broadening and combined absorption coefficients at T = 2000°K.
3. BASIC
EQUATION
OF THE
RADIATION
FIELD
The following assumptions are made : (1) the radiation field is in local thermodynamic equilibrium ; (2) continuum radiation is negligible near the resonance line ; (3) atoms and electrons in the plasma are in thermal equilibrium at the temperature 7’; (4) the population distribution of excited levels of radiating atoms is Maxwellian at a temperature T and the transitions of the atomic state are caused mainly by electron collisions ; (5) the temperature has the spatial distribution T = C+(T,-
LJI-($9
(0 S f(x) I I),
(6)
where T, and T’ are temperatures at the plasma center and the wall (surface of the plasma), respectively. The temperature range is restricted between 1500 and 2000°K and the spatial distribution function f(x) is symmetric with respect to the plasma center. Under these assumptions, the equation of the radiation field may be written as
f.%+$ =k,(B,(T)-I,), where I,, B, are specific intensity and Plank function and k, is the absorption coefficient expressed by equation (l), since induced radiation can be neglected. At the steady state, equation (7) can be integrated with the result IL(l) = Z,(O) e-‘icl*o)+
s 0
k,(T)B,(T)
e-rr(l.s) ds,
(8)
KIYOSHI YOWIKAWAand ITARU MICHIY~SI-II
1676
where I,(O) = I, (s = 0). Here the optical thickness of the layer rl(s, s’) is defined as s
TJS, s’) =
4. THE
MEASURING
GEOMETRY
I S’
(9)
k,(T) ds.
AND
EXPERIMENTAL
METHOD
The measuring configuration for the line-reversal method is illustrated in Fig. 3. The plasma is contained in the region between s = 0 and s = 1,where the direction of s denotes the direction of observation. The W-lamp of known brightness temperature is set on the line s (the optical axis) and the light from the W-lamp goes through the plasma into the inlet slit of the spectrometer. The light from the W-lamp is focused at the plasma center
0 Q Slit of I?M.(A)
rE
-
Ts or BA(Ts)
X
FIG. 3. Measuring configuration of the line-reversal method. The specific intensity A@,) corresponds to the equivalent specific intensity expressed byequation (11) and B(&) corresponds to equation (lo) for 1,. The point S gives the,line-reversal temperature.
(s = 1/2) and the inlet slit of the spectroscope. The light emitted from the W-lamp is received by photomultipliers P.M.1 and P.M.2 whose outputs are connected to the X- and Y-axes of an X-Y recorder, respectively. The spectral shape on the focal plane of the spectroscope is sharp in ideal cases but, in practice, because of the finite width of the inlet slit of the spectroscope, distortion occurs. The distorted spectral shape characterized.by a superposed width 6 can be calculated by integrating the spectral shape with no distortion. The superposed width, 6 and the wave length region (A, -A/2,&, + A/2) covered by P.M.2 are shown in Fig. 4.
Influence of spatial temperature distribution and measuring configuration on line-reversal temperature
1677
FIG. 4. A distorted spectral shape (dotted curve) on the focal plane of the spectroscope, where 6 and A are the superposed width and the spectral region covered by a photomultiplier, respectively. Intensities at points a, b, c, d and e are calculated by integrating a non-distorted spectral shape (region of oblique lines).
4.1 s=oA The specific intensity of the light at the wave length 1 emitted from the W-lamp (the brightness temperature Ts) through the plasma is written as I,(K) = B,(~)e-‘a”*o’+
s
’ k,(T)B,(T)e-‘“(‘,“)ds.
(IO)
0
In the absence of the plasma, the specific intensity is given by I,(D) = B*(E).
(II)
When the specific intensities in equations (10) and (11) are equal, the following equation holds : 1
B,( Ts) = B,( Ts) e-rA(l*o) +
I
k,( T)B,( T) e-rz(f,s) ds.
(12)
0
The temperature satisfying equation (12) is defined as the line-reversal temperature (LRT) at the wave length 1. At a constant plasma temperature, equation (12) reduces to B,Vs) = B,(T),
(13)
1678
KIYOSHI YOSHIKAWA
and ITNW
MICHIYOSHI
showing that the LRT is equal to the plasma temperature. over the region (A,,-A/2, &, + A/2) as lo-VA/Z
We define the average LRT( T,)
lo+A/Z
JAdk
s
lo-612
lo-A/2
where Ja =
s
k,(T)B,( T) e-‘A(‘pa)ds.
