Computational Spectroscopy of C-Like Mg VII

Transcription

1 Atlanta Univrsity Cntr W. Wdruff Library, Atlanta Univrsity Cntr Elctrnic Thss & Dissrtatins Cllctin fr Atlanta Univrsity & Clark Atlanta Univrsity Clark Atlanta Univrsity Fall Cmputatinal Spctrscpy f C-Lik Mg VII Salh Allhabi salh.allhabi@studnts.cau.du Fllw this and additinal wrks at: Part f th Atmic, Mlcular and Optical Physics Cmmns, and th Quantum Physics Cmmns Rcmmndd Citatin Allhabi, Salh, "Cmputatinal Spctrscpy f C-Lik Mg VII" (2018). Elctrnic Thss & Dissrtatins Cllctin fr Atlanta Univrsity & Clark Atlanta Univrsity This Thsis is brught t yu fr fr and pn accss by th Clark Atlanta Univrsity at DigitalCmmns@Rbrt W. Wdruff Library, Atlanta Univrsity Cntr. It has bn accptd fr inclusin in Elctrnic Thss & Dissrtatins Cllctin fr Atlanta Univrsity & Clark Atlanta Univrsity by an authrizd administratr f DigitalCmmns@Rbrt W. Wdruff Library, Atlanta Univrsity Cntr. Fr mr infrmatin, plas cntact cwisman@auctr.du.

2 ABSTRACT DEPARTMENT OF PHYSICS ALLEHABI, SALEH B.S. TAIBAH UNIVERSITY, 2007 COMPUTATIONAL SPECTROSCOPY OF C-LIKE Mg VII Cmmitt Chair: Swaraj S. Tayal, Ph.D. Thsis datd Dcmbr 2018 In this thsis, nrgy lvls, liftims, scillatr strngths and transitin prbabilitis f Mg VII hav bn calculatd. Th Hartr-Fck (HF) and Multicnfiguratin Hartr-Fck (MCHF) mthds wr usd in th calculatins f ths atmic prprtis. W hav includd rlativistic pratrs mass crrctin, spin-rbit intractin, n bdy Darwin trm and spin-thr-rbit intractin in th Brit-Pauli Hamiltnian. Th cnfiguratins, (1s 2 ),, 2p 4, 2s 2 2p3s, 2s 2 2p3p, 2s2p 2 ( 4 P)3s and which crrspnd t 52 fin-structur lvls, wr includd in th atmic mdl fr th Mg VII ins. Th prsnt rsults hav bn cmpard with NIST cmpilatin and thr thrtical rsults, and gnrally a gd agrmnt was fund.

3 COMPUTATIONAL SPECTROSCOPY OF C-LIKE Mg VII A THESIS SUBMITTED TO THE FACULTY OF CLARK ATLANTA UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE BY SALEH ALLEHABI DEPARTMENT OF PHYSICS ATLANTA, GEORGIA DECEMBER 2018

4 2018 SALEH ALLEHABI All Rights Rsrvd

5 ACKNOWLEDGMENTS It is a plasur t xprss my dp sns f thanks and gratitud t my advisr, Dr. Swaraj S. Tayal, fr his ddicatin and kn intrst in hlping m cmplt my prjct. I thank him fr his tim and ffrt that h dvtd t supprt m during this prjct. I wuld lik t giv spcial thanks t all my graduat prfssrs, spcially Drs. Wang, Mszan, and Mickns fr thir ncuragmnt and supprt. I w a dp sns f gratitud t Drs. Mszan and Wang wh wr a part f my thsis cmmitt. Thy ffrd guidanc thrughut my ntir jurny. I als thank Ms. Hard fr hr ncuragmnt and fr vrything sh did t assist m. I wuld lik t thank my classmat, Mhamd Krwat, fr his hlp and supprt. I am thankful t all thr faculty and staff mmbrs in th dpartmnt f physics fr thir kind cpratin and assistanc. I wuld lik t acknwldg with gratitud, my parnts fr taching m arly in my childhd and I apprciat thir cntinud ncuragmnt and supprt. Finally, I thank my wif fr hr cntinuus supprt thrughut th duratin f my studis. Sh always std by m and kpt m ging. ii

6 TABLE OF CONTENTS ACKNOWLEDGMENTS... ii LIST OF FIGURES... iv LIST OF TABLES...v LIST OF ABBREVIATIONS... vi CHAPTER 1. INTRODUCTION COMPUTATIONAL METHODS Nn-rlativistic Mthd Smi-rlativistic Mthd RESULTS AND DISCUSSION Enrgy Lvls Liftims Oscillatr Strngths and Transitin Prbabilitis Oscillatr Strngths and Transitin Prbabilitis fr sm E1 Transitins Transitin Prbabilitis fr Frbiddn E2 and M1 Transitins CONCLUSION REFERENCES iii

7 LIST OF FIGURES Figur 1. Fin-structur and trm splitting f th 1s 2 cnfiguratin Cmparisn btwn th prsnt and Kllhr and Pdbdva [9] calculatd transitin prbabilitis fr sm E1 transitins Cmparisn btwn lngth and vlcity frms f prsnt scillatr strngths fr sm E1 transitins Cmparisn f calculatd lngth frm f scillatr strngths fr sm E1 transitins with Frs Fischr and Tachiv [13] Cmparisn f lngth and vlcity frms f th prsnt scillatr strngths as a functin f lngth frm f prsnt valus fr sm E1 transitins iv

8 LIST OF TABLES Tabl I. Cmparisn f prsnt xcitd nrgy lvls f Mg VII (Ry) with thr calculatins...20 II. Cmparisn f liftims (s) f Mg VII lvls with thr calculatins...24 III. Cmparisn f scillatr strngths and transitin prbabilitis (s -1 ) fr sm E1 transitins with thr calculatins...28 IV. Cmparisn f lngth and vlcity scillatr strngths fr sm E1 transitins with Frs Fischr and Tachiv [13]...33 V. Cmparisn f transitin prbabilitis (s -1 ) fr frbiddn E2 and M1 transitins with thr calculatins...41 v

9 LIST OF ABBREVIATIONS Mg VII HF MCHF SS GRASP CIV3 EUV CSFs LSJ a.u. Ry fl fv Al Av E1 E2 M1 Mg 6+ ins Hartr-Fck Multicnfiguratin Hartr-Fck SUPERSTRUCTURE Gnral Purps Rlativistic Atmic Structur Prgram Th Gnral Cnfiguratin Intractin Cd Extrm Ultravilt Cnfiguratin Stat Functins Atmic Trms Symbl Atmic Units Rydbrg Oscillatr Strngths in Lngth Frmulatin Oscillatr Strngths in Vlcity Frmulatin Transitin Prbabilitis in Lngth Frmulatin Transitin Prbabilitis in Vlcity Frmulatin Elctric Dipl Transitin Elctric Quadrupl Transitin Magntic Dipl Transitin vi

