Estimation of optimum electron temperature for maximum xray laser gain from 3p–3stransitions of neonlike ions in laser plasmasG. P. Gupta and B. K. Sinha Citation: Journal of Applied Physics 79, 619 (1996); doi: 10.1063/1.360804 View online: http://dx.doi.org/10.1063/1.360804 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/79/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Effect of nonequilibrium ionization process on gain of neon-like argon x-ray laser J. Appl. Phys. 95, 434 (2004); 10.1063/1.1633986 XRay Spectroscopy On NeonLike Heavy Ions AIP Conf. Proc. 652, 291 (2003); 10.1063/1.1536388 New Insights into the Xray Spectra of Heliumlike and Neonlike Ions AIP Conf. Proc. 635, 135 (2002); 10.1063/1.1516304 Gain measurements and spatial coherence in neonlike xray lasers AIP Conf. Proc. 332, 483 (1995); 10.1063/1.48009 Parametric dependence of xray laser gain in laser plasmas for 3p3s transitions in neonlike krypton ions J. Appl. Phys. 77, 2287 (1995); 10.1063/1.358817

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Estimation of optimum electron temperature for maximum x-ray laser gainfrom 3 p –3s transitions of neonlike ions in laser plasmas

G. P. Gupta and B. K. SinhaLaser and Plasma Technology Division, Bhabha Atomic Research Centre, Trombay, Bombay 400 085, India

~Received 8 June 1995; accepted for publication 6 September 1995!

The fraction of neonlike ions and the average ionic charge state in laser plasmas formed at thesurface of the solid targets of germanium, selenium, and krypton have been obtained at severalelectron temperatures using the steady-state collisional radiative ionization model. Dielectronicrecombination has been considered. The effect of consideration of dielectronic recombination ofions of one or more charge states on these parameters has been studied. It is observed that the valuesof electron temperature corresponding to the maximum abundance of neonlike ions in differentcases of dielectronic recombination are significantly different from those obtained withoutconsideration of dielectronic recombination. The optimum electron temperature of the plasma isestimated from the maximization of the x-ray laser gain through temperature-dependent variables ofthe gain coefficient and is observed to be the electron temperature at which the abundance ofneonlike ions is maximum. ©1996 American Institute of Physics.@S0021-8979~95!03024-3#

I. INTRODUCTION

X-ray lasers using laser plasmas as the active mediumhave been extensively reported following both the electroncollisional and recombination schemes.1–10 Collisionallypumped x-ray lasers in the neon isoelectronic sequence are3p–3s transitions in laser plasmas of high atomic numberelements and have been demonstrated by various workers.2–6

As a number of problems concerning the collisional pumpingof the Ne-like ions and the difficulties in achieving largegain-length product remain unexplained, a better understand-ing of atomic physics and plasma kinetics is essential formodeling x-ray lasers.4,7 An important unexpected anomalywas reported by Elton, Lee, and Molander7 who could notdemonstrate the x-ray laser gain from 3p–3s transitions ofNe-like ions in copper plasmas, although the plasma condi-tions were created as per the scaling model of the plasmaparameters.8 As reported by Elton and co-workers7 them-selves, the unexpected result from 3p–3s transitions in cop-per plasmas is ‘‘puzzling’’ and no suitable explanation hasbeen provided so far.

The two plasma parameters, electron densityne and tem-peratureTe , are of paramount importance to the optimizationof gain for collisionally pumped x-ray lasers. In order toachieve the maximum population inversion between the rel-evant levels for transition and, hence, the gain, it is essentialto have optimumne andTe for a given target plasma. Theoptimumne derived from the collisional thermalization limitof the 3p–3s levels is around 531020 cm23 and the opti-mum Te obtained by maximizing the collisional excitationfrom Ne-like ground state~3s state! to the excited one~3pstate! without causing the ionization of Ne-like ions isaround 1 keV, as reported by Key,1 irrespective of the targetelements useful for the 3p–3s transitions. Feldman andco-workers8 have presented the scaling of the optimum elec-tron density and the laser gain with atomic numberZa of thetarget element for collisionally pumped x-ray lasers in theneon isoelectronic sequence. In doing so, they have assumedthe optimumTe equal to half of the ionization potentialXI of

Ne-like ions from the consideration of their maximum abun-dance which, according to them, is expected to occur atTe/XI'0.5. Moreover, they have assumed arbitrarily con-stant values for the average ionic charge stateZ̄(Te) and thefraction d(Te) of Ne-like ions in laser plasmas of differentelements in their laser gain calculations, irrespective of thevalues ofTe andZa . Subsequently, Whitten and co-workers9

have accounted for detailed atomic models to calculate theionization balance and the atomic level populations and re-ported the scaling of Ne-like collisionally pumped x-ray la-sers using exploding foils. They have considered dielectronicrecombination in the ionization balance only for recombina-tion from Ne-like to Na-like ions and neglected that for re-combination from F-like to Ne-like ions, although the rel-evant x-ray laser plasma contains Ne-like and F-like ionspredominantly.10 Holden et al.10 have recently calculatedd(Te) in a germanium plasma by considering dielectronicrecombination only for recombination from F-like to Ne-likeions in the ionization balance.

