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Determination of electron density from line merging
J. Qumr.
.Sp~cmsc.
Radar.
Transfer.
Vol.
6. pp. 575-578..
Pergamon Press Ltd., 1966. Printed in Great Britain
DETERMINATION FROM
OF ELECTRON LINE MERGING
DENSITY
C.-R. VIDAL Max-Planck-Institut fiir Physik und Astrophysik, Miinchen, Germany The following article is a brief summary of the results already published in JQSRT
and simple estimate it is a very convenient method to get the electron density from the principal quantum number n, of the last discernible line in a series by means of the well known INGLIS-TELLER relation”’ FOR A ROUGH
log(N cm3) = 23.26 - 7.5 log n, + 4.5 log z,
(1)
where z = effective nuclear charge. The density N is equal to the sum of the ion and electron density, N = Ni+N,, as long as the condition
T “Kc
4.6. 105. z n,
is fulfilled (see UNSOLD@)). This frequently used relation yields the electron density only within a factor of two because of three reasons. Firstly, the “last discernible” line is not precisely defined. Secondly,_the splitting of the Stark levels is very much simplified by using an average splitting for all Stark components together. Thirdly, Inglis and Teller use a fixed mean value of the microfields neglecting that the actual distribution function of the microfields depends on the mutual interaction of the perturbing particles or, in other words, that the mean value of the microfields depends on tempezature and density. In order to avoid the preceding deficiencies improved calculations have been performed and a new simple method for determining the electron density from the line merging has been suggested as follows. If one plots a curve Zminthrough the minima of the line merging and a curve I,, through the maxima, the ratio of these envelope curves R(n) = k Ill‘” is a unique function of the electron density. 515
(2)
576
C.-K.
VIDAL
The calculations start with the determination of the differential oscillator strength dj;!dl’ in the region of the merging. Similar to PANNEKOEK ‘3) this has been performed by summing up the partial line intensities of each line profile giving a contribution to the line merging at the position of interest. For this purpose the single quasistatic line profiles have been approximated by a proper expression, which is able to describe the line profiles with an,error of less than 1 per cent. By means of the differential oscillator strength one can immediately obtain the emission coefficient, which is given by (3) where N,* is the population density of the ground term according to the Saha equation. Calculating the ratio of the envelope curves on the basis of equation (3) it turns out of the maxima that a number of quantities cancel, because only the relative intensities and minima of the line merging are important. Therefore besides the constant factors we can drop the population density expressed by N,* . exp( - hv/kT), the frequency dependence i’,, - v3 and the oscillator strength of the lines concerned. Moreover the frequency dependence of the measuring device can be neglected. Furthermore the ratio of the envelope curves can also be applied to atoms and ions other than hydrogen, because in the region of the merging, where the Stark broadening gets comparable with the distance of neighbouring terms with different principal quantum numbers, all the lines involved‘ are to a very good approximation hydrogenic.
FIG. I. The ratios of the envelope curves R,(Q, ro/D) for three different
parameters
of rO/D.
Determination
of electron
density
571
from line merging
The numerical evaluation of the ratio R(n) simpZties appreciably in the case of low electron densities. Figure 1 contains the results as a function of the dimensionless variable Q, which is given by
Q = 1.07. 1Or6 cm-’
z3 (4)
- s/n)3 .
N??(l
N is again the sum of ion and electron density and 6 the Rydberg the very important influence of the different microfield distribution by the shielding parameter r,,/D r0 D
-
N?16
OK) l/.7 ,
O.O898(T;“‘(cm
=
correction. One notices functions characterized
where r. = mean particle
distance
D = Debye
and
length.