10 CHAPTER 1 INTRODUCTION Magnsium is th ighth mst abundant lmnt in th arth crust, whr it cnstituts abut 1.93 prcnt f th Earth primarily fund in minral dpsits [1]. It has th ability t cmbin with thr lmnts as it is nvr fund in th pur frm [2]. Magnsium was discvrd in 1755 by Jsph Black [3]. It has an atmic numbr f 12, and its atmic wight is (24.304, ) u [4]. Mg atm has grund stat lctrn cnfiguratin f 1s 2 2s 2 2p 6 3s 2 with twlv lctrns. Fr ur atmic systm, Mg VII (Mg 6+ ) lss six lctrns whil kping th cmpnnts f th nuclus. Thus, th grund stat lctrn cnfiguratin fr Mg VII will b 1s 2. W nt that Mg VII is islctrnic with carbn atm. Thrfr, Mg VII is calld carbn-lik magnsium. Carbn, nitrgn, and xygn ar th plntiful lmnts fund in th univrs and cmprising th Earth s atmsphr thy ar als frquntly bsrvd in astrphysical bjcts. Emissin lins f carbn lik ins ar bnficial fr th studis f astrphysical plasmas, slar transitin rgin, plantary nbula, and fusin plasmas and prcis atmic data ar rquird fr th intrprtatin f thir spctra [5]. Th Natinal Arnautics and Spac Administratin (NASA) spac missins hav bn bsrving astrphysical bjcts spctra with high rslutin in a brad wavlngth rgin ranging frm ultravilt t X- ray. On f th rcnt xprimnts was flwn n th Extrm Ultravilt Explrr 1

11 2 spaccraft and n th Advancd X-Ray Astrphysical Facility. Fr stars, ths spctra f high rslutin can b btaind in X-ray wavlngth and xtrm ultravilt wavlngth rangs. Fr intrprting th larg xpctd data frm ths missins, it is ssntial t hav prcis atmic data, fr instanc radiativ dcay rats and cllisin strngths fr many csmically abundant ins [6]. In plantary nbula NGC 7207, ultravilt and infrard missin lins fr Mg ins hav als bn bsrvd [7]. Fractinal abundancs frmd by magnsium ar 1% in charg stat f Mg 6+ in cllisinally inizd plasmas (CPs) at spcific lctrn tmpratur T with kbt in th rang f V and phtinizd plasmas (PPs) at kbt in th rang f V, whr kb is Bltzman cnstant [8]. Amng grund stat cnfiguratin f Mg VII, a frbiddn transitin is f grat imprtanc rgarding astrphysical applicatins prviding infrmatin n cmpsitin, dnsity and tmpratur frm lin intnsity ratis and lvl ppulatins. Mg VII ins cnstitut abut 7.60 f slar crna n a scal f lg n[h] = Th grund stat cnfiguratin f Mg VII 1s 2 givs ris t 10 transitins: 3 P 0 1, 3 P 0 2, 3 P 1 2, 3 P 0 2,, 3 P 1 2,, 3 P 2 2, 3 P 0 1 S 0, 3 P 1 1 S 0, 3 P 2 1 S 0, 1 D 2 1 S 0, and ths ar calld frbiddn transitins. Th significant transitin that has bn sn and idntifid in th EUV slar spctrum is 3 P 1 1 S 0 with wavlngth À [6]. Transitin prbabilitis fr th intrcmbinatin r smi-frbiddn lins ar fund t b snsitiv t th chic f wav functins and rquir sufficint justificatin f crrlatin and mixing ffcts. W hav studid Mg VII scillatr strngths f diffrnt bsrvd spctral faturs wing t Mg VII ins in this thsis.

12 3 Effcts f cnfiguratin intractin ar s prminnt that vn th mst sphisticatd calculatins availabl fr sm transitins dmnstrat strng disagrmnts. T dtrmin th individual lins strngth, n f th critical factrs is rlativistic ffct cnsidratin fr calculatd data, particularly th trm mixing f angular prtin rlatd t th wav functins. Ths ffcts significantly nhanc acrss th pridic tabl hrizntally. Fr highly xcitd atms, LS cupling is usually lss valid. With incrasing nuclar charg (Z), th rlativistic ffcts als bcm imprtant. In intrmdiat cupling, individual lin strngths hav bn cmputd by diffrnt thrists thrugh th us f Brit-Pauli trms. Ths calculatins ar attributd t b cmputr intnsiv t variabl dgrs. As cmpard t missin xprimnts, ths calculatins smtims rsult in largr dviatins frm LS cupling. Th Opacity prjct is spcifically limitd t nnrlativistic multiplt data; LS-cupling schm ar applid fr btaining individual lin data [9]. In th prsnt study, varius calculatins rgarding lctric dipl transitins such as nrgy lvls, transitin prbabilitis, scillatr strngths and lin strngths wr invstigatd. Th btaind rsults f calculatins wr usd t cmpar with th data frm Lstinsky t al. [8], Kllhr and Pdbdva [9], NIST cmpilatin [10], Bhatia and Dschk [11], Aggarwal [12], Frs Fischr and Tachiv [13], and Zhang and Sampsn [14]. In th fild f atmic cmputatinal atmic structur rsarch study, n f th frmst cntributrs is Frs Fischr wh dvlpd th MCHF atmic structur cds. Cmputatinal mthds such as MCHF and HF wr usd t valuat diffrnt calculatins in this thsis. Smi-rlativistic MCHF cd basd n th Brit-Pauli

13 4 Hamiltnian has bn usd. This mthdlgy uss th prturbatin thry t includ rlativistic crrctins. Our findings and cmputatinal rsults hav bn cmpard with thr availabl rsults and ar fund in vry wll agrmnt xcpt fr a fw transitins. As a rsult f cancllatin ffcts, incnsistncis ar spcifically gratr with wakr scillatr strngths f sm smi-frbiddn transitins.

