Goel, B.

Yabe, T.

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KfK 4176 März 1987 Relativistic Calculations of Level Energies of a High-Z Atom T. Yabe, B. Goel Institut für Neutronenphysik und Reaktortechnik Kernforschungszentrum Karlsruhe KERNFORSCHUNGSZENTRUM KARLSRUHE Institut für Neutronenphysik und Reaktortechnik KfK 4176 Relativistic Calculations of Level Energies of High-Z Atom Takashi Yabe* and Balbir Goel *Permanent address: Institute of Laser Engineering, Osaka University, Yamada-Oka 2-6, Suita, Osaka 565, Japan Kernforschungszentrum Karlsruhe GmbH, Karlsruhe Als Manuskript vervielfältigt Für diesen Bericht behalten wir uns alle Rechte vor Kernforschungszentrum Karlsruhe GmbH Postfach 3640, 7500 Karlsruhe 1 ISSN 0303-4003 Abstract Level energies are calculated using relativistic and non-relativistic Hartree-Fock-Siater (RHFS & HFS) models for an Au-atom in various charge states. The Ievei splitting due to orbital angular momentum and electron spin is investigated in comparison to a simple analytical model. This model is basedonshell modeland is an extension of previous such models in that it takes overlap of orbitals into account. The model can satisfactorily replicate the results of RHFS and HFS calculations. This model can be applied to other atoms as weil. Relativistische Berechnung des Energie-Niveaus der Atome mit hoher Ladungszahl Zusammenfassung Niveau-Energien für Au-Atome in verschiedenem Ladungszustand werden mit Hilfe der relativistischen und der nicht relativistischen Hartree-Fock-Siater-Methode (RHFS & HFS) berechnet. Die Niveau-Aufspaltung durch Bahndreh-impuls und Elektronenspin wird im Vergleich zu einem einfachen analytischen Modell untersucht. Dieses Modell basiert auf dem Schalen-Modell und ist eine Erweiterung der bisherigen Modelle, indem die überlappungen der Elektronen-Bahnen berücksichtigt werden. Dieses einfache Modell gibt die Ergebnisse der RHFS und HFS Rechnungen wieder. Dieses kann daher auch für andere Atome angewandt werden. -1-Recently, the non-L TE average-ion model coupled with a hydrodynamic code and radiation transports has been extensively used for interpreting soft X-ray spectra from laser-irradiated high-Z plasmas l-4 . Although for Ievei calculations only principal quantum number and a screened hydrogenic model were used, the model replicated the experimental results both qualitatively and quantitatively , except a few discrepancies. One of these discrepancies is probably attributed to the ionic charge distribution : the line locations of the fictitious " average ion" are different from those of "real ions" being in different charge states ; even if they are in the same charge state, the Ievei electron population is different which causes the change of Ievei energies5 . In order to eliminate th is defect we developed a hybrid-atom model6 , in wh ich the charge state of ions and their excited state population can be simultaneously calculated. The calculation time is comparable to that of the average-ion model. We have applied this model to analyze X- ray spectrum from a Iaser- heated cavity plasm/. By introducing the charge state distribution and hence treating the "real ions" instead of an average ion, line transport can be correctly performed and the line locations in energy space are correctly calculated. As pointed out in a previous paper4 , some of the discrepancies can also be attributed to the neglect of line splitting due to angular momentum and spin-orbit interaction. Although preliminary calculations show that introduction of orbital quantum numbers can make the line spectra shift towards the lower energy, its implementation into the hydrodynamic code -2-was until now beyond our reach, because a simple formula for energy Ieveis and oscillator strenths of high-Z plasmas were not available. The aim of this