Asymmetries in plasma line broadening

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Asymmetries in plasma line broadening
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vii, 114 leaves : ill. ; 28 cm.
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Joyce, Robert Foster, 1953-
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Plasma radiation   ( lcsh )
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Thesis:
Thesis (Ph. D.)--University of Florida, 1986.
Bibliography:
Bibliography: leaves 112-113.
Statement of Responsibility:
by Robert Foster Joyce.
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Typescript.
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Vita.

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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. ION-QUADRUPOLE INTERACTION............. ......................1

A. Formalism with lon-Quadrupole 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 HELIUM-LIKE 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
HELIUM-LIKE IONS ..........................................108

REFERENCES .......................................... .......... 112

BIOGRAPHICAL SKETCH ............................ .................. 114

iii















LIST OF FIGURES


FIGURE PAGE

1 Energy levels of the n-2 states of hydrogenic ions in the
presence of a uniform electric field and in the presence
of a field gradient........................................... 4

2 Ly-a lines of Ar+17, with and without the ion-quadrupole
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 ion-quadrupole
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 cm-3............................28

4 Ly-Y lines of Ar17, with and without the ion-quadrupole
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 Ly-8 line of Ne+9 at an electron density of 3 x 1023 cm-3
and a temperature of 800 eV with and without the
ion-quadrupole interaction...................................32

6 Ly-a 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 cm-3.........................38

7 Ly-8 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 cm-3.............................40

8 Ly-Y 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 cm-3...a........................42










9 Ly-8 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 Ly-a 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 cm-3............. ....... .....49

11 Ly-8 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 Ly-Y 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 Ly-8 of Ne+9 at an electron density of 3 x 1023 cm-3 and
a temperature of 800 eV, with and without the quadratic
Stark effect ................................... .......... 55

14 J(w,E) of a Ly-a line for various values of E .................57

15 Energy levels of the n-2 states of hydrogen as a function
of perturbing field strength................................ 60


16 Ly-a lines of
effects, at a
(a) Electron
(b) Electron
(c) Electron

17 Ly-B lines of
effects, at a
(a) Electron
(b) Electron
(c) Electron

18 Ly-Y 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 cm-3 .............................65

Ar 17, with and without the three asymmetry
temperature o 800 eV.
density 103 c3 -3
density 3 x 10 3cm
density 102 cm-3 ............. ............. 67


19 Asymmetry of Ly-B, defined as 2(Ib-I )/(Ib +I), as a
function of density...........................................70









20 He um-l1ke a-line of Ar+17 at an electron density of
10 cm- and a temperature of 800 eV........................77

21 He ium-l ke S-line of Ar+17 at an electron density of
10 cm and a temperature of 800 eV.........................79

22 He ium-l ke Y-line 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. Non-hydrogenic spectral lines are

also asymmetric. In Chapter Four, a procedure for calculating the

special lines from Helium-like ions is outlined.


vii
















CHAPTER I
ION4QUADRUPOLE INTERACTION

A. Formalism with lon-Quadrupole 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) x-E(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 = is the dipole moment of the radiator and

QiJ = is the quadrupole tensor of

the radiator.



This multiple expansion is basically an expansion in the atomic

radius divided by the average inter-lon 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 Ly-B

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 n-2 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 well-known Stark states

with a perturbing field E-e/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 Ly-a, 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 ensemble-averaged radiation emitted by one ion of

that type as





P(w) 3 1 I12 a (wb) a 4 1(w), (I.A.5)
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 ae dt. (I.A.6)
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 n-2 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 i2w-w ta
(w) Re f I 2 pe (-ab dt
0 aba


= Re f et.dt
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 radiator-plasma 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 full-Coulomb radiator-plasma interaction, V ap which is

given by a sum of pair-wise 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 (Z-1)e /r,




-0-


v1 (a,j) -Z e2 a + P (cosaj),
1+ aj
2-i 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(c-Z 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(T-0 ) = 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 c-number. 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) 1-Re 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) ,



