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ASYMMETRIES IN PLASMA LINE BROADENING BY ROBERT FOSTER JOYCE A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1986 ACKNOWLEDGEMENTS I would like to thank Dr. C. F. Hooper, Jr., for suggesting this problem and for his guidance during the course of this work. I would also like to thank Dr. J. W. Dufty, Dr. L. A. Woltz and Dr. C. A. Iglesias for many helpful discussions and Dr. R. L. Coldwell for his computational expertise. Finally, I would like to thank my parents, Col. Jean K. Joyce and Dorothy F. Joyce, for their continued support and encouragement during the course of this work. TABLE OF CONTENTS Page ACKNOWLEDGEMENTS ...........................................ii LIST OF FIGURES.................................................... iv ABSTRACT............................. ......................................vii CHAPTER I. IONQUADRUPOLE INTERACTION............. ......................1 A. Formalism with lonQuadrupole Interaction............. B. Calculation of the Constrained Plasma Average........21 C. Trends and Results...................................24 II. OTHER ASYMMETRY EFFECTS..................................... 34 A. Fine Structure........................ ............. 34 B. Quadratic Stark Effect..............................46 III. RESULTS..................................................... 62 IV. SPECTRAL LINES FROM HELIUMLIKE IONS........................73 APPENDICES A. ZWANZIG PROJECTION OPERATOR TECHNIQUES......................83 B. PROOF THAT THE OCTAPOLE EFFECT IS SYMMETRIC TO FIRST ORDER IN PERTURBATION THEORY.................................90 C. CALCULATION OF Qzz...................................... 94 D. THE INDEPENDENT PERTURBER MICROFIELD.......................99 E. CALCULATION OF AN IMPORTANT INTEGRAL.......................102 F. NUMERICAL PROCEDURES ...................................... 103 G. ELECTRIC FIELD BEHAVIOR OF THE ENERGY LEVELS OF HELIUMLIKE IONS ..........................................108 REFERENCES .......................................... .......... 112 BIOGRAPHICAL SKETCH ............................ .................. 114 iii LIST OF FIGURES FIGURE PAGE 1 Energy levels of the n2 states of hydrogenic ions in the presence of a uniform electric field and in the presence of a field gradient........................................... 4 2 Lya lines of Ar+17, with and without the ionquadrupole interaction, at a temperature of 800 eV. (a) Electron density 10 3 cT3  (b) Electron density 3 x 10 cm3 (c) Electron density 10 cm 3.............................25 3 Ly$ lines of Ar+17, with and without the ionquadrupole interaction, at a temperat e of 800 eV. (a) Electron density = 10 3 c 3 (b) Electron density = 3 x 10 cm (c) Electron density 1024 cm3............................28 4 LyY lines of Ar17, with and without the ionquadrupole interaction, at a temperat re of 800 eV. (a) Electron density 10 3 c3  (b) Electron density 3 x 10 cm (c) Electron density 102 cm3 ............................30 5 Ly8 line of Ne+9 at an electron density of 3 x 1023 cm3 and a temperature of 800 eV with and without the ionquadrupole interaction...................................32 6 Lya lines of Ar17, with and without fine structure, at a temperature of 800 eV (a) Electron density 10 3 c3 3 (b) Electron density = 3 x 10 cm (c) Electron density 1024 cm3.........................38 7 Ly8 lines of Ar17, with and without fine structure at a temperature of 800 eV (a) Electron density 103 3 3 (b) Electron density 3 x 10 cm (c) Electron density 1024 cm3.............................40 8 LyY lines of Ar+17, with and without fine structure at a temperature of 800 eV (a) Electron density 102 3 c3 (b) Electron density 3 x 10 cm (c) Electron density = 1023 cm3...a........................42 9 Ly8 line of Ne+9 at an electron density of 3 x 1023 and a temperature of 800 eV, with and without fine structure..................................................... 44 10 Lya of Ar+17, with and without the quadratic Stark effect, at a temperature of 800 V. (a) Electron density 10 cm3  (b) Electron density 3 x 10 cm (c) Electron density 1024 cm3............. ....... .....49 11 Ly8 of Ar+17, with and without the quadratic Stark effect, at a temperature of 800 eV. (a) Electron density = 10 c3 (b) Electron density = 3 x 10 cm (c) Electron density 1024 cm3........................ ...51 12 LyY of Ar+17, with and without the quadratic Stark effect, at a temperature of 800 eV. (a) Electron density 10 c3 e5 (b) Electron density 3 x 10 cm (c) Electron density 1024 cm3.............................53 13 Ly8 of Ne+9 at an electron density of 3 x 1023 cm3 and a temperature of 800 eV, with and without the quadratic Stark effect ................................... .......... 55 14 J(w,E) of a Lya line for various values of E .................57 15 Energy levels of the n2 states of hydrogen as a function of perturbing field strength................................ 60 16 Lya lines of effects, at a (a) Electron (b) Electron (c) Electron 17 LyB lines of effects, at a (a) Electron (b) Electron (c) Electron 18 LyY lines of effects, at a (a) Electron (b) Electron (c) Electron Ar+17, with and without the three asymmetry temperature of 800 eV. density 10 c3 3 density 3 x 10 cm density 10 4 cm 3 ........ ..... ..............63 Ar+17, with and without the three asymmetry temperature fC 800 eV. density 10" cm3 3 density 3 x 10 cm density 102 cm3 .............................65 Ar 17, with and without the three asymmetry temperature o 800 eV. density 103 c3 3 density 3 x 10 3cm density 102 cm3 ............. ............. 67 19 Asymmetry of LyB, defined as 2(IbI )/(Ib +I), as a function of density...........................................70 20 He uml1ke aline of Ar+17 at an electron density of 10 cm and a temperature of 800 eV........................77 21 He iuml ke Sline of Ar+17 at an electron density of 10 cm and a temperature of 800 eV.........................79 22 He iuml ke Yline of Ar+17 at an electron density of 10 3cm and a temperature of 800 eV.........................81 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy ASYMMETRIES IN PLASMA LINE BROADENING Robert Foster Joyce May 1986 Chairman: Charles F. Hooper, Jr. Major Department: Physics Spectral lines have long been used as a density diagnostic in dense plasmas. Early theories produced hydrogenic spectral lines that were symmetric. At higher plasma densities diagnostics could be improved if asymmetries were included. In this work, several sources of line asymmetry are considered. In Chapter One, the ion quadrupole interaction is studied. This is the interaction between the radiator quadrupole tensor and the electric field gradients of the perturbing ions. The field gradients are calculated in an Independent Perturber Model. Two other asymmetry sources, fine structure splitting and quadratic Stark effect, are considered in Chapter Two. Results of the combination of these effect follow in Chapter Three. Nonhydrogenic spectral lines are also asymmetric. In Chapter Four, a procedure for calculating the special lines from Heliumlike ions is outlined. vii CHAPTER I ION4QUADRUPOLE INTERACTION A. Formalism