14 CHAPTER 2 COMPUTATIONAL METHODS In rdr t undrstand th atmic prprtis f Mg VII, w us tw mthds, first is calld nn-rlativistic mthd and th scnd is calld smi-rlativistic mthd. Mg VII is a multi-lctrn in with 6 lctrns, whr 2 lctrns ar in th 1s cr-shll, and 4 lctrns ar utsid th cr. Bcaus w hav mr than 1 lctrn, w hav t apply an apprximat mthd which hlps us in slving th Schrödingr quatin. Thrugh this chaptr, w will shw tw cupling schms f calculatins in dpth and giv an vrviw f th tchniqus applid. 2.1 Nn-rlativistic Mthd Th starting pint f th calculatins n many lctrn atms is cntral fild apprximatin, accrding t which it is assumd that ach lctrn in a many lctrn cmplx atms is actd upn by a cntral fild prducd by th nuclus which is assumd vry small and infinitly havy. This apprximatin hlds bcaus th dviatin f th cntral ptntial V(r) du t th cls passag f all th thr lctrns is rlativly small. This is s bcaus th nuclar ptntial is largr than th ffct f th abv fluctuating ptntial. Nw in th cas f tw lctrn atms, it is pssibl t trat th mutual intractin btwn th tw lctrns as a prturbatin which is addd t th cntral ptntial du t 5

15 6 th nuclus. T d th sam in th cas f th many lctrn atms w hav t writ th unprturbd Hamiltnian s as t includ th ttal ffctiv cntral ptntial as statd abv. Th prturbatin thn cntains th rmaining sphrical and all th nn-sphrical parts f th lctrnic intractins. Nw if th rlativistic intractin, that is, prturbatin is nglctd thn th cntral fild Hamiltnian fr th many lctrn systms can b writtn as [15]: N H = ( ħ2 2m i 2 i=1 Z 2 4πε 0 r i ) N + 2 4πε 0 r ij i<j (1) Th symbl H is knwn as th Hamiltnian pratr f th atmic systm. Z dscribs th atm s nuclar charg. Hr r i dnts th rlativ c-rdinat f th i th lctrn with rspct t th nuclus, r ij = r i r j, and th last summatin is vr all pairs f lctrns. Nw it is cnvnint t us atmic units s that th Hamiltnian bcms N N H = ( 1 2 i 2 Z ) + 1 r i r ij i=1 i<j (2) Nw w can dfin any N-lctrn systm using th wav functin (qi,, qn) whr qi = (ri, σi) and ri knwn as th spac and σi knwn as th spin crdinats f th lctrn catgrizd i. Th wav functin is usd t slv th tim indpndnt Schrödingr wav quatin [16] Hψ(q1,, qn) = ψ(q1,, qn). (3)

16 7 Th symbl E is knwn as th ignvalu f th Hamiltnian pratr. Th ignvalu E rfrs t th vrall nrgy f th systm. Th Eign-functin is th atmic stat wav functin (q1,, qn). Th Hamiltnian mntind in th abv quatin is nly accurat if it is assumd that th rlativistic ffcts can b disrgardd. Additinally, it is suppsd that th nuclus is a pint charg with a cuntlss mass. Th wav functins r ignfunctins ar dscribd using M ψ(ls) = a i i (α i ; LS). (4) i=1 Symbls L and S ar th ttal rbital angular mmntum and ttal spin angular mmnta, M rfrs t numbr f th cnfiguratins and α i rfrs t th angular mmnta cupling schm f th i th cnfiguratin, a i is th cnstant f nrmalizatin, and i is th cnfiguratin stat functin. Ths wav functins ar nrmalizd: ψ(q 1,., q N ) 2 dq 1, dq N < ψ ψ >= 1. (5) q Taking int cnsidratin that th intgratin is vr all spac and spin crdinats rspctivly, th apprximat wav functins can b btaind by substituting th full Hamiltnian H with a sparabl Hamiltnian which is givn by N H H = ( 1 2 i 2 z r i + V(r i )), (6) i=1 whr th ptntial V(r i ) is an stimat fr th Culmb rpulsin btwn th lctrns.

17 8 Th sparabl Hamiltnian lik th full Hamiltnian cmmuts with th ttal angular mmntum pratrs: L 2, Lz, S 2, and Sz. W can als slct th wav functins f th sparabl Hamiltnian H0 t bcm th wav functins f ths pratrs. H 0 ψ 0 (q 1,, q N ) = E 0 ψ 0 (q 1,, q N ). (7) W can writ E 0 and ψ 0 as [14]: btwn th tw lctrns as a prturbatin which is addd t th cntral ptntial du t N E = E i i=1 (8) ψ (q 1,., q N ) = Π N i=1 φ(α i ; q i ). (9) Als, th 1-lctrn ignfunctin can b writtn as: φ(α i ; q i ) = 1 r P(nl; r)y lm l (θ, φ)x ms (σ), (10) whr P(nl; r) is radial ignfunctin, Y lml (θ, φ) is th sphrical harmnics, and n, l and m l can assum th valus n = 1,2,3, l = 0,1,2,. (n 1) m l = 0, ±1, ±2,, ±l

18 9 Whn th spin f th lctrn is takn int accunt, ach stat f th lctrn is thn charactrizd by th fur quantum numbrs n, l, m l and m s, whr m s can hav valus ± 1. Th Hamiltnian H ds nt chang with rspct th variatins f lctrn c- 2 rdinats. Accrdingly, any variatin f th crdinats in th prduct functin lads t th wav functin: Φ(q 1,. q N ) = A Π N i=1 φ(α i ; q i ) (11) Whr A rfrs t anti-symmtric functin. W can xprss this functin using th Slatr dtrminant: Φ(q 1,. q N ) = 1 φ(α 1 ; q 1 ) φ(α 1 ; q 2 ) φ(α 1 ; q N ) N! φ(α 2 ; q 1 ) φ(α 2 ; q 2 ) φ(α 2 ; q N ) (12) φ(α N ; q 1 ) φ(α N ; q 2 ) φ(α N ; q N ) Sinc th dtrminant vanishs whn tw clumns r rws ar qual, w hav th rsult that n tw individual lctrns can hav all th fur quantum numbrs n, l, m l and m s qual. This is th statmnt f th Pauli xclusin principl. In th MCHF apprach, th wav functins ar stimatd using a linar cmbinatin f rthnrmal cnfiguratin stat functins [16] givn by M Ψ(γLS) = c i Φ(γ i LS) (13) i=l M c i 2 = 1 (14) i=l

19 Smi-rlativistic Mthd T calculat th rlativistic impact in atmic systm Mg VII, w apply th Brit- Pauli Hamiltnian t crrct ur nnrlativistic Hamiltnian. Th Brit-Pauli Hamiltnian is spcifid by [16] H BP = H NR + H RS + H FS (15) H NR rfrs t th nnrlativistic many-lctrn Hamiltnian. H RS rfrs t th rlativistic shift pratr, and ths vary with L and S which is givn by th quatin H RS = H MC + H D1 + H D2 +H OO + H SSC (16) Symbl H MC rfrs t th mass crrctin trm and is givn by N H MC = α2 8 ( i 2 2 ) + i i=1 (17) H D1 and H D2 ar th n and 2-bdy Darwin trms which ar stimatd by and N H D1 = α2 Z 8 i 2 ( 1 ), (18) r i i=1 N H D2 = α2 4 i 2 ( 1 r i j ) i<j (19) rspctivly.