In this article we have obtained the values ofZ̄(Te) andd(Te) in laser plasmas of germanium (Za532), selenium(Za534), and krypton (Za536) at several values ofTe us-ing the collisional-radiative~CR! ionization model in thesteady state and incorporating dielectronic recombination.The values ofZ̄(Te) andd(Te) have been obtained consid-ering dielectronic recombination from Ne-like to Na-likeions or from F-like to Ne-like ions or both from Ne-like toNa-like ions and F-like to Ne-like ions using the correspond-ing recombination coefficients given by Hagelstein, Rosen,and Jacobs.11 These parameters have also been evaluated bytaking into account dielectronic recombination for all ioniza-tion stages through a parameterd as considered byEidmann.12 Results ond(Te) and Z̄(Te) for different casesof dielectronic recombination in the ionization balance arecompared and discussed. An interesting observation aboutthe computations presented here is that they provide a rea-sonably good description of the steady-state ionization bal-ance using the simple model. We have also estimated thevalues of optimumTe in laser plasmas of different elements

619J. Appl. Phys. 79 (2), 15 January 1996 0021-8979/96/79(2)/619/6/$6.00 © 1996 American Institute of Physics [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

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from the maximization of the x-ray laser gain throughtemperature-dependent variables of the gain coefficient. TheoptimumTe obtained is the value ofTe at which the fractionof Ne-like ions is maximum. The values of optimumTe areobserved to be different for different cases of dielectronicrecombination.

II. GAIN FORMULA

The gain coefficientG for a Doppler-broadened lasertransition, dominant in high-temperature plasmas, is ex-pressed as8

G5l3

8pAulS M

2pKTiD 1/2guSNu

gu2Nl

glD , ~1!

wherel is the wavelength of the transition between the up-per and lower levels,Aul is the rate of spontaneous emissionfrom the upper levelu to the lower levell , Nj is the ionpopulation density in the levelj having statistical weightgj ,K is the Boltzmann constant,M is the ion mass, andTi is theion temperature. ExpressingNj in terms of electron and iondensitiesne and ni (5ne/Z̄) of a plasma formed from atarget element of atomic numberZa as

Nj5~Nj /nI !~nI /ni !~ni /ne!ne5~Nj /nI !~d/Z̄!ne , ~2!

wherenI is the number density of Ne-like ions in all levels,d(5nI /ni) is the fraction of Ne-like ions,ni is the total iondensity, andZ̄ is the average ionic charge state; Eq.~1! canbe rewritten as

G~cm21!51.7431026lcm

3 m1/2

Ti K1/2 AulguSNu8

gu2Nl8

glD S d

Z̄D ne .~3!

HereNj8 (5Nj /nI) is the fractional level population densityof Ne-like ions,m is the atomic weight of the target element,lcm is given in units of cm,Ti K in K, Aul in s21, andne incm23.

III. IONIZATION BALANCE

The estimation ofG from Eq. ~3! requires the values ofd(Te) and Z̄(Te) at a givenTe . These are obtained from asuitable plasma ionization model. In the steady-state CR ion-ization model including dielectronic recombination, the iondensities in two consecutive charge states are related as13,14

nZ11

nZ5

SZaZ111DZ111nebZ11

, ~4!

whereSZ is the collisional ionization coefficient correspond-ing to the ionic charge stateZ, andaZ11, DZ11, andbZ11are, respectively, the radiative, dielectronic, and three-bodyrecombination coefficients corresponding to the ionic chargestateZ11. In our recent work14 on x-ray laser gain calcula-tion, we have considered three expressions of the ionizationcoefficient in estimatingd(Te) of Ne-like Kr ions and con-cluded that the expression ofSZ due to McWhirter15 is themost suitable one. This expression is given as

SZ52.4331026jZTeV23/2@exp~2u!/u7/4# cm3 s21, ~5!

whereu5XZ/TeV . XZ is the ionization potential in eV andjZ is the number of electrons in the outmost layer corre-sponding to the ionic charge stateZ. The charge states withZ5Za29, Za210, andZa211 represent F-like, Ne-like,and Na-like ions, respectively.TeV is the value ofTe in eV.The values of the ionization potential for different ionizationstates up toZ5Za23 are taken directly from the tabulatedresults of Fraga, Karwowski, and Saxena16 and those for thecharge states ofZa22 andZa21 are obtained on extrapola-tion of their tabulated values given for charge states up toZa23. The values ofXI for Ge, Se, and Kr are 2177, 2537,and 2925 eV, respectively.