Comparing the ratio R,(Q) for the limit of low electron densities with the exact ratio R(n) at higher electron densities it turns out that both functions coincide within 3 per cent as long as the conditions N ‘k z~.‘~O’~ cmm3 and R(n) < 3 are fulfilled. Therefore R,(Q) is a useful quantity in nearly all applications. In Fig. 1 the value of Q = 3.35 is distinguished by the Inglis-Teller relation inserting equation (1) into (4). It yields the ratio of the envelope curves, by which the “last discernible” line as proposed by Inglis and Teller, is defined. As this ratio can become very large, Fig. 1 demonstrates that in general the value of the electron density determined by the Inglis-Teller relation is considerably too small. The final method of evaluation can now be given by the following steps: (a) determination of R(n) from the measured line merging (b) determination of ro/D and the appropriate ratio R,(Q, r,/D). For the most unfavourable case of large r,/D the maximum error of the temperature should not exceed about 30 per cent in order to get the highest attainable accuracy of the method. For the density Ni it is sufficient to use the rough estimate according to the INGIS-TELLER relation”‘. (c) determination of Q by comparing the experimental and theoretical ratios R(n) = Ro(Q, ro/D) (d) determination of N = Ni + N, by the relation (4). In summary it can be stated that providing sufficient measuring accuracy the suggested method for determining the electron density from the merging of the lines gives the electron density with an error of about 5 per cent as long as the conditions N e < z~.‘~O’~ cme3 9 are fulfilled. The error increases series of ionized atoms. The new suggested method
T
4.6 . 10’~
“Kc
n,
.to about has been
and
R(n) < 5
10 per cent if one employs applied
to measurements
the merging performed
of the with
a
578
C.-R.
VIDAL
stationary r.f. discharge producing homogeneous and optically thin plasmas of hydrogen or helium. In both cases the electron density determined by means of the ratio of the envelope curves differs less than 3 per cent from the results obtained by independent measurements of the very distant quasistatic line wings of the first series members. REFERENCES
I. D. INGLISand E. TELLER, Astrophp. J. 90, 439 (1939). 2. A. UNSOLD, Physik der Sternatmosphiiren, Springer (1955). 3. A. PANNEKOEK, Mon. Not. R. Astr. Sot. 98, 694 (1938).
.Sp~cmsc.
Radar.
Transfer.
Vol.
6. pp. 575-578..
Pergamon Press Ltd., 1966. Printed in Great Britain
DETERMINATION FROM
OF ELECTRON LINE MERGING
DENSITY
C.-R. VIDAL Max-Planck-Institut fiir Physik und Astrophysik, Miinchen, Germany The following article is a brief summary of the results already published in JQSRT
and simple estimate it is a very convenient method to get the electron density from the principal quantum number n, of the last discernible line in a series by means of the well known INGLIS-TELLER relation”’ FOR A ROUGH
log(N cm3) = 23.26 - 7.5 log n, + 4.5 log z,
(1)
where z = effective nuclear charge. The density N is equal to the sum of the ion and electron density, N = Ni+N,, as long as the condition
T “Kc
4.6. 105. z n,
is fulfilled (see UNSOLD@)). This frequently used relation yields the electron density only within a factor of two because of three reasons. Firstly, the “last discernible” line is not precisely defined. Secondly,_the splitting of the Stark levels is very much simplified by using an average splitting for all Stark components together. Thirdly, Inglis and Teller use a fixed mean value of the microfields neglecting that the actual distribution function of the microfields depends on the mutual interaction of the perturbing particles or, in other words, that the mean value of the microfields depends on tempezature and density. In order to avoid the preceding deficiencies improved calculations have been performed and a new simple method for determining the electron density from the line merging has been suggested as follows. If one plots a curve Zminthrough the minima of the line merging and a curve I,, through the maxima, the ratio of these envelope curves R(n) = k Ill‘” is a unique function of the electron density. 515
(2)
576
C.-K.