with lonQuadrupole Interaction In this chapter, we will consider the effect on spectral line shapes of the interaction of the radiator quadrupole tensor with the electric field gradient due to perturbing ions. The form of this interaction is derived naturally from a multiple expansion of the radiator as a localized charge distribution p(x) in an external potential O(x) caused by the perturbing ions (Jackson 1972). The electrostatic energy of the system is W = fp(x)O()d3x. (I.A.1) The potential can be expanded in a Taylor series with the origin taken as the radiator center of mass. 1 x 2(O) (x) (0) + V(O) + I xx + ... (I.A.2) ij 1 j ia Div(E(O) = 4ZrZ e6(x ), where the sum is over all perturbing ions and zie i is the charge of ion "i". Since Coulombic repulsion will keep the perturbing ions from reaching the origin, (1/6)r2Div(E(0)) can be subtracted, giving aE (x) = (0) xE(0) I (x ) (0) + ... (I.A.3) ij j or, aE 1 1 W qj(0) d.E(O) i ( Q E (0) + ..., (I.A.4) ij j where q is the total charge of the radiator, d = QiJ = the radiator. This multiple expansion is basically an expansion in the atomic radius divided by the average interlon spacing, or (n2 a/Z)/Ri where n is the principle quantum number of the radiator, a0 is the Bohr radius, Z is the radiator nuclear charge and Ri0 is the average distance between ions (Sholin 1969). The dipole interaction is second order in this parameter, the quadrupole interaction is third order, etc. For a LyB line emitted from an Ar+17 plasma with an electron density of 1024 cm"3, this smallness parameter is .15. To understand qualitatively the asymmetry due to this effect, consider the perturbation on the n2 levels of a hydrogen atom located at the origin, by a single ion of charge +1 located at z. If we initially consider the perturbation upon the isolated hydrogenic energy levels to be due only to the electric field produced by the ion, the new states are, to first order in perturbation theory, the wellknown Stark states with a perturbing field Ee/z2. Now, if we take these as our new basis states and include the effect of the field gradients due to the I perturbating ion, the result is shown in Figure 1. The higher energy levels, which give rise to the blue wing intensity, (w > 0w, where hw0 is the energy difference between the unperturbed initial and final state) are brought in closer while the lower energy levels, corresponding to the red wing (w < w0), are spread further out. These shifts, when averaged over all possible perturber positions, will lead to a blue asymmetry for Lya, at least out to the wings. Similar arguments can be made for higher Lyman series lines. Now that we have some idea of what to expect from the quadrupole interaction, we proceed with the general theoretical formulation (Griem 1974). The power spectrum emitted by one type of ion in a plasma can be written in terms of the ensembleaveraged radiation emitted by one ion of that type as P(w) 3 1 I 3c3 ab 3 where pa is the probability that the system is in state la>, d is the radiator dipole moment, and Wab (E Eb)/h. The second equation defines the line shape function I(w). Using the integral definition for the Dirac delta function, we obtain, S "lbl2 i(u`hab)t (w) I 2* ab a Note that the integrand for negative values of t is equal to the complex conjugate of the integrand for positive values of t. This enables us to FIGURE 1 Energy levels of the n2 states of hydrogenic ions in the presence of a uniform electric field and in the presence of a field gradient. A2 IA2 Al \A2 APPLY APPLY ELECTRIC FIELD FIELD GRADIENT 6 write I(w) as an integral from zero to infinity: 1 i2ww ta (w) Re f I 0 aba = Re f et 0 ab S dt eiwt + iLt( SRe f dt e Tr(dpe d). (I.A.7) 0 L and p are the Liouville and equilibrium density matrix operators for the radiatorplasma system, respectively. The trace is over states of the radiator and plasma. Thus far, Doppler broadening has been neglected. We will include this approximately at the end of the calculation by convoluting the Stark line shape with a Doppler profile derived using a Maxwell velocity distribution. This approximation assumes that the change in momentum of the radiator, due to the emission of a photon, is negligible and therefore the Doppler and Stark broadening are independent. Rather than immediately factoring the density matrix, we follow the general formalism of Iglesias (1984). This will show, in a more natural way, how the electron screening of ion interactions comes about. We consider the fullCoulomb radiatorplasma interaction, V ap which is given by a sum of pairwise additive terms. Furthermore, we will treat a two component plasma, which consists of electrons and only one type of ion, hydrogenic ions of nuclear charge Ze. The Coulomb potential between the radiator and the perturbing ion can then be written as N N 2 2 a a Z Ze Z e V 1 v (a,j)= I [ o a , ap j=1 a Sa J (I.A.8) where a signifies perturber type (in this case, electrons and one type of ion); N and Z e are the number and charge of perturbers of type o ao o respectively. Expanding this expression in Legendre polynomials, we find, where CO r< vo(a,j) = Z e [ Z S=0 r> 6 o]P(cose a), r< smaller of Ira and Ir r = larger of r al and Ir j 6 angle between r and r.. aj a J This expression may be regrouped as v(a,j) 00(j) + v1 (a,j) + v2a(a,j), (I.A.9) (I.A.10) where oa(j) is the monopole term given by Z (Z1)e /r, 0 v1 (a,j) Z e2 a + P (cosaj), 1+ aj 2i r. jo and 2 r. r Z e2 j [0 a ]P(coseaj) r 5 r a 1+1 1+1 1 aj a v2 (a,J) a 0 rj > ra 0 0 0 The sum of v + 0 is identical to v in the region r > ra the no penetration region, but extends this form to all values of rj. In this form, v10 may be written as a "product" of an operator on perturber coordinates times an operator on radiator coordinates, in the following manner, v10(a,j) = M(a)0(j) = UK(a)oK (J) K=1 (.(j) + 1 X Qmnmn(j) + ...). (I.A.11) mn Here M(a) stands for the set {IK(a)l which depends only on the radiator state. For example, U a, which is the radiator dipole moment, and Y2 = /6, which is the radiator quadrupole moment. The perturber operator, 0a(j), depends only on the jth perturber of type a. The term ao(j) represents the set 14kk(j)}, where 1 (j) = Eo(j) is the electric field at the radiator produced by the jth perturber of type a, *2 (J) am n(j) is the electric field gradient at the radiator produced by the jth perturber of type o, and so forth. By design, pKo couples to k. We next generalize the usual definition for an electric microfield distribution, W(e) = Tr p(cZ Ei), to include the field gradients and i higher order derivatives which are used in equation (I.A.11): W(T) = Tr p6(y *), (I.A.12) where 6(T0 ) = I 6(' 0 ). K K K K=1 The YK and QK are analagous to ik in equation (I.A.11). That is, they denote the value of the electric field, field gradient tensor, etc. at the radiator, produced by the perturbers. In the above, 4 is an, as yet, arbitrary function of only ion coordinates and Y is a cnumber. Now equation (I.A.7) can be written as 1 r iwt I(w) Re f dt et Tr{d Tr pjd'6(V *)d(t)} 0 a p fdV W(T) 1Re J dt eit Tr{d.Tr p6(* r)/W(Y)a(t)} 0 a p fdy W(Y)J(w,,), (I.A.13) where 10 J(w,') 1 Re dt eit Tri .Tr p(4)d(t)} 0 a p p() P p6('4 )/w('). We may rewrite J(w,v) as (I.A. 14) (I.A.15) J(w,Y) Re fdt eit Tr d.f(a,)D(t), 0 a where D(t) P f(a,S) Tr p(.). p P The Hamiltonian associated with the Liouville operator can be written as the sum of a free atom Hamiltonian, an isolated plasma Hamiltonian, and an interaction term. Noting equations (I.A.8)(I.A.11) for the interaction, we have a H H + H + ~ X v(a,j) a p a j1 H +H + a p o 40(J) + v2 (a,j) + M(a)O(j). j1 The atomic Hamiltonian, Ha, includes the center of mass motion of the (I.A.16) atom and the internal degrees of freedom. The plasma Hamiltonian, Hp, includes the kinetic energy of the perturbing electrons and ions as well as their Coulombic interactions. We now make the static ion approximation, which can be expressed as eiLt e p(d) p(C). This property, stationarity, implies that the kinetic energy of the perturbing ions plays no role in the line shape problem and may be integrated out. Therefore, the delta function in p() allows us to make the replacement 0 0 + T 4 in H, giving N o * H H + M(a)Y + H + 0 () + 0 v 2(a,j) + M(a)(I I t'(j)_) a p J o a j1 H + H + V2 + V ( ), (I.A.17) where H = Ha + M(a)T, p Hp + L oI 1 0(J), ap p oJ V2 I I vo(a,j), and I * V1(4 ) M(a)( I *O(j) ). oj 12 We have combined Ha and M(a)Y because these involve only atomic coordinates. Similarly, the monopole term, 0 1 40(j), does not depend oj on internal atomic coordinates and therefore is combined with Hp. Let L(a,T), L L2, and L 1( ) be the Liouville operators corresponding to Ha, p V2, and V1( ) respectively. Further let a L L I( ) = L1($ ) + L2 be the Liouville operator corresponding to the radiatorplasma interaction. Now use the Zwanzig projection operator technique to derive a resolvent expression for J(w,T). The details are in Appendix A and the result is J(w,') =  Im Tr{d f(a,')[wL(a,')B('i), (w,')]1da, (I.A.18) a where B() / (,,) f (a,')Tr L (0 )p(*)(wQLQ) QL ( ), p and Q = 1P, where P is a projection operator given by P(...) = <(...)>. We wish to cast the expression for I(w) in a more familiar form, one which uses the standard microfield. To this end, we must separate out the electric field dependence from '. Let {E,Y'}, that is, V' is all of except for c. Now, 13 W(Y) = Q(E)W(cI'() (I.A.19) where Q(E) is the usual microfield (Tighe 1977; Hooper 1968) and W(EI) is the conditional probability density function for finding Y' given c. Then, I(M) = fd Q(E)J(w,C) (I.A.20) which is the desired form, but now J(W,c) = fdT' W(Y'")J(w,'). (I.A.21) At this point, some simplifications must be made in order to arrive at a calculable scheme. 1. Neglect ion penetration. With this approximation, the ion interaction can be treated entirely in the microfield fashion above. Error is only introduced for configurations in which a perturbing ion is closer to the radiator than the radiator electron is. This is extremely rare due to the large Coulombic repulsion of these highly charged ions. 2. Choose 0 which has been arbitrary up to this point, to force  .L4 e i * 0 Tr p(,)(,e + i _ ) p Tr 6(* *){Tr p(e + 1) *)/W(), (I.A.22) i e where p Tr p. e To satisfy the above, it is sufficient to define 1 e i 0 pi Tr p( e + ) e Si + pi Tr pe. (I.A.23) e Obviously, 4 is equal to the pure ion term plus a shielding term due to the electrons. This is not an approximation. 3. Assume that the entire effect of the electronion interaction is to screen the ion interaction, 4 in the manner above. With this approximation, the ion parts of V 1( ) cancel, leaving e V ( ) M(a) e 4e(j). To lowest order in the plasma j1 parameter, this screening is given by the DebyeHuckel interaction (Dufty and Iglesias 1983) and that form will be assumed here. (Even if the DebyeHuckel form were not used, the procedure would still be applicable as long as the potential is a a the sum of single particle terms.) Note that V1(0 ) and LI(* ) no longer depend on 4 ; therefore they will henceforth be denoted simply V1 and LI, respectively. Neglect of electronion 15 interactions also allows the density matrix to be factored. Thus f(a,V) can be reduced to Tr p.6(*4 )Tr pae 1 e f(a,*) i e Tr p 6(*4 )Tr Pae i ae Tr Pae e a f(a). (I.A.24) Tr pae ae 4. For the plasma conditions and Lyman lines which we are considering, most of the line is within the electron plasma frequency; therefore the electron broadening is primarily due to weak electron collisions and a treatment of V(w,~,) which is second order in LI is appropriate (Smith, Cooper, and Vidal 1969). This is accomplished by replacing the L in /(mw,) by LO. Further simplification concerning the projection operators is given in Appendix A, with the result (m(.i) f(a)Tr L p(*)(wLO) !L (I.A.25) p Note that the LO in the last equation has an ion part, however this ion Liouville operator acts only on functions of radiator and electron coordinates and therefore contributes nothing. The trace over ions is now trivial, leaving 16 1  V(wp) f (a)Tr Llp (wL(a,Yi)L ) LI e f1 (a)Tr I L (j)pae(mL(a,*) I Le(k))~1 LI(), e j k (I.A.26) where the Liouville operators have been written as sums of single electron operators. Terms with jOi are zero by angular average and terms with ku are zero because L (k) is zero unless acting on functions of electron k. 1 1 Y(w,() f (a)n Tr L ,1(a,1)f(a,1)(L(a,*) Le(1)) L (a,1), 1 e where nf(a,1) = Tr Pae (I.A.27) e2...eN 5. The No Lower State Broadening Assumption states that there is no broadening of the final radiator state by perturbing electrons. This is a very good approximation for Lyman series lines because the final state (the ground state) has no dipole interaction with the plasma. 6. The No Quenching Approximation assumes that electron collins do not cause the radiator to make nonradiative transitions between states of different principal quantum number. This approximation is best for low level Lyman series lines since the levels are well separated. Although approximations (5) and (6) are not necessary, they are made here for the sake of numerical ease. These reduce I (W) to (Tighe 1977) 17 A() in Trf dtLI e ,,eiH(1)t/ h 1 0 i" SVii'eH(1)t/f(1), (I.A.28) where the trace is over states of a single electron, H(1) is the Hamiltonian for that single electron including its kinetic energy and monopole interaction with the radiator, Am = w (EaE )/f and subscripts i (f) stand for initial (final) radiator states. 7. Equation (I.A.27) includes the ion shift of radiator states in '/(w,') through L(a,'). Dufty and Boercker (1976) found that 9 (w,Y') is fairly constant within the electron plasma frequency. For this reason, we will neglect the ion shift in the electron broadening operator by replacing L(a,Y) by L(a). 8. The real part of (w) is called the dynamic shift operator. It has been calculated by Woltz (1982) and found to give a very small red shift. It will be ignored here. Most of these approximations are standard in calculable line broadening theories. In fact, if we were now to neglect completely, replacing L(a,') by L(a,E) and J(w,') by J(w,e), this reduces to the full Coulomb (electron interactions) of Woltz et al. (1982). Instead, limit F' to the quadrupole interaction. Higher order terms should be successively smaller in magnitude (Sholin 1969) and the next higher term, the octapole term, is a symmetric effect (Appendix B.) to first order in 18 perturbation theory and therefore should not have a large effect on the line shape. This yields ) d dxxyydzzdxydxz dyzQ() 0 W('IExx yy,...)J(W,E,E xxyy,...), (I.A.29) where ac  ii ax. We now make the following approximation W(IEXXEyy,...) 