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The asymmetry of Ly-α and Ly-β
J. Quant. Spectrosc. Radiat. Transfer. Vol. 17, pp. 501-512. Pergamon Press 1977. Printed in Great Britain
THE ASYMMETRY OF Ly-a AND Ly-/3 M. E. BACON~ Department of Physics, University of British Columbia, Vancouver B.C., Canada (Received 19 July 1976)
Abstract--The asymmetryof the first two spectral lines of the Lyman series of hydrogenis calculated. Detailed profiles,whichinclude the effects of time orderingand asymmetry,are presented for a regionof the line profilefrom the center to the line wings, where the impactapproximationfor the electrons and the quasi static approximationfor ions are reasonablyapplicable. Deviationsin the line wingsfrom "Unifiedtheories" are of the order 0-15%for the profile parameter range considered. Suchdeviations shouldhave only a minor effect on the computed asymmetry, The far red wing of both Ly-a and Ly-fl is seen to dominate the blue wing. In the near wings the blue intensity becomesequal to the red intensity. From here into the line center, the blue intensity is larger than the red intensity. In the case of Ly-/3,there is the expectedenhancementof the blue peak and reductionof the red peak, The calculated profiles are compared with previous calculations and recent experimental results. I. INTRODUCTION IN RECENTyears there has been renewed experimental interest in the asymmetry of the hydrogen spectral lines. "-'~ This is due not only to its intrinsic interest as a check on line broadening calculations but also because of possible astrophysical significance/2"5'6~ In addition, such studies are related to, and have a certain relevance to, asymmetries and shifts of hydrogenic ion lines in high density laser produced plasmas/7~ The asymmetry of the hydrogen lines most frequently referred to in the literature is the characteristic asymmetry of the blue and red peaks of Ha, primarily because this line has received the most attention. Reference to this work and earlier experimental results dealing with the asymmetry of other hydrogen lines can be found in Ref. 8. There have been a number of attempts to account theoretically for the asymmetry of the hydrogen iinesff -''~ with particular attention paid to the Ly-a asymmetry. For the parameter range investigated experimentally to date, the Ly-a asymmetry can be traced to the effects of ion-field inhomogeneity, as outlined by SHOLIN."3~ Numerical asymmetry computations for one-electron density and temperature, which include ion-field inhomogeneity and the quadratic Stark effect, have been performed previously by the author, "5~ who, however, slightly overestimated the effects of the quadratic Stark effect. Recently, DEMURAand SHOLINt~s~have presented some analytic results for the asymmetry of the hydrogen spectral lines and compared them with the experimental results of BOLDT and C O O P E R . tiT) In the present paper, we confine our attention to the Ly-a and Ly-/3 lines and present detailed calculations which include the effects of time ordering, ion-field inhomogeniety and the quadratic Stark effect. Although these two lines are difficult to measure experimentally, °'tT"ts~ they are the easiest to treat theoretically and should give an indication of the expected behaviour of the more experimentally accessible lines, such as the Balmer Series lines. In the following section, a brief outline of the computational details used in the present calculations of Ly-a and Ly-/3 are presented. The relevant asymmetry corrections to the wave functions and energy levels for the n = 1, 2, and 3 levels are given in the appendix. Also given are the dipole moments for Ly-a and Ly-/3. In Section 3, we compare the present profiles with previously calculated profiles and with the most recent experimental results. The final section contains a discussion and conclusion. 2. C O M P U T A T I O N A L
DETAILS
In the present calculation, we used the impact approximation in treating the electrons and the quasi static approximation in treating the ions. In this sense, therefore, the present calculations tOn leave from the University of Natal, Durban, S. Africa. 501
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M.E. BACOS
can be considered to be a "modified impact" calculation. The use of the above approximations should be valid for the range of profile parameters considered and deviations from unified calculations should be small. The treatment of the quasi-static ion perturbation was carried out in a similar manner to that described by S8OLI#'3) with only minor differences. Firstly, a number of sign changes are apparent in the evaluation of some of the ion-atom multipole matrix elements. This is due to the choice of ion field direction. Secondly, there is a difference in the second order ion-atom quadrupole contribution to the energy as mentioned in Ref. 15. This has very little effect on the calculations, since the major contribution to the asymmetry is due to the first-order quadrupole correctionsY 3~ The various matrix elements of the ion-atom multipole perturbations were checked against the appropriately modified electron values used in Refs. 19 and 20 in treating the electron impact broadening of the n = 2 and n = 3 states, respectively. The expressions [eqn (48)] given by NCtrVEN-HOEet al. <2"were used to evaluate the contribution of the quadratic Stark effect to the dipole moments. This differs from the calculations of Ref. 15, where (n 2- 1) was used instead of (7n 2 - 1) in the above mentioned equations. This resulted in a slight overestimation in the significance of the quadratic Stark effect at the larger separations from the line center. For Ly-a, contributions from the quadrupole ion-atom interactions were included in calculating the asymmetry. For Ly-fl, the contribution from the octupole ion-atom interaction was also included. This was done in an approximate way by assuming that the ion-fields were produced by one nearest neighbor. In other words, the )t s (not to be confused with wavelength) in the expressions for the wave functions, energies and dipole moments were given by A = n2a([Fo/e) I/2, where n is the principal quantum member of the level in question, a