paper is to provide one of such formula suitable for the average ion model as weil as for hybrid atom model. For this purpose, we extend the simple shell model 8 proposed by Parker and compare the result with a relativistic Hartree-Fock-Siater model, which was used in interpreting the M-shell line spectra in laser-produced Au 9 plasmas. Calculations have been done by peeling off electrons one by one from ground Ieveis. In Fig. 1, Ievei energies of the R-=0 state are plotted for various n ; R- and n mean the orbital and principal quantum numbers, respectively. In Figs. 2 and 3, Ievei energies measured from R-=0, that is E n -E 0 are plotted both for non-relativistic and n,x. n, relativistic cases, respectively. Surprisingly, these values remain constant over a wide range of the charge state although E 0 changes largely as seen from Fig. 1. n, ln the followings, we try to find a simple scaling law of these Ievei energies using Parker's shell model, which has been modified for open shells ; the electron density is written in terms of contributions from each shell, ~ ~ 2 p(r)= r pp(r)= r P(p)ö (r)/4Tir (1) r--~ r-: ~ P where P(p) is the electron population in the p-th shell and is 2p2 for a closed shell and pp ( r) is the density of p-shell electron at radius r. Each 6 ( r) is a fucntion p -3-which sharply peaks at r = p2/z (atomic units are p p used). The effective charge Zn can be calculated to be n·l\ Z = Z - P(n)/2 -n I P(p) P-:.A Then the Ievei energy is given by E = n 2 2 ~ Z /2n +P(n)/2r + I P(p)/r n n p,. .... -+", p + liENC + liEEX + liERE' where liENC' liEEX' and liERE are the (2) (3) non-Coulombic, exchange, and relativistic corrections, respectively. The exchange correction was given by Parker as follows; (4) An estimate of the non-Coulombic potential can be obtained by expanding the electron density araund r=r n liV ~ - 2nr p(r )(r-r ) 2/r (5) n n n 8 and th us The density in the n-th state is approximated as p (r)=A r 211 - 2exp(··2Z r/n), (G) n n n where the normalization constant A is n -4-This profile is found to be a good approximation to that obtained with RHFS. However, Parker used pn(rn) for p(rn) assuming non-overlapping of orbits. Using this expression we obtain = AE =-z P(n)/41T112n712 n,Q. n x [ n2-Q.(Q.+l) ] (7) This equation distinguishes from Parker's equation in that it is not confined to closed shell. Th is model suggests that AE n - AE 0 = n,)(, n, ZnP(n)Q.(Q.+l)/411 112n712 remains constant for fixed n and Q, if Z P(n) is constant. As is easily n seen, this is not the case. ZnP(n) can change by an order of magnitude from a closed shell to an open shell. The result with this model( Eqs. (3) and (7) ) is shown in Figs. 1, 2, and 3 by the dashed lines and is compared with calculations with a non-relativistic Hartree-Fock-Siater (HFS) model. Although Eq. (7) and HFS agree with each other for the lower charge state, the difference becomes significant when the number of electron in the shell in wh ich the c!!nergy is being calculated decreases. This is due to inadequacy of the shell model as shown below. ln order to clarify this, we take the ratio of electron densities in the n-th and ( n-1) -th s'hells at r=r n ; = r=r n -5-P(n-1) Zn_ 1n [--P(n) Zn(n-1) Zn-ln X exp{ -2n[ ----1] } . 2n-1 ] n(n-1/2) lf we assume Zn ~ Zn-l and n=4, then p3/p4 "' 7.3 P(3)/P(4) and hence the density of the third shell cannot be neglected even at r=r 4 . This behavior is easily understood if the orbit radius calculated by r =n2/z n n and Eq. (2) is plotted for various ionic charge states as in Fig. 4 ; as the eh arge state becomes h igher, all the shells come closer, and the density profile of various shells overlap with each other. We find that this overlap can be easily taken into account by modifying the density to give; where the contribution from different shells is included in the last term in Eq. (8), where densities are estimated at r=r* n wh ich is taken to be somewhat smaller than r because of n the finite width of the distribution Eq. (6). We find that where -6-A=l.O , 8=0.93333 for r n <0. 