6(VXX That is, the field gradients are replaced by their constrained averages. This form is exact in the nearest neighbor limit. It is, in general, exact to linear order in 2 2 errors which are proportional to ij This yields I(w) = fdE Q(4)J(WC), with J(W,c) W Im Tr{d.f(a)(wL(a,c, a 19 The Hamiltonian corresponding to L(a,, H aP* Qij /6 where p is the radiator dipole operator and Q is ij the radiator quadrupole operator. By taking advantage of the fact that the ground state has no dipole or quadrupole moments, we can rewrite the expression for J(w,e) as J(w,e) Im Tr D{w(H ) p Q < >/6 SJ(r ,E)0 ij r ij (I.A.32) where 0 is the radiator ground state energy and D is given by dd, and the radiator dipole operator, d, is restricted to have nonzero matrix elements only between initial and final states of the Lyman line to be calculated. The DebyeHuckel interaction was chosen for i Therefore, SE (Ze) x er/ 1 3 ) (I.A.33) x z 2 2 2 r3 r Ar \r r and r/A[O 1 1 2( 3 1 3)] B3E (Ze)e  [(r ) 2( +  + 3)] (I.A.34) r Xr r 2r r 3 where v cos(8) and A is the screening length. All offdiagonal terms are of the form of equation (I.A.33) and all diagonal terms are of the form of equation (I.A.34). Since only the z direction is special (c is along z), xy c xz E yz c LU SQij ij Ei Q xx xx xx e yy yye ZZ zz e (Q + Q ) xx yy xx e zz zz e Q 3 Q ( 2 zz e ZE z3 For the DebyeHuckel form, 1 nA V*E I (Ze) e /r. 2 (I.A.35) (I.A.36) r/A 2 1 e r r 2 zz 3 E 3 X 32 E Now J(w,c) may be written J(W,E) Im Tr Dw(Hrw0) P Qz (I.A.37) (I.A.38) The quadrupole moment Qzz is easily calculated as in Appendix C. All that is needed now is to calculate the constrained plasma average B. Calculation of the Constrained Plasma Average From equation (I.A.33), the constrained plasma average can be written C zz 3 E r/X 2 SZe r3 3 2 2 Obviously, f can be written as a sum of single ion terms, f = Ze I F(r)(13p2), (I.B.2) ion where r/A 2 e r 1r F(r) = (1 + + ) 3 A 3 2 Therefore, we have Jd3Nr (f f )6(E"E)e fd r 6(4E)e8V S r 3N + BV Njd3r f 6E)e *' 1 _____ (I.B.3) ZN where Q(e) is a microfield and ZN is the partition function for this system. By separating out the integral over perturber number one, we obtain LL N 3 3(N1) N O.B. ZNQ(E) 1=2 Within the square brackets, V can be considered to be an N1 particle potential. Then [ ] has the form of a microfield for this potential, times its partition function,ZN' In symbols, [ ] = Q'(EE1)Z 1 Note that ZN1 g() (I.B.5) zN " where g(r) is the radial distribution function and R is the system volume. Now C where n, is the ion density. The Independent Perturber Model (Iglesias et al. 1983) is used for Q' and Q. See Appendix D. In this model nh (k) Q() =  J dk k sin(kE) e 2w c 0 where h1(k) 4r 0 dr r2 e (j(kE(r))1). (I.B.7) Therefore, Zeni k2e hl (k)  S 2irQ(C) 0 0  . where the integral over angles, I is given by (I.B.9) This integral is calculated in Appendix E with the result I = 8wr j2(ke)j2(ke1). Substituting this into equation (I.B.8) yields 0 2 nh (k) ke j2(kc)f dr o r2g(r)F(r)j2(k 1(r)) J dk k e sin (k) (I.B.11) This can be expressed in terms of dimensionless variables as 6 dk k 2 1 nhl(k)j2 (kE) dx 0 0e nhl(k) Sdk f dk k e ax e ( g(x) (1 x 22 ax + ax + )j (J 3 2 1 sin(ke) (I.B.12) where (I.B.10) e E  Jd 32jo2 kl ). I fdQ(13P ) 0 (k1= 1 ). 24 E is scaled in terms of EO = e/rO 2 4 0r, n 1, 1 k is scaled in terms of E 3 f is scaled in terms of e/r 0 a = r /) and x r/rO. Numerical procedures are given in Appendix F. We used the DebyeHuckel radial distribution function, g(r), in this calculation. In order to estimate the error introduced in this way, we ran one case (ne  1024 cm3, T 800 eV) using a Monte Carlo generated radial distribution function. At the value of e corresponding to the peak of the microfield, the difference in zz E C. Trends and Results As seen at the conclusion of Appendix B, the quadrupole interaction gives rise to an asymmetric effect, while the dipole interaction is symmetric. A measure of the relative importance of the quadrupole interaction on the line broadening problem is given by the ratio of the quadrupole interaction to the dipole interaction. This is the smallness 2 2 1/3 4/3 parameter mentioned earlier, (n a0/Z)/Ri n ne /Z . That is, the asymmetry due to the ionquadrupole interaction will increase with principal quantum number and with density but will decrease with radiator nuclear charge. These trends can be seen in the calculated profiles. Figures 2a, 2b, and 2c show the Lya lines of hydrogenic Argon, with and without the quadrupole interaction, at densities of 1023, 3 x 1023, and 1024 electrons per cubic centimeter and FIGURE 2 Lya lines of Ar+17, with and without the ionquadrupole interaction, at a temperature of 800 eV. a) Elec on density 1023 cm 3; b) Electron density 3 x 10 cm 3 c) Electron density 10 cm3. /o u.um i 0.80 0.67 0.53 ( 0.400.270.13 000 0.13 0.27 0. 40 0.53 0.67 0.80 Q80 067 053 040 027 0.27 0.40 0.53 0.67 0.80 I 0.60 0.50 I 0.40  I t 0.30 I 0.20  O.10  000 i 0.80 0.670530.40 027 0.13 0.00 0.13 0.27 0.40 0.53 0.67 080 DELTA OMEGA (RYD) a temperature of 800 eV. These figures show a blue asymmetry (as predicted in section I.A) which increases with density. Figures 3 and 4 show the LyB and LyY lines at the same densities. Again the increase in asymmetry with density is observed. By comparing Figures 24 one can also notice the increase in asymmetry with increasing initial principal quantum number. Figure 5 shows the LyB line of hydrogenic Neon at an electron density of 3 x 1023. Comparison with Figure 3b demonstrates the expected decrease in asymmetry with increasing radiator nuclear charge. The quadrupole interaction in line broadening is the major thrust of this work, however, before comparison with experiment can be made, other sources of asymmetry must be investigated. FIGURE 3 Ly8 lines of Ar+17, with and without the interaction, at a temperature of 800 eV. density 1023 cm ; b) Electron density c) Electron density 10 cm3. ionquadrupole a) Electon 3  3 x 10 cm ; 1.20100 0.800.600.400.20 0.00 0.20 0.40 0.60 0.80 1.00 1.20 1.20 1.60 2.00 2.40 2.00 2.50 300 DELTA OMEGA (RYD) 0.60 0.50 0.40  0.30 n FIGURE 4 LyY lines of Ar+17, with and without the lonquadrupole interaction, at a temperature of 800 eV. a) Elect on density 1023 cm3; b) Eectron density a 3 x 10 cm ; c) Electron density 10 cm. 0.60 0.50 0.40 0.30 0.20 0.10 I 0.50 I I 0.40 0.30 / 020 0.10 000 4.804.00 3.20 2.40I.600.80 0.00 0.80 1.60 2.40 120 4.00 4.80 It 0.60 \ 0.50 \ 0.40 \ 0.30  020 0.10 .005.004.00 300 20 1.00 0.00 10 2.00 100 4.00 5.00 600 DELTA OMEGA (RYD) V x 0 e DI C L ct 0 0 L M 4 * 4 0Oo 0 ) 0) 0 OC e3 C > tO0 2 0) 0 3C >* C ~~,3 0 0 0 0 (m)I o o 0 8 cJ 0 (.D O 0 o 8C; 'o 00 d 0 < 8. 