20 11 Als, H OO is th rbit-rbit trm which is givn by N H OO = α2 2 [p i p j i<j r ij + r ij(r ij p i )p j r ij 3 ] (20) Lastly, H SSC rfrs t th spin-spin cntact factr which can b stimatd by N H SSC = 8πα2 3 (s i i<j s j ) δ(r i r j ) (21) Th fin structur pratr H FS dfin th intractins amng th spin and rbital angular mmnta f th systm. Additinally, th H FS ds nt vary with L and S, but this pratr nly varis with th ttal angular mmntum pratr J = L + S [16]. H FS = H SO + H SOO + H SS (22) Whr H SO is th spin-rbit trm that can b stimatd by N H SO = α2 Z r l i s i i i=1 (23) H SOO is th spin-thr-rbit trm givn by N H SOO = α2 2 r ij p i 3 (s i + 2s j ) (24) r ij i<j

21 12 H SS is th spin-spin paramtr and is givn by N H SS = α r [s i s j 3 (s i r ij )(s j r ij ) 2 ] (25) ij r ij i<j Th Brit-Pauli wav functins shuld b writtn as a linar cmbinatin M ψ(γjm J ) = c i Φ(γ i L i S i JM J ), (26) i=1 whr vry M singl-cnfiguratin functins Φ is cmpsd f 1-lctrn functins, γ dscribs th cupling f angular mmnta f th lctrns, and c i is mixing cfficint. Φ(γ i L i S i JM J ) = LM L SM S LSJM J Φ(γLM L SM S ) (27) M L M S LSJ cupld CSFs ar nticd t b dscribd as (γilisijmj). CSFs hav dissimilar LS trms which ar includd within th abv xpansin as L and S ar nt gd quantum numbrs. In th intrmdiat cupling, th wav quatin is givn by [16] Hc=Ec (28) H ij = γ i L i S i JM J H BP γ j L j S j JM J (29) Starting frm this pint, w cnsidr fin structur lvls f th systm. This assumptin will assist in cmprhnding hw th rlativistic nrgy crrctins cm int play. Whn th xpansin mntind in quatin (26) includs th rlativistic trms, w can xprss th nrgy as fllws [16]: E = E NR + E RS + E FS (30)

22 13 Th trm E NR dscribs th rdinary nnrlativistic nrgy E NR = γlsjm J H NR γlsjm J, (31) and trm E RS dscribs th rlativistic shift cntributin t th crrctin f rlativistic nrgy. Als, it stands fr a shift f th nnrlativistic nrgy trm E NR as E RS is indpndnt f J and M J. Th rlativistic shift pratrs nly chang with L and S: E RS = γlsjm J H RS γlsjm J (32) E FS is knwn as th cntributin f fin structur crrctin t th rlativistic nrgy E FS = γlsjm J H FS γlsjm J (33) On th thr hand, w can xprss th fin structur as fllws: E FS = E SO + E SOO + E SS (34) And trms: E SO, E SOO and E SS ar knwn as th nrgis rlatd t th spin-rbit, spinthr-rbit and spin-spin pratrs, rspctivly [16]. Th nrgy dpnds n th J quantum numbr lading t th splitting f th nnrlativistic LS trm E NR nrgy int th fin structur lvls. Nxt, w shw additin f angular mmnta giving th prbabl valus f J fr knwn valus f L and S. L S, L S + 1,. L + S 1, L + S, (35) whr th numbr f lvls in th trm is spcifid by th multiplicity 2S+1 if L S and by 2L+1 if L< S.

23 E SO and E SOO ar th spin-rbit and spin-thr-rbit trms rspctivly, and ach is prduct f rank n and spatial tnsr pratrs. 14 E SO = γlsjm J H SO γlsjm J ( 1) L+S+J L L 1 { S S J } (36) E SOO = γlsjm J H SOO γlsjm J ( 1) L+S+J L L 1 { S S J } (37) Basd n th cncpt f th Wignr Eckart [16] in additin t knwing that nrgy frm th spin-spin pratr is a scalar prduct f 2 rank and 2 tnsr pratrs E SS is as fllws: E SS = γlsjm J H SS γlsjm J ( 1) L+S+J L L 2 { S S J } (38) Rfrring t th xprssins xplicitly in 6-j symbls w gt: ( 1) L+S+J L L 1 { } J(J + 1) L(L + 1) S(S + 1) (39) S S J ( 1) L+S+J L L 2 { S S J } 3 C(C + 1) L(L + 1) S(S + 1) (40) 4 And assum C=J(J+1) L(L+1) S(S+1), hnc w can writ th fin-structur nrgis as fllws: E SO = {J(J + 1) L(L + 1) S(S + 1)}ζ SO (γls) (41)

24 15 E SOO = {J(J + 1) L(L + 1) S(S + 1)}ζ SOO (γls) (42) E SS = {J(J + 1) L(L + 1) S(S + 1)}ζ SS (γls) (43) And ζ SO (γls), ζ SOO (γls)and ζ SS (γls) ar ach indpndnt n J. Ignring th spin-spin trm lads t th nrgy diffrnc btwn tw squntial fin structur lvls J and J-1 ΔE FS = 2ζJ, (44) and th symbl ζ rfrs t th rul f Landé intrval fr th fin-structur and is givn by ζ = ζ SO (γls) + ζ SOO (γls), (45) whr th fin structur may b ithr nrmal r invrtd. It is nrmal whn it is psitiv, and th fin structur nrgy riss as th valu f J incrass. Whn it is ngativ, w say that th fin structur is invrtd [16]. Whn th spin-spin trm can t b ignrd th rul f Landé intrval braks dwn, and th fin structur shws unbalancd bhavir. This bhavir is nticd whn dissimilar CSFs with dissimilar L and S, whr ach n cupls t th sam ttal J, ar invlvd in th quatin [16]. Figur 1 shws an xampl f th fin structur and trm splitting f th 1s 2 cnfiguratin in ur prpsd Mg VII systm.