15, A=O. 9275 , 8=0.45 for 0. 15<r <0.45, n A=O. 725 , 8=0. for 0.45<r <0.8 , n A=0.8797 , 8=0. 1934 for 0.8 <r n <2. 17, A=0.46 ' 8=0. for 2. 17<r n can reasonably describe the behavior obtained with HFS. This comparison is given in Figs. 1 and 2 ; the results with the above model Eq. (8) are shown by the solid lines. As clear from Figs.2 and 3, a significant difference exists between RHFS and HFS, particularly in the inner shells. lf the effective eh arge Z from Eq. (2) is used in the expression of n the relativistic correction ; the large difference can not be explained. lnstead of trying to find an alternative expression, we try to adjust the effective eh arge Z* n to fill th is difference with Eq. ( 10). calculated so that AERE I j* - AERE I j=l/2 fits to Z* is n the corresponding difference between RHFS and HFS ; j* is 3/2, 5/2, and 7/2 for n = 3, 4, and 5, respectively, and other Ieveis are calculated with the corresponding same Z* and Eq. ( 10). As is n shown in Fig.3, this modification can satisfactorily replicate the RHFS results. The Z*n thus obtained is plotted in Fig.5, which shows Z* is significantly larger than Z except for the n n lower n (~3). a reasonable explanation of this large deviation of -7-Z* from Z has not yet been found. One possibility is the n n non-Coulombic potential ; in deriving Eq. (10), the Coulombic potential Z e2 Ir was used. lf we extend it to include the n non-Coulombic potential such as Eq. (5), the relativistic cor·rection also changes. In summary, we performed the relativistic and non-relativistic Hartree-Fock-Siater calculations for an Au atom and tried to get a simple scaling law for Ievei energies. We found that for the estimation of the non-Coulombic potential the shell model should be modified to include the overlap of orbits. This aim was partly accomplished. Although it is somewhat empirical, we have derived a formula to incorporate orbit overlap in a simple form and furthermore obtained an empirical formula for the relativistic corrections. Further improvement of the formula is expected for th is relativistic correction. The authors would like to thank Mr. S. Kiyokawa at Nara Women's University for his help to make computer programs of RH FS and H FS and Dr. R. Fröh lieh at KfK for discussion and encouragement. One of the authors (T. Y.) would also like to thank Professor G.Keßler and Dr.R.Fröhlich at KfK for giving him a chance to stay at KfK as a guest scientist. -8-References 1. W.C.Mead et al., Phys.Fiuids 27,1301(1984). 2. S.Kiyokawa, T.Yabe, and T.Mochizuki, Jpn.J.Appi.Phys. 22, L 772( 1983). 3. T.Mochizuki, T. Yabe, K.Okada, M.Hamada, N.lkeda, S.Kiyokawa, and C.Yamanaka, Phys.Rev. A33,525(1986). 4. T.Yabe and C.Yamanaka, Comm.Piasma Phys. Controlled Fusion 9,169(1985). 5. We remind that an average ion is an ion having in principle non-integral occupation numbers of electrons in a Ievei. A plasma of N "average ions" represents N ions in different charge and excitation states, but conserves number of free and bound electrons. Number of bound electrons in a particular Ievei is also cons(.~rved. For example an average ion of two ions with one neutral atom and one in charge state X will be in average charge state X/2 and all energy Ieveis will belong to this charge state. ln reality however, 50",.b of the Ieveis correspond to a neutral atom and 50% to an atom in charge state X. The energies of these Ieveis can be completely different than those of the average atom. 6. M.ltoh, T.Yabe, and S.Kiyokawa, Jpn.J.Appi.Phys. 25, L45 ( 1986) ; M.ltoh, T.Yabe, and S.Kiyokawa, Phys.Rev. in print. 