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Approximation for the probabilities of the realization of atomic bound states in a plasma
J. Quant, Spectrosc. Rndiot. Transfer Vol. 59, Nos l/2, pp. 65-69. 1998 0 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain 0022~4073/98 $19.00 + 0.00 PII: S0022-4073(!H)00137-4
Pergamon
APPROXIMATION FOR THE PROBABILITIES OF THE REALIZATION OF ATOMIC BOUND STATES IN A PLASMA
Institute
for
L. G. D’YACHKOV High Temperatures,RussianAcademyof Sciences, Izhorskaya 13/19, 127412 Moscow, Russia (Received 17 February 1997)
Abstract-A
simple analytical approximation for the cumulative of the Holtsmark and Hooper microfield distribution functions, which estimates the probability of the realization of atomic bound states in a plasma, is presented. 0 1998 Elsevier Science Ltd. All rights reserved
1.
INTRODUCTION
Higher excited states of an atom immersed in a plasma are destroyed due to the interaction with surrounding particles. This leads to the truncation of the internal partition function, dissolution of spectral lines and transformation of the upper members of spectral series to a continuum. Although the real physical situation is complex, a rather simple microfield model is usually used It is based on the assumption that an atomic for the description of the state destruction.‘-” bound state can be realized in a plasma if the value of the quasi-static (ionic) electric microfield F at the point where the atom is placed, is less than a certain critical value F,, and is destroyed (dissolved) if F> F,. Then the probability of finding the atom in this state relative to that of finding it in the case of an unperturbed atom, i.e., the realization probability, is F, w= J0
P(F ) dF
(1)
where P(F) is the quasi-static microfield distribution function. For non-correlated charges in an ideal plasma the distribution P(F) has been obtained by Holtsmark.” An accurate rational approximation for the Holtsmark distribution and its cumulative (Equation (1)) has been given by Hummer.12 Hooper13 has taken into account the influence of the interaction between charges. He has calculated the distribution function in neutral and charged points for a low-frequency electric microfield arising from singly charged ions which interact with each other through a shielder Coulomb potential. It is helpful to get a simple approximation for the integral (l), since the direct numerical calculation is too cumbersome in the case of a plasma of complex chemical composition.” Such an approximation is presented here.
2. APPROXIMATION We present a simple approximate expression for the integral (1) taking into account dependence on both the critical microfield Fc and the parameter c( = q-l”, where q is the number of ions in a Debye sphere of radius rD= (kT/4ne2ni) ‘I2 . It is written as the sum of main and asymptotic terms w = win + ( wa- Wnl)f(P) with the main term in the form 65
(2)
66
L. G. D’yachkov
(3)
where x = log fi, /3 = F,/Fo is the reduced critical field and Fo = e(4nnJ3)2’3. Parameters A, B, x0 and Ax depend on CIand are approximated by polynomials. Since the expression (3) does not provide the accurate behaviour of W for B --, 0 and fi + 00, we add the second term in Equation (2) with
J?nB1(y!+$?‘+pg Blexp
x0
w, =
(4) l-
---$zB~/~(Do
+ $j?-3f2
+ +fi-‘)
fi > exp x0
/
and
‘(‘) = 1exp(-q44/@) exp(-P4/p4)
PI cxp x0 /I > exp x0
where p = exp(xo - 2Ax)
(6)
and exp(xo + 2Ax) ’ = I exp[xo + (2 + cr/2)Ax]
for neutral point for charged point
(7)
Function (4) with Ca=l, Cl= -0.463, C2=0.123, Do=l, Di=5.11 and D2=14.4 gives the accurate asymptotic behaviour of the integral (1) of the Holtsmark distribution function (a = 0) for /?-=~l and B-1. In general case (a # 0) we take the same form of the asymptotic expressions, but the coefficients C and D will depend on a. The function (5) provides a rapid transition from W,to W,,so WrW,,,forpq. The a dependence of the parameters involved in Equations (3) and (4) is presented by following polynomials. For a neutral point A = 0.966 - 0.023a + 0.028a2 - 0.009a3 B = 0.966 + 0.033a + 0.01 la2 - 0.017a3
