Asymmetries in plasma line broadening

asymmetriesinpla00joyc.pdf

AA00004864_00001.pdf

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




ASYMMETRIES IN PLASMA LINE BROADENING
4
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


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.
ii


TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ii
LIST OF FIGURES iv
ABSTRACT vii
CHAPTER
I. I ON-QU ADR UPOLE INTERACTION 1
A. Formalism with Ion-Quadrupole Interaction 1
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
1
2
3
4
5
6
7
8
PAGE
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
+ 1 7
Ly-a lines of Ar with and without the ion-quadrupole
interaction, at a temperature of 800 eV.
(a) Electron density = 1023 cm~3
(b) Electron density = 3 *10 cm
Electron density = 1 02 cm3
(c)
.25
Ar+1?,
Ly-B lines of
interaction, at a
(a) Electron dens
(b) Electron density
(c) Electron density
with and without the ion-quadrupole
emperature of 800 eV.
ty = 1023 Cn^3
= 3
10
,10
24 -3
cm ->
cm
.28
+ 17
with and without the
a temperature of 800 eV.
nsity = 1023 cbj~3
Ly-Y lines of Ar
interaction, at
(a) Electron density = UT-' cm- __
(b) Electron density = 3 *..10 _cm
ion-quadrupole
(c) Electron density = 10
,+9
24 -3
cm
.30
Ly-B line of NeT;7 at an electron density of 3 x 10
and a temperature of 800 eV with and without the
ion-quadrupole interaction
23 -3
cm
,32
Ly-a
at a
(a)
(b)
(c)
,+17
lines of Ar
temperature of
Electron density
with and without fine structure,
800 eV.
1 0^3 cd) 3
Electron density = 3 10 cm
Electron density 1 02 ^m"3
cm
,38
+ 17
Ly-B lines of Ar with and without fine structure
at a temperature of 800 eV_,
Electron density
(a)
(b)
(c)
Electron density
Electron density
1C3
3 x,10
1024 cm-3
cm
-3
.40
Ly-Y
at a
(a)
(b)
(c)
lines of Ar+1^i
temperature of
with and without
800 eV.
fine structure
Electron density = 10^3 Cm~3
Electron density =3*10 cm 3
2*3 "3
Electron density 10 J cm .....
,42
iv


9
10
11
12
13
14
15
16
17
18
19
44
49
51
53
55
57
60
63
65
67
70
_ Q Q
Ly-B line of Ne+' at an electron density of 3 10
and a temperature of 800 eV, with and without fine
structure
Ly-a of Ar+1^> with and without the quadratic Stark
effect, at a temperature of 800 eV.
(a) Electron density = 102^ cm ^
(b) Electron density = 3 *,10 cm
(c) Electron density = 10 cm
Ly~B of Ar+1^, with and without
effect, at a temperature of 800
(a) Electron density 102-* ci
(b) Electron density = 3 *10
(c) Electron density =10 cm 0
the
eV.
quadratic Stark
?3 -3
cm
Ly-Y of Ar+1^, with and without the quadratic Stark
effect, at a temperature of 800 eV.
(a) Electron density = 102-* cm~^
(b) Electron density = 3 *10 cm
(c) Electron density = 1 02 cm ^
Ly-B of Ne+^ at an electron density of 3 10 cm and
a temperature of 800 eV, with and without the quadratic
Stark effect
J(cu,e) of a Ly-a line for various values of e
Energy levels of the n=2 states of hydrogen as a function
of perturbing field strength
Ly-a lines of
effects, at a
(a) Electron
(b) Electron
(c) Electron
|1 7
Ar ', with and without the three asymmetry
temperature of
density = 102^
density = 3 *J
density = 102 cm
800 eV.
-3
Ly-B lines of
effects, at a
(a) Electron
(b) Electron
(c) Electron
xl 7
Ar ', with and without the three asymmetry
temperature of 800 eV.
density 102^ cm3
density = 3 *10 cm
density = 102 cm-^
Ly-Y lines of
effects, at a
(a) Electron
(b) Electron
(c) Electron
+1 T
Ar with and without the three asymmetry
temperature of 800 eV.
density = 1 O2^ cm~3 -
density = 3 10 0111
density 10cm-^
Asymmetry of Ly-B, defined as 2db-Ip)/db+Ip), as a
function of density
v


20
Helium-1;
10: cm :
Lke a-line of Ar+1^
> and a temperature
21
Helium-lj
10*3 cm*-1
.ke g-line of Ar+1^
* and a temperature
22
Helium-1j
10:> cm :
ke Y-line of Ar+1^
* and a temperature
at an electron density of
of 800 eV 77
at an electron density of
of 800 eV 79
at an electron density of
of 800 eV 81
vi


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
IONQUADRUPOLE INTERACTION
A. Formalism with Ion-Quadruple 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 multipole expansion of the radiator as a
localized charge distribution p(x) in an external potential 4>(x) caused
by the perturbing ions (Jackson 1972). The electrostatic energy of the
system is
W = Jp (x )(x )d^x
(I.A.1)
The potential can be expanded in a Taylor series with the origin
taken as the radiator center of mass.
324>( 0)
4>(x) = $(0) + x V4>(0) + \ l x x a
2 ij 1 J 3Xi9Xj
(I.A.2)
Div(E(0) = 4n^Z.e6(x. ), where the sum is over all perturbing ions and z^e
i 1 1
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
9E.
4>(x) = *(0)
or,
x-E(O) ¡ ^ OXjXj r28lJ) jji(0) *
(I.A.3)
-1-


-2-
W = q>( 0) d-(O)
2
6
I
ij
3E.
i
3E.
J
(0) +

(I.A.4)
where
q is the total charge of the radiator,
d = <\}j|~er [ijj> is the dipole moment of the radiator and
Q = is the quadrupole tensor of
the radiator.
This multipole expansion is basically an expansion in the atomic
2
radius divided by the average inter^ion spacing, or (n aQ/Z)/Ri0 where n
is the principle quantum number of the radiator, aQ is the Bohr radius, Z
is the radiator nuclear charge and R^q 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 plasma with an electron density of 10 cm
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/zS Now, if we take these as our new basi3
states and include the effect of the field gradients due to the


perturbating ion, the result is shown in Figure 1. The higher energy
levels, which give rise to the blue wing intensity, (id > where hm^
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 < m^) are spread further out. These shifts, when averaged
over all possible perturber positions, will lead to a blue asymmetry
for Ly~ct, at least out to the wings. Similiar 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 er.semble^averaged radiation emitted by one ion of
that t ype as
P(w) ~ I | | 2 p 6(uuahI(m), (I.A.5)
3c3 ab a aD 3c3
where p is the probability that the system is in state |a>, d is the
El
radiator dipole moment, and m = (E *E. )/h. The second equation
ab a b
defines the line shape function Km). Using the integral definition for
the Dirac delta function, we obtain,
. , i(m*m )t
Km) y- j l I I p_e dt. (I. A.6)
*< ab
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 o'" a uniform electric field and in the presence o
field gradient.


-5-
APPLY
ELECTRIC
FIELD
\ A2
J }A2
\} A2
APPLY
FIELD
GRADIENT


-6-
write l(u) as an integral from zero to infinity:
, o i(u-) )t
I(u) = Re l I| p e D dt
IT K a
0 ab
^ Re j l elwtdt
0 ab
- Re J dt el)tTr(d*pe iLtd).
(I.A.7)
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 (198*0. 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 which is
ap
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
Vap 1 v 0 J-1
No Z Ze2
I I [-*
a>1 I?, I
Z e
o
i "*
r -r.
' a 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
a a
respectively. Expanding this expression in Legendre polynomials, we
find,
where
oo r i
(a.j) -Zoe I [-^j- Wcoseaj>*
t-0 r>-
(I.A.9)
r^ = smaller of Ir I and |r.
< 1 a1 1 j
r> = larger of |ra| and |r
6 = angle between r and r.
aj a j
This expression may be regrouped as
v(a,j) = 0( j ) + v1 (a,j ) + v2(a,j),
Cl.A.10)
where
£
4>q (j) is the monopole term given by Z (Z-1)e /r,