is the Bohr radius, Fo = 2.61 eN 2s3is the Holtsmark normal field strength, and f is the reduced field strength. The Hooper ion field distribution functions ~2~ were used throughout, except for values of f > 30 where the Holtsmark values were used. Although this is rather a naive procedure for the region of the line profile around the line center, it would seem likely that a more exact treatment, which would include the many-body aspects of the problem, would not yield significantly different results. In any event, the blue-red peak asymmetry of Ly-/3 calculated using the above procedure should be better than, say, a calculation using the normal Holtsmark field and the dipole moments given in the appendix The effects of time ordering have been included in the treatment of the electron impact perturbation and are based on earlier numerical computations. "9'2°~ In addition, in the far wings frequency dependent cut-offs~23~have been used in treating the electron impact broadening operator. In the case of Ly-a, the reduced line Profiles S(a) were calculated for various values of electron density and temperature, and are given in Tables 1-5. In these tables, the electron density is in units of cm -3, and the Holtsmark normal field is in cgs field strength units. Alpha is given by (to - coo)Ao2/21rcFo,where COois the angular frequency at the unshifted line center and oJ is the angular frequency at the point of interest on the line profile. An estimation of the wavelength separation from the line center for a particular value of S can be obtained by multiplying a by Fo. A more exact expression for the wavelength is given by h = AdO + Foa/ho), where ho is the wavelength at the line center. The integers in the S(a) columns represent the power of ten associated with the preceeding number. An absorption profile can be obtained from these values of S(a) by multiplying by a factor CO/COo= (1 + Foalho). An emission profile can be obtained by multiplying S(a) by (1 + Foa/ho) 4. Thermal Doppler broadening has not been included in these calculations. The reduced field strength, f ranged from 0-100 in steps of 0.1. The reduced line profiles for Ly-/3 were calculated in a similar manner and are given in Tables 5-10. In these calculations, however, f ranged from 0-50 in steps of 0.2. Using these tables and interpolating for an electron density of 7.2 x 1016cm -3 and an electron temperature of 1.22 x 104 K yielded profiles for Ly-a and Ly-fl, which agreed to within < 2% With calculations using the original computer program. The correspondingly evaluated asymmetry agreed to within 5%. 3. COMPARISON WITH PREVIOUS CALCULATIONS AND RECENT EXPERIMENTS Considering the Ly-a profiles first and comparing these with the calculations of KEPPLEand GRIEM~24)and VIOALet al. <2~we note the dominant time ordering effect at the line center, namely a
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lowering of the peak intensity. "9'2° In the far wings, the profiles have generally a somewhat larger value than that of the "unified" calculation of Vidal et al. This is again due to time ordering and is compatible with calculations by GODFREY et a/. t2" and VOSLAMBERc2s) [see also GREENE and COOPER(29q.Generally, if the asymmetry is neglected, the present profiles differ from the unified results of Vidal et al. by ~< 10% in the far wings. In the far wings, the red wing ( - c~) is seen to be of higher intensity than the blue wing (+ a). This asymmetry decreases as we approach the line center. At [al ~ 10-3, the wing intensities become equal and from there into the line center, the blue side dominates the red. This behavior is consistant with what one might expect from a consideration of Fig. 1 given by SHOLIN."~ The asymmetry described above can be traced primarily to the quadrupole ion-atom interaction with an insignificant role played by the quadratic Stark effect, as already mentioned. In Fig. 1, we compare the results of the present calculations with the recent experimental data o f FUSSMANN. °) In this figure, R is the ratio of the calculated and experimental results to the Holtsmark values. The experimental curve is taken from Ref. 3. The agreement between calculation and experiment in terms of absolute values is seen to be rather good. Figure 2 . . . . . .
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The asymmetryof Ly-a and Ly-g
509
compares the theoretical and experimental asymmetry, as measured by (AA)I = I/2(AR -- AB) and (AA)2 = l/2(h~ + AB) -- ho; AR and As are the wavelength values on the red and blue wings, for equal intensity points. This method of measuring the asymmetry is relatively insensitive to differences in absolute intensities as determined by various calculations, unified or improved impact. The experimental points are given by the crosses and the dashed curve is a smooth curve drawn through these points. ~3~The open circles represent the calculated asymmetry, including that due to frequency to wavelength conversion and the frequency factor for the absorption profile. The solid curve is a smooth curve fit to these calculated data. The agreement between experiment and theory is relatively good, considering the complexities involved in the experimental measurements. Significant differences are perhaps the lower experimental asymmetry at (AA~) > 6/~, the different crossover point (0.9/~ experimentally and - 0.5/~ theoretically) and the larger blue asymmetry near the line center. Turning now to Ly-/3 and comparing the present calculations with those of KEPPLE and GRIEMt24)and VIDALet al. ~25~we again see the effect of time ordering at the line center, viz a filling in of the central dip between the two maxima. Figure 3 illustrates the typical differences between the present calculations and those of Refs. 24 and 25. The profiles of both Kepple and Griem and Vidal et al. are symmetric and, hence, only half of each profile is shown, as in the case of Ly-a, the red wing is seen to dominate the blue wing and there is a crossover point as one approaches the line center, this time at [a I ~ 3 x 10-3. For as smaller than this, the blue dominates the red. Again, this behavior is as one might expect on the basis of Fig. 1, Ref. 13.