x0 = 0.739 - 0.661a + 0.049a2 Ax = 0.359 + 0.152a + 0.008a2 Co = (I + 1.188a + 0.973a2 + 0.529a3)’
Ci = -(0.773 + 1.318a + 0.266a2 + 1.745a3)3 C2 = (0.59 + 0.865a + 0.61a2 + 1.396a3)4 DO = 1 - 0.026a + 0.066a2 - 0.084a3 D1 = 5.11 + 0.76a - 19.22a2 + 13.07a3
02 = 14.4 - 172.3a + 356.2a2 - 207.7a3
Probabilities of the realization of atomic bound states
67
and for a charged point A = 0.966 - 0.03~ + 0.119~~ - 0.056~~ B = 0.966 + 0.023~ + 0.102~~ - 0.056~~ x0 = 0.139 - 0.686a - 0.06~~ Ax = 0.359 + 0.14411 Co = (1 + 1.329~ + 0.613~~ + 1.373a3)* c, = -(0.773 + 1.303LY + 0.502M2+ 1.991a3)3 C2 = (0.59 + 0.8371x+ 0.831~1~+ 1.512~r~)~ DO = 1 - 0.029~ - 1.791a2 + 1.08cr3 DI = 5.11 + 0.41cr+ 8.34~~ - 5.25cr3 D2 = 14.4 - 173x + 211.21~~- 108.2~~ They give the approximation of the cumulative of the Hooper distribution with accuracy within 0.003 (within 0.002 for the Holtsmark distribution) which is quite enough for most of applications. We show W versus F,/Fo in Fig. 1, where our approximation is compared with results of accurate numerical integration of the Holtsmark and Hooper distributions for neutral point. Both the accurate and approximate curves for a = 0, 0.2, 0.4, 0.6 and 0.8 coincide. The calculation in the present approximation is possible up to CI= 1. Extrapolation of the Hooper results to the region a > 1 is not correct, since the Debye radius becomes less than the ion-sphere one r0 = (4LZ?Zi/3)-“3and loses the mining of the screening radius. However, formal extrapolation of the results is possible up to a r 1.5 if for c(> 1 one replaces Ax in Equations (6) and (7) by aAx (curves 1.2 and 1.4 in Fig. 1).
0
1
2
3
4
5
F/F, Fig. 1. Comparison of the present approximation (- - -) with accurate cumulative of the Holtsmark (a = 0) and Hooper (a>O) distributions (---) for neutral point. For a = 0 to 0.8 the accurate and approximate curves merge.
68
L. G. D’yachkov 0.6
0.1
0.0
1
0
2
3
4
5
F/F, Fig. 2. Comparison
of the present approximation (- --) with the Holtsmark (a = 0) and Hooper (a > 0) distributions (---) for neutral point.
An important problem in the calculations of the realization probabilities is the estimation of the critical microfield F,. It is a rather complex problem which has no final solution at present. The simplest models, uniform field (UF) and nearest neighbour (NN), which give considerably different values F, = E2/4e3 (UF) and F, = E2/16e3 (NN) for a state with ionization energy E in the unperturbed atom (atomic core and nearest neighbour ion are assumed singly charged), are frequently applied. ‘-lo For one-electron atoms UF is reasonable,10T’4 while for many-electron atoms it is possible that NN is more appropriate.’ Weaker dissolution of hydrogenic states can apparently be explained by the fact that the classical electron orbits in a Coulomb field are closed.‘5 Using the approximation Equations (2)-(7) and differentiating W, we can also easily obtain an analytical approximation for the microfield distribution function P(F). A comparison of this approximation with Holtsmark and Hooper distributions is shown in Fig. 2. Obviously, the accuracy is lower than for their cumulatives, but appropriate for most of applications. The error for a = 0 is 0.0025 and it increases up to 0.02 for LY= 0.8. 3. CONCLUSIONS
The analytical approximation for the cumulative of the Holtsmark and Hooper microfield distributions is presented. It gives a simple expression for the probability of the bound state realization in a plasma as a function of both the critical microfield F,, at which the state begins to be destroyed, and the parameter tl taking into account the degree of the plasma non-ideality. From the differentiation of this expression, the approximation for the microfield distribution depending on ~1is also obtained. The accuracy of the approximations is appropriate for most applications. Acknowledgements-The author thanks G. A. Kobzev, Yu. K. Kurilenkov and Y. Vitel for the discussions of some questions considered in this note. This research was supported in part by the Russian Foundation for Basic Research under Grant No. 96-02-18830 and NATO International Scientific Exchange Program under Linkage Grant No. HTECH.LG 960803.