8 N 0 NO It I \ ^1 CHAPTER II OTHER ASYMMETRY EFFECTS A. Fine Structure In this chapter the effects of fine structure and quadratic Stark effect, as presented by Woltz and Hooper (1985), are discussed. As a starting point, we take equation (I.A.38). The quadrupole interaction may be included or not by retaining or omitting the Qzz term. In this chapter we will be interested in the characteristic features of the other asymmetry effects and so will not include the quadrupole interaction. In Chapter III, all three effects will be considered together. First we consider the fine structure effects. The Hamiltonian, Hp, of equation (I.A.38) is often taken to be H 2V/2V Ze2/r (II.A.1) r where y is the reduced mass of the electron and nucleus. If the Schrodinger equation is solved with this Hamiltonian the eigenstates are the usual Inam> states where n is the principal quantum number, I is the orbital quantum number, and m is the magnetic quantum number. The eigenvalues, given by 4 2 r A2h n are, for a given principal quantum number, all degerate. Consider the states with a given principal quantum number. If a uniform electric field 34 35 is applied, the new energy levels can be derived from the linear Stark effect, which is the result of first order degenerate perturbation theory. Some of the states are raised in energy and others are lowered, maintaining the symmetry about the original energy level. This symmetry manifests itself in a symmetric line profile. If the original Hamiltonian did not give rise to completely degenerate states, an asymmetry would be introduced. The degeneracy is actually lifted by two effects neglected in equation (II.A.1), the relativistic effects and the spinorbit interaction. These effects are of the same order of magnitude and together are known as fine structure. The exact solution of the Dirac formula gives a result which is diagonal in the representation Inljj >. The result is (Bethe and Salpeter 1977), n(J+1/2) + J(j+1/2)2 Z( where a is the fine structure constant (1/137), and j is the quantum number associated with the total angular momentum. Little error is introduced in expanding this result to first order in (aZ) 4 2 n j+1/2 2h n The first term is just the result given in equation (II.A.2) and the second term is the fine structure correction. The Dirac theory does not contain radiative corrections, but these are of order aln(a) smaller than the last term kept above. If we calculate J(w,e) using the basis Intm m >, we have J(W,E) = Im I m'ms 'm 'm 'm EIs SIm Tr D() R( IT (II.A.5) If we neglect nonradiative transtitions between states of different principal quantum number and consider only the linear Stark effect, the matrix R(n) is obtained by inverting the matrix having elements (II.A.6) The operator ww0. (0w) is usually calculated in an Inim > representation. The only difference here is that it will have twice as many rows and columns (for the two values of ms), with the elements being equal to zero if ms m '. Equation (II.A.3) gives Hr in the representation Inljjz>. It is a simple matter to transform to the Inim ms> basis using the unitary transformation +1/2jz . (2j+1)1/2 ( A 1/2 m s j ) (II.A.7) 0 The symbol m1 2 m3 m1 m2 m3 is the Wigner 3j symbol and is easily calculated (Edmonds 1957). We are now able to calculate lines with fine structure. For convenience, we take the energy level of the state with the largest value of j to be our zero reference point. Figures 6a, 6b, and 6c show the Lya lines of hydrogenic Argon, with and without fine structure, at densities of 1023, 3 x 102, and 102 electrons per cubic centimeter and a temperature of 800 eV. Figures 7 and 8 give the same cases for the LyB and LyY lines. Figure 9 shows a LyB line of Neon at a density of 3 x 1023 electrons per cubic centimeter. From equation (II.A.4), we see that the fine structure correction, and therefore the asymmetry of the line shape, increases with nuclear charge, Z, but decreases with increasing principal quantum number, n. Comparison of Figure 9 and Figure 8b shows the expected dependence on nuclear charge and a comparison of the appropriate members of Figures 68 shows decreasing asymmetry with increasing principal quantum number. The decrease in asymmetry with increasing density (see Figure 6, 7, or 8) is also to be expected. As the density increases, so does the average ion field perturbation of the energy levels. The fine structure correction is purely a result of atomic Physics and does not change with density. At low densities, the fine structure correction is larger than the average ion field perturbation and therefore a large asymmetry exists. At high densities the reverse is true and the line is more symmetric. FIGURE 6 Lya lines of Ar+17, with and without fine structte, at a temperature of 800 eV. a) E "ctro density = 10 cm ; b) Electron density = 3 x 10 cm ; c) Electron density . 10 cm 3. 0.27 0.40 0.53 '0670.53 040 027 013 Q0 0.13 0.27 0.40 0.53 00270.13 QO Q13 0.27 Q40 0.53 0.67 0.80 DELTA OMEGA (RYD) 0.67053 FIGURE 7 LyB lines of Ar+17, with and without fine structure at a temperature of 800 eV. a) Elctro2 density 1023 cm' ; b) electron density 3 x 10 cm ; c) Electron density 10 cm3. 41 1.000.80 060 040 0.20 000 020 0.40 060 0.80 1.00 1.20 DELTA OMEGA (RYD) FIGURE 8 LyY lines of Ar 17, with and without fine structure at a temperature of 800 eV. a) Elctroi density = 1023 cm3; b) ectron density = 3 x 10 cm ; c) Electron density = 10 em3cm I 0.60 0.50 040 030 020 0.10 2.00 1.50 1.00 0.50 000 0.50 100 1.50 2.00 2.50 3.00 4.804.00 3202401600.80 0.00 080 160 240 3.20 4.00 480 0.60 0.50 040 0.30 0.20 0.10 .0r5.o4.00oo.02.0 I'0 0.00 1.00 2o00 3. 4oo s'oo 60 DELTA OMEGA (RYD) V CO L x 4 04 oJ4 0 mb , 0) ti 0 04 C 4) + 00 + CO o CO o o !1 0 0 O 0 8 O (0 O O oo O 0 > o 0060 o. o Q I ooc* m tD (nr)i 0 0 0 '+ J B. Quadratic Stark Effect The quadratic Stark effect causes asymmetry in hydrogenic spectra. This effect is one order higher in our smallness parameter, (n a0/Z)/R0 than the quadrupole interaction (Sholin 1969). Consider again the Hamiltonian (II.A.1) for an isolated hydrogenic ion of nuclear charge Z. This Hamiltonian has eigenstates Inlm> and eigenvalues 4 2 Sie Z E nm (II.B.1) ntm 2 2 21' n which are completely degenerate in i and m If a uniform electric field in the z direction is applied, the total Hamiltonian is now H = Hr + H1, where H1 = ezE, E being the magnitude of the electric field. Degenerate perturbation theory provides a systematic procedure for finding the new energy levels. To first order, this procedure amounts to diagonalizing H1, within each block of degenerate states, that is, within each block of states having a given principal quantum number. The results are the usual Stark states Inqm> with energy levels given by E) Higher order corrections are increasingly more difficult to calculate. They involve the mixing of states with different principal quantum numbers and therefore require infinite sums. The sums for the second order correction, the quadratic Stark effect, can be done analytically (Bethe and Salpeter 1977) with the result 47 (2) (1) 1 3 4 2 2 2 2 SE E a () E (17 n3q 9m +19) ( .3) nqm nqm 16 0Z where a0 is the Bohr radius. Consider now, J(W,E) as given by equation (I.A.38), neglecting, for the moment, the quadrupole term. Let us refer to the operator that is to be inverted as the resolvent matrix. If we consider E constant, J(m,e) as a function