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510
M. E; BACON
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theory and the experimental results of Fussmann for AA > 6 ~ and AA < 1 A. In the latter region, this may be due to the nearest neighbor approximation, although Doppler broadening, which was not included in the present calculations, may also play a role. In the former region, there would appear to be a number of possible theoretical reasons for the discrepancy. Firstly, the so called dissolution effect, treated by LAscZOS°°~ and considered by VuJsOVlC~12) in the context of hydrogen Stark broadening, is known to produce a sudden weakening of the red shifted Stark components at high field strengths. °l~ This is then followed by a weakening of the blue components at even higher fields. Based on calculations°2-34~ of this effect, however, field strengths in excess of 1.5 × 107 V/cm would be necessary to make the ionization probability of the red component, due to tunneling comparable to the radiative decay probability [see Fig. 7 of Ref. 30]. Even at an electron density of 10~8cm -3, field strengths of this order only occur for f ~ 100 and, hence, this effect is completely negligible for the experimental conditions of Fussmann. A second possibility is the occurrence of satellites of the type considered by STEWARTet al. ~'> and LE QUANGRANGand VOSLAMBEX.°6~ However, since the red satellites are predicted to occur closer to the line center than the blue satellites, any such effect would tend to enhance the red wing. In any event, the satellites are predicted to occur at AA > 20 A and it is, therefore, unlikely that they would effect the region of the line profile under consideration here (AA < 10 A). The suggestion that screening of the nuclear charge may perhaps play a role,<7}seems also to be untenable at the relatively low density of 7.2 × 10~6cm -3. A more viable explanation of the discrepancy between theory and experiment, is the effect of quasi-static electrons, as suggested by GPaEM° " (see also OKS and SHOLIN<~4>).In the present calculations, no account is taken of quasi static electrons, except, insofar as they are excluded from the electron impact broadening via the frequency dependent cut-offs. Since the quadrupole ion-atom interaction is the dominant' source of asymmetry, and since this depends on the sign of the charged perturbers, the effect of these quasi static electrons would be to reduce the asymmetry. Additional experiments at densities - 10~sand temperatures -< 2 x 104 K, where the effect should be more pronounced, could substantiate this point. The case of Ly-/~ is somewhat different from that of Ly-a. In the latter case, as we have seen above, there is relatively good agreement between calculated and experimental wing intensities and that the relatively small deviations from the observed asymmetries have a possible theoretical explanation. Ly-fl, on the other hand, shows a measured red wing which is substantially less than predicted by theory in the range 0.3 A < AA < 2/~. In addition, the wing asymmetry inferred from Fig. 4 is completely at odds with the calculated asymmetry. The possible theoretical short-comings mentioned above in connection with Ly-a, should be insignificant for the density, temperature and wavelength range under consideration for Ly-fl.
The asymmetry of Ly-a and Ly-//
511