= f-1(a,') Tr p(y)X for any X, and
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 j-1


-H +H +
a p


o 40(J) + v2 (a,j) + M(a)O(j).
j-1


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



e-iLt
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 j-1


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

radiator-plasma 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,')[w-L(a,')-B('i)-, (w,')]-1da, (I.A.18)
a


where



B()
  • ,



    / (,,) f (a,')Tr L (0 )p(*)(w-QLQ) -QL ( ),
    p
    and

    Q = 1-P, 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

  • = 0. This implies that




    - .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 electron-ion 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
    j-1
    parameter, this screening is given by the Debye-Huckel

    interaction (Dufty and Iglesias 1983) and that form will be

    assumed here. (Even if the Debye-Huckel 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 electron-ion





    -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(*)(w-LO) !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 (w-L(a,Yi)-L ) -LI
    e


    f1 (a)Tr I L (j)pae(m-L(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 non-radiative 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 ,,e-iH(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 (Ea-E )/-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- ,...). (I.A.30)



    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 E but introduces second order
    2 2
    errors which are proportional to These should be
    ij three orders higher in the smallness parameter n2aO/Z/ROi.

    This yields



    I(w) = fdE Q(4)J(WC),

    with



    J(W,c) -W Im Tr{d.f(a)(w-L(a,c, ) ( m))1 }. (I.A.31)
    a





    -19-


    The Hamiltonian corresponding to L(a,, ) is

    H a-P* 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 d-d, 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 Debye-Huckel interaction was chosen for i Therefore,



    SE (Ze) x e-r/ 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 off-diagonal 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), 0. Therefore,
    xy c xz E yz c




    -LU-


    SQijE
    ij Ei


    Q xx + Q < > + Q zz
    xx xx e yy yye ZZ zz e


    (Q + Q ) + Q (e >
    xx yy xx e zz zz e


    -Q + Q zzzz >



    3 Q ( 1 ).
    2 zz e ZE z3



    For the Debye-Huckel form,


    -1 -nA
    V*E -I (Ze) e /r.
    2


    (I.A.35)


    (I.A.36)


    -r/A 2
    1 e r r 2
    (Ze )< (1 *+ (1-3 )>
    zz 3 E 3 X 32 E


    .



    Now J(w,c) may be written


    J(W,E) Im Tr Dw-(Hr-w0) P- Qz r


    (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



    = < V-C>
    C zz 3 E

    -r/X 2
    SZe, (I.B.1)
    r3 3 2 2


    Obviously, f can be written as a sum of single ion terms,



    f = Ze I F(r)(1-3p2), (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-
    3N
    fd r 6(4-E)e-8V
    S r 3N + -BV




    Njd3r f 6-E)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(N-1) N O.B.
    d3r fl[d r 6((-E1)- I Ei )e ]. (I.B.4)
    ZNQ(E) 1=2


    Within the square brackets, V can be considered to be an N-1 particle

    potential. Then [ ] has the form of a microfield for this potential,

    times its partition function,ZN' In symbols, [ ] = Q'(E-E1)Z -1

    Note that


    ZN-1- g() (I.B.5)
    zN "


    where g(r) is the radial distribution function and R is the system

    volume. Now can be written
    C


    nJd3r flg(r)Q'(e-E )/Q( ) (I.B.6)



    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) -
    Ze1 dk k2e dr r2g(r)F(r)I (I.B.8)
    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(1-3P ) 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 Debye-Huckel

    radial distribution function, g(r), in this calculation. In order to

    estimate the error introduced in this way, we ran one case (ne -

    1024 cm-3, 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 was less than 5%.
    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 ion-quadrupole 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 Ly-a 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

    Ly-a lines of Ar+17, with and without the ion-quadrupole
    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 cm-3.