REFERENCES 1. 2. 3. 4. 5. 6.
Unsaid,
A., Z. Astrophys.,
1948, 24, 355.
Avilova, I. V. and Norman, G. E., Teplofiz. Vys. Temp., 1964, 2, 517. 1971, 11, 1. 1971, 34, 310. Gurovich, V. Ts. and Engelsht, V. S., Sov. Phys. JETP, 1977,45,232. Ruzdjak, V. and Vujnovic, V., Astron. Astrophys., 1977, 54, 751. Giindel,
H., Beitr. Pfusmuphys.,
Kobzev, G. A., Sov. Phys. JETP,
Probabilities of the realization of atomic bound states 7. Kobzev, G. A. and Kurilenkov, Yu. K., High Temp., 1978, 16, 385. 8. Sevastyanenko, V., Contrib. Plasma Phys., 1985, 25, 151. 9. Gavrilov, V. E. and Gavrilova, T. V., Opt. Spectrosc., 1987, 63, 429. 10. Hummer, D. G. and Mihalas, D., Astrophys. J., 1988, 331, 794. 11. Holtsmark, J., Ann. Phys., 1919, 58, 577. 12. Hummer, D. G., JQSRT, 1986,36, 1. 13. Hooper, C. F. Jr., Phys. Rev., 1968, 165, 215. 14. Seaton, M. J., J. Phys. B, 1990, 23, 3255. 15. Kazanskii, A. K., Opt. Spectrosc., 1988, 64, 448; 1988, 65, 826.
69
Pergamon
APPROXIMATION FOR THE PROBABILITIES OF THE REALIZATION OF ATOMIC BOUND STATES IN A PLASMA
Institute
for
L. G. D’YACHKOV High Temperatures,RussianAcademyof Sciences, Izhorskaya 13/19, 127412 Moscow, Russia (Received 17 February 1997)
Abstract-A
simple analytical approximation for the cumulative of the Holtsmark and Hooper microfield distribution functions, which estimates the probability of the realization of atomic bound states in a plasma, is presented. 0 1998 Elsevier Science Ltd. All rights reserved
1.
INTRODUCTION
Higher excited states of an atom immersed in a plasma are destroyed due to the interaction with surrounding particles. This leads to the truncation of the internal partition function, dissolution of spectral lines and transformation of the upper members of spectral series to a continuum. Although the real physical situation is complex, a rather simple microfield model is usually used It is based on the assumption that an atomic for the description of the state destruction.‘-” bound state can be realized in a plasma if the value of the quasi-static (ionic) electric microfield F at the point where the atom is placed, is less than a certain critical value F,, and is destroyed (dissolved) if F> F,. Then the probability of finding the atom in this state relative to that of finding it in the case of an unperturbed atom, i.e., the realization probability, is F, w= J0
P(F ) dF
(1)
where P(F) is the quasi-static microfield distribution function. For non-correlated charges in an ideal plasma the distribution P(F) has been obtained by Holtsmark.” An accurate rational approximation for the Holtsmark distribution and its cumulative (Equation (1)) has been given by Hummer.12 Hooper13 has taken into account the influence of the interaction between charges. He has calculated the distribution function in neutral and charged points for a low-frequency electric microfield arising from singly charged ions which interact with each other through a shielder Coulomb potential. It is helpful to get a simple approximation for the integral (l), since the direct numerical calculation is too cumbersome in the case of a plasma of complex chemical composition.” Such an approximation is presented here.