7. T.Mochizuki, T.Yabe, H.Azechi, K.A.Tanaka, T.Boehly, -9-N.Miyanagal H.Nishimura 1 S.ldo 1 M.Yamanaka, K.Nishihara, T. Norimatsu 1 T .Jitsuno 1 M. Nakatsuka 1 K. Mima 1 S. Nakai 1 C.Yamanaka 1 R.Sigel 1 G.D.Tsakiris 1 K.Eidmannl P.Herrmann 1 R. Pakula 1 P. Sachsenmaier 1 S. Sakabe 1 and S. Witkowski I 11-th lnt. Conf. Plasma Phys, Contr. Nucl. Fusion, Kyoto( 1986), IAEA-CN-47 /B-1-3. 8. G.W.Parker 1 Phys.Rev. A33 1 799(1986). 9. S.Kiyokawa 1 T.Yabe 1 N.Miyanaga, K.Okada 1 H.Hasegawa, T. Mochi:zuki 1 T. Yamanaka, C. Yamanaka 1 and T. Kagawa, Phys.Rev.Lett. 54, 1999(1985). r;J 1 > QJ 0 c w 104 ~_,..... . .,.,...~~. :;,....-"::;::::.-" e.,...J- • -::.- ;.....-n r 3 , • •> _._,_- """" -------- .. 411 !9-.!i--' ---· 0 0 0 • 0 • • • • _OT ___ ..__...,__.~ o o o 0 0 0 0 0 0 0 - 0 103 • 102 0 10 20 30 40 50 60 70 80 CHARGE ST A TE Level energies of l=O state for various charge states. Filled circles: RHFS. Open circles: HFS. Dashed lines: modified Parker's model Eqs.(3),(4), and (7) without the relativistic correction ~ERE. Solid lines: the present model Eqs.(3),(4), and (8) without ~ERE. Dash-dot lines : the present model with 6ERE. ..... 0 I l{72{n) SOOr-----.-----.-----.-----~----~----~----~ HFS 700 o- o===.=o - o:::o;;;;..-=..;;:====== 600 I o D > CU 500 0• 400 m w I --300 m w 100 0 _p ___ P.,_ ___ P._p _____ ~ p~ ' 10 20 30 40 50 60 CHARGE ST A TE Non-relativistic level energies measured from t=O state versus charge states for (a) n=3, (b) n=4, and (c) n=S. P,D, F, and G denote the results calculated with HFS for t=l, 2,3, and 4, respectively. The solid and dashed lines represent the results with the present modeland the modified Parker's model, repectively. 600~----~----~----~----~----~----~--~ TJ 2(b J > QJ 400 0_ 300 --.1-w I -200 --.1-w D 100 p- p r-0 see Figure Caption 2(a) D p 10 IFI' llil'-D 0 D p p p 20 "", ', D D D '\.),. D '\ \ \ ........ \ " p p p p ......... IPI 30 40 CHARGE ST A TE HFS D p 50 60 ..... 1\) \ 500 I . HFS I > 400 QJ 300 0 -"' G U'"l w -I 200 ~ G -- F U'"l w D D D ~ ~ 100 (.) D I D p p I:_ ""-P.._~ p p 0 10 20 30 40 50 60 CHARGE ST A TE J,v 2r'c) see Figure Caption 2(a) 1400 I ' I ' I ' ' RHFS 1200 0 D D D 0 0 I ~ u D D-D- D I 1000 > QJ 800 0 1- p p p p p p ----1 - p rrl , ft.A I w I --rrl w 400 ~ p p p p p p 200 0 10 20 30 40 50 60 CHARGE ST A TE J,j. 3(7) Relativistic level energies similar to Fig.2 P,P D,D, F, and F denote the results with RHFS ' for (t,j) = (1,1/2), (1,3/2), (2,3/2), (2,5/2) (3,5/2), and (3,7/2), respectively. The solid lines are the results with the present model Eqs.(3), (4), (8), and (10); in Eq.(lO) z* is used instead of Z · (a) n=3, (b) n=4. n n ~ ~ 800~--~----~--~----~--~----~--~ RHFS (i=) ,,.", 1\ /IJV 600 t----F \F) ·" ' > QJ 500 (ö) 0 -...:t 400° w D . ö_LÖ ~ ~--01 • D 0 ()'I -...:t w 300 ~ to) I (P) -- P'- p 20t r I p p p p p I 100 e_ p p p p p p p 0 30 40 50 60 C RGE ST TE ''J- 3/b) see Figure Caption 3(a) 1.5 ~~~~~~~~~.-~~~ c: L. 1. 0 V1 :::> -D <{ 0::: ' 1- o.sf= n=4 - / ~ 1 a:l 0::: 0 ~ n=3 ~ K"'n=2 0 10 20 30 40 50 60 70 1-} L/ CHARGE ST ATE The orbit radii of various shells versus charge states calculated with rn = n 2jz and Eq.(2). n ~ Ol I TJ. s-c * c 100~----~----~----~----~----~----~----~ n=3 --- z ___ ,_!liiiiiJ!I!Il'""' ~---- --- __ :.:,~ -------- .....""~ ------ ~/--n=5 /~ :.,_.....-_.,........-------------so n=4 j ___ _ zn* 0 10 20 30 40 50 C ARGE ST TE The dashed lines: the effective charge Z calculated with Eq~(2). The solid lines: the effective charge z*n required to get rea~onable relativistic corrections with Eq.(IO) by fitting it to the difference between RHFS and HFS. ~ .... 

Relativistic calculations of level energies of a high-Z atom

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Relativistic calculations of level energies of a high-Z atom

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