(15)
0
Since A/n, CC1, the left-hand side of equation (14) can be rewritten as &,+A/2
&+A/2
B,(TA)x(l-e-‘~(‘*o) s lo-A/2
)dJ=B,,(T,)
j+
(l-e-‘“(l*o))drl.
(16)
Lo-A/2
For large r,(l, 0), equation (12) reduces to = B,(Tw).
g,(Ts) = B,(T)
(17)
s=l
For small z,(l, 0), on the other hand, equation (12) becomes
Note that the LRTs satisfying equation (18) correspond to the asymptotic IA-lo/ -+ co and are independent of the interaction length 1.
LRT(T,,)
at
4.2 S # 0 A Because of the finite width of the slit, the distorted spectral shape appears as shown in Fig. 4 (dotted curve). In this case, the LRT at the wave length 1 is defined as 2+a/2
1 s s
B,( 7%)x (1 - e -n(l*o)) drl = Z,(n),
(19)
1-812
where I+aj2 W
=
f
J,
1
dl.
(20)
1 --a/2
The average LRT(T,) over the region (A, - A/2,1, + A/2) is similarly defined as lo +A/Z lo + A/2 s 10-A/2
4,UJ
dA =
s do-A/2
I,@)
dl,
(21)
Influence of spatial temperature distribution and measuring configuration on line-reversal temperature
1679
where
4,(T,)
= f
BI(TA) x
J
(1 -e-‘2”Vo’) dl.
(22)
a-a/t
5. RESULTS
OF CALCULATIONS
All calculations were made for the potassium resonance line (no = 7699 A) at E = 10m3 and atmospheric pressure. Under these conditions, the specific intensities I, and the LRTs may be regarded as being almost symmetric with respect to the resonance line center. 5.1 Constant temperature At a constant temperature (T = 2OOO”K),spectral intensities emitted from the plasma alone are shown for I = 1 and 10 cm in Fig. 5 (solid curves).
01 -12
I
’
I
-8
I
I
-4
I
I
0 A-&
I 4
FIG.5. Intensity profiles at constant temperature for l
I
I 8
I
I
(A) '2
= 1 cm and 10 cm.
Since the intensity emitted from the plasma is proportional to [l -exp(- k,o], the spectral shape shows a sharp peak around the line center for small 1. With increasing 6, the shape becomes flattened and the intensity also decreases. The dotted curves show intensities of the light emitted from the W-lamp at Ts = T, and 6 = 0 A after passing through the plasma. 5.2 Variable temperatures Three spatial distribution
functions of the temperature f&4 = II-
2x/4,
f&x) = (l-11 f&x)
=
-2x/9)“‘,
sin(xx/l).
are chosen as follows: (23) (24)
(25)
1680
Kt~wn
YOSHIKAWA and ITARU Mtc~rvasm
The spatial distribution function fi(x), often encountered in an MHD experiment, changes markedly at the boundary of the plasma; f3(x) is chosen for comparison with JJx); fi(x) is chosen for the special case where the temperature at the boundary is highest. Spectral intensity profiles corresponding to jr(x) and Jz(x) for 1 = 1 and 10 cm are shown in Figs. 6 and 7, respectively. The solid and dotted curves refer to the same conditions as in Section 5.1. It is obvious that the intensity near the line center is independent of the intensity emitted from the W-lamp because of the large optical thickness of the plasma. It may also be seen from Fig. 7 that self-reversal occurs for distributions having a maximum temperature at the plasma center, and, with increasing 6, the concave shapes that are characteristic of self-reversal diminish. The LRTs for each distribution function are shown in Figs. 8 and 9. For the function j-i(x), the LRT has a maximum at the line center for small 6 and is equal to the plasma temperature Tw for 6 = OA, as predicted by equation (17). Similarly, the LRT for fi(x) and&(x) has a minimum at the line center. These figures show 4 .... .. ..__ ._ A..
3
"E 5 52 A -: l-
! !
\
P=lOcm
i
i
0 -8
-12
-4
FIG. 6. Intensity
0 A-L
profiles for fi(x) and
3
0
A\
_..-
..