-o-
-,"(a,3) -Ze2 I Pecse ),
£=1 r.
j
and
v2 (a, j )
r. I
00 J p
-Z e2 l [2_ ]p.(cose .)
0 lo r 1+1 r *+1 aj
a Jo
O
r. ir
Jo 3
r. > r
Jo 3
The sum of v 0 + is identical to v in the region r. > r the
i U j a
no penetration region, but extends this form to all values of rj. In
this form, v^0 may be written as a "product" of an operator on perturber
coordinates times an operator on radiator coordinates, in the following
manner,
v^ia.J) = M(a)$(j)
ao
- I UK(a)*K(j)
K-1 K K
= (d*e (j ) + g- £ mn^mGn ^ J ) + ) (I.A.11)
mn
Here M(a) stands for the set {y (a)} which depends only on the radiator
state. For example, y^ = d, which is the radiator dipole moment, and
l¡2 Q/6, which is the radiator quadrupole moment. The perturber
operator, 4>a(j), depends only on the jth perturber of type o. The
term 4>( j ) represents the set {k(j)}, where 4> 1 a (j) e(j) is the


-9-
electric field at the radiator produced by the jth perturber of
type o, 4>2(j) 3m£n(j) is the electric field gradient at the radiator
produced by the jth perturber of type o, and so forth. By design,
couples to $k.
We next generalize the usual definition for an electric microfield
distribution, W(e) = Tr p(e-I ), to include the field gradients and
i
higher order derivatives which are used in equation (I.A.11):
W(Â¥) Tr pfi(Â¥-*), (I.A.12)
00
* #
where 5(Y- ) = n ).
K-1 K K
The y and 4> are analagous to $ 0 in equation (I.A.11). That is,
K K K
they denote the value of the electric field, field gradient tensor, etc.
*
at the radiator, produced by the perturbers. In the above, $ is an, as
yet, arbitrary function of only ion coordinates and f is a c-number. Now
equation (I.A.7) can be written as
00
I(m) - Re f dt el)t Tr{d*Tr pfd¥5(F-* )d(-t)}
7T J J
0 a p
00
= fd¥ W(¥) ^Re [ dt el)t Tr|d*Tr p5(¥-**)/W(lF)3(-t)}
* o a p
- JdY W(V) J(aj,y), (I.A. 13)
where


-10-
J(u>,Â¥) - Re dt el)t Tr|d-Tr p(y)d(-t)},
u 0 a p
and
p(Y) = p5(Y-**)/W(Y).
(I.A.14)
We may rewrite J(,'P) as
J(u,Â¥)
dt e
iiot
Tr d*f(a,V)D(t),
a
(I.A.15)
where
D (t) = ,
= f ^(a,40 Tr p(Y)X for any X, and
P
f(a,Â¥) = Tr 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
N
o
H = H H + I l v(a,j)
3 P o J-1
N
H + H + l l 4>na (j) + v (a,j) + M(a)$(j).
3 P o j-1
(I.A.16)
The atomic Hamiltonian, Ha, includes the center of mass motion of the


-.Li
atn) 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'iLtp(40 = p(Y).
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(y) allows us to make
*
the replacement $ = $ + y i> in H, giving
N
H = Hg + M(a)Y + H + I I ) + I l v2(a,j) + M(a)(£ I 4>(j )-**)
o j o j o j-1
H + H
V2 +
V^* ),
(I.A.17)
where
H = H + M(a)Y,
a a
H H II *00(J)-
0 j
v2 I I v2(a,j),
V (*) M(a)(I l *(J) *).
o j
and


-12-
We have combined Ha and M(a)'F because these involve only atomic
coordinates. Similiarly, the monopole term, £ £ 4>q (j) does not depend
o j
on internal atomic coordinates and therefore is combined with Hp.
*
Let L(a,4'), Lp, L2> and ) be the Liouville operators corresponding
_ *
to H H V0, and V.($ ) respectively. Further let
3 P eL
# #
L ($> ) = ) + 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(m,4'). The details are
in Appendix A and the result is
Jiw,?) = -n1Im Tr{d*f (a,'r)[w-L(a,Â¥)-B('F)-^ (w,Â¥) ]"1d }, (I.A.18)
a
where
BO?) = ,
(to'?) = f"1(a,Â¥)Tr Lj(**)p(i|>) (u-QLQ)1QLI(**),
P
and
Q = 1-P, where P is a projection operator given by
P(...) <(...)>.
We wish to cast the expression for I(co) in a more familiar form, one
which uses the standard microfield. To this end, we must separate out
the electric field dependence from v. Let 4* that is, V' is
all of V except for e. Now,


-13-
W(Â¥) = Q(e)W(e|r)
(I.A.19)
where Q(e) is the usual microfield (Tighe 1977; Hooper 1968) and
W(e|4') is the conditional probability density function for finding
4" given e. Then,
(I.A.20)
which is the desired form, but now
(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 4> which has been arbitrary up to this point, to force
This implies that


-L *-t
6 ^
0 = Tr p(\Jj)($ + 4> 4> )
P
= Tr 6(i|)-4>*){Tr p(4>6 + 41) p )/W(ip), (I.A.22)
i e
where p^ = Tr p.
e
To satisfy the above, it is sufficient to define
* -1 e i
$> 2 p. Tr p (4> + 4> )
e
= 4-1 + pi1 Tr pO. (I.A.23)
e
*
Obviously, $ 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>i, in the manner above. With
*
this approximation, the ion parts of V^(* ) cancel, leaving
e
# 0
V.($ ) = M(a) l 4 (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
* *
the sum of single particle terms.) Note that V^(# ) and )
*
no longer depend on $ ; therefore they will henceforth be denoted
simply V1 and Lj, respectively. Neglect of electron-ion


-15-
interactions also allows the density matrix to be factored.
Thus f(a,f) can be reduced to
*
Tr p. 6 (ipi> )Tr p
.1 36
f(a,i|0 ^
Tr PjCiM )Tr pag
i ae
Tr
e
Tr
ae
f (a).
(I.A.24)
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 % () which is
second order in Lj is appropriate (Smith, Cooper, and Vidal
1969). This is accomplished by replacing the L in w(u),$>) by
Lq. Further simplification concerning the projection operators
is given in Appendix A, with the result
£/(>,*) f1(a)Tr Lip(*)(m-L0)-1Li (I.A.25)
P
Note that the Lq 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-
- f 1 (a)Tr LTp (-L(a,iJ))-L )
1 30 0 1
e
= f"1(a)Tr l Lt (j )p (c-L(a,iJ/) l L (k))"1 l LTU),
e j 1 ae k 6 i 1
(I.A.26)
where the Liouville operators have been written as sums of single
electron operators. Terms with are zero by angular average
and terms with k*i, are zero because L (k) is zero unless acting
on functions of electron k.
= f1(a)n Tr L (a, 1 )f (a, 1) (u-L(a,i|i) L (1) )~1 L (a, 1),
1 i
where nf(a,1) = Tr p (I.A.27)
ae
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 collions 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 & (oj) to (Tighe 1977)


-1/-
00
e
-iH( 1 )t/fi
h 1 0 i"
ii"
i H (1 )t/fi
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, Aw = w (E^E^/fi
and subscripts i (f) stand for initial (final) radiator states.
7. Equation (I.A.27) includes the ion shift of radiator states
in ^(w.Y) through L(a,Y). Dufty and Boercker (1976) found that
(w,40 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,40 by L(a).
8. The real part of fl^(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 4 completely,
replacing L(a,40 by L(a,e) and J(w,40 by J(w,e), this reduces to the
full Coulomb (electron interactions) of Woltz et al. (1982). Instead,
limit 4M 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
I(a)) de de de de de de de Q(e)
1 xx yy zz xy xz yz
W(e I e e .)tJ(co,e,c e ,...),
1 xx yy xx yy
(I.A.29)
3E
i
where z. = ~
1J 3xj
We now make the following approximation
W(e|e ,e ,...) + 6(e - e ~
1 xx yy xx xx e yy yy e
(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 £, but introduces second order
2 2
errors which are proportional to These should be
ij e ij e
2
three orders higher in the smallness parameter n Aq/Z/Rq^.
This yields
Kin) Jde Q(e) J(u>,e),
with
J(oi,e) = -it 1Im Tr {d *f (a) (oi-L(a ,e , ) -#( to)) 1d}. (I.A.3D
- ^ J ^


-19-
The Hamiltonian corresponding to L(a ,e,<£_>e) *s
H -p*e Y Q. , /6 where p is the radiator dipole operator and Q is
a F ij ij e
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{oj (H - it r u
r
l Q ,/6 # (u)
ij J J
i-1
(I.A.32)
where is the radiator ground state energy and D is given by d*d, and
4
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 *. Therefore,
9 E
x z
(Ze) e
r
zx _-r/A f-3
F?-
Ar
-L-2-)
X2r r3
(I.A.33)
and
9 E
z z
(Ze)e
-r/A
[R + + + ]
Ar
Ar
A2r
(I.A.31)
where y = cos(9) 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.31*). Since only the z direction is special
. is along a), £ £ £ 0.
Therefore,