2. APPROXIMATION We present a simple approximate expression for the integral (1) taking into account dependence on both the critical microfield Fc and the parameter c( = q-l”, where q is the number of ions in a Debye sphere of radius rD= (kT/4ne2ni) ‘I2 . It is written as the sum of main and asymptotic terms w = win + ( wa- Wnl)f(P) with the main term in the form 65
(2)
66
L. G. D’yachkov
(3)
where x = log fi, /3 = F,/Fo is the reduced critical field and Fo = e(4nnJ3)2’3. Parameters A, B, x0 and Ax depend on CIand are approximated by polynomials. Since the expression (3) does not provide the accurate behaviour of W for B --, 0 and fi + 00, we add the second term in Equation (2) with
J?nB1(y!+$?‘+pg Blexp
x0
w, =
(4) l-
---$zB~/~(Do
+ $j?-3f2
+ +fi-‘)
fi > exp x0
/
and
‘(‘) = 1exp(-q44/@) exp(-P4/p4)
PI cxp x0 /I > exp x0
where p = exp(xo - 2Ax)
(6)
and exp(xo + 2Ax) ’ = I exp[xo + (2 + cr/2)Ax]
for neutral point for charged point
(7)
Function (4) with Ca=l, Cl= -0.463, C2=0.123, Do=l, Di=5.11 and D2=14.4 gives the accurate asymptotic behaviour of the integral (1) of the Holtsmark distribution function (a = 0) for /?-=~l and B-1. In general case (a # 0) we take the same form of the asymptotic expressions, but the coefficients C and D will depend on a. The function (5) provides a rapid transition from W,to W,,so WrW,,,forpq. The a dependence of the parameters involved in Equations (3) and (4) is presented by following polynomials. For a neutral point A = 0.966 - 0.023a + 0.028a2 - 0.009a3 B = 0.966 + 0.033a + 0.01 la2 - 0.017a3
x0 = 0.739 - 0.661a + 0.049a2 Ax = 0.359 + 0.152a + 0.008a2 Co = (I + 1.188a + 0.973a2 + 0.529a3)’
Ci = -(0.773 + 1.318a + 0.266a2 + 1.745a3)3 C2 = (0.59 + 0.865a + 0.61a2 + 1.396a3)4 DO = 1 - 0.026a + 0.066a2 - 0.084a3 D1 = 5.11 + 0.76a - 19.22a2 + 13.07a3
02 = 14.4 - 172.3a + 356.2a2 - 207.7a3
Probabilities of the realization of atomic bound states
67
and for a charged point A = 0.966 - 0.03~ + 0.119~~ - 0.056~~ B = 0.966 + 0.023~ + 0.102~~ - 0.056~~ x0 = 0.139 - 0.686a - 0.06~~ Ax = 0.359 + 0.14411 Co = (1 + 1.329~ + 0.613~~ + 1.373a3)* c, = -(0.773 + 1.303LY + 0.502M2+ 1.991a3)3 C2 = (0.59 + 0.8371x+ 0.831~1~+ 1.512~r~)~ DO = 1 - 0.029~ - 1.791a2 + 1.08cr3 DI = 5.11 + 0.41cr+ 8.34~~ - 5.25cr3 D2 = 14.4 - 173x + 211.21~~- 108.2~~ They give the approximation of the cumulative of the Hooper distribution with accuracy within 0.003 (within 0.002 for the Holtsmark distribution) which is quite enough for most of applications. We show W versus F,/Fo in Fig. 1, where our approximation is compared with results of accurate numerical integration of the Holtsmark and Hooper distributions for neutral point. Both the accurate and approximate curves for a = 0, 0.2, 0.4, 0.6 and 0.8 coincide. The calculation in the present approximation is possible up to CI= 1. Extrapolation of the Hooper results to the region a > 1 is not correct, since the Debye radius becomes less than the ion-sphere one r0 = (4LZ?Zi/3)-“3and loses the mining of the screening radius. However, formal extrapolation of the results is possible up to a r 1.5 if for c(> 1 one replaces Ax in Equations (6) and (7) by aAx (curves 1.2 and 1.4 in Fig. 1).