Acknowledgements--The financial assistance of the Atomic Energy Board of South Africa and the National Research Council of Canada is gratefully acknowledged. REFERENCES 1. W. L. WIESE, D. E. KELLE,ERand D. E. PAQUEI"rE,Phys. Rev. 6A, 1132 (1972). 2. R. D. BENGS"rONand G. R. CHESTEL Appl. J. 178, 565 (1972). 3. G. FUSSMANN,JQSRT 15, 791 (1975). 4. R. D. BENGSTONand G. R. CHESTER,Phys. Rev. 13A, 1762 (1976). 5. W. L. WIESE and D. E. KELLEHER,Appl. J. 166, L59 (1971). 6. V. TRIMnLEand J. L. GREENSTEIN,Appl. J. 177, 441 (1972). 7. J. C. WElSHEITand B. F. ROZSNVAI,J. Phys. B: Atom Molec. 9, L63 (1976). 8. H. R. GRIEM, Spectral Line Broadening by Plasmas. Academic Press, New York (1974). 9. L. P. KUDRINand G. V. S,OLIN, Soy. Phys. DOgLADY7, 1015 (1%3). 10. NGUVES-HoL H. W. DRAWINand L. HERMAN,JQSRT 4, 846 (1964). 11. H. R. GRIEM,Phys. Rev. 140, All40 (1%5). 12 V. VUJNOVlC,Int. J. Electronics 18, 411 (1%5). 13. G. V. S,OLIh, Opt. Spectros. 26, 275 (1%9). 14. E. A. OKS and G. V. SHOt.Is, Opt. Spectros. 33, 217 (1972). 15. M. E. BACON,JQSRT 13, 1161 (1973). 16. A. B. DEMURAand G. V. SHOLtN,JQSRT 15, 881 (1975). 17. G. BOLI~Tand W. S. COOPER,Z. Naturforsch. 19a, 968 (1964). 18. R. C. ELTON and H. R. GPaEM,Phys. Rev. 135, 6A (1964). 19. M. E. BACON,K. Y. SHEN and J. COOPER,Phys. Rev. 188, 50 (1%9). 20. M. E. BACON,JQSRT 12, 519 (1972). 21. NGUYEN-HOE,E. BANERJEA,H. W. DRAWLSand L. HERMAN,JQSRT 5, 835 (1%5). 22. C. F. HOOPER,Phys. Rev. 165, 215 0%8); 169, 193 (1968). 23. H. R. GPaEM, Appl. 3". 147, 1092 0%7). 24. P. KEPPLEand H. R. GRIEM,Phys. Rev. 173, 317 (1968); University of Maryland report No. 831. 25. C. R. VIDAL,J. COOPERand E. W. SMrra, Appl. J. Supp. 25, 37 (1973). 26. H. PFENNIG,JQSRT 12, 821 (1972). 27. T. GODFREY,C. R. V]DAL,E. W. SMrm and J. COOPER,Phys, Rev. A3, 1543 (1971). 28. D. VOSLAMBER,Phys. Left. 42A, 469 (1973). 29. R. L. GREENEand J. COOPER,JQSRT 15, 991 (1975). 30. C. LANCZOS,Z. Physik 68, 204 (1931). 31. H. RAUSCH,V. TRAURENBERG,R. GEBAUERand G. LEWIN, Naturwiss. 18, 417 (1930). 32. M. H. RICE and R. H. GOOD,JE., J. Opt. Soc. Am. 52, 239 (1%2). 33. D. S. BAILEY,J. R. HISKESand A. C. RIVlERE,Nucl. Fusion 5, 41 (1965). 34. M. H. ALEXANDER,Phys. Rev. 178, 34 (1%9). 35. J. C. STEWART,J. M. PEEK and J. COOPER,Appl. J. 179, 983 (1973). 36. LE QU^NG RANGand D. VOSLAMnER,J. Phys. B: Atom Molec. 8, 337 (1975). APPENDIX In this appendix, we list the wave functions and energies for the n = 1, 2 and 3 levels of hydrogen, which are used in the calculations outlined in Section 2. This is followed by a list of the relevant dipole moments for Ly-a and Ly-/3. Atomic units are used in these expressions. The wave functions and energies for the n = 1 and n = 2 states have been given previously in Ref. 15. They are reproduced here, with a few changes, for completeness. In Ref. 15, A ---4a(fFole)m for both the n = I and n = 2 levels. Here we use X -= n2a(fFole)'12.The wave functions are denoted [n,n2m) where n, and n2 are the usual electric quantum numbers and m is the magnetic quantum number. The energies are denoted by E,,,2,. The double primes indicate quantities evaluated to second order in the asymmetry corrections, (a) Wavefunctions n = 1 states:
Iooo)"= Iooo) n = 2 states:
1100)"= IlO0)+ (M4)[010)- (A~/32)[100) 1010)"= 1010)- (A/4)1100)- (A2/32)1010) [00-+ 1)"= IO0+ 1) n = 3 states: I110)" = [110) - (A13)[200)+ (AI3)[020)- (A5/9)[110) + (A2519)1200)+ (A55/9)[020)
1200)" = 1200) + (A/3)1110)- (A~/18)1200) + (A=/18)1020)- (A25/9)[110)
Io2o)"= 1020)- (x/3)[110) - (A2/18)1020)+ (A2/18)1200)- (A55/9)ll10) [10 +- 1)"= I10---1)+ (X/3)[Ol +- 1)-(A2/18)110 z 1)
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n = 2 states E';oo= E2 + A2(3/16)- A'(3/32) + A'(3/128) - A'(21/64)
512
M.E.