of w has the form of a Lorentzian, whose peaks correspond to the hydrogenic energy levels for that fixed c. A common procedure is to numerically invert the resolvent within the subspace of initial radiator states (those states with principal quantum number = n). This is clearly equivalent to the linear Stark effect of degenerate perturbation theory, which was outlined above. For this reason, the approximation of tracing only over initial radiator states is known as the linear Stark effect. This will be a good approximation as long as the ion field perturbations are small compared to the separation between states of different principal quantum number. It is not clear how to systematically improve the line broadening theory to include the quadratic Stark effect. We shall outline here two quite different approximate treatments of the quadratic Stark effect in line broadening. These two different approximations give results that are in good agreement, suggesting that both work well for the plasma conditions discussed here. The states that are closest in energy to the states with principal quantum number n, (and therefore most likely to mix with them) are those states with principal quantum number n+1. The first method, then, is to include in the calculation of the resolvent, not only all states with principal quantum number n, but also n+1. The second method uses Stark states Inqm> of only one principal quantum number, but artificially uses the quadratic energy shift (II.B.3) instead of the linear term (II.B.2) for the diagonal matrix elements of Hr + H1. Figure 10a, b, and c show Lyc lines of hydrogenic Argon, with and without the quadratic Stark effect, at densities of 1023, 3 x 1023, and 1024 electrons per cubic centimeter and a temperature of 800 eV. Figures 11 and 12 give the LyB and LyY lines for the same conditions. Figure 13 shows the LyB line of hydrogenic Neon at 3 x 1023 electrons per cubic centimeter. From equation (II.B.3), we see the effect of the quadratic Stark effect should increase with principal quantum number and with density (since increasing the density increases the average ion field) and should decrease with increasing nuclear charge. These trends are born out by the figures. The LymanB line is particularly interesting because it is usually intense enough to be seen when the Lymana line is observable, but it is not as optically thick. The asymmetry of the LymanB induced by the quadratic Stark effect can be explained as follows: In the linear Stark effect, the electric field causes the initial states of the LymanB line to be split symmetrically above and below their unperturbed energy level, the splitting being proportional to the field strength. The quadratic Stark effect slightly lowers the energies of all these states; and the lowering becomes greater as the field strength increases. This gives rise to an increased density of states contributing to the blue peak of the line profile and a decreased density of states contributing to the red peak, causing a blue asymmetry. It is a simple matter to combine the quadratic Stark effect, as outlined here, with the fine structure. In figure 14, we show FIGURE 10 Lya of Ar*17, with and without the quadratic Stark effect, at a temperature of 800 eV. a) ~lectgon density = 1023 m3; b) ectron density 3 x 10 cm ; c) Electron density  10 om 3. 50 0.60 050 040 030 020 0 10 00m 0.80 0670.53 0.601 053 0400.270.13 0.00 0.13 0.27 0. 40 0.53 0.67 0.80 DELTA OMEGA (RYD) " '  I V FIGURE 11 LyBof Ar+17, with and without the quadratic Stark effect at a temperature of 800 eV. a) 21lectson density = 10 3 cm ; b) Eectron density = 3 x 10 cm ; c) Electron density = 10 cm"3. 52 1.00 080 0.60 0400.20 000 0.20 0.40 0.60 0.80 1.20 1.60 50 1.00 0.50 0.00 0.50 1.00 1.50 2.00 2.50 DELTA OMEGA (RYD) FIGURE 12 LyY of Ar+17, with and without the quadratic Stark effect, at a temperature of 800 eV. a) 2 lect5on density 10 cm3 ; b) Eectron density = 3 x 10 cm ; c) Electron density = 10R cm3. 54 0.60 050 0.30 020 0.10 0.00 i 3002.50 200 1.50 1.00 0.50 000 0.50 1.00 1.50 2.00 2.50 3.00 A 0.60 0.50 0.40 * 0.30 0.20  0.10 0. 04.804.00320 2.40 1.600.8 0.00 0.80 1.60 2.40 3.20 4.00 4.80 A I 0.60 0.50 0.40 0.30 020 0.10 1 0.00 6.00500 400 3.002001.00 000 1.00 2.00 3.00 4.00 5.00 6.00 DELTA OMEGA (RYD) cn o (L m i 0 > .C 0 54 co 0 0 VCC o S.0 cO + *. OL3 CO ) 0 oal 56 0 /i8 0 0o o o O S8. (m) I FIGURE 14 J(W,E) of a Lya line for various values of e. 1.20 0.60 0.00 0.60 1.20 1.80 2.40 0.80 0700.600.50 0.40 Q30 0.20010 000 0.10 0.20 0.30 0.40 0.50 0.60oo 0.40 0.50 0.60 0.70 080 0.40 0.50 060 0.70 0.80 58 IL A 0.40 0.20 0.80 1.40 2.00 2.60 0 0.70 0600.50 0.400.30 020 0.10 000 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 DELTA OMEGA I I I I i  I I I 59 J(w,c), for a Lya line, as a function of w, for various values of ion field, e. The peaks should correspond to the energy levels of the hydrogenic states for different values. In Figure 15 we show the energy levels of the n = 2 states of hydrogen as a function of electric field (Luders 1951). Comparison of Figures 14 and 15 show good agreement, giving confidence in the methods outlined here. FIGURE 15 Energy levels of the n=2 states of hydrogen as a function of perturbing field strength. 61 0.4 0.2 0.0 0.2 0.2  u 2 0.4  0.6  0.8 0.0 0.4 0.8 1.2 1.6 E (A.U.) CHAPTER III RESULTS It is a simple matter to combine the three asymmetry effects dis cussed in the previous chapters. The quadrupole interaction is included in J(w,e) as in equation (I.A.38). Fine structure is included by calculating Hr as in equation (II.A.4). The resolvent matrix is calculated only for states of a given principal quantum number. It has 2n2 rows and columns, twice the number it would have if spin were ignored. This result can be transformed to the Inqm> representation so that the quadratic Stark effect may be included. In this representation, the operator p.E is assumed diagonal and is replaced by 3cn I 3n42 2 2 2 2Zje 1 a 0 (y (17n 3q9m2+19) (III.A.1) as in equations (II.B.2) and (II.B.3). In Figures 1618 we show line shapes with and without all three asymmetry effects included. Figure 16a, b, and c show Lya lines at densities of 1023, 3 x 1023, and 1024 electrons per cubic centimeter and a temperature of 800 eV. Figures 17 and 18 show the LyB and LyY lines under the same conditions. As mentioned earlier, LymanB lines are particularly useful as density diagnostics because they are nearly as intense as Lymana lines, but are not nearly so optically thick. Although inclusion of the three asymmetry effects discussed here will cause significant changes in Lyman line profiles, for most temperature and density regimes the LymanB line 62 FIGURE 16 Lya lines of Ar+17, with and without the three asymmetry effects, at a temperature of 800 eV. a) Electron23 density = 1023 cm 3; b) Eectron density = 3 x 102 cm c) Electron density = 10 cm3. 