25 16 FIG. 1. Th fin-structur and trm splitting f th 1s 2 cnfiguratin in Mg VII. In th Brit-Pauli calculatins, w hav ignrd th spin-spin, rbit-rbit, and scnd Darwin cntact trms. Th ignfunctins frm quatin (6) can b usd t dscrib th lngth and vlcity structurs f th scillatr strngths and transitin pssibilitis fr transitins btwn th fin structur lvls. Th scillatr strngths can b indicatd as absrptin r missin scillatr strngth. W can writ th absrptin scillatr strngths as: f πk (γj, γ J ) = 1 α C k[α(e γ J E γj)] 2k 1 Sπk (γj, γ J ) g J, (46) whr th atm in th lwr stat absrbs a phtn, thn it is xcitd t an uppr stat. Hr g J is th statistical wight f th uppr lvl which can b writtn as:

26 17 g J = 2J + 1, (47) And C k = (2k + 1)(k + 1) k((2k + 1)!!) 2 (48) S πk (γj, γ J ) rfrs t th lin strngth givn by S πk (γj, γ J ) = γjm O π(k) q α J M 2 M,M,q (49) Whr O q π(k) is a gnral transitin pratr f rank k and parity π. Fr lctric multipl transitin (π =( 1) k ) and magntic multi-pl transitin (π =( 1) k+1 ) th lctric multi-pl transitin pratr is calculatd by N E (k) q = r k (k) (i)c q (i), (50) i=1 and th magntic multi-pl transitin pratr is givn by M (k) 1 q = α k(2k + 1) [ k + 1 MA (k) q g smb (k) q ] (51) N MA (k) q = r k 1 (i) [c (k 1) (i) L (1) (k) (i)] q i=1 (52) MB q (k) = N i=1 r k 1 (i) [c (k 1) (i) S (1) (k) (i)] q (53)

27 W bgin with th transitin intgral t cmprhnd hw th transitin happns amng an uppr stat and a lwr stat 18 L q πk (γjm, γ J M ) = γjm O q π(k) γ J M, (54) and th cmpnnt strngth is givn as fllws: S πk (γjm, γ J M ) = L πk q (γjm, γ J M ) 2 (55) q Furthrmr, th transitin pssibility frm th uppr lvl t th lwr lvl can b calculatd by A πk (γj, γ J ) = 2C k [α(e γ J E γj)] 2k+1 Sπk (γj, γ J ), (56) g J whr th liftim, τγ J is ppsit f th summatin f transitin pssibilitis vr th multi-pl transitins t vry lwr nrgy lvl [16-17]: τ γ J = 1 A πk (γj, γ J ) πk,γj (57)

28 CHAPTER 3 RESULTS AND DISCUSSION 3.1 Enrgy lvls In Tabl I, w hav prsntd th first fifty-tw (52) nrgy lvls f Mg VII arising frm th (1s 2 ),, 2p 4, 2s 2 2p3s, 2s 2 2p3p, 2s2p 2 ( 4 P)3s and cnfiguratins, and w hav cmpard ur calculatd nrgis with NIST cmpilatin [10]. In th sam tabl w hav als cmpard ur rsults with th thrtical lvl nrgis f Bhatia and Dschk [11] btaind using th SS prgram f Eissnr t al. (1974) Aggarwal [12] wh usd th CIV3 cd f Hibbrt (1975), Frs Fischr and Tachiv [13] btaind frm th MCHF mthd and als Zhang and Sampsn [14] wh usd th GRASP prgram f Dyall t al. (1989). W nt that th NIST rsults ar availabl nly fr th 42 nrgy lvls; thus, th NIST valus ar nt availabl fr 10 lvls. Ths lvls ar 2s 2 2p3p 1 P 1, 2s 2 2p3p 3 D 1, 2s 2 2p3p 3 D 2, 2s 2 2p3p 3 D 3, 2s 2 2p3p 3 S 1, 2s 2 2p3p 2, 2s 2 2p3p 1 S 0, 2s2p 2 ( 4 P)3s 1, 3 F 3 and 3 F 4. Furthrmr, w bsrv that th thr thrtical rsults ar cmputd fr 46 nrgy lvls, thus w nt that six nrgy lvls cmputd in th prsnt wrk wr nt givn in th thr thrtical rsults. Ths lvls ar 2s2p 2 ( 4 P)3s 1, 2s2p 2 ( 4 P)3s 5 P 2, 2s2p 2 ( 4 P)3s 5 P 3, 2s2p 2 ( 4 P)3s 0, 2s2p 2 ( 4 P)3s 3 P 1 and 2s2p 2 ( 4 P)3s 2. 19

29 20 TABLE I. Cmparisn f prsnt xcitd nrgy lvls f Mg VII (Ry) with NIST cmpilatin and thr thrtical calculatins. Indx CFS LSJ Enrgy Lvls (Ry) Prsnt [NIST] [MCHF] [SS] [CIV3] [GRASP] [Diff] 1. 3 P P P D S S D D D P P P D S P p 4 3 P P P S p 4 1 S s 2 2p3s 3 P P

30 21 TABLE I. (Cntinud). Indx CFS LSJ Enrgy Lvls (Ry) Prsnt [NIST] [MCHF] [SS] [CIV3] [GRASP] [Diff] P s 2 2p3s 1 P s 2 2p3p 1 P s 2 2p3p 3 D D D s 2 2p3p 3 S s 2 2p3p 3 P P P s 2 2p3p 1 D s 2 2p3p 1 S s2p 2 ( 4 P)3s 5 P F s2p 2 ( 4 P)3s 5 P s2p 2 ( 4 P)3s 5 P F D F D D D P

31 22 TABLE I. (Cntinud). Indx CFS LSJ Enrgy Lvls (Ry) Prsnt [NIST] [MCHF] [SS] [CIV3] [GRASP] [Diff] P P s2p 2 ( 4 P)3s 3 P F s2p 2 ( 4 P)3s 3 P P s2p 2 ( 4 P)3s 3 P [NIST] Kramida, Ralchnk, Radr and NIST ASD Tam [10]. [MCHF] Frs Fischr and Tachiv [13]. [SS] Bhatia and Dschk, [11]. [CIV3] Aggarwal [12]. [GRASP] Zhang and Sampsn [14]. [Diff] Diffrnc btwn th prsnt calculatin and NIST cmpilatin. Th agrmnt f th prsnt calculatd nrgis with th NIST and thr calculatd nrgis is xcllnt. Th avrag dviatin is f arund Ry fr all lvls, with th xcptins f th 2p 4 1 S 0 and 1 F 3 lvls that dviat arund Ry and Ry, rspctivly frm th NIST valus. W hav als fund diffrnc fr th 5 S 2 lvl whr th prsnt valu is lwr than th NIST cmpilatin by arund Ry. Th prsnt valus agr vry wll with th NIST rsults fr th first twlv (12) nrgy lvls. Th prsnt rsults als agr vry wll with th MCHF calculatins f Frs Fischr and Tachiv fr all lvls. Th avrag dviatin is f arund Ry, with th xcptin f th 2p 4 1 D 2 and 2p 4 1 S 0 lvls that