BACON
Egto = E2 - A :(3/16) - ~.3(3/32) - A4(3/128) - :t'(21/64) Eoo+, = E2 + :t 3(3/32) - x ' ( 3 9 / 1 2 8 ) n = E'~,o = # E2oo = E~oo =
3 states E3 + A3(4/81) - A'(43/324) E3 + A2/9 - A3(8/81) + ,~'(1/81) + A4(5/81) E3 + ~. 219 - ~t 3(8/81) + ~.4(1181) + ~.4(5/81) E','o~, = E3 + :t 2/18 + ,~ 3(1/81) + ~4(1/81) - ,~'(5/81) Eg~, = E~ - X2/18 + A3(1/81) - A'(1/81) + )t'(5/81) E~o~2 = E3 + ,~3(4/81) - ,~'(34/324)
A'(40/324) ~.'(40/324) - ,~'(40/324) - A'(40/324)
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THE ASYMMETRY OF Ly-a AND Ly-/3 M. E. BACON~ Department of Physics, University of British Columbia, Vancouver B.C., Canada (Received 19 July 1976)
Abstract--The asymmetryof the first two spectral lines of the Lyman series of hydrogenis calculated. Detailed profiles,whichinclude the effects of time orderingand asymmetry,are presented for a regionof the line profilefrom the center to the line wings, where the impactapproximationfor the electrons and the quasi static approximationfor ions are reasonablyapplicable. Deviationsin the line wingsfrom "Unifiedtheories" are of the order 0-15%for the profile parameter range considered. Suchdeviations shouldhave only a minor effect on the computed asymmetry, The far red wing of both Ly-a and Ly-fl is seen to dominate the blue wing. In the near wings the blue intensity becomesequal to the red intensity. From here into the line center, the blue intensity is larger than the red intensity. In the case of Ly-/3,there is the expectedenhancementof the blue peak and reductionof the red peak, The calculated profiles are compared with previous calculations and recent experimental results. I. INTRODUCTION IN RECENTyears there has been renewed experimental interest in the asymmetry of the hydrogen spectral lines. "-'~ This is due not only to its intrinsic interest as a check on line broadening calculations but also because of possible astrophysical significance/2"5'6~ In addition, such studies are related to, and have a certain relevance to, asymmetries and shifts of hydrogenic ion lines in high density laser produced plasmas/7~ The asymmetry of the hydrogen lines most frequently referred to in the literature is the characteristic asymmetry of the blue and red peaks of Ha, primarily because this line has received the most attention. Reference to this work and earlier experimental results dealing with the asymmetry of other hydrogen lines can be found in Ref. 8. There have been a number of attempts to account theoretically for the asymmetry of the hydrogen iinesff -''~ with particular attention paid to the Ly-a asymmetry. For the parameter range investigated experimentally to date, the Ly-a asymmetry can be traced to the effects of ion-field inhomogeneity, as outlined by SHOLIN."3~ Numerical asymmetry computations for one-electron density and temperature, which include ion-field inhomogeneity and the quadratic Stark effect, have been performed previously by the author, "5~ who, however, slightly overestimated the effects of the quadratic Stark effect. Recently, DEMURAand SHOLINt~s~have presented some analytic results for the asymmetry of the hydrogen spectral lines and compared them with the experimental results of BOLDT and C O O P E R . tiT) In the present paper, we confine our attention to the Ly-a and Ly-/3 lines and present detailed calculations which include the effects of time ordering, ion-field inhomogeniety and the quadratic Stark effect. Although these two lines are difficult to measure experimentally, °'tT"ts~ they are the easiest to treat theoretically and should give an indication of the expected behaviour of the more experimentally accessible lines, such as the Balmer Series lines. In the following section, a brief outline of the computational details used in the present calculations of Ly-a and Ly-/3 are presented. The relevant asymmetry corrections to the wave functions and energy levels for the n = 1, 2, and 3 levels are given in the appendix. Also given are the dipole moments for Ly-a and Ly-/3. In Section 3, we compare the present profiles with previously calculated profiles and with the most recent experimental results. The final section contains a discussion and conclusion. 2. C O M P U T A T I O N A L
DETAILS
In the present calculation, we used the impact approximation in treating the electrons and the quasi static approximation in treating the ions. In this sense, therefore, the present calculations tOn leave from the University of Natal, Durban, S. Africa. 501
502
M.E. BACOS
can be considered to be a "modified impact" calculation. The use of the above approximations should be valid for the range of profile parameters considered and deviations from unified calculations should be small. The treatment of the quasi-static ion perturbation was carried out in a similar manner to that described by S8OLI#'3) with only minor differences. Firstly, a number of sign changes are apparent in the evaluation of some of the ion-atom multipole matrix elements. This is due to the choice of ion field direction. Secondly, there is a difference in the second order ion-atom quadrupole contribution to the energy as mentioned in Ref. 15. This has very little effect on the calculations, since the major contribution to the asymmetry is due to the first-order quadrupole correctionsY 3~ The various matrix elements of the ion-atom multipole perturbations were checked against the appropriately modified electron values used in Refs. 19 and 20 in treating the electron impact broadening of the n = 2 and n = 3 states, respectively. The expressions [eqn (48)] given by NCtrVEN-HOEet al. <2"were used to evaluate the contribution of the quadratic Stark effect to the dipole moments. This differs from the calculations of Ref. 15, where (n 2- 1) was used instead of (7n 2 - 1) in the above mentioned equations. This resulted in a slight overestimation in the significance of the quadratic Stark effect at the larger separations from the line center. For Ly-a, contributions from the quadrupole ion-atom interactions were included in calculating the asymmetry. For Ly-fl, the contribution from the octupole ion-atom interaction was also included. This was done in an approximate way by assuming that the ion-fields were produced by one nearest neighbor. In other words, the )t s (not to be confused with wavelength) in the expressions for the wave