64 I 0.6 0.50 0.40 0.30 0.20 0.10 0.00 0800670.530.400.270,13 0.00 0.13 0.27 0.40 053 0.67 0.80 0.50 0.50 I 0.40 030 0.20 \ 0.10 0.800.670.530.400270.13 0.00 0.13 0.27 0.40 0.53 0.67 0.80 I0 ""I 30.670.530.400270.13 0.00 0.13 0.27 0.40 0.53 0.67 0.80 DELTA OMEGA (RYD) FIGURE 17 LyS lines of Ar+17, with and without the effects, at a temperature of 800 eV. a) density 1023 cm3; b) Eectron density c) Electron density 10 cm . three asymmetry Electron23  = 3 x 102 cm 66 1.60 1201 200 1.50 1.00 0.50 0.00 0.50 1.00 DELTA OMEGA (RYD) 0.60 0.40 0.30 0.10 0.40  0.30 0.20 0.10 060 0.50 FIGURE 18 LyY lines of Ar+17, with and without the effects, at temperature of 800 eV. a) density = 103 cm ; b) Electron density c) Electron density 10 cm3. three asymmetry Electron23 = 3 x 10 e m 3 68 0.60 0.50 0.40 0.30 0.20 0.10 0.00 30 2.50 2.00 1.50 1.00 0.50 0.00 0.50 1.00 1.50 2.00 2.50 3.00 0.60 0.50  0.40   0.30 /  \\ 4.804.00 320 240 2.40 3.20 400 4.80 6.00 5.004.00 3.00 2.00 1.00 0.00 1.00 200 3.00 4.00 DELTA OMEGA (RYD) 69 will still be a two peaked function. Let us define a measure of asymmetry for the LymanB line as 2(Ib Ir)/(Ib +Ir) where Ib and Ir are the blue and red peak intensities, respectively. In Figure 19, we plot this asymmetry as a function of density for an Ar+17 plasma at 800 eV. Lines which include only the fine structure splitting have a negative (red) asymmetry which decreases in magnitude with increasing density. Both the quadratic Stark effect and the ion quadrupole interaction cause positive (blue) asymmetries which increase with increasing density. These two effects are shown together in Figure 19. Also shown is the asymmetry of lines which are calculated using all three effects. The asymmetry of these lines is a rapidly changing monotonic function of density (at least in the range of experimental interest) and therefore could provide diagnostics if the effects of opacity are not to great. It is interesting to note that, while each effect, taken alone, would cause significant asymmetry at 4 x 1023 cm3, their combined effect is to yield a nearly symmetric line profile. Since this is the region of current experiments, these asymmetries will be very difficult to observe. The table below shows the separate contributions to the asymmetry at several densities. O 0 O .,.4 a, 4 ,o .0 t O L 0 c\ 4 L * EU, 0,) >> C 71 xO 0 0 W  WI 0Z 8 ow oO W ci  o N  d A813lN1NASV I 72 TABLE 1. Contributions to the asymmetry of the Ly8 line of Ar+17 1023 3 x 1023 1024 3 x 10 FINE STRUCTURE .187 .112 .053 .030 QUADRATIC STARK .023 .034 .062 .064 QUADRUPOLE .028 .051 .098 .151 TOTAL .142 .022 .085 .174 If this procedure is performed for different elements, the trends are the same, but the point of zero asymmetry will be different. For example, Ne+9 lines are nearly symmetric at 1022 cm3. The asymmetry of experimental emission line profiles can be affected by more than just the three effects outlined here. For example, the background radiation slope must be suitably accounted for and pertinent overlapping spectral lines must be included in the calculation (Delamater 1984). 4 CHAPTER IV SPECTRAL LINES FROM HELIUMLIKE IONS It has been conjectured (H.R.Griem, private communication) that for high temperatures and densities, a product of hydrogenic wavefunctions would provide a satisfactory approximation for the heliumlike wave functions of highZ ions. A strict interpretation of this statement, however, would imply a degeneracy in the orbital quantum number, "t", a degeneracy which is not present is heliumlike ions. This nondegeneracy causes an asymmetry in the heliumlike lines even without considering the ion quadrupole interaction, the fine structure splitting, or the quadratic Stark effect which give rise to additional asymmetries, as seen in earlier chapters. Because of this qualitative difference between the hydrogenic and heliumlike wavefunctions, we take the following as our model. One electron is to be in the n1 level and the other initially is in a higher level. The ground state electron has only two effects (in our approximation). First, it causes the idegeneracy for the other electron to be broken, and, secondly, and less importantly, it screens the nucleus so that matrix elements are calculated using hydrogenic wavefunctions with a nuclear charge of Z1. Initially, the two electrons may couple into either a singlet or a triplet state, that is, a state of total spin zero or one. The ground state, however, must be a singlet by the Pauli exclusion principal. Transitions from triplet states to singlet states, such as 1s2p P 1s2 iS, give rise to lines called intercombination lines. These lines have low transition probabilities since they are dipole 73 74 forbidden. For this reason, we will completely ignore the triplet states in this calculation and only consider singletsinglet transitions. In actual experiments, the intercombination lines may be observable. In these cases they may be approximated by a Lorentzian or Voight profile (Delamater 1984). Following the formalism of Chapter I, but neglecting the quadrupole interaction, we have I(W) = fd Q(C)J(W,c), J(w,e) = 1 Im Tr D{w(H w) p* ()}1. (IV.A.1) a The trace is taken with respect to singlet states of a given principal quantum number. The Hamiltonian, Hr,is assumed to have eigenvalues (Z1) Enm Z2 + w(n,Z,Z), (IV.A.2) n where the energy has been expressed in Rydbergs. The first term is just the energy of an isolated hydrogenic ion of nuclear charge Z1 and the second term is an adependent energy shift. This shift represents the effect that the ground state electron has on the energy levels of the excited electron. Clearly, w(n,t,Z) removes the idegeneracy. The values for w(n,t,Z) must be input to our code from an outside source. For the heliumlike alines, it is the n2 heliumlike energy levels that are required. These have been conveniently calculated by Knight and Scherr (1963) as an expansion in powers of (1/Z). For higher 75 series members, the energy levels must be obtained elsewhere, either from theory or experiment (Bashkin and Stoner 1975; A.Hauer, private communication). If the ion field perturbation is small compared to the energy difference between states of different orbital quantum number, w(n,L1,Z) w(n,t2,Z), then these states will not mix, and there will be no linear Stark effect. The energy levels will shift quadratically with electric field. For large ion field perturbations, the states mix substantially and the levels shift linearly with electric field. Appendix G shows this change from quadratic to linear dependence for the simplest case, the aline. It is interesting to note that, in the case of the aline given in Appendix G, the shifted components shift symmetrically about their initial mean, sp/2, and yet the line is markedly asymmetric. There are two reasons for this asymmetry. First, the unshifted components are centered about zero (the 1 energy level) rather than the symmetry point  sp/2. Secondly, the transition probabilities for the shifted components are not equal. The shifted components are linear combinations of the states 1200> and 1210>. In the absence of any field these states are not mixed and have relative energy levels wsp and zero. This gives rise to asymmerty because the state 1200> is dipole forbidden. Only for large electric fields is the situation symmetric, as the eigenvalues are then (1200> 1210>)//i. While the presence of more different initial energy levels introduces additional asymmetry to the lines of higher series members, it is