    -/o-


    u.um i


    -0.80 -0.67 -0.53 -(


    0.40-0.27-0.13 000 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 -0.67-053-0.40 -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 Ly-B and Ly-Y lines at the same densities. Again the increase

    in asymmetry with density is observed. By comparing Figures 2-4 one can

    also notice the increase in asymmetry with increasing initial principal

    quantum number. Figure 5 shows the Ly-B 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


    Ly-8 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.


    ion-quadrupole
    a) Electon -3
    - 3 x 10 cm ;


























    -1.20-100 -0.80-0.60-0.40-0.20 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

    Ly-Y lines of Ar+17, with and without the lon-quadrupole
    interaction, at a temperature of 800 eV. a) Elect on
    density 1023 cm-3; 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
    -4.80-4.00 -3.20 -2.40-I.60-0.80 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-

    -.00-5.00-4.00 -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
    -e (II.A.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 spin-orbit

    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),



    [1 + (aZ )212 1,/
    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
    2 2 [1 + n J71/ 4)]. (II.A.4)
    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





    S-Im 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 w-w0-.-- (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)


    -+1/2-jz .
    (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 3-j 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 Ly-a 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 Ly-B

    and Ly-Y lines. Figure 9 shows a Ly-B 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 6-8

    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

    Ly-a 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


    '-067-0.53 -040 -027 -013 Q0 0.13 0.27 0.40 0.53


    0-027-0.13 QO Q13 0.27 Q40 0.53 0.67 0.80
    DELTA OMEGA (RYD)


    -0.67-053




























    FIGURE 7

    Ly-B 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-


    -1.00-0.80 -060 -040 -0.20 000 020 0.40 060 0.80 1.00 1.20


    DELTA OMEGA (RYD)




























    FIGURE 8

    Ly-Y lines of Ar 17, with and without fine structure at a
    temperature of 800 eV. a) Elctroi density = 1023 cm-3;
    b) ectron density = 3 x 10 cm ; c) Electron density
    = 10 em-3cm





    -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


    -4.80-4.00 320-240-160-0.80 0.00 080 160 240 3.20 4.00 480





    0.60

    0.50

    040

    0.30

    0.20

    0.10


    .0r-5.o-4.-00o-o.0-2.0 -I'0 0.00 1.00 2o00 3. 4oo s'oo 60
    DELTA OMEGA (RYD)




































    V
    CO




    L
    x 4



    0--4


    oJ-4
    0
    mb








    -,
    0)




    ti

    0

    0-4

    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
    S-ie 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) nE 2n n


    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 n-3q -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 Ly-c 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 Ly-B and Ly-Y lines for the same conditions. Figure 13

    shows the Ly-B 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 Lyman-B line is particularly interesting because it is usually

    intense enough to be seen when the Lyman-a line is observable, but it is

    not as optically thick. The asymmetry of the Lyman-B 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 Lyman-B 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

    Ly-a of Ar*17, with and without the quadratic Stark effect, at
    a temperature of 800 eV. a) ~lectgon density = 1023 m-3;
    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 -067-0.53-


    0.601


    -053


    040-0.27-0.13 0.00 0.13 0.27 0.


    40 0.53 0.67 0.80


    DELTA OMEGA (RYD)


    " '| | I


    V




























    FIGURE 11

    Ly-Bof 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 -040-0.20 000 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

    Ly-Y 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 cm-3.






    -54-


    0.60

    050



    0.30

    020-

    0.10

    0.00 i
    -300-2.50 -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.
    0-4.80-4.00-320 -2.40 -1.60-0.8 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
    -6.00-500 -400 -3.00-200-1.00 000 1.00 2.00 3.00 4.00 5.00 6.00
    DELTA OMEGA (RYD)






































    cn
    o

    (L

    m -i




    0- >

    .C
    0


    5-4
    co






    0 0

    VCC








    o
    S.0




    cO

    + *.
    OL3







    CO )
    0
    oal
    C1 0




    -56-


    0

    /i-8



    0







    0o







    o o
    O
    S8.








    (m) I































    FIGURE 14

    J(W,E) of a Ly-a line for various values of e.