32 23 dviat by Ry and Ry, rspctivly. Th prsnt rsults agr vry wll with Frs Fischr and Tachiv rsults fr th 1, 2s 2 2p3p 0, 2s 2 2p3p 3 P 1 and 2s 2 2p3p 3 P 2 lvls with dviatin f Ry, Ry, Ry and Ry, rspctivly. As sn frm th tabl, ur MCHF calculatin displays slightly bttr agrmnt with th MCHF rsults f Frs Fischr and Tachiv calculatin. Fr th SS and GRASP calculatins, th agrmnt is similar amng th rsults fr th lwst 20 lvls, with avrag dviatin f arund Ry. Thrfr, th agrmnt f th prsnt calculatd nrgis with th SS and GRASP calculatd nrgis is gd with th first twlv (12) lvls, with th xcptin f th 5 S 2 with dviatin f Ry. Th agrmnt amng th SS nrgy lvls and NIST rsults ar bttr in svn lvls frm th first 12 lvls. On th thr hand, th prsnt nrgis ar in bttr agrmnt with th NIST rsults, particularly fr th last 25 lvls fr which th SS nrgy lvls ar cnsistntly highr. Likwis, th agrmnt amng th GRASP nrgy lvls and NIST rsults ar bttr in fur lvls frm th first 12 lvls. Hwvr, th prsnt nrgis ar in bttr agrmnt with th NIST rsults, spcially fr th last 17 lvls fr which th GRASP nrgy lvls ar cnsistntly lwr. Th prsnt rsults gnrally agr with th CIV3 calculatin fr mst lvls. Th prsnt rsults agr vry wll with th CIV3 rsults fr th 1, 2s 2 2p3p 1 P 1, 2s 2 2p3p 3 D 1, 2s 2 2p3p 3 D 2, 2s 2 2p3p 3 D 3, and 2s 2 2p3p 3 S 1, with avrag dviatin f arund Ry. Th CIV3 nrgis fr th last tn lvls ar in bttr agrmnt with NIST rsults, whras th prsnt nrgy lvls ar much clsr f th NIST nrgy lvls fr mst f th first 27 lvls fr which th CIV3 nrgy lvls ar highr.

33 Liftims W hav calculatd transitin prbabilitis f transitins btwn th lvls f th grund cnfiguratin (1s 2 ) and th xcitd cnfiguratins, 2p 4, 2s 2 2p3s, 2s 2 2p3p, 2s2p 2 ( 4 P)3s and. In Tabl II, w hav listd th liftims and cmpard thm with Lstinsky t al. [8] and Frs Fischr and Tachiv [13]. As sn frm th tabl that sm transitins calculatd in th prsnt wrk wr nt givn in th thr thrtical rsults. Th agrmnt f ur liftims rsults with th thr tw thrtical rsults is xcllnt. Th prsnt liftims agr vry wll with Frs Fischr and Tachiv fr all lvls, spcially vry gd agrmnt fr th 2, 1 P 1, 2s 2 2p3p 0, 2s 2 2p3p 3 P 1 and 2s 2 2p3p 3 P 2 lvls. W nt frm this tabl that Mg VII has sm lng-livd mtastabl lvls: 1, 2, 2, 1 S 0, and 5 S 2. Ths fiv lvls hav cnsidrably highr liftims than th thr lvls. This is bcaus th transitins btwn ths fur lvls ar nt allwd transitins; thy ar frbiddn transitins, and th transitin t th fifth lvl is calld smi-frbiddn r intrcmbinatin transitin. On th thr hand, th rmaining transitins ar allwd transitins. TABLE II. Cmparisn f liftims (s) f Mg VII lvls with thr calculatins. Indx CFS LSJ Liftims (s) Prsnt [CFF] [L] 1. 3 P P E E E P E E E+01

34 25 TABLE II. (Cntinud). Indx CFS LSJ Liftims (s) Prsnt [CFF] [L] 4. 1 D E E E S E E E S E E E D E E E D E E E D E E E P E E E P E E E P E E E D E E E S E E E P E E E p 4 3 P E E E P E E E P E E E p 4 1 D E E E p 4 1 S E E E s 2 2p3s 3 P E E P E E P E E s 2 2p3s 1 P E E s 2 2p3p 1 P E E s 2 2p3p 3 D E E-10 -

35 26 TABLE II. (Cntinud). Indx CFS LSJ Liftims (s) Prsnt [CFF] [L] D E E D E E s 2 2p3p 1 S E E s 2 2p3p 3 P E E P E E P E E s 2 2p3p 1 D E E s 2 2p3p 1 S E E s2p 2 ( 4 P)3s 5 P E P E F E E s2p 2 ( 4 P)3s 5 P E F E E D E E F E E D E E D E E D E E P E E P E E P E E s2p 2 ( 4 P)3s 3 P E P E

36 27 TABLE II. (Cntinud). Indx CFS LSJ Liftims (s) Prsnt [CFF] [L] F E E P E E s2p 2 ( 4 P)3s 3 P E [CFF] Frs Fischr and Tachiv [13]. [L] Lstinsky t al. [8]. 3.3 Oscillatr Strngths and Transitin Prbabilitis Oscillatr Strngths and Transitin Prbabilitis fr Sm E1 Transitins In Tabl III, w hav mad a cmparisn f th lngth valus f scillatr strngths and transitin prbabilitis fr sm E1 transitins amng th trms f th grund (1s 2 ) and th xitd, 2s 2 2p3s, and cnfiguratins. Includd in this tabl ar th rsults f Kllhr and Pdbdva [9] calculatin, alng with ur prsnt rsults. W hav shwn in th tabl that gi and gk, whr gi is th statistical wight f th initial (lwr) stat, can b calculatd by (2Ji+1). Th gk is th statistical wight f th final (uppr) stat. Likwis, it can b calculatd by (2Jk+1). Hr, J mans Ttal Angular Mmntum Quantum Numbr.