the a line which is qualitatively the most different from its hydrogenic 76 counterpart. Figures 2022 show heliumlike a, B, and Y lines of Argon in regimes of experimental interest. Because the a line is so narrow, we show it on a log scale to bring out the features. The ability to calculate heliumlike spectra, as well as hydrogenic line shapes, greatly increases the possibility of using several lines to analyze a single experimental shot. A problem with this, however, is that the peak hydrogenic line intensity and the corresponding peak heliumlike line intensity will, in general, occur at different temperatures and densities that arise during the course of the experiment. Therefore the choice of temperature and density conditions which best fit a hydrogenic line in a given experiment will not necessarily fit a heliumlike line occurring in the same experiment. This problem may be greatly reduced through the timeresolved spectroscopy techniques presently available. FIGURE 20 Heliumlike aline of Ar+17 at an electron density of 1023 cm3 and a temperature of 800 eV. 78 1.60 0.80 1.33 1.07 0.800.53 0.27 0.00 0.27 0.53 0.80 "0.80 0.00 0 1 0.80 I.bU 1.5 1 r 0.800.55 0.27 0.00 0.27 0.53 0.80 1.07 1.33 1.60 2.40 1.60 0.80 0.00 0.80  1.60 2.40 1.601.33 I7 0.80 0.530.27 000 0.27 0.53 080 1.07 1.33 1.60 DELTA OMEGA (RYD) FIGURE 21 Helum'l ke Bline of Ar+17 at an electron density of 10 J cm and a temperature of 800 eV. 80 1.67 1.33 2.402.00 L60 1.20 1.60 200 2.40 2.50200 L50 1.00 0.50 0.00 0.50 1.00 1.50 2.00 2.50 3.00 DELTA OMEGA (RYD) FIGURE 22 He~umli~ke Yline of Ar+17 at an electron density of 10J cm" and a temperature of 800 eV. 82 2.502.001.50 1.000.50 0.000.50 1.00 1 4.00 320 2.40 2.50 3.00 )02.001.00 000 1.00 200 300 4.00 500 6.00 DELTA OMEGA (RYD) 0.60r 0.60 0.50 APPENDIX A ZWANZIG PROJECTION OPERATOR TECHNIQUES From Eqn. (I.A.15) we have I(W) = fdr W(/)J(w,,), J(w,Y) r where f(a,') = Tr P w 1Re f dt e itTr 0 a (a,') Tr p(y), P p 6(y$ ), W(Y) Tr p 6(14 ), and p(v) = p 6 (0 )/W(y). Define an operator P by PX a P Note that 83 and (A.1) (A.2) 84 P X fl(a,T) Tr [p()fl (a,T) Tr p(T)X] P P = '(a,T)(Tr p(')) f' (a,T) Tr p(T)X P P = PX (A.3) so P2 P. This property, idempotence, defines a projection operator (Reed and Simon 1972). We can now write J(w,T) as 1 J(w,') nt Re Tr a r iwt d.f(a,T) f dt e Pd(t) 0 1   iT Re Tr df(a,') 1(0), a where P1 (w) P4(o), operator technique can Let D(t) t)). Let D(t) = d(t). and fO(w) J dt e d(t) The Zwanzig Projetlon 0 be used to obtain a general expression for We have i D(t) LD(t). (A.5) Let PD D1 and (1P)D D2 so that D1 + D2 D. Then (A.4) 85 1 2 = (1P)L(D1 + D2), and i D PL(D1 + D2). Therefore (t + i (1'P)L)D2(t) *i(1P)LD(1t), which has the solution D2(t) = e'it(1P)LD2() i For our choice of P, D2(0) 0. Therefore t S (t) PLD (t) i f PL eiS(1P)L (1P)LD1 (ts)ds. 0 This can be transformed using the Convolution Theorem, i e Wt61(t)dt 0 SPL f e itD1(t)dt IPL J e ite it(1P)Ldt 0 0 0 (O) ) PL ) + PL P) ei D1(P)L)ds, M (0) + W& () PLO (w) + PL (1 P)LO (W). JL1 1i 1 + w(1P)L1 1(w) = i[I[PLPL(wQL)IQL]1D1(0), (A.6) (A.7) (A.8) (A.9) (A.10) (A.11) e (1P)LD (ts)ds. (A.12) 86 where Q 1P. Putting this in (A.4) yields J(,) Im Tr (a,) [ L> L(QL d. J(wf) W Im Tr df(a,y) [w (A.13) Let L = L0 + LI(1 ) and L0 L(a,) + L where LO is the Liouville operator for the isolated atom and Lp is the Liouville operator for the plasma. Then, (A. 14) The first term is 1 a a a P and the second term is p a pa because LpXa = 0, so functions. Also '1 QLoXa = LOXa 1(a,V) Tr p(Y)LOXa P 1 LX f (a,i) Tr p()L X aa a P LaX a L = 0. aa aa (A.15) 87 This allows us to replace the last L by LI ( ) using also Q2 Q, we can write J(W,')  J(m,P) = w in l(w). Im Tr df(a,')[wL(a,) a  Now we work on the first L in /(w). Consider PLQX = f' (a,') Tr p('Y)LQX P = f (a,) Tr Lp(')QX P 1  (a,T.) Tr P (L(a,') + L + L I( ))p(C)QX. The first term is 1 f1(a,V) Tr L(a,)p(P)QX p P '1 f1(a,T)L(a,P) Tr p(')X p P f Ia,')L(a,P) Tr p(Y)f la,1) Tr p(T)X P P fi1(a,y)L(a,T) Tr p(Y)X f (a,')L(a,V) Tr p(F)X 0. The second term on the right hand side of (A.17) is Now, (A.16) (A.17) (A.18) 88 1 f (a,) Tr p But Tr L M = Tr p p states and these cyclicly permute ~'(a,') Tr P 1 L p(V)QX f1 (a,r) Tr L pP ()QX. p H M MH Since Hp is an operator only on perturber states are being traced over, it is legitimate to Hp, i.e. Tr H M = Tr MH Therefore, P p L p(Y)QX = 0. p (A.19) This leaves 1 + J(wm,) = 1 Im Tr d.*(a,y)[wL(a,y) a 1 ]I f(a,' ) Tr L (0 )p(Y)(wQLQ) QL (*)] d. p P This is the result shown in equation (I.A.18). Now consider (w,) = f1(a) Tr L p(Y)(aurQL Q)QL = '1(a) Tr LIp(V) I ( QLoQ)QLI. p n0 (A.20) (A.21) From equation (A.18) we have PLOQX 0 or QLOQX = LOQX for arbitrary X. Using this, one may write 1 1 1 n (w,Y) f (a) Tr L p(T) 1 ( L Q) QL p nO = f1(a) Tr LIp(Y)(wLO) QLI. P (A.22) 89 Noting that (w,'Y) acts only on atomic functions, we can write 1 * f (a) Tr pi6(VT ) Tr PaeL Xa PLIXa e (A.23) Tr p 6(YI ) Tr p i 1 ae This yields (,?) f (a) Tr Lp(Y)(wLo) L (A.24) P which is the result shown in eqn. (I.A.25). APPENDIX B PROOF THAT THE OCTAPOLE EFFECT IS SYMMETRIC TO FIRST ORDER IN PERTURBATION In this appendix, it will be shown that terms of the multiple expansion with odd "1" values contribute a symmetric effect on the line shape, to first order in perturbation theory. Consider hydrogenic states Inqm> in the parabolic representation. The presence of a uniform electric field removes the q degeneracy but not the m degeneracy. Upon perturbation by an im mode, the energy shift of levels with given n and q values will be a linear combination of terms like M = im Switching to the spherical representation by Inqm> Ijntm> where n1 n1 mq mq 'm 2 2 gives 90 91 Si1/2(1+m qn) Im 1/2(1+M2qn 2 /(21 +1)(2 2+1) n1 n1 1 n1 2 2 1 2 m1q m1+q m q 2 2 ml 2 Int2 = J RniRR n1 2 2+q 2 r+2dr r dr d Y* d Y d 1 m 1 The angular integral is (Edmonds 1957) S (2l +1)+1 2 +1)(21+1) (1) i/ 2 4 r 0 Let R I2 represent the radial integral. 2.2. m1+m m (2 qn M 1 R 1(1) (1) an1 21.2 n1 2 2 n1 n1 2 1 2 m1q m2q 2 1) 2 (2 +1)2 (22 +1)2 (21+1) nl1 2 m2q 2 z2 m2 R22 M I 1 12 9 (1) Now, (B.3) Y Y 2 im 2m2 (B.4) (B.5) 2 m m2 12 2 S12 1 1 0 0) (m, 92 0 0 0) ( 0 smfI. m m2 (B.6) Now consider M' = am the state which is shifted opposite to the original state, by a uniform electric field. m +m2 ml 2 +qn (2 +1)2 (21 +1 )(2e+1) R ( '1) (1) RI12( 4 n1 n1 2 1 2 m1q 2+q 2m  2 1 2 n1 t2 / 2 I1 m2 q m2 2  0 0 m 1 Sm2 m m (B.7) Recalling the symmetry property C o o\ t 1 0 t\ 8 +t +4 z1 2 ~3 %2 1 3 ) Y1 2 3 = ( I(21) 23 1 "2 m3 m2 m m3 and q is an integer so (I1)+q (1)"q, one obtains M' 1 it2 n1 2 2 (B.8) 93 m +m   m1 +2 2 2 ml 2 qn (2 +1) (22 +1) (21+1) R (1) (1) J 4w 11 2. n1 n1 n1 2 2 1 2 m q ml +q m2q S 2 m 2 2 2 1 2 n1 n1 2 2 1 8 (1) The last phase factor is n1 2 m +q 2 2 m2 0 0 O 1 2 0 0 )(1 m m 2 2 n n1l + + + 2 2 2 (B.9) (1) 1 2 S But/ (0 9 +.  Sis even, so (1) 2 ( 1 + z2 + Z is even, so (1) . 2) 0 unless I 0 The result is M' 1 (1) M. If i is odd (=1, dipole; =3, octapole), states with given qlj shift symmetrically, either both out or both in towards the center. If I is even, states with given Iq shift the same way (either all to the blue or all to the red) and so an asymmetry results. M' = 11i2 