VIDAL
The calculations start with the determination of the differential oscillator strength dj;!dl’ in the region of the merging. Similar to PANNEKOEK ‘3) this has been performed by summing up the partial line intensities of each line profile giving a contribution to the line merging at the position of interest. For this purpose the single quasistatic line profiles have been approximated by a proper expression, which is able to describe the line profiles with an,error of less than 1 per cent. By means of the differential oscillator strength one can immediately obtain the emission coefficient, which is given by (3) where N,* is the population density of the ground term according to the Saha equation. Calculating the ratio of the envelope curves on the basis of equation (3) it turns out of the maxima that a number of quantities cancel, because only the relative intensities and minima of the line merging are important. Therefore besides the constant factors we can drop the population density expressed by N,* . exp( - hv/kT), the frequency dependence i’,, - v3 and the oscillator strength of the lines concerned. Moreover the frequency dependence of the measuring device can be neglected. Furthermore the ratio of the envelope curves can also be applied to atoms and ions other than hydrogen, because in the region of the merging, where the Stark broadening gets comparable with the distance of neighbouring terms with different principal quantum numbers, all the lines involved‘ are to a very good approximation hydrogenic.
FIG. I. The ratios of the envelope curves R,(Q, ro/D) for three different
parameters
of rO/D.
Determination
of electron
density
571
from line merging
The numerical evaluation of the ratio R(n) simpZties appreciably in the case of low electron densities. Figure 1 contains the results as a function of the dimensionless variable Q, which is given by
Q = 1.07. 1Or6 cm-’
z3 (4)
- s/n)3 .
N??(l
N is again the sum of ion and electron density and 6 the Rydberg the very important influence of the different microfield distribution by the shielding parameter r,,/D r0 D
-
N?16
OK) l/.7 ,
O.O898(T;“‘(cm
=
correction. One notices functions characterized
where r. = mean particle
distance
D = Debye
and
length.
Comparing the ratio R,(Q) for the limit of low electron densities with the exact ratio R(n) at higher electron densities it turns out that both functions coincide within 3 per cent as long as the conditions N ‘k z~.‘~O’~ cmm3 and R(n) < 3 are fulfilled. Therefore R,(Q) is a useful quantity in nearly all applications. In Fig. 1 the value of Q = 3.35 is distinguished by the Inglis-Teller relation inserting equation (1) into (4). It yields the ratio of the envelope curves, by which the “last discernible” line as proposed by Inglis and Teller, is defined. As this ratio can become very large, Fig. 1 demonstrates that in general the value of the electron density determined by the Inglis-Teller relation is considerably too small. The final method of evaluation can now be given by the following steps: (a) determination of R(n) from the measured line merging (b) determination of ro/D and the appropriate ratio R,(Q, r,/D). For the most unfavourable case of large r,/D the maximum error of the temperature should not exceed about 30 per cent in order to get the highest attainable accuracy of the method. For the density Ni it is sufficient to use the rough estimate according to the INGIS-TELLER relation”‘. (c) determination of Q by comparing the experimental and theoretical ratios R(n) = Ro(Q, ro/D) (d) determination of N = Ni + N, by the relation (4). In summary it can be stated that providing sufficient measuring accuracy the suggested method for determining the electron density from the merging of the lines gives the electron density with an error of about 5 per cent as long as the conditions N e < z~.‘~O’~ cme3 9 are fulfilled. The error increases series of ionized atoms. The new suggested method
T
4.6 . 10’~
“Kc
n,
.to about has been
and
R(n) < 5
10 per cent if one employs applied
to measurements
the merging performed
of the with
a
578
C.-R.
VIDAL
stationary r.f. discharge producing homogeneous and optically thin plasmas of hydrogen or helium. In both cases the electron density determined by means of the ratio of the envelope curves differs less than 3 per cent from the results obtained by independent measurements of the very distant quasistatic line wings of the first series members. REFERENCES
I. D. INGLISand E. TELLER, Astrophp. J. 90, 439 (1939). 2. A. UNSOLD, Physik der Sternatmosphiiren, Springer (1955). 3. A. PANNEKOEK, Mon. Not. R. Astr. Sot. 98, 694 (1938).