functions, energies and dipole moments were given by A = n2a([Fo/e) I/2, where n is the principal quantum member of the level in question, a is the Bohr radius, Fo = 2.61 eN 2s3is the Holtsmark normal field strength, and f is the reduced field strength. The Hooper ion field distribution functions ~2~ were used throughout, except for values of f > 30 where the Holtsmark values were used. Although this is rather a naive procedure for the region of the line profile around the line center, it would seem likely that a more exact treatment, which would include the many-body aspects of the problem, would not yield significantly different results. In any event, the blue-red peak asymmetry of Ly-/3 calculated using the above procedure should be better than, say, a calculation using the normal Holtsmark field and the dipole moments given in the appendix The effects of time ordering have been included in the treatment of the electron impact perturbation and are based on earlier numerical computations. "9'2°~ In addition, in the far wings frequency dependent cut-offs~23~have been used in treating the electron impact broadening operator. In the case of Ly-a, the reduced line Profiles S(a) were calculated for various values of electron density and temperature, and are given in Tables 1-5. In these tables, the electron density is in units of cm -3, and the Holtsmark normal field is in cgs field strength units. Alpha is given by (to - coo)Ao2/21rcFo,where COois the angular frequency at the unshifted line center and oJ is the angular frequency at the point of interest on the line profile. An estimation of the wavelength separation from the line center for a particular value of S can be obtained by multiplying a by Fo. A more exact expression for the wavelength is given by h = AdO + Foa/ho), where ho is the wavelength at the line center. The integers in the S(a) columns represent the power of ten associated with the preceeding number. An absorption profile can be obtained from these values of S(a) by multiplying by a factor CO/COo= (1 + Foalho). An emission profile can be obtained by multiplying S(a) by (1 + Foa/ho) 4. Thermal Doppler broadening has not been included in these calculations. The reduced field strength, f ranged from 0-100 in steps of 0.1. The reduced line profiles for Ly-/3 were calculated in a similar manner and are given in Tables 5-10. In these calculations, however, f ranged from 0-50 in steps of 0.2. Using these tables and interpolating for an electron density of 7.2 x 1016cm -3 and an electron temperature of 1.22 x 104 K yielded profiles for Ly-a and Ly-fl, which agreed to within < 2% With calculations using the original computer program. The correspondingly evaluated asymmetry agreed to within 5%. 3. COMPARISON WITH PREVIOUS CALCULATIONS AND RECENT EXPERIMENTS Considering the Ly-a profiles first and comparing these with the calculations of KEPPLEand GRIEM~24)and VIOALet al. <2~we note the dominant time ordering effect at the line center, namely a
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lowering of the peak intensity. "9'2° In the far wings, the profiles have generally a somewhat larger value than that of the "unified" calculation of Vidal et al. This is again due to time ordering and is compatible with calculations by GODFREY et a/. t2" and VOSLAMBERc2s) [see also GREENE and COOPER(29q.Generally, if the asymmetry is neglected, the present profiles differ from the unified results of Vidal et al. by ~< 10% in the far wings. In the far wings, the red wing ( - c~) is seen to be of higher intensity than the blue wing (+ a). This asymmetry decreases as we approach the line center. At [al ~ 10-3, the wing intensities become equal and from there into the line center, the blue side dominates the red. This behavior is consistant with what one might expect from a consideration of Fig. 1 given by SHOLIN."~ The asymmetry described above can be traced primarily to the quadrupole ion-atom interaction with an insignificant role played by the quadratic Stark effect, as already mentioned. In Fig. 1, we compare the results of the present calculations with the recent experimental data o f FUSSMANN. °) In this figure, R is the ratio of the calculated and experimental results to the Holtsmark values. The experimental curve is taken from Ref. 3. The agreement between calculation and experiment in terms of absolute values is seen to be rather good. Figure 2 . . . . . .
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The asymmetryof Ly-a and Ly-g
509
compares the theoretical and experimental asymmetry, as measured by (AA)I = I/2(AR -- AB) and (AA)2 = l/2(h~ + AB) -- ho; AR and As are the wavelength values on the red and blue wings, for equal intensity points. This method of measuring the asymmetry is relatively insensitive to differences in absolute intensities as determined by various calculations, unified or improved impact. The experimental points are given by the crosses and the dashed curve is a smooth curve drawn through these points. ~3~The open circles represent the calculated asymmetry, including that due to frequency to wavelength conversion and the frequency factor for the absorption profile. The solid curve is a smooth curve fit to these calculated data. The agreement between experiment and theory is relatively good, considering the complexities involved in the experimental measurements. Significant differences are perhaps the lower experimental asymmetry at (AA~) > 6/~, the different crossover point (0.9/~ experimentally and - 0.5/~ theoretically) and the larger blue asymmetry near the line center. Turning now to Ly-/3 and comparing the present calculations with those of KEPPLE and GRIEMt24)and VIDALet al. ~25~we again see the effect of time ordering at the line center, viz a filling in of the central dip between the two maxima. Figure 3 illustrates the typical differences between the present calculations and those of Refs. 24 and 25. The profiles of both Kepple and Griem and Vidal et al. are symmetric and, hence, only half of each profile is shown, as in the case of Ly-a, the red wing is seen to dominate the blue wing and there is a crossover point as one approaches the line center, this time at [a I ~ 3 x 10-3. For as smaller than this, the blue dominates the red. Again, this behavior is as one might expect on the basis of Fig. 1, Ref. 13.