I’
P=lcm
*
(A)
'2
(A)
l2
I = 1cm or 10 cm.
I I i
I
/
1
/
;
-.
\
P=lOcm
I
0 -12
4
-8
FIG. 7. Intensity
-4
r-Ox
4
8
profiles for h(x) and 1 = 1 cm or 10 cm.
Influence of spatial temperature distribution and measuring configuration on line-reversal temperature
T4778.5 -12
i -8
-4
0 X-L
4
1681
I 8
(A)
l2
FIG.8. Line-reversal temperatures for ii(x) (A = 0 A) and I = 1 cm or 10 cm.
that the LRT depends strongly on both the spatial-distribution function and on the superposed width. The LRT obtainedt5’ experimentally has a behavior that is similar to that shown in Fig. 9 at moderate 6. The temperatures T,,including effects of both 6 and A, are shown in Figs. 10-12. With increasing A, every curve approaches the curve with 6 = 0 A, and TAwill be almost independent of 6 if A is taken to be at least 20 A. Comparing these average LRTs at large A, TAfor fi(x) is closest to the temperature at the plasma center because it maintains higher temperatures in the vicinity of the plasma boundary. In an MHD experiment, it is not desirable that the observed LRTs depend strongly on 6 or A. This conclusion follows because it is impossible to explain an observed LRT on the basis of the coupled functions 6 or A with an unknown spatial distribution of the plasma temperature. Furthermore, it is desired to measure the LRT close to the plasma center, from which a theoretical electrical conductivity can be determined. Note that the superposed width S and the spectral region covered by the photomultiplier A are interchangeable when either of them is small.
FIG.9. Line-reversal temperatures for j*(x) (A = 0 A) and I = 1 cm or 10 cm.
1682
KIYOSHIYOSHIKAWAand ITARU MICHIYCSHI
16OOl
I
I
L
6
I 12 A
16
20
,
J
24 (8, 26
FIG. 10. Average line-reversal temperatures for fI(x) and
1= 1 cm or 10 cm.
1600
6
12
A
16
20
FIG. 11. Average line-reversal temperatures forf,(x)
2& (A, 26
and 1 = 1 cm or 10 cm.
Influence of spatial temperature distribution and measuring configuration on line-reversal temperature
1683
A
u
4
u
12 A
16
20
24 (A, 28
FIG. 12. Average line-reversal temperatures for fa(x) and I = 1 cm.
6. CONCLUSION
The absorption coefficient is well approximated by collision broadening. For 6 = 0 A, the intensity profile is very sharp ; with increasing 6, its variation with wavelength becomes small. Self-reversal cannot be identified for large 6. The line-reversal temperatures (73) become Tw at the resonance line center for 6 = 0 A in an optically thick plasma ; with an increase of 6, T, vs. A-I, curves become flattened, thereby leading to a constant temperature at 6 = 20 A near the resonance-line center and they approach Tsco. Including effects of 6 and A, the average LRT should be obs,erved by widening 6 or A in order to get as much information as possible from the inner region of the plasma. It is, however, very difficult to guess the temperature in the inner region of the plasma from these observed LRTs since they are very sensitive to the spatial distribution of the plasma temperature and to the measuring configuration. Acknowledgement-The authors are much indebted to Dr. M. Numano, Department of Nuclear Engineering, Kyoto University, for suggesting this study.
REFERENCES 1. Ch. F~?RY, Compf. rend. 137,909 (1903). 2. H. KOHN,Ann. Physik 44,749 (1914). 3. E. GRIFFITH and J. H. AWBERY,Proc. R. Sot. (Land.) 123,401 (1929). 4. H. M. STRONG and F. P. BUNDY, J. Appl. Phys. 25,1521 (1954). 5. W. RIEDM~LLEN, G. Banonmow and M. SALVAT, Z. Nafurforsch 23a, 731 (1968). 6. A. C. G. MITCHELL and M. W. ZBMANSKY, Resonance Radiation and Excited Atoms. Cambridge University Press, New York (1961). 7. H. R. GRIEF,Plasma Spectroscopy. McGraw-Hill, New York (1964). 8. T. HIRAMOTO, J. Phys. Sot. Japan 26,785 (1969). 9. S. S. PEN-, Quantitative Molecular Spectroscopy and Gas Emissivities. Pergamon Press, London (1959).