-zu-
y Q <0 > =Q+Q+Q
jj lj ij 0 xx xx 0 yy yy 0 zz zz e
(Q + Q ) + Q
xx yy xx 0 zz zz 0
-Q <0 > + Q <0 >
zz xx 0 zz zz 0
1 Q (<0 > 4 )
2 zz zz 0 3 e
(I.A.35)
For the Debye-Huckel form,
So
V*0 = -1 (Ze) e'rA/r.
A
<0
zz
1 ~r/A 2 _
5 V*0> = (Ze)<^-r (1 + y ^-r)(1-3y )>
j 0 3A e
= .
0
(I.A.36)
(I.A.37)
Now J(ai,0) may be written
J(u,0) - Im Tr d{w-(H
V -
p*0 Q
/U #( ZZ 0
-1
(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


zz 3 e
-r/X 2 -
^e r i r N 2N v
Ze< r (1 + + = r)(1_3y )>
r A 5 X
(I.B.1)
Obviously, f can be written as a sum of single ion terms,
f Ze l F(r)(1-3u ),
ion
(I.B.2)
where
\ e"r/X ,, r 1 r2,
F(r) = r3 d + x + 3 a2}*
Therefore, we have
Jd3Nr (I f1)6(ei-E)e"6V
£'
N/d3Nr f 5(e-E)e~BV
= (I.B.3)
ZNQ(e)
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


-zz-

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,Z' .. In symbols, [ ] = Q'(e-E, )Z.l ..
Note that
where g(r) is the radial distribution function and n is the system
volume. Now can be written
e
(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
ao
nh.(k)
2n e 0
where
CO
(I.B.7)
Therefore,
Zen
00
nh,(k)
(I.B.8)
0


J
where the integral over angles, 1^, is given by
In s /dn(1-3y2)j0(k|e-e1|).
(I.B.9)
This integral is calculated in Appendix E with the result
In = J 2^ke ^ 2^ke1 ^ *
(I.B.10)
Substituting this into equation (I.B.8) yields

oo (k) 00
-8TrZeri.eJ dk k2e 1 j_(ke)J dr r2g(r)F(r)j (ke (r))
1 0 0
nh^k)
J dk k e sin(ke)
0
(I.B.11)
This can be expressed in terras of dimensionless variables as
, -ax 2 2
1 ;j 2(ke ) J dx g (x) ^ (1 + ax + )j 2(J e -j)
nh (k)
dk k e sin(ke)
0
-6eJ dk k2e
qh
=
e
I
(I.B.12)
where


-24-
z is scaled in terms of z
f is scaled in terms of e/r
k is scaled in terms of e
-1
0
-o3.
a = Tq/X and
x = r/rQ.
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 =
10^ 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 z
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 21/3 4/3
parameter mentioned earlier, (n ag/Z)/R.Q 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
pi 23
densities of 10 -5, 3*10, and 10 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) Electron u
density = 102^ cm~^; b) Electron density = 3 1cm
c) Electron density = 1 02 cm-^.


-zo-
A


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
23
electron density of 3 x 10 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
+1 T
Ly-B lines of Ar with and without the ion-quadrupol
interaction, at a temperature of 800 eV. a) Electron
density = 10'
23
cm
3.
b)
ectron density
c) Electron density = 10£
cm
-3
3*10
cm


-z?-
DELTA OMEGA (RYD)


FIGURE H
Ly-Y lines of
interaction, at
1023
Ar*1?,
with and without the
a temperature of 800 eV.
ion-quadruple
a)
density
Electron
cm-3; b) Electron density =3*10 cm
c) Electron density = "\0 cm~3.


"Jl
I
A
i \


FIGURE 5
LyB line of Ne+^ at an electron density of 3 lO'0 cm and
a temperature of 800 eV with and without the ion3quadrupole
interaction.


DELTA OMEGA (RYD)


CHAPTER II
OTHER ASYMMETRY EFFECTS
A. Fine Struct ore
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, Hr,
of equation (I.A.38) is often taken to be
2 2 2
H = -fi V /2y Ze 7r
r
(II.A.1)
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 |nlm> states where n is the principal quantum number, , is the
orbital quantum number, and m is the magnetic quantum number. The
eigenvalues, given by
=
(II.A.2)
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 |nljjz>.
The result is (Bethe and Salpeter 1977),
[1 + (
ctZ
2-1-1/2
n
n-(j + 1/2) + /(j+1/2)2~o2Z2
- 1,
(II.A.3)
where a is the fine structure constant (-1/137), and j is the quantum
number associated with the total angular momentum. Little error is
2
introduced in expanding this result to first order in (aZ) ,
[t [J7T72 -
2n 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(o>,e) using the basis |n£m m >, we have
X s
J(,e) - In I
tmlm3
l'W

is11 £ s
Q
i n u uu i i WU a
£ s 1L r 0
£ s
1 _ ^(n)(n)
lit Tr D R .
n
r
(II.A.5)
If we neglect nonradiative transitions between states of different
principal quantum number and consider only the linear Stark effect, the
matrix R^ is obtained by inverting the matrix having elements
.
£ s 1 r 0 1 £ s
(II.A.6)
The operator u-oj -p*e-^ (w) is usually calculated in an |n£m^>
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 m m '. Equation (II.A.3) gives Hp in the
s s
representation |nljJz> It is 3 simple matter to transform to the
|n£m m > basis using the unitary transformation
X S
= (-1)
z x, s
-£+1/2-J £ 1/2
Z^-)1/2 L .
j
-j2)6
££*
(II.A.7)


The symbol
JL l
2
l
3
m
m
2
m
3
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 102^, 3 x 10^, and 10214 electrons per cubic centimeter and a
temperature of 800 eV. Figures 7 and 8 give the same cases for the Ly~8
and Ly-Y lines. Figure 9 shows a Ly~B line of Neon at a density of
23
3*10 electrons per cubic centimeter.
From equation (II. A. i), 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 structure, at a
temperature of 800 eV. a) Electron density = 1023 cm"3;
b) Electron density =3*10 cm ; c) Electron density
= 1 0cuT^.


(') I
- jy-


FIGURE 7
, + 17
Ly-B lines of Ar ', with and without
temperature of 800 eV. a) Electron density
b) Electron density =3*10 cm
= 10^ cm3.
ine structure
1023 cm
c) Electron density
at a
n-3.


-41-


FIGURE 8
+ 17
1, with and without fine structure
temperature of 800 eV. a) Electron density = 102^
Ly-Y lines of Ar
at a
-3.
b)
10"
ectron density =
cm" 3.
10 :
cm
cm
c) Electron density


*T J


FIGURE 9
Ly~B line of Ne+" at an electron density of 3 10 and
temperature of 800 eV, with and without fine structure.


I (A/)
DELTA OMEGA (RYD)
r


B. Quadratic Stark Effect
The quadratic Stark effect causes asymmetry in hydrogenic spectra.
This effect is one order higher in our smallness parameter,
2
(n a^/D/R^, 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 |nlm> and
eigenvalues
E
ni,m
(II.B.1)
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 = Hp + 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 |nqm> with energy levels given by
C 3^]. II.B.2)
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-
s2)
nqm
(1) 1
E 77
nqm 16
a03(|)V(17n2-3q2-9m2+19)
(II.B.3)
where a0 is the Bohr radias.
Consider now, J((D,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(oj,e) as a function of oj has the form of a Lorentzian, whose peaks
correspond to the hydrogenic energy levels for that fixed e. 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 |nqm> 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-a lines of hydrogenic Argon, with and
23
without the quadratic Stark effect, at densities of 10 3*10, and
pii
10 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
23
shows the Ly-B line of hydrogenic Neon at 3 10 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+1^, with and without the quadratic Stark effect, at
a temperature of 800 eV. a) Electron density = 102^ cm~3;
b) Electron density = 3 10 cm ; c) Electron density
= 10^ cm~^.


I (O
-50-


FIGURE 11
Ly-Bof Ar + 1^, with and without the quadratic Stark effect, at
a temperature of 800 eV. a) Electron density = 102^ cm~3;
b) Electron density = 3 10 cm ; c) Electron density
= 102^ cm~^.


(") I
-52-


FIGURE 12
, + 17
with
and
Ly-Y of Ar
a temperature of 800 eV.
b) Electron density = 3
10
cm
~3
wit hout
a) 2§
10
the quadratic Stark
10
af,fect
cm"
23 -3
' r*m J
lectron density =
cm ^ ; c) Electron density
at


-54-



FIGURE 13
Ly-B of Ne+9 at an electron density of 3 10^ cm ^ and a
temperature of 800 eV, with and without the quadratic Stark
effect.