The expressions fora and b used in the earlierworks14,17 are due to Kolb and McWhirter18 as

aZ1155.2310214~Z11!u1/2~0.42910.5 ln u

10.469u21/2! cm3 s21, ~6!

bZ1152.97310227jZ /@TeVxZ2~4.8811/u!# cm6 s21.

~7!

The dielectronic recombination rates are difficult to obtainprecisely but are available with variations in the literature.Hagelstein and co-workers11 have calculated the density-dependent effective dielectronic recombination coefficientsas a function ofTe at ne5331020 cm23 for recombinationfrom Ne-like to Na-like ions and from F-like to Ne-like ionsin a selenium plasma and compared them with those reportedearlier in the zero-density limit. Table I shows the dielec-tronic recombination coefficients obtained from the work ofHagelstein and co-workers.11 We have also shown in thetable the corresponding radiative and three-body recombina-tion coefficients estimated from Eqs.~6! and~7! for the sakeof comparison. As seen from the table, the radiative recom-bination coefficient is smaller by an order of magnitude andthe three-body recombination coefficient is smaller by fiveorders of magnitude as compared to the dielectronic recom-bination coefficient at a givenTe , showing that dielectronicrecombination is the dominant recombination mechanism at

TABLE I. Comparison of various recombination coefficients~dielectronicD, radiativea, and three-bodyneb! at several electron temperaturesTe in aselenium plasma with an electron density ofne5331020 cm23.

Te~eV!

D ~1.0310211 cm3/s! a ~1.0310212 cm3/s! neb ~1.0310216 cm3/s!

Ne-liketo

Na-like

F-liketo

Ne-like

Ne-liketo

Na-like

F-liketo

Ne-like

Ne-liketo

Na-like

F-liketo

Ne-like

400 1.90 1.80 2.40 5.04 3.95 4.12500 2.00 1.75 2.01 4.24 3.10 3.27600 1.98 1.70 1.73 3.68 2.54 2.71700 1.95 1.65 1.53 3.27 2.14 2.30800 1.90 1.60 1.38 2.94 1.84 2.00900 1.85 1.45 1.25 2.68 1.61 1.761000 1.80 1.30 1.15 2.46 1.42 1.571100 1.70 1.25 1.07 2.28 1.27 1.421200 1.60 1.20 1.00 2.13 1.17 1.291300 1.45 1.18 0.94 2.00 1.04 1.191400 1.35 1.15 0.88 1.88 0.95 1.091500 1.25 1.00 0.84 1.78 0.88 1.01

620 J. Appl. Phys., Vol. 79, No. 2, 15 January 1996 G. P. Gupta and B. K. Sinha [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

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the plasma conditions relevant to x-ray laser. As dielectronicrecombination is nearly independent ofZa ,

9 we have consid-ered the values ofD given in Table I for selenium as well asgermanium and krypton plasmas in order to include the di-electronic recombination process approximately in the ion-ization balance for Ne-like and F-like ions which are pre-dominantly abundant10 in the relevant x-ray laser plasma.Eidmann12 has accounted for dielectronic recombination byusingDZ5daZ with d as a free parameter owing to uncer-tainties in the calculation ofDZ . By comparing his estimatesof the average ionic charge state of gold with those of themore accurate atomic model which takes into account dielec-tronic recombination in a consistent manner, he concludesthat dielectronic recombination is suitably taken into accountwith d'10 in the temperature range&1000 eV. Followinghis work, we have further considered dielectronic recombi-nation for all charge stages usingDZ equal to 10 timesaZ inestimating the values ofd(Te) and Z̄(Te).