37 TABLE III. Cmparisn f prsnt scillatr strngths and transitin prbabilitis (s -1 ) fr sm E1 transitins in Mg VII with DEK rsults. 28 Transitin Prsnt [DEK] Initial Lvl Final Lvl g i g k f l A l f l A l 5 S E E E E+04 5 S E E E E+04 3 D E E E E+09 3 D E E E E+08 3 D E E E E+09 3 D E E E E+09 3 D E E E E+08 3 D E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E+07 3 S E E E E+09 3 S E E E E+09 3 S E E E E+10

38 29 TABLE III. (Cntinud). Transitin Prsnt [DEK] Initial Lvl Final Lvl g i g k f l A l f l A l 1 P E E E E+05 1 P E E E E+07 1 P E E E E+06 2s 2 2p3s E E E E+10 2s 2 2p3s E E E E+10 2s 2 2p3s E E E E+10 2s 2 2p3s E E E E+10 2s 2 2p3s E E E E+10 2s 2 2p3s E E E E+10 2s 2 2p3s 1 P E E s 2 2p3s 1 P E E s 2 2p3s 1 P E E F E E F E E F E E E E E E D E E E E+11 3 D E E E E+11 3 D E E E E+11 3 D E E E E+09

39 30 TABLE III. (Cntinud). Transitin Prsnt [DEK] Initial Lvl Final Lvl g i g k f l A l f l A l 3 D E E E E+10 3 D E E E E E E E E E E E E E E E E E E E E E E E E E E E E+11 1 F E E P E E P E E P E E S E E D E E E E+05 3 D E E E E+05 3 D E E E E E E E E E E E E E E E E+10 3 S E E E E+06 1 P E E E E+10

40 31 TABLE III. (Cntinud). Transitin Prsnt [DEK] Initial Lvl Final Lvl g i g k f l A l f l A l 2s 2 2p3s E E s 2 2p3s E E s 2 2p3s 1 P E E E E+10 3 F E E E E+10 3 F E E E E E E+11 3 D E E D E E D E E E E E E F E E E E+11 1 P E E E E+10 1 S 3 D E E E E+04 1 S E E E E+05 1 S 3 S E E E E+06 1 S 1 P E E E E+09 1 S 2s 2 2p3s E E S 2s 2 2p3s 1 P E E E E+10 1 S 3 D E E

41 32 TABLE III. (Cntinud). Transitin Prsnt [DEK] Initial Lvl Final Lvl g i g k f l A l f l A l 1 S E E S 1 P E E E E+11 [DEK] Kllhr and Pdbdva [9]. In Figur 2, w hav cmpard th lngth valus f transitin prbabilitis frm th tw calculatins. As sn frm th figur, th agrmnt btwn th prsnt and th thr thrtical transitin prbabilitis is vry gd with th xcptin fr th 3 P 2 3 D 1 transitin. Th avrag rati btwn th tw sts f data is (Al (Prsnt)/Al (DEK)) = 1.02; thus, it shws that ur calculatd transitin prbabilitis can b xpctd t b rliabl t within 10 % fr mst f th transitins. FIG. 2. Cmparisn btwn th calculatd transitin prbabilitis btaind in th prsnt wrk (Al) and th thrtical valus (Al) rprtd by Kllhr and Pdbdva [9].

42 33 In Tabl IV, w hav cmpard th prsnt rsults fr sm E1 transitins amng th trms f grund stat (1s 2 ) and th xcitd stats, 2s 2 2p3s and with Frs Fischr and Tachiv [5]. A vry gd agrmnt was btaind btwn th tw calculatins. TABLE IV. Cmparisn f prsnt f lngth and vlcity scillatr strngths fr sm E1 transitins in Mg VII with CFF rsults. Transitin Prsnt [CFF] Initial Lvl Final Lvl g i g k f l fv f l fv 5 S E E E E-06 5 S E E E E-06 3 D E E E E-02 3 D E E E E-02 3 D E E E E-02 3 D E E E E-04 3 D E E E E-03 3 D E E E E E E E E E E E E E E E E E E E E E E E E-02

43 34 TABLE IV. (Cntinud). Transitin Prsnt [CFF] Initial Lvl Final Lvl g i g k f l fv f l fv E E E E E E E E E E E E-04 3 S E E E E-01 3 S E E E E-01 3 S E E E E-01 1 P E E E E-06 1 P E E E E-04 1 P E E E E-06 2s 2 2p3s 2s 2 2p3s 2s 2 2p3s 2s 2 2p3s 2s 2 2p3s 2s 2 2p3s 2s 2 2p3s 1 P 2s 2 2p3s 1 P 2s 2 2p3s 1 P E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-05

44 35 TABLE IV. (Cntinud). Transitin Prsnt [CFF] Initial Lvl Final Lvl g i g k f l fv f l fv 2s2 2p3d 3 F 2s2 2p3d 3 F 2s2 2p3d 3 F 2s2 2p3d 2s2 2p3d 2s2 2p3d 3 D 2s2 2p3d 3 D 2s2 2p3d 3 D 2s2 2p3d 3 D 2s2 2p3d 3 D 2s2 2p3d 3 D 2s2 2p3d 2s2 2p3d 2s2 2p3d 2s2 2p3d 2s2 2p3d 2s2 2p3d 2s2 2p3d 1 F E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E-05

45 36 TABLE IV. (Cntinud). Transitin Prsnt [CFF] Initial Lvl Final Lvl g i g k f l fv f l fv 2s2 2p3d 1 P E E E E-03 2s2 2p3d 1 P E E E E-04 2s2 2p3d 1 P E E E E-05 5 S E E E E-09 3 D E E E E-06 3 D E E E E-05 3 D E E E E E E E E E E E E E E E E-01 3 S E E E E-05 2s 2 2p3s E E E E-04 2s 2 2p3s E E E E-05 2s 2 2p3s 1 P E E E E-02 2s2 2p3d 3 F E E E E-02 2s2 2p3d 3 F E E E E-03 2s2 2p3d E E E E-01 2s2 2p3d 3 D E E E E-05

46 37 TABLE IV. (Cntinud). Transitin Prsnt [CFF] Initial Lvl Final Lvl g i g k f l fv f l fv 3 D E E E E-04 1 S 1 S 1 S 3 D E E E E E E E E E E E E-03 1 F E E E E-01 1 P E E E E-03 3 D E E E E E E E E-05 3 S E E E E-05 1 S 1 S 1 S 1 S 1 S 1 P E E E E-01 2s 2 2p3s E E E E-04 2s 2 2p3s 1 P E E E E-02 3 D E E E E E E E E-04 1 S 1 P E E E E+00 [CFF] Frs Fischr and Tachiv [5].