0
1
2
3
4
5
F/F, Fig. 1. Comparison of the present approximation (- - -) with accurate cumulative of the Holtsmark (a = 0) and Hooper (a>O) distributions (---) for neutral point. For a = 0 to 0.8 the accurate and approximate curves merge.
68
L. G. D’yachkov 0.6
0.1
0.0
1
0
2
3
4
5
F/F, Fig. 2. Comparison
of the present approximation (- --) with the Holtsmark (a = 0) and Hooper (a > 0) distributions (---) for neutral point.
An important problem in the calculations of the realization probabilities is the estimation of the critical microfield F,. It is a rather complex problem which has no final solution at present. The simplest models, uniform field (UF) and nearest neighbour (NN), which give considerably different values F, = E2/4e3 (UF) and F, = E2/16e3 (NN) for a state with ionization energy E in the unperturbed atom (atomic core and nearest neighbour ion are assumed singly charged), are frequently applied. ‘-lo For one-electron atoms UF is reasonable,10T’4 while for many-electron atoms it is possible that NN is more appropriate.’ Weaker dissolution of hydrogenic states can apparently be explained by the fact that the classical electron orbits in a Coulomb field are closed.‘5 Using the approximation Equations (2)-(7) and differentiating W, we can also easily obtain an analytical approximation for the microfield distribution function P(F). A comparison of this approximation with Holtsmark and Hooper distributions is shown in Fig. 2. Obviously, the accuracy is lower than for their cumulatives, but appropriate for most of applications. The error for a = 0 is 0.0025 and it increases up to 0.02 for LY= 0.8. 3. CONCLUSIONS
The analytical approximation for the cumulative of the Holtsmark and Hooper microfield distributions is presented. It gives a simple expression for the probability of the bound state realization in a plasma as a function of both the critical microfield F,, at which the state begins to be destroyed, and the parameter tl taking into account the degree of the plasma non-ideality. From the differentiation of this expression, the approximation for the microfield distribution depending on ~1is also obtained. The accuracy of the approximations is appropriate for most applications. Acknowledgements-The author thanks G. A. Kobzev, Yu. K. Kurilenkov and Y. Vitel for the discussions of some questions considered in this note. This research was supported in part by the Russian Foundation for Basic Research under Grant No. 96-02-18830 and NATO International Scientific Exchange Program under Linkage Grant No. HTECH.LG 960803.
REFERENCES 1. 2. 3. 4. 5. 6.
Unsaid,
A., Z. Astrophys.,
1948, 24, 355.
Avilova, I. V. and Norman, G. E., Teplofiz. Vys. Temp., 1964, 2, 517. 1971, 11, 1. 1971, 34, 310. Gurovich, V. Ts. and Engelsht, V. S., Sov. Phys. JETP, 1977,45,232. Ruzdjak, V. and Vujnovic, V., Astron. Astrophys., 1977, 54, 751. Giindel,
H., Beitr. Pfusmuphys.,
Kobzev, G. A., Sov. Phys. JETP,
Probabilities of the realization of atomic bound states 7. Kobzev, G. A. and Kurilenkov, Yu. K., High Temp., 1978, 16, 385. 8. Sevastyanenko, V., Contrib. Plasma Phys., 1985, 25, 151. 9. Gavrilov, V. E. and Gavrilova, T. V., Opt. Spectrosc., 1987, 63, 429. 10. Hummer, D. G. and Mihalas, D., Astrophys. J., 1988, 331, 794. 11. Holtsmark, J., Ann. Phys., 1919, 58, 577. 12. Hummer, D. G., JQSRT, 1986,36, 1. 13. Hooper, C. F. Jr., Phys. Rev., 1968, 165, 215. 14. Seaton, M. J., J. Phys. B, 1990, 23, 3255. 15. Kazanskii, A. K., Opt. Spectrosc., 1988, 64, 448; 1988, 65, 826.
69