IV. CALCULATION OF d(Te) AND Z̄(Te)

We have computednZ/ni whereni 5 (k51Za nk and Z̄ ~

5 (k51Za knk /ni) for laser plasmas of Ge, Se, and Kr at

ne5331020 cm23 and having different values ofTe , withand without dielectronic recombination in the ionization bal-ance. The value ofnZ/ni for Z5Za210 corresponds to thevalue of the fractiond(Te) of Ne-like ions. Figures 1–3showd(Te) as a function ofTe in Ge, Se, and Kr plasmas,respectively. In these figures, curves labeled with A are ob-

tained when the ionization balance takes into account dielec-tronic recombination only for recombination from Ne-like toNa-like ions. Curves B are obtained when the ionization bal-ance takes into account dielectronic recombination for re-combination from Ne-like to Na-like ions as well as from

FIG. 1. Fraction of neonlike ions as a function of electron temperature ingermanium plasma atne5331020 cm23 with and without dielectronic re-combination in the ionization balance. The curve A refers to dielectronicrecombination from Ne-like ions, curve B that from Ne-like and F-like ions,curve C that from F-like ions, and curve D that from ions of all chargestates.

FIG. 2. Fraction of neonlike ions as a function of electron temperature inselenium plasma atne5331020 cm23. Other details are the same as inFig. 1.

FIG. 3. Fraction of neonlike ions as a function of electron temperature inkrypton plasma atne5331020 cm23. Other details are the same as in Fig. 1.

621J. Appl. Phys., Vol. 79, No. 2, 15 January 1996 G. P. Gupta and B. K. Sinha [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

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F-like to Na-like ions. Curves C are obtained when dielec-tronic recombination only from F-like to Ne-like ions is ac-counted for in the ionization balance. Curves D are obtainedwhen dielectronic recombination is taken into accountthroughDZ510aZ for all ionic charge states. As seen fromFigs. 1–3, the values of the electron temperature at which theabundance of Ne-like ions is maximum, obtained for differ-ent cases of dielectronic recombination in the ionization bal-ance, are significantly different from that obtained when nodielectronic recombination is considered. For example, thecalculated electron temperature in Ge plasma correspondingto maximumd is 400 eV in case of no dielectronic recombi-nation and is equal to 500, 600, 500, and 800 eV correspond-ing to curves A, B, C, and D, respectively, which take intoaccount dielectronic recombination differently. The corre-spondingTe value in Se~Kr! plasma is 500 eV~600 eV!without dielectronic recombination and is equal to 600~800!,800 ~1000!, 600 ~800!, and 1000 eV~1300 eV! with dielec-tronic recombination in the ionization balance considereddifferently as per curves A, B, C, and D, respectively. Thenature of curves D is substantially different from that ofcurves A, B, and C. One further notes that the curves D havebroader distribution around the peak, which will cause largeerror in the estimation of optimum value ofTe . Since thecurves B take into account dielectronic recombination forcharge states of F-like and Ne-like ions predominantlypresent in the plasma, we consider the predictions from thecurves B as the most justified ones. Thus, theTe values cor-responding to maxima of curves B are equal to 600, 800, and1000 eV in Ge, Se, and Kr plasmas, respectively. The corre-sponding maximum fraction of Ne-like ions is equal to 0.43,0.40, and 0.37 in Ge, Se, and Kr plasmas, respectively, asobtained from curves B in Figs. 1–3. The curves D, whichare based on Eidmann’s suggestion,12 give the values ofTefor maximum values ofd(Te) as 800, 1000, and 1300 eVwith the corresponding value ofd(Te) as 0.55, 0.52, and0.49 for Ge, Se, and Kr plasmas, respectively, which arequite far off the values given by the curves B. As statedearlier, the profiles of curves D are quite different from thoseof A, B, and C. Because of these two reasons the resultsobtained from the parameterd as suggested by Eidmann arenot satisfactory.

For comparison of the present results with the literature,we consider the theoretical estimates of Lee19 who has cal-culated the ionic charge-state distribution of a Se plasma byusing the ionization balance model accounting for detailedconfiguration of atomic kinetics and those of Daidoet al.20

who have calculated the values ofd(Te) in a Ge plasma byusing the CR model including dielectronic recombinationwith its coefficients given by Postet al.21 From these worksthe values ofTe at whichd(Te) is maximum are obtained asabout 600 and 800 eV in Ge and Se plasmas, respectively, atne5331020 cm23. The corresponding maximum value ofd(Te) is about 0.5 in both plasmas. Thus, the present esti-mates from the curves B are in reasonably good agreementwith those from the above theoretical works, showing theimportance of the simple model providing a reasonably gooddescription of the steady-state ionization balance.

Figures 4–6 show the average ionic charge stateZ̄ as a

function ofTe in Ge, Se, and Kr plasmas, obtained by solv-ing numerically the steady-state CR ionization model equa-tions with dielectronic recombination~depicted by the solidlines! and by using the empirical relationZ̄5(2/3)(ZaTeV!