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510
M. E; BACON
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theory and the experimental results of Fussmann for AA > 6 ~ and AA < 1 A. In the latter region, this may be due to the nearest neighbor approximation, although Doppler broadening, which was not included in the present calculations, may also play a role. In the former region, there would appear to be a number of possible theoretical reasons for the discrepancy. Firstly, the so called dissolution effect, treated by LAscZOS°°~ and considered by VuJsOVlC~12) in the context of hydrogen Stark broadening, is known to produce a sudden weakening of the red shifted Stark components at high field strengths. °l~ This is then followed by a weakening of the blue components at even higher fields. Based on calculations°2-34~ of this effect, however, field strengths in excess of 1.5 × 107 V/cm would be necessary to make the ionization probability of the red component, due to tunneling comparable to the radiative decay probability [see Fig. 7 of Ref. 30]. Even at an electron density of 10~8cm -3, field strengths of this order only occur for f ~ 100 and, hence, this effect is completely negligible for the experimental conditions of Fussmann. A second possibility is the occurrence of satellites of the type considered by STEWARTet al. ~'> and LE QUANGRANGand VOSLAMBEX.°6~ However, since the red satellites are predicted to occur closer to the line center than the blue satellites, any such effect would tend to enhance the red wing. In any event, the satellites are predicted to occur at AA > 20 A and it is, therefore, unlikely that they would effect the region of the line profile under consideration here (AA < 10 A). The suggestion that screening of the nuclear charge may perhaps play a role,<7}seems also to be untenable at the relatively low density of 7.2 × 10~6cm -3. A more viable explanation of the discrepancy between theory and experiment, is the effect of quasi-static electrons, as suggested by GPaEM° " (see also OKS and SHOLIN<~4>).In the present calculations, no account is taken of quasi static electrons, except, insofar as they are excluded from the electron impact broadening via the frequency dependent cut-offs. Since the quadrupole ion-atom interaction is the dominant' source of asymmetry, and since this depends on the sign of the charged perturbers, the effect of these quasi static electrons would be to reduce the asymmetry. Additional experiments at densities - 10~sand temperatures -< 2 x 104 K, where the effect should be more pronounced, could substantiate this point. The case of Ly-/~ is somewhat different from that of Ly-a. In the latter case, as we have seen above, there is relatively good agreement between calculated and experimental wing intensities and that the relatively small deviations from the observed asymmetries have a possible theoretical explanation. Ly-fl, on the other hand, shows a measured red wing which is substantially less than predicted by theory in the range 0.3 A < AA < 2/~. In addition, the wing asymmetry inferred from Fig. 4 is completely at odds with the calculated asymmetry. The possible theoretical short-comings mentioned above in connection with Ly-a, should be insignificant for the density, temperature and wavelength range under consideration for Ly-fl.
The asymmetry of Ly-a and Ly-//
511