47 38 Th ratis f th prsnt scillatr strngths in lngth and vlcity frmulatins hav bn plttd in Figur 3. As sn frm th figur, w hav a vry gd agrmnt btwn th tw frmulatins. Th gd pint distributin alng th straight lin is btaind fr th strngr lins. Th avrag rati amng th tw sts f data is (fl /fv) = 0.96, accrdingly this mans that ur calculatins can b xpctd t b rliabl t within 10% fr mst transitins. Th prsnt lngth scillatr strngths hav bn cmpard with th rsults f Frs Fischr and Tachiv [5] in Figur 4. Th agrmnt btwn th tw MCHF calculatins is vry gd fr all transitins, with th xcptins f th 3 P 2 1 F 3, 3 P 1 3 F 2 and 1 D 2 3 F 3 transitins. Th avrag rati amng th tw sts f data is (fl (Prsnt) /fl (CFF)) = 0.96, hnc; again indicating an accuracy f 10% fr mst f th transitins. FIG. 3. Cmparisn btwn lngth and vlcity f th prsnt scillatr strngths fr sm E1 transitins.

48 39 FIG. 4. Cmparisn btwn lngth frms f th prsnt and CFF [5] scillatr strngths fr sm E1 transitins. In Figur 5, w hav rprsntd th rati f th prsnt vlcity and lngth f th scillatr strngths as a functin f th prsnt lngth scillatr strngths. Th tw lins in this figur display a dviatin. As sn frm th figur that th agrmnt btwn th prsnt vlcity and lngth f th scillatr strngth valus is within 20 % fr mst f th transitins.

49 40 FIG. 5. Th rati btwn lngth and vlcity frms f th prsnt scillatr strngths as a functin f lngth frm f prsnt scillatr strngths fr sm E1 transitins has bn shwn Transitin Prbabilitis fr (E2 and M1) Frbiddn Transitins In this part, w hav calculatd transitin prbabilitis fr frbiddn E2 and M1 transitins and prsntd in Tabl V. Th prsnt valus hav bn cmpard with Kllhr and Pdbdva [9] and with Frs Fischr and Tachiv [13]. Th agrmnt btwn th prsnt valus and th tw calculatins is xcllnt, with xcptin fr n transitin 3 P 1 2p 4 1 S 0.

50 41 TABLE V. Cmparisn f prsnt transitin prbabilitis (s -1 ) fr th frbiddn transitins in Mg VII with CFF and DEK rsults. Transitin Prsnt [CFF] [DEK] Initial Lvl Final Lvl g i g k Typ Al Al Al 1 5 E2 2.17E E E E2 4.45E E E M1 2.41E E E M1 7.83E E E E2 1.29E E E E2 3.68E E E E2 2.05E E E M1 1.24E E E M1 3.25E E E+00 1 S 5 1 E2 3.89E E E-02 1 S 3 1 M1 3.72E E E+01 1 S 5 1 E2 4.24E E E+00 2p 4 2p E2 1.47E p 4 2p E2 1.13E p 4 2p M1 3.36E p 4 2p M1 1.91E p 4 2p E2 3.43E

51 42 TABLE V. (Cntinud). Transitin Prsnt [CFF] [DEK] Initial Lvl Final Lvl g i g k Typ Al Al Al 2p 4 2p E2 1.20E p 4 2p 4 2p 4 2p 4 2p 4 2p E2 1.36E p M1 6.67E p 4 1 S 5 1 E2 4.26E p 4 1 S 3 1 M1 4.73E p 4 1 S 5 1 E2 1.00E p E2 6.94E E+03 2p E2 3.47E E+04 2p E2 1.54E E+04 2p E2 2.59E E+04 2p E2 8.74E p E2 1.19E p M1 4.63E E-01 2p M1 1.61E E+00 2p M1 4.83E p M1 1.56E E+00 2p M1 2.34E E+00 2p M1 1.26E

52 43 TABLE V. (Cntinud). Transitin Prsnt [CFF] [DEK] Initial Lvl Final Lvl g i g k Typ Al Al Al 2p E2 3.73E E-01 1 S 1 S 1 S 2p E2 2.71E E+01 2p M1 2.31E p M1 6.97E E-01 2p E2 1.94E p 4 1 S 5 1 E2 1.27E E+01 2p 4 1 S 3 1 M1 4.04E E-01 2p E2 2.75E E-02 2p M1 5.98E E-01 2p E2 4.02E p E2 4.95E E+00 2p E2 2.66E E+01 2p E2 4.53E p 4 1 S 5 1 E2 5.83E E+04 [CFF] Frs Fischr and Tachiv [13]. [DEK] Kllhr and Pdbdva [9].

53 CHAPTER 4 CONCLUSION W hav prsntd ur calculatins f nrgy lvls, liftims, and scillatr strngths fr th dipl allwd transitins and frbiddn transitins f Mg VII arising frm th (1s 2 ),, 2p 4, 2s 2 2p3s, 2s 2 2p3p, 2s2p 2 ( 4 P)3s and cnfiguratins. Th prsnt nrgy rsults hav bn cmpard with th rsults frm th NIST cmpilatin [10], th SS prgram calculatin by Bhatia and Dschk [11], th CIV3 cd calculatin by Aggarwal [12], MCHF calculatin by Frs Fischr and Tachiv [13], and th GRASP prgram calculatin by Zhang and Sampsn [14]. Th prsnt liftims rsults hav bn cmpard with Lstinsky t al. [8] and Frs Fischr and Tachiv [13]. Th prsnt rsults f scillatr strngths and transitin prbabilitis cmpard with Kllhr and Pdbdva [9] and Frs Fischr and Tachiv [5,13]. Cmparisns btwn nrgy lvls calculatd by th diffrnt atmic mthds indicat that th agrmnt with NIST rsults is satisfactry, with th xcptin f fw nrgy lvls. Unfrtunatly, svral nrgy lvls ar nt listd in th NIST atmic databas. Th cmparativ studis f atmic structur data validat ur rsults. Transitin prbabilitis and scillatr strngths fr transitins btwn th (1s 2 ),, 2p 4, 2s 2 2p3s, and cnfiguratins hav bn cmputd by many scintists. W hav cmpard ur rsults fr bth allwd (E1) and frbiddn (M1 44

54 45 and E2) typs f transitins. Th cmparisn btwn th prsnt valus and th availabl thr tw calculatins displays vry gd agrmnt fr mst f th transitins. Th diffrncs btwn th prsnt rsults and thr calculatins ar f th rdr f 10% r bttr fr strng transitins.

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