1/3

given in Ref. 17~depicted by the broken lines!. The curvesA, B, C, and D in these figures refer to several cases ofdielectronic recombination in the same manner as discussed

FIG. 4. Average ionic charge stateZ̄ as a function of electron temperature ingermanium plasma atne5331020 cm23. The solid line shows the resultsobtained from the numerical solution of the CR ionization model equationswith dielectronic recombination. The broken line shows the results obtainedfrom the empirical relation given in Ref. 17. The curves A, B, C, and D havethe same meaning as in Fig. 1.

FIG. 5. Average ionic charge stateZ̄ as a function of electron temperature inselenium plasma atne5331020 cm23. Other details are the same as inFig. 4.

622 J. Appl. Phys., Vol. 79, No. 2, 15 January 1996 G. P. Gupta and B. K. Sinha [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

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earlier in the context of Figs. 1–3. As seen from Figs. 4–6,the values ofZ̄ are considerably different for different elec-tron temperatures and target elements. One may further notethat the empirical relation does not fit satisfactorily in thevicinity of the electron temperature associated with the maxi-mum abundance of Ne-like ions, whereas the change inZ̄due to different cases of dielectronic recombination is smallnear this temperature.

V. OPTIMUM ELECTRON TEMPERATURE

In order to maximize the gain coefficient for estimatingthe optimumTe , let us look at the temperature-dependentvariables affecting the gain coefficient expressed by Eq.~3!.These are the fractional population inversion densityDN( 5 Nu8/gu 2 Nl8/gl), the fractiond of Ne-like ions,Z̄, andTi . The laser gain becomes maximum at the optimumTe atwhich the term [dDN/(Z̄Ti

1/2)] is maximum. Feldman andco-workers8 calculatedDN atTe/XI50.25, 0.5, and 0.75 andfound that it is approximately proportional toTe . This tem-perature dependence ofDN nearly cancels the temperaturedependence of the termZ̄Ti

1/2 which is also approximatelyproportional toTe . Hence, the termdDN/(Z̄Ti

1/2) is maxi-mum at the value ofTe at which d is maximum. Thus, theoptimumTe for x-ray laser gain is estimated to be the valueof Te at which the abundance of Ne-like ions is maximum.As discussed in Sec. IV, the value ofTe at whichd is maxi-mum is strongly dependent on how many charge states ofions in dielectronic recombination are taken into account inthe ionization balance. Hence, the optimumTe for the maxi-mum laser gain is precisely estimated by suitably consider-ing dielectronic recombination in the ionization balance. Thevalues of optimumTe obtained from the curves B of Figs.1–3 are around 600, 800, and 1000 eV for laser plasmas ofGe, Se, and Kr, respectively. As there is a paucity of experi-

mental and theoretical data on the conditions existing in theplasmas at the time of maximum x-ray laser gain, it has notbeen possible to compare the present results of optimumTewith the literature.

It is worth pointing out that a considerable change inTeon either side of the optimumTe reduces exponentially theabundance of Ne-like ions, thereby reducing the gain coeffi-cient substantially. As the optimum election density range ina laser plasma produced at the surface of a solid target isalways satisfied during the expansion of the plasma, the cre-ation of the plasma with a value ofTe around its optimumvalue is essential for optimum operation of collisionallypumped x-ray lasers. If the laser-plasma experiments fordemonstrating x-ray laser gain in the neon isoelectronic se-quence are not carried out with an electron temperaturearound the correct optimum value ofTe , the laser gain maynot be observed. We attribute this reason as the likely causefor the lack of distinguishable gain from Ne-like ionic tran-sitions in x-ray laser experiments using copper plasmas byElton and co-workers7 who generated the plasmas followingtheZa scaling model

8 which, as discussed above, has a num-ber of flaws.

VI. CONCLUSION

We have investigated the values of optimumTe formaximum x-ray laser gain from 3p–3s transitions of Ne-like ions in laser plasmas of Ge, Se, and Kr. The optimumTeis estimated to be the electron temperature at which theabundance of Ne-like ions is maximum. The values of thistemperature are substantially different for different cases ofdielectronic recombination in the ionization balance. It isconcluded that the production of a laser plasma with an op-timum electron temperature corresponding to the maximumx-ray laser gain from collisionally pumped 3p–3s transi-tions in Ne-like ions is a necessity to successfully demon-strate the laser gain. The on-line electron temperature mea-surement is required for the analysis of the experimentalresults on x-ray laser gain measurements.

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FIG. 6. Average ionic charge stateZ̄ as a function of electron temperature inkrypton plasma atne5331020 cm23. Other details are the same as in Fig. 4.

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