Acknowledgements--The financial assistance of the Atomic Energy Board of South Africa and the National Research Council of Canada is gratefully acknowledged. REFERENCES 1. W. L. WIESE, D. E. KELLE,ERand D. E. PAQUEI"rE,Phys. Rev. 6A, 1132 (1972). 2. R. D. BENGS"rONand G. R. CHESTEL Appl. J. 178, 565 (1972). 3. G. FUSSMANN,JQSRT 15, 791 (1975). 4. R. D. BENGSTONand G. R. CHESTER,Phys. Rev. 13A, 1762 (1976). 5. W. L. WIESE and D. E. KELLEHER,Appl. J. 166, L59 (1971). 6. V. TRIMnLEand J. L. GREENSTEIN,Appl. J. 177, 441 (1972). 7. J. C. WElSHEITand B. F. ROZSNVAI,J. Phys. B: Atom Molec. 9, L63 (1976). 8. H. R. GRIEM, Spectral Line Broadening by Plasmas. Academic Press, New York (1974). 9. L. P. KUDRINand G. V. S,OLIN, Soy. Phys. DOgLADY7, 1015 (1%3). 10. NGUVES-HoL H. W. DRAWINand L. HERMAN,JQSRT 4, 846 (1964). 11. H. R. GRIEM,Phys. Rev. 140, All40 (1%5). 12 V. VUJNOVlC,Int. J. Electronics 18, 411 (1%5). 13. G. V. S,OLIh, Opt. Spectros. 26, 275 (1%9). 14. E. A. OKS and G. V. SHOt.Is, Opt. Spectros. 33, 217 (1972). 15. M. E. BACON,JQSRT 13, 1161 (1973). 16. A. B. DEMURAand G. V. SHOLtN,JQSRT 15, 881 (1975). 17. G. BOLI~Tand W. S. COOPER,Z. Naturforsch. 19a, 968 (1964). 18. R. C. ELTON and H. R. GPaEM,Phys. Rev. 135, 6A (1964). 19. M. E. BACON,K. Y. SHEN and J. COOPER,Phys. Rev. 188, 50 (1%9). 20. M. E. BACON,JQSRT 12, 519 (1972). 21. NGUYEN-HOE,E. BANERJEA,H. W. DRAWLSand L. HERMAN,JQSRT 5, 835 (1%5). 22. C. F. HOOPER,Phys. Rev. 165, 215 0%8); 169, 193 (1968). 23. H. R. GPaEM, Appl. 3". 147, 1092 0%7). 24. P. KEPPLEand H. R. GRIEM,Phys. Rev. 173, 317 (1968); University of Maryland report No. 831. 25. C. R. VIDAL,J. COOPERand E. W. SMrra, Appl. J. Supp. 25, 37 (1973). 26. H. PFENNIG,JQSRT 12, 821 (1972). 27. T. GODFREY,C. R. V]DAL,E. W. SMrm and J. COOPER,Phys, Rev. A3, 1543 (1971). 28. D. VOSLAMBER,Phys. Left. 42A, 469 (1973). 29. R. L. GREENEand J. COOPER,JQSRT 15, 991 (1975). 30. C. LANCZOS,Z. Physik 68, 204 (1931). 31. H. RAUSCH,V. TRAURENBERG,R. GEBAUERand G. LEWIN, Naturwiss. 18, 417 (1930). 32. M. H. RICE and R. H. GOOD,JE., J. Opt. Soc. Am. 52, 239 (1%2). 33. D. S. BAILEY,J. R. HISKESand A. C. RIVlERE,Nucl. Fusion 5, 41 (1965). 34. M. H. ALEXANDER,Phys. Rev. 178, 34 (1%9). 35. J. C. STEWART,J. M. PEEK and J. COOPER,Appl. J. 179, 983 (1973). 36. LE QU^NG RANGand D. VOSLAMnER,J. Phys. B: Atom Molec. 8, 337 (1975). APPENDIX In this appendix, we list the wave functions and energies for the n = 1, 2 and 3 levels of hydrogen, which are used in the calculations outlined in Section 2. This is followed by a list of the relevant dipole moments for Ly-a and Ly-/3. Atomic units are used in these expressions. The wave functions and energies for the n = 1 and n = 2 states have been given previously in Ref. 15. They are reproduced here, with a few changes, for completeness. In Ref. 15, A ---4a(fFole)m for both the n = I and n = 2 levels. Here we use X -= n2a(fFole)'12.The wave functions are denoted [n,n2m) where n, and n2 are the usual electric quantum numbers and m is the magnetic quantum number. The energies are denoted by E,,,2,. The double primes indicate quantities evaluated to second order in the asymmetry corrections, (a) Wavefunctions n = 1 states:
Iooo)"= Iooo) n = 2 states:
1100)"= IlO0)+ (M4)[010)- (A~/32)[100) 1010)"= 1010)- (A/4)1100)- (A2/32)1010) [00-+ 1)"= IO0+ 1) n = 3 states: I110)" = [110) - (A13)[200)+ (AI3)[020)- (A5/9)[110) + (A2519)1200)+ (A55/9)[020)
1200)" = 1200) + (A/3)1110)- (A~/18)1200) + (A=/18)1020)- (A25/9)[110)
Io2o)"= 1020)- (x/3)[110) - (A2/18)1020)+ (A2/18)1200)- (A55/9)ll10) [10 +- 1)"= I10---1)+ (X/3)[Ol +- 1)-(A2/18)110 z 1)
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n = 2 states E';oo= E2 + A2(3/16)- A'(3/32) + A'(3/128) - A'(21/64)
512
M.E.
BACON
Egto = E2 - A :(3/16) - ~.3(3/32) - A4(3/128) - :t'(21/64) Eoo+, = E2 + :t 3(3/32) - x ' ( 3 9 / 1 2 8 ) n = E'~,o = # E2oo = E~oo =
3 states E3 + A3(4/81) - A'(43/324) E3 + A2/9 - A3(8/81) + ,~'(1/81) + A4(5/81) E3 + ~. 219 - ~t 3(8/81) + ~.4(1181) + ~.4(5/81) E','o~, = E3 + :t 2/18 + ,~ 3(1/81) + ~4(1/81) - ,~'(5/81) Eg~, = E~ - X2/18 + A3(1/81) - A'(1/81) + )t'(5/81) E~o~2 = E3 + ,~3(4/81) - ,~'(34/324)
A'(40/324) ~.'(40/324) - ,~'(40/324) - A'(40/324)
(c) Dipole moments
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