    1.20

    0.60

    0.00

    -0.60

    -1.20

    -1.80

    -2.40


    -0.80 -070-0.60-0.50 -0.40 -Q30 -0.20-010 000 0.10 0.20 0.30 0.40 0.50 0.60o-o


    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 -060-0.50 -0.40-0.30 -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 Ly-a 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 -3q-9m2+19) (III.A.1)


    as in equations (II.B.2) and (II.B.3).

    In Figures 16-18 we show line shapes with and without all three

    asymmetry effects included. Figure 16a, b, and c show Ly-a 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 Ly-B and Ly-Y lines

    under the same conditions.

    As mentioned earlier, Lyman-B lines are particularly useful as

    density diagnostics because they are nearly as intense as Lyman-a 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 Lyman-B line

    -62-





























    FIGURE 16

    Ly-a 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
    -080-067-0.53-0.40-0.27-0,13 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


    -0.80-0.67-0.53-0.40-027-0.13 0.00 0.13 0.27 0.40 0.53 0.67 0.80




    I0
    ""I


    3-0.67-0.53-0.40-027-0.13 0.00 0.13 0.27 0.40 0.53 0.67 0.80
    DELTA OMEGA (RYD)





























    FIGURE 17


    Ly-S lines of Ar+17, with and without the
    effects, at a temperature of 800 eV. a)
    density 1023 cm-3; b) Eectron density
    c) Electron density 10 cm .


    three asymmetry
    Electron23 -
    = 3 x 102 cm






    -66-


    -1.60 -120-1


    -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


    Ly-Y 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 /
    - \\


    -4.80-4.00 -320 -240


    2.40 3.20 400 4.80


    -6.00 -5.00-4.00 -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 Lyman-B 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 cm-3, 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 Ly-8 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 cm-3.

    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 HELIUM-LIKE 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 helium-like wave-

    functions of high-Z 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 helium-like ions. This non-degeneracy

    causes an asymmetry in the helium-like 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 helium-like wavefunctions, we take the following as our model. One

    electron is to be in the n-1 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 i-degeneracy 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 Z-1.

    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 singlet-singlet 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


    (Z-1)
    Enm Z-2 + 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 Z-1 and the

    second term is an a-dependent 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 i-degeneracy.

    The values for w(n,t,Z) must be input to our code from an outside

    source. For the helium-like a-lines, it is the n-2 helium-like 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

    a-line.

    It is interesting to note that, in the case of the a-line 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 20-22 show helium-like 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 helium-like 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 helium-like

    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 helium-like line

    occurring in the same experiment. This problem may be greatly reduced

    through the time-resolved spectroscopy techniques presently available.






























    FIGURE 20

    Helium-like a-line 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 -0.80-0.53 -0.27 0.00 0.27 0.53 0.80


    "0.80

    0.00
    0
    -1 -0.80


    -I.bU -1.5 -1 r -0.80-0.55 -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
    -1.60-1.33 -I7 -0.80 -0.53-0.27 000 0.27 0.53 080 1.07 1.33 1.60
    DELTA OMEGA (RYD)





























    FIGURE 21

    Helum'l ke B-line of Ar+17 at an electron density of
    10 J cm and a temperature of 800 eV.





    -80-


    -1.67 -1.33


    -2.40-2.00 -L60


    1.20 1.60 200 2.40


    -2.50-200 -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 Y-line of Ar+17 at an electron density of
    10J cm" and a temperature of 800 eV.





    -82-


    -2.50-2.00-1.50 -1.00-0.50 0.000.50 1.00 1


    -4.00 -320 -2.40


    2.50 3.00


    )0-2.00-1.00 000 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-



    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(1-4 ),


    and


    p(v) = p 6 (-0 )/W(y).


    Define an operator P by



    PX a = f' (a,y) Tr p(Y)X.
    P


    Note that


    -83-


    and


    (A.1)


    (A.2)





    -84-


    P X fl(a,T) Tr [p()f-l (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 d-f(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 (1-P)D D2 so that D1 + D2 D. Then


    (A.4)





    -85-


    1 2 = (1-P)L(D1 + D2),


    and

    i D PL(D1 + D2).