-2.40 -2.00 -1.60 -1.20-0.80-0.40 0.00 0.40 0.80 1.20 1.60 2.00 2.40
DELTA OMEGA (RYD)


FIGURE 14
J(w,e) of a Ly-a line for various values of e.


LOG INTENSITY
-58-
DELTA OMEGA


-59-
J(o),e), for a Ly-a line, as a function of to, 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.


AE(Ryd)
i i i


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(>,e) as in equation (I.A.38). Fine structure is included by
calculating Hp as in equation (II.A.4). The resolvent matrix is
calculated only for states of a given principal quantum number. It has
p
2n rows and columns, twice the number it would have if spin were
ignored. This result can be transformed to the |nqm> representation so
that the quadratic Stark effect may be included. In this representation,
the operator p*e is assumed diagonal and is replaced by
3fi2enq 1
2Zpe 16
aQ3(l)1 (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
p p 23 2 4
densities of 10 3*10, and 10 electrons per cubic centimeter and a
temperature of 800 eV. Figures 17 and 18 show the Ly-0 and Ly-Y lines
under the same conditions.
As mentioned earlier, Lyraan-S 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+1^t with and without the three asymmetry
effects, at a temperature of 800 eV. a) Electron.. .
density = 10 -> cm b) ]
c) Electron density = 10'
ectron density = 3 x 1 0 cm
1 cm~^.


-64-
DELTA OMEGA (RYD)


FIGURE 17
Ly-B lines of Ar
+ 17
effects, at a temperature
density = 10^3 cm-^; t>)
c) Electron density
with and without the
of 800 eV. a)
lectron density
^ cm~3
three asymmetry
Electron
= 3 10:5 cm~^
10£


-66-
DELTA OMEGA (RYD)


FIGURE 18
Ly~Y lines of
effects at a
Ar
+ 17
density = 10
with and without the
temperature of 800 eV. a)
density
,23 crafc3.
b)
c) Electron density = 1 0
Electron
' cm-3.
three asymmetry
Electron
= 3 10' cm 5


-68-
DELTA OMEGA (RYD)


-69-
will still be a two peaked function. Let us define a measure of asymmetry
for the Lyman-6 line as 2(Ib -Ip)/(Ib +Ip) where Ib and Ip are the blue
and red peak intensities, respectively. In Figure 19, we plot this
asymmetry as a function of density for an Ar+1^ 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
23 -3
asymmetry at 4 x 10 cm 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.


FIGURE 19
Asymmetry of Ly~B. defined as 2(1 -1^)/(I^+I^), as a function of
density.


ASYMMETRY
3x10 10
ELECTRON DENSITY
(cm-3)


-72-
TABLE 1. Contributions to the asymmetry of the Ly-B line of Ar+1^
1023
3 x 1023
1024
O
X
m
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,
+Q pp O
Ne J lines are nearly symmetric at 10 cm
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).


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, "i,", 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 -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
3 2 1
1s2p P -* 1s S, 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(id) = Jde Q(e)J(w,e),
J(co,e) = ~ Im Tr D{id-(Hp-uQ) pe ^( a
The trace is taken with respect to singlet states of a given principal
quantum number. The Hamiltonian, Hr,is assumed to have eigenvalues
2
E 0 = + w(n,2,Z), (IV. A. 2)
n2m 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 2,-dependent energy shift. This shift represents the
effect that the ground state electron has on the energy levels of the
excited electron. Clearly, Kn.i.Z) removes the 2-degeneracy.
The values for iD(n,2,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,i,.j,Z) )(n,£2,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
o-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, -w /2, and yet the line is markedly asymmetric. There are two
sp
reasons for this asymmetry. First, the unshifted components are centered
about zero (the 4-1 energy level) rather than the symmetry point
-uj /2. Secondly, the transition probabilities for the shifted
sp
components are not equal. The shifted components are linear combinations
of the states 1200> and |210>. In the absence of any field these states
are not mixed and have relative energy levels -uj and zero. This gives
sp
rise to asymmerty because the state |200> is dipole forbidden. Only for
large electric fields is the situation symmetric, as the eigenvalues are
then (I200> |210>)//2.
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 o, 8, 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
occuring in the same experiment. This problem may be greatly reduced
through the time-resolved spectroscopy techniques presently available.


FIGURE 20
4.I 7
Helium-like a-line of Ar at an electron density o
1023 cm~3 and a temperature op 800 eV.


-78-


FIGURE 21
Helium-like B-line of Ar+17 at an electron density of
10^3 cm~3 and a temperature of 800 eV.


IUO
-80-
DELTA OMEGA (RYD)


FIGURE 22
+1 T
Helium-like Y-lir.e of Ar at an electron density o
10criT^ and a temperature of 800 eV.


'") I
-82-


APPENDIX A
ZWANZIG PROJECTION OPERATOR TECHNIQUES
From Eqn. (I.A.15) we have
Km) = Jdf W(f) J(oj,4*) (A. 1)
and
Jiu.'f) = ir1Re / dt elltTr d? (a ),
0 a
where = f 1(a,f) Tr p('V),
P
f(a.'i') = Tr p 5(4'-$*),
P
W(4') = Tr p 5(4'-$*),
and
p(40 = p 5(4'-$*)/W(4').
Define an operator P by
PX s = f1 (a ,4*) Tr p(f)X.
P
Note that
(A.2)
-83-


-84-
P2X f1(a,Â¥) Tr [p (f )f~ 1 (a ,Â¥) Tr p (y )x]
P P
.-1
f1(a,4')(Tr p(f)) f"1(a,4') Tr p(f)X
PX
(A.3)
so P2 = P.
This property, idempoter.ee, defines a projection operator (Reed and Simon
1972). We can now write J(oj,y) as
oo
J(oo,y) = u~1Re Tr d-f(a,f) / dt e1(i)tPd(-t)
a 0
=* ir 1Re Tr df(a,'F) (A.4)
a
00
where o& (oo) = P&iu), and ?<5"(io) = J dt elu,td(-t) The Zwar.zig Projetion
1 0
operator technique can be used to obtain a general expression for
u>).
Let D(t) = d(-t). We have
i 5 (t) = LD (t).
(A.5)
Let PD = D-| and (1 P)D = D2 so that D-| D2 =
D. Then


-85-
i D2 = C1-P)L(D1 + D2),
and (A.6)
i D1 = PL(D1 + D2).
Therefore
Ot + i (1i-P)L)D2(t) = *! (1-P)LD^ (t) ,
(A.7)
which has the solution
DpCt) .lt('-PV<0) i S
0
(1P)LD1(t-s)ds. (A.8)
For our choice of P, D2(0) = 0. Therefore
i D
51 (t) = PLD] (t) i I PL
e",l3(1'P)L(1-P)LDl(t-s)ds. (A.9)
This can be transformed using the Convolution Theorem,
. r lout* .. r i)t_ ... ... r iu)t -lit (1-P)L
i j e D.| (t )dt = PL J e D^itjdt iPL J e e
dt
9 (1aP)L J el)SD (s)ds,
(A.10)
or
-10,(0) + m&Au) = PL (a)) + PL 7¡--Vr (WLdfr, U). (A.11)
So
c0>1 (uj) = i[u-PL-PL(ctf*QLr1QL]1D1(0),
(A.12)


-86-
where Q = 1-P. Putting this in (A.4) yields
) = -tt"1 Im Tr d-f(a,?) [u- ]d. (A.13)
a
*
Let L = Ln + Lt($ ) and L_ = L(a.'j') + L where Ln is the Liouville
OI 0 p u
operator for the isolated atom and Lp is the Liouville operator for the
plasma. Then,
= + + *) >. (A. Hi)
The first term is
X = f~1(a,Â¥) Tr p(t)L(a )X = L(aff)Xo ,
cl cL a
P
and the second term is
Xa = f~1(a ,Y) Tr p(T)(LpXa) = 0,
because LpXa = 0, so = Lta.Y) + when acting on atomic
functions. Also
QL0Xa Va Tr p(*)L0Xa
LaXa Tr l P
LX *- L X =
a a a a
0
(A.15)