    Therefore



    (t + i (1'P)L)D2(t) *i(1-P)LD(1t),


    which has the solution


    D2(t) = e'it(1-P)LD2() i


    For our choice of P, D2(0) 0. Therefore


    t
    S (t) PLD (t) i f PL eiS(1P)L (1-P)LD1 (t-s)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-(1-P)L1


    1(w) = i[-I[PL-PL(w-QL)IQL]-1D1(0),


    (A.6)


    (A.7)


    (A.8)


    (A.9)


    (A.10)


    (A.11)


    e- (1-P)LD (t-s)ds.


    (A.12)





    -86-


    where Q 1-P. Putting this in (A.4) yields



    J(,) Im Tr (a,) [ L> L(-QL d.
    J(wf) -W Im Tr d-f(a,y) [w- ]d.


    (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
    Xa = f- (a,) Tr p(O)L(a,')Xa L(a,Y)X ,
    a a a
    P


    and the second term is



    X f- (a,F) Tr p(Q)(L X ) = 0,
    p a pa


    because LpXa = 0, so = L(a,Y) +
  • when acting on atomic

    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,')[w-L(a,)-
    a


    -]- d.



    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
    f-1(a,V) Tr L(a,)p(P)QX
    p
    P
    '-1
    f-1(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)[w-L(a,y)

  • a

    --1 ]-I
    f(a,' ) Tr L (0 )p(Y)(w-QLQ) QL (*)] d.
    p
    P


    This is the result shown in equation (I.A.18).

    Now consider



    (w,) = f-1(a) Tr L p(Y)(aurQL Q)-QL


    = '1(a) Tr LIp(V) I ( QLoQ)QLI.
    p n-0


    (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 n-O


    = f1(a) Tr LIp(Y)(w-LO) -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(Y-I ) Tr p
    i 1 ae

    This yields



    (,?) f (a) Tr Lp(Y)(w-Lo) 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 = . (B.1)
    im


    Switching to the spherical representation by



    Inqm> Ijntm>



    where

    n-1 n-1

    = (. 1)1/2(1+m qn) 2,2 (B.2)
    m-q mq '-m
    2 2

    gives


    -90-





    -91-


    Si1/2(1+m -q-n)
    (-1) 1
    Im


    1/2(1+M2-q-n
    2


    /(21 +1)(2 2+1)


    n-1 n-1 1 n-1
    2 2 1 2
    m1-q m1+q m -q
    2 2 -ml 2




    Int2 = J RniRR


    n-1
    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 -q-n
    M 1 R 1(-1) (-1)
    an1 21.2


    n-1
    2


    2


    n-1 n-1
    2 1 2
    m1-q m2-q
    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' = . This is the same perturbation on
    am
    the state which is shifted opposite to the original state, by a uniform

    electric field.


    m +m2
    ml 2 +q-n (2 +1)2 (21 +1 )(2e+1)
    R ( '1) (-1)
    RI12( 4


    n-1 n-1
    2 1 2


    m1-q 2+q
    2-m -
    2 1 2


    n-1 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(2-1) 23
    1 "2 m3 m2 m m3

    and q is an integer so (-I1)+q (-1)"q, one obtains


    M' 1
    it2


    n-1
    2


    2


    (B.8)





    -93-


    m +m ------ -
    m1 +2 2 2
    ml 2 -q-n (2 +1) (22 +1) (21+1)
    R (-1) (-1) J 4w
    11 2.


    n-1 n-1 n-1
    2 2 1 2

    m -q ml +q m2-q
    S 2 -m 2
    2 2 1 2


    n-1 n-1
    2 2 1
    8 (-1)


    The last phase factor is


    n-1
    2

    m +q
    2
    2


    -m2 0


    0- O- 1
    2


    0 0 )(1


    m m 2
    2


    n- n-1l
    + -+ --+
    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