-87-
This allows us to replace the last L by 1^(4 ) in ^(to). Now,
O
using also Q = Q, we can write
J (uj.y) = -it 1 Im Tr d-f (a ,Â¥) [oo-L(a )-
a
(A.16)
-]"1 d.
Now we work on the first L in Consider
PLQX = f1(a,T) Tr p (4)LQX
P
= f~1(a,4) Tr Lp (4* )QX
P
= f~1 (a ,4*) Tr (L(a,40 + L + 1^(4*) )p (4* )QX (A.17)
P P
The first term is
f~1 (a ,f) Tr L(a )p (4* )QX
P
=* f~1 (a ,4* )L(a ,4*) Tr p (4 )X
P
- f *"|a ,4* )L(a ) Tr p(4-)f ""la,?) Tr p(4')X
P P
- f"1 (a,4')L(a,4>) Tr p (4 )X f-1 (a ,4-)L(a ,4-) Tr p(4-)X = 0. (A.18)
P P
The second terra on the right hand side of (A.17) is


-88-
f ~1 (a ,4*) Tr L p(f)QX = f~1(a,40 Tr L p(40QX.
P P P
But Tr L M = Tr H M MH
P P P
P P
Since Hp is an operator only on perturber
states and these states are being traced over, it is legitimate to
cyclicly permute Hp, i.e. Tr H M = Tr MH Therefore,
P P P P
.*1
(a,>f) Tr L p(40QX = 0.
P
(A.19)
This leaves
<1(10,40 = -it1 Im Tr d*f (a ,40[to-L(a ,40
a
- f~1(a,40 Tr LI(i*)p(4')((o-QLQ)~1QLI($*)]'1d. (A.20)
P
This is the result shown in equation (I.A.18).
Now consider
(¡0,40 = f 1 (a) Tr L p (4-) (ur-QLQQ )"1QL].
P
CO
= f1(a) Tr Ltp(40 l ( QL.Q)nQLx. (A.21)
p 1 113 r.-0 1
From equation (A.18) we have PLqQX = 0, or QLqQX = LqQX for arbitrary X.
Using this, one may write
( I ¡o ¡o 0 I
p n=0
= f~1(a) Tr LIp(4')( P
(A.22)


-89-
Noting that (10,40 acts only on atomic functions, we can write
PLtX
I a
This yields
f1 (a) Tr p 6(4'-$*) Tr p LTXo
i e
Tr p.5(4'-4>*) Tr p
i ae
i ae
(A.23)
(io,40 = f~1 (a) Tr LIp(4')((o-L0)1LI ,
P
(A.24)
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 multi pole
expansion with odd "," values contribute a symmetric effect on the line
shape, to first order in perturbation theory.
Consider hydrogenic states |nqm> 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 o terms
like
M = .
Im
(B. 1)
Switching to the spherical representation by
|nqm> = £ |ni,m>
l
where
(B. 2)
gives
-90-


-91-
M = l 1m1 |r i |n£,2m2>(~1)
" Im
1/2(1+m^-q-n)
l'\l2
1/2(1+M -q-n)
9 M) /(2£ +1)(2l +1)
/ n-1 n-1
2 2
/ml ml \
2 2 2
\
n1q n^+q
~r ~~t mi,
ra2-q m2+q
-m
2/
Now,
j R R r1' dr- /do Y Y..Y .
Hm 0 1 2 *'11 Urn 2 2
The angular integral is (Edmonds 1957)
m, (21 +1) (2Z +1) (22.+1) / ^1 1 Z *1 1 l2\
(-1) V 1
4ir
0 0 0 / V -m1 m m2y
Let R represent the radial integral.
*1*2
mi+m2,
M l Ro o (-1) 1(-1)
4^2 12
m. (-^-)',q~n (2.+1)2(2H +1)2(2+1)
/
ml ml J
2 2
m1 ~q n^-q
~2~ T
A
-mj
ii mi
2 2
m2~q ra2_q
\ 2 2 ~m2
(B.3)
(B.4)
(B. 5)


-92-
A1 l l2
(l1
i *2\
o
o
o
Vmi
m m 2 /
(B.6)
Now consider M' = . This is the same perturbation on
1 i
the state which is shifted opposite to the original state, by a uniform
electric field.
m.
m1 +m2
I Ro o ("1) 1(1)
i}i2 12
, +q~n (2t1+1)2(2l,+1)2(2l+1)
2 / l 2
4 ir
n-1
sil .A IkX %z\ h. I
2 1
m, +q m1 -q
\-V -W v
2 2
m2+q m2-q
T T
w
^ l l2
o cyl-i
m. m m
2*
(B.7)
Recalling the symmetry property
('
M
, A2 M
^1+^2+^3
(-1) 5
(B.8)
\m1
m2 m31
^ m2 ml m3/
and q is an integer so (~1)+C1 = (-1)^, one obtains

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FILES


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

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.
ii

TABLE OF CONTENTS
Page
ACKNOWLEDGEMENTS ii
LIST OF FIGURES iv
ABSTRACT vii
CHAPTER
I. ION-QUADRUPOLE INTERACTION 1
A. Formalism with Ion-Quadrupole Interaction 1
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
1
2
3
4
5
6
7
8
PAGE
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
+ 1 7
Ly-a lines of Ar , with and without the ion-quadrupole
interaction, at a temperature of 800 eV.
(a) Electron density = 1023 cm~3 __
(b) Electron density = 3 *10 _ cm
Electron density = 1 02 cm”3
(c)
.25
Ar+1?,
Ly-B lines of
interaction, at a
(a) Electron dens
(b) Electron density
(c) Electron density
with and without the ion-quadrupole
emperature of 800 eV.
ty = 1023
= 3
10
,10
24 -3
cm ->
cm
.28
+ 17
with and without the
a temperature of 800 eV.
nsity = 1023 cdj~3
Ly-Y lines of Ar
interaction, at
(a) Electron density = UT-' cm,
(b) Electron density = 3 *,.10 _cm
ion-quadrupole
(c) Electron density = 10
24 -3
cm
.30
+ Q
Ly-B line of Ne at an electron density of 3 x 10
and a temperature of 800 eV with and without the
ion-quadrupole interaction
23 -3
cm
,32
Ly-a
at a
(a)
(b)
(c)
,+17
lines of Ar
temperature of
Electron density
with and without fine structure,
800 eV.
1 0^3 cd) 3
Electron density = 3 «10 cm
Electron density - 1 02
cm
,38
+ 17
Ly-B lines of Ar , with and without fine structure
at a temperature of 800 eV_,
Electron density
(a)
(b)
(c)
Electron density
Electron density
1C¿3
3 x,10
1024 cm-3
cm
-3
.40
Ly-Y
at a
(a)
(b)
(c)
lines of Ar+1^i
temperature of
with and without
800 eV.
fine structure
Electron density = 10^3 Cm~3
Electron density =3*10 cm 3
2? — "3
Electron density « 10 J cm
,42
iv

9
10
11
12
13
14
15
16
17
18
19
44
49
51
53
55
57
60
63
65
67
70
_ Q Q
Ly~B line of Ne+' at an electron density of 3 x 10
and a temperature of 800 eV, with and without fine
structure
Ly-a of Ar+1^> with and without the quadratic Stark
effect, at a temperature of 800 eV.
(a) Electron density = 1 O2^ cm ^
(b) Electron density = 3 *10 cm
(c) Electron density = 10 cm
Ly~B of Ar+1^, with and without
effect, at a temperature of 800
(a) Electron density - 102-^ ci
(b) Electron density = 3 *,10 „
(c) Electron density =10 cm 0
the
eV.
quadratic Stark
?3 -3
cm
Ly-7 of Ar+1^, with and without the quadratic Stark
effect, at a temperature of 800 eV.
(a) Electron density = 102-* cm~^
(b) Electron density = 3 *10 _ cm
(c) Electron density = 1 02 cm ^
Ly-B of Ne+^ at an electron density of 3 * 10 cm and
a temperature of 800 eV, with and without the quadratic
Stark effect
J(cu,e) of a Ly-a line for various values of e
Energy levels of the n=2 states of hydrogen as a function
of perturbing field strength
Ly-a lines of
effects, at a
(a) Electron
(b) Electron
(c) Electron
|1 7
Ar ', with and without the three asymmetry
temperature of
density = 102^
density = 3 *1
density = 102 cm
800 eV.
2? -3
Ly-B lines of
effects, at a
(a) Electron
(b) Electron
(c) Electron
xl 7
Ar ', with and without the three asymmetry
temperature of 800 eV.
density • 102^ cm”3
density = 3 *10 cm
density = 102 cm-^
Ly-T lines of
effects, at a
(a) Electron
(b) Electron
(c) Electron
+1 T
Ar , with and without the three asymmetry
temperature of 800 eV.
density = 1 O2^ cm~3 __
density = 3 * 10¿ 0111
density » 102i4 cm-^
Asymmetry of Ly-B, defined as 2db-Ir)/db+Ip), as a
function of density
v

20
Helium-1;
10¿:í cm :
Lke a-line of Ar+1^
* and a temperature
21
Helium-1;
10*5 cm*-1
.ke g-line of Ar+1^
' and a temperature
22
Helium-1;
10¿:> cm :
ke Y-line of Ar+1^
* and a temperature
at an electron density of
of 800 eV 77
at an electron density of
of 800 eV 79
at an electron density of
of 800 eV 81
vi

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
ION»QUADRUPOLE INTERACTION
A. Formalism with Ion-Quadrupóle 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 multipole expansion of the radiator as a
localized charge distribution p(x) in an external potential 4>(x) caused
by the perturbing ions (Jackson 1972). The electrostatic energy of the
system is
W = Jp (x )(x )d^x
(I.A.1)
The potential can be expanded in a Taylor series with the origin
taken as the radiator center of mass.
324>( 0)
4>(x) = 4>(0) + x• V4>(0) + \ l x x . a
2 ij 1 J 3Xi9Xj
(I.A.2)
Div(E(0) = 4ir£z.e6(x. ), where the sum is over all perturbing ions and z^e
i 1 1
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
9E.
4>(x) = 4»(0)
or,
x-E(O) - ¡ ^ OXjXj - r28iJ) jji(0) *
(I.A.3)
-1-

-2-
W = qí>( 0) " d-É(O)
2
6
I
ij
3E.
i
3E.
J
(0) +
• • • »
(I.A.4)
where
q is the total charge of the radiator,
d = <\}j|~er [i|>> is the dipole moment of the radiator and
Q = is the quadrupole tensor of
the radiator.
This multipole expansion is basically an expansion in the atomic
2
radius divided by the average inter^ion spacing, or (n aQ/Z)/Ri0 where n
is the principle quantum number of the radiator, aQ is the Bohr radius, Z
is the radiator nuclear charge and R^q 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 ' plasma with an electron density of 10 cm
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/zS Now, if we take these as our new basi3
states and include the effect of the field gradients due to the

perturbating ion, the result is shown in Figure 1. The higher energy
levels, which give rise to the blue wing intensity, (id > m0> where hm^
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 < m^) , 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. Similiar 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 er.semble^averaged radiation emitted by one ion of
that t ype as
P(w) « ~ I | | 2 p 6(u«wahI(io), (I. A.5)
3c3 ab a aD 3c3
where p is the probability that the system is in state |a>, d is the
El
radiator dipole moment, and m . = (E *E. )/h. The second equation
ab a b
defines the line shape function Km). Using the integral definition for
the Dirac delta function, we obtain,
. ,°° - Í(ü)'O) h)t
I(w) * J l I I p_e dt. (I. A.6)
*<» ab
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 o'" a uniform electric field and in the presence o
field gradient.

-5-
APPLY
ELECTRIC
FIELD
\ ÍA2
J }A2
\} A2
APPLY
FIELD
GRADIENT

-6-
write l(u) as an integral from zero to infinity:
, ® o i(u-ü) . )t
Km) = - Re Í l I | p e D dt
IT K a
0 ab
^ Re j l elwt•dt
0 ab
- Re J dt elü)tTr(d*pe iLtd).
(I.A.7)
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 (198*0. 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 , which is
ap
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
Vap " 1 * v°(a'J)
0 J = 1
No Z Ze2
I I [-*—
oj-1 I?, I
Z e
o
i *
r -r.
' a 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
a a
respectively. Expanding this expression in Legendre polynomials, we
find,
where
oo r i
’(a.j) - -Zoe I [-^j- - WC03V’
d-0 r>-
(I.A.9)
r^ = smaller of Ir I and |r.
< 1 a1 1 j
r> = larger of |ra| and |r
6 . = angle between r and r.
aj a j
This expression may be regrouped as
v°(a,j) = 0°(j) + V1 °(a, j ) + v2°(a,j),
(I.A.10)
where
£
4>0 (j) is the monopole term given by Z (Z-1)e /r,

-o-
■ -V2 X VC03V’
£=1 r.
and
v2 (a, j )
r. I
00 J p
-Z e2 I [—2_ —]p.(cose .)
0 lío r £+1 r *+1 * aj
a Jc
0
r. ir
Jo 3
r. > r
Jo a
The sum of v 0 + n° is identical to v° in the region r. > r , the
i U j a
no penetration region, but extends this form to all values of rj. In
this form, v^0 may be written as a "product" of an operator on perturber
coordinates times an operator on radiator coordinates, in the following
manner,
v^ia.J) = M(a)$°(j)
ao
- - I y (a)* °(J)
K-1 K K
= - (d*e (j ) + g- £ Snn2mGn + **'^* (I.A.11)
mn
Here M(a) stands for the set {y (a)} which depends only on the radiator
state. For example, y^ = d, which is the radiator dipole moment, and
l¡2 - Q/6, which is the radiator quadrupole moment. The perturber
operator, 4>a(j), depends only on the jth perturber of type o. The
term 4>°( j ) represents the set |k°(j)}, where 4> 1 a (j) - e°(j) is the

-9-
electric field at the radiator produced by the jth perturber of
type o, 4>2°(j) ” 9men°(j) is the electric field gradient at the radiator
produced by the jth perturber of type o, and so forth. By design,
couples to $k°.
We next generalize the usual definition for an electric microfield
distribution, W(e) = Tr p(e-I ), to include the field gradients and
i
higher order derivatives which are used in equation (I.A.11):
W(y) » Tr pfi(¥-•*), (I.A.12)
00
* #
where 5(Y- ) = n 5(T.,-4> ).
K-1 K K
The y and 4> * are analagous to $ 0 in equation (I.A.11). That is,
K K K
they denote the value of the electric field, field gradient tensor, etc.
*
at the radiator, produced by the perturbers. In the above, $ is an, as
yet, arbitrary function of only ion coordinates and f is a c-number. Now
equation (I.A.7) can be written as
00
I(u) - - Re f dt elü)t Tr{d*Tr pfd¥5(’F-* )d(-t)}
7T J J
0 a p
00
*= fdy W(y) ^Re [ dt elü)t Tr|d*Tr p5(y-$*)/W(y)d(-t)}
* o a p
- Jdy w(y)j(uj,y), (1.A.13)
where

-10-
J(u,¥) - - Re í dt elü)t Tr|d-Tr p(»)3(-t)},
u 0 a p
and
p(Y) = p5(Y-**)/W(Y).
(I.A.14)
We may rewrite J(,'P) as
J(cú,’1')
dt e
iuit
Tr d*f(a ,V)D(t) ,
a
(I.A.15)
where
D (t) = ,
= f ^(a,40 Tr p(Y)X for any X, and
P
f(a,Â¥) = Tr 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
N
o
H = H ♦ H + I I v°(a,J)
3 P o J-1
N
H + H + l l 4>na(j) + v °(a,j) + M(a)$°(j).
3 P o j-1 ü ¿
(I.A.16)
The atomic Hamiltonian, Ha, includes the center of mass motion of the

-.Li¬
atón) 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'iLtp('F) = p(Y).
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(y) allows us to make
*
the replacement = $ + y - i> in H, giving
N
H - H + M(a)y + H + I I + I l v°(a,j) + M(a)(£ I 4>°(j )-$*)
a P 0 j ° c j 2 o j-1
H + H
V2 +
V^* ),
(I.A.17)
where
H = H + M(a)Â¥,
a a
»p ■ HP * H »00(J)'
0 j
v2 - I I v2°(a,j),
V (♦*) - M(a)(I l *°(J) - ♦*).
o j
and

-12-
We have combined Ha and M(a)'F because these involve only atomic
coordinates. Similiarly, the monopole term, £ £ 4>0 ° (j)» does not depend
o j
on internal atomic coordinates and therefore is combined with Hp.
*
Let L(a,4'), Lp, L2> and ($ ) be the Liouville operators corresponding
_ _ *
to H , H , M0, and V.(i> ) respectively. Further let
3 P eL
# #
L ($ ) = 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(o>,'F). The details are
in Appendix A and the result is
Jtw.Y) = -Ti”1Im Tr{d-f (a ,Y) [u»-L(a )-B(V)-^ (w,’F) ]"1d }, (I. A.18)
a
where
and
B(?) = ,
V U,40 = f 1 (a,V)Tr L (**)p(^) (u-QLQ)”1QLI(**) ,
P
Q = 1-P, where P is a projection operator given by
P(...) - <(...)>.
We wish to cast the expression for I(co) in a more familiar form, one
which uses the standard microfield. To this end, we must separate out
the electric field dependence from v. Let 4* - {e,4**}, that is, V' is
all of V except for e. Now,

-13-
W(Â¥) = Q(e)W(e|r)
(I.A.19)
where Q(e) is the usual microfield (Tighe 1977; Hooper 1968) and
W(g |y) is the conditional probability density function for finding
Â¥' given e. Then,
(I.A.20)
which is the desired form, but now
(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 4> , which has been arbitrary up to this point, to force
This implies that

— -L *-t —
6 Í *
0 = Trp(\J))($ + $ - $> )
P
= Tr 6(i|)-4>*){Tr p(*e + 4»1) - p * )/W(t|>), (I.A.22)
i e
where p^ = Tr p.
e
To satisfy the above, it is sufficient to define
* -1 e i
$> 2 p^ Trp( + 4>)
e
= 4-1 + Pi"1 Tr p$®. (I.A.23)
e
*
Obviously, $ 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>i, in the manner above. With
*
this approximation, the ion parts of ) cancel, leaving
e
V ($ ) = M(a) l $ (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
* *
the sum of single particle terms.) Note that V^(# ) and Lji* )
*
no longer depend on $ ; therefore they will henceforth be denoted
simply V1 and Lj, respectively. Neglect of electron-ion

- 15-
interactions also allows the density matrix to be factored.
Thus f(a,f) can be reduced to
*
Tr p 6(^-4» )Tr p
.1 36
f(a,i|0 - s-5
Tr p )Tr pag
i ae
Tr
e
Tr
ae
f (a).
(I.A.2H)
H. 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 (w,4>) which is
second order in Lj is appropriate (Smith, Cooper, and Vidal
1969). This is accomplished by replacing the L in w(o),4>) by
Lq. Further simplification concerning the projection operators
is given in Appendix A, with the result
£/(ü>,*) - f"1(a)Tr . (I.A.25)
P
Note that the Lq 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-
- f 1 (a)Tr LTp (íü-L(a,iJ))-L )
1 30 0 1
e
= f"1(a)Tr l Lt(j )p (cú~L(a,iJ/) - l L (k))"1 l LAl),
e j 1 ae k 6 i 1
(I.A.26)
where the Liouville operators have been written as sums of single
electron operators. Terms with are zero by angular average
and terms with k*i, are zero because L (k) is zero unless acting
on functions of electron k.
= f~1 (a )n Tr L (a, 1 )f (a, 1) (u-L(a,ip) - L (1)) ~1L (a, 1),
1 i
where nf(a,1) = Tr p . (I.A.27)
Kae
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 collions 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 (oj) to (Tighe 1977)

-1/-
00
e
-iH( 1 )t/fi
h 1 0 i"
ii"
i H (1 )t/fi
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, Aw = w - (E^E^/fi
and subscripts i (f) stand for initial (final) radiator states.
7. Equation (I.A.27) includes the ion shift of radiator states
in “^(w.Y) through L(a,Y). Dufty and Boercker (1976) found that
(w,40 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,40 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 4” completely,
replacing L(a,40 by L(a,e) and J(w,40 by J(w,e), this reduces to the
full Coulomb (electron interactions) of Woltz et al. (1982). Instead,
limit 4M 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
I(oj) = íde de de de de de de Q(e)
1 xx yy zz xy xz yz
W ( E 1 E , E , • • • ) J ( QJ, £ , £ fE , . . . ) ,
1 xx yy xx yy
(I.A.29)
3E
i
where e.. s ,,
We now make the following approximation
W(e|e ,e ,...) ■+ 6(e - , £ -
1 xx yy xx xx e yy yy e
(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 £, but introduces second order
2 2
errors which are proportional to - . These should be
ij e ij e
2
three orders higher in the smallness parameter n Aq/Z/Rq^.
This yields
Kin) - Jde Q(e) J(üj,e) ,
with
J(oi,e) = -n 1Im Tr {d *f (a) (oi-L(a ,e , ) -&( to)) 1d}. (I.A.3D
- ^ J ^

-19-
The Hamiltonian corresponding to L(a ,e,<£_>e) *s
H -p*e - Y Q. . /6 where p is the radiator dipole operator and Q is
a F ij ij e
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{oj— (H ~o)n) - p*e -
it r u
r
l Q ,/6 - # (u)
ij J J
i-1
(I.A.32)
where 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 *. Therefore,
9 E
x z
(Ze) — e
r
zx _-r/X f-3
\r
-L-2-)
X2r r3
(I.A.33)
and
9 E
z z
(Ze)e
-r/X
[(^a + “4> _ + 4- + \>]
Xr
Xr
X2r
(I.A.3^)
where y = cos(9) and X 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.3*0. Since only the z direction is special
(J is along a), £ - £ - £ - 0.
Therefore,

-zu-
y Q. , =Q+Q+Q
lj ij e xx xx e yy yy e zz zz e
(Q + Q ) + Q
xx yy xx e zz zz e
-Q + Q
ZZ XX E ZZ ZZ E
1 Q ( - 4 )
2 zz zz e 3 e
(I.A.35)
For the Debye-Huckel form,
So
V•e = (Ze) e'rA/r.
A
< E
ZZ
1 *r/A 2 _
V• e> = (Ze)<^-r— (1 + y + )>
j e 3A e
= .
E
(I.A.36)
(I.A.37)
Now J(ai,e) may be written
J(w,e) * - - Im Tr d{oj-(H
V -
P * E - Q
/i| - #(«)}
ZZ E
-1
(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


zz 3 e
-r/X . 2 -
„ .e ,, r irw„_2N.
Ze< r— (1 + — + rr —-) ( 1 ~3w )> ■
rd A J X
(I.B.1)
Obviously, f can be written as a sum of single ion terms,
f - Ze I F(r)(1-3u ),
ion
(I.B.2)
where
N e"r/X ,, . r . 1 r2,
F(r) = r3 d + x + 3 a2}*
Therefore, we have
Jd3Nr (l fi)6(e-E)e~6V
£ ' fd^r 5!j4)e-8V
N/d3Nr f 5(e-E)e~6V
= - (I.B.3)
ZNQ(e)
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

-zz-

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,Z' .. In symbols, [ ] = Q'(e-E, )Z.\ ..
N~1 I
Note that
where g(r) is the radial distribution function and n is the system
volume. Now can be written
e
(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
ao
nh.(k)
2it £ 0
where
00
(I.B.7)
Therefore,
Zen
00
nh.(k) ®
(I.B.8)
0

J—
where the integral over angles, 1^, is given by
In s /dn(1-3y2)j0(k|e-e1|).
(I.B.9)
This integral is calculated in Appendix E with the result
IP = J 2 ^ke ^j 2 ^ke 1
(I.B.10)
Substituting this into equation (I.B.8) yields
oo (k) 00
-8irZeTi.eJ dk k2e 1 j?(ke)J dr r2g(r)F(r)j (ke (r))
1 0 0

® hh. (k)
dk k e sin(ke)
0
(I.B.11)
This can be expressed in terras of dimensionless variables as
Hkj'dk k2entl1 (k,j2(kc)/o
-ax 2 2
e ,, ax..,. .
—— (1 + ax + —
=
® ph, (k)
J dk k e sin(ke)
0
(I.B.12)
where

-24-
z is scaled in terms of z
f is scaled in terms of e/r
k is scaled in terms of e
-1
0 ’
-o3.
a = Tq/X and
x â–  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 =
10^ 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 z
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 21/3 4/3
parameter mentioned earlier, (n Sq/ZJ/R ^ - 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
pi 23
densities of 10 -5, 3*10, and 10 electrons per cubic centimeter and

FIGURE 2
Ly-a lines of Ar*1”7, with and without the ion-quadrupole
interaction, at a temperature of 800 eV. a) Electron u
density = 102^ cm~^; b) Electron density = 3 * 1cm
c) Electron density = 1 02 cm-^.

-zo-
A

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
23
electron density of 3 x 10 . 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
+1 T
Ly-B lines of Ar , with and without the ion-quadrupol
interaction, at a temperature of 800 eV. a) Electron
density = 10'
23
cm
â– 3.
b)
ectron density
c) Electron density = 10£
cm
-3
3 x 10°
cm

-z?-
DELTA OMEGA (RYD)

FIGURE H
Ly-Y lines of
interaction, at
1023
Ar*1?,
with and without the
a temperature of 800 eV.
ion-quadrupóle
a)
density
Electron
cm-3; b) Electron density =3*10 cm
c) Electron density = "\0¿ cm~3.

- Jl-
I
A
i \

FIGURE 5
Ly“B line of Ne+^ at an electron density of 3 * 'm'5 cm and
a temperature of 800 eV with and without the ion3quadrupole
interaction.

DELTA OMEGA (RYD)

CHAPTER II
OTHER ASYMMETRY EFFECTS
A. Fine Struct ore
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, Hr,
of equation (I.A.38) is often taken to be
2 2 2
H = -fi V /2y - Ze 7r
r
(II.A.1)
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 |nlm> 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
=
(II.A.2)
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 |nljjz>.
The result is (Bethe and Salpeter 1977),
- [1 + (
ctZ
2-1-1/2
n
n-(j + 1/2) + /(j+1/2)2~o2Z2
- 1,
(II.A.3)
where a is the fine structure constant (-1/137), and j is the quantum
number associated with the total angular momentum. Little error is
2
introduced in expanding this result to first order in (aZ) ,
- ¿[t * —[J7T72 -
2n 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(m,e) using the basis |n£m m >, we have
X» s
J(«,e) - - - Im I
tmlm3
l'W

is11 £ s
9 ) ] 1 |n£m m >
i n ui uu i i WU a
£ s 1L r 0
£ s
1 _ _(n)_(n)
lit Tr D R .
n
r
(II.A.5)
If we neglect nonradiative transitions between states of different
principal quantum number and consider only the linear Stark effect, the
matrix R^ is obtained by inverting the matrix having elements
.
£ s 1 r 0 1 £ s
(II.A.6)
The operator u)-toQ-p*e-^ (to) is usually calculated in an |n£m^>
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 m * m '. Equation (II.A.3) gives Hp in the
s s
representation |nljjz>. It is a simple matter to transform to the
|n£m m > basis using the unitary transformation
X# S
= (-1)
Z Xj s
-£+1/2-j 1/? £ 1/2
Z^-)1/2 L .
j
-j2)6
££’*
(II.A.7)

The symbol
JL l
2
l
3
m
m
2
m
3
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 102^, 3 * 10^, and 10214 electrons per cubic centimeter and a
temperature of 800 eV. Figures 7 and 8 give the same cases for the Ly~8
and Ly-Y lines. Figure 9 shows a Ly~B line of Neon at a density of
23
3*10 electrons per cubic centimeter.
From equation (II. A. *0 , 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 structure, at a
temperature of 800 eV. a) Electron density = 1023 cm"3;
b) Electron density =3*10 cm ; c) Electron density
= 102^ cm"3.

('») I
- jy-

FIGURE 7
, + 17
Ly-B lines of Ar ', with and without
temperature of 800 eV. a) Electron density
b) Electron density =3*10 cm
= 10^ cm”3.
ine structure
1023 cm
c) Electron density
at a
n~3.

-41-

FIGURE 8
+ 17
1, with and without fine structure
temperature of 800 eV. a) Electron density = 102^
Ly-Y lines of Ar
at a
-3.
b)
10"
ectron density =
cm" 3.
10 :
cm
cm
c) Electron density

— n j

FIGURE 9
. o Q
Ly~B line of Ne+" at an electron density of 3 * 10 and
temperature of 800 eV, with and without fine structure.

I (A/)
DELTA OMEGA (RYD)
r *->

B. Quadratic Stark Effect
The quadratic Stark effect causes asymmetry in hydrogenic spectra.
This effect is one order higher in our smallness parameter,
2
(n a^/D/R^, 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 |nlm> and
eigenvalues
E
ni,m
(II.B.1)
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 = Hp + 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 |nqm> with energy levels given by
C ‘ - * 3^]. ÍII.B.2)
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-
sí2)
nqm
„(1) 1
E - 77
nqm 16
a03(|)4e2(17n2-3q2-9m2+19)
(II.B.3)
where a0 is the Bohr radias.
Consider now, Jiw.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(ü>,e) as a function of to has the form of a Lorentzian, whose peaks
correspond to the hydrogenic energy levels for that fixed e. 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 |nqm> 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 Hp + H1.
Figure 10a, b, and c show Ly-a lines of hydrogenic Argon, with and
23
without the quadratic Stark effect, at densities of 10 , 3*10, and
pii
10 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
23
shows the Ly-B line of hydrogenic Neon at 3 * 10 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) Electron density = 102^ cm~^;
b) Electron density =» 3 * 10 cm ; c) Electron density
= 10^ cm~^.

I («O
-50-

FIGURE 11
Ly-Bof Ar + 1^, with and without the quadratic Stark effect, at
a temperature of 800 eV. a) ^Electron density = 102^ cm~3;
b) Electron density = 3 * 10¿ cm ; c) Electron density
= 102^ cm~^.

(") I
-52-

FIGURE 12
, + 17
with
and
Ly-Y of Ar
a temperature of 800 eV.
b) Electron density = 3
10‘
cm
~3
wit hout
a) 2§
10°
the quadratic Stark
10
af,fect
cm"
23 -3’
—' r*»m J •
lectron density =
cm J ; c) Electron density
at

-54-
á

FIGURE 13
Ly-B of Ne+9 at an electron density of 3 » 10^ cm ^ and a
temperature of 800 eV, with and without the quadratic Stark
effect.

-2.40 -2.00 -1.60 -1.20-0.80-0.40 0.00 0.40 0.80 1.20 1.60 2.00 2.40
DELTA OMEGA (RYD)

FIGURE 1U
j(to,e) of a Ly-a line for various values of e.

LOG INTENSITY
-58-
DELTA OMEGA

-59-
J(ü),e), for a Ly-a line, as a function of to, 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.

AE(Ryd)
i i i

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(ü>,e) as in equation (I.A.38). Fine structure is included by
calculating Hp as in equation (II.A.4). The resolvent matrix is
calculated only for states of a given principal quantum number. It has
p
2n rows and columns, twice the number it would have if spin were
ignored. This result can be transformed to the |nqm> representation so
that the quadratic Stark effect may be included. In this representation,
the operator p*e is assumed diagonal and is replaced by
3fi2enq 1
2Zpe ” 16
aQ3(l)1 (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
p p 23 p ii
densities of 10 3*10, and 10 electrons per cubic centimeter and a
temperature of 800 eV. Figures 17 and 18 show the Ly-0 and Ly-Y lines
under the same conditions.
As mentioned earlier, Lyraan-S 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+1^t with and without the three asymmetry
effects, at a temperature of 800 eV. a) Electron.. .
density = 10 -> cm b) ]
c) Electron density = 10
ectron density = 3 * 10 cm
1 cm~^.

-64-
DELTA OMEGA (RYD)

FIGURE 17
Ly-B lines of Ar
+ 17
effects, at a temperature
density = 10^3 cm-^; t>)
c) Electron density
with and without the
of 800 eV. a)
lectron density
^ cm~3
three asymmetry
Electron
= 3 * 10¿:S cm~^
10£

-66-
DELTA OMEGA (RYD)

FIGURE 18
Ly~Y lines of
effects , at a
Ar
+ 17
density = 10
with and without the
temperature of 800 eV. a)
density
,23 crafc3.
b)
c) Electron density = 1 0‘
Electron
' cm-3.
three asymmetry
Electron
= 3 * 10'° cm 5

-68-
DELTA OMEGA (RYD)

-69-
will still be a two peaked function. Let us define a measure of asymmetry
for the Lyman-6 line as 2(Ib -Ip)/(Ib +Ip) where Ib and Ip are the blue
and red peak intensities, respectively. In Figure 19, we plot this
asymmetry as a function of density for an Ar+1^ 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
23 -3
asymmetry at 4 x 10 cm , 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.

FIGURE 19
Asymmetry of Ly~B. defined as 2(I^-I^)/(I^+I^), as a function of
density.

ASYMMETRY
3x10 10
ELECTRON DENSITY
(cm-3)

-72-
TABLE 1. Contributions to the asymmetry of the Ly-B line of Ar+1^
1023
3 x 1023
1024
O
X
m
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,
+Q pp —O
Ne J lines are nearly symmetric at 10 cm
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).

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, "i,", 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 ¿-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
3 2 1
1s2p P -*• 1s S, 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(iu) = Jde Q(e)J(u,e),
J(w,e) = - Im Tr D{u-(Hp-uQ) - p«e - ^(to)}~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
2
E 0 = - ♦ w(n,2,Z), (IV.A.2)
n2m 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 2,-dependent energy shift. This shift represents the
effect that the ground state electron has on the energy levels of the
excited electron. Clearly, ü)(n,2,Z) removes the 2-degeneracy.
The values for io(n,2,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, win.i^.Z) - io(n, 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, -id /2, and yet the line is markedly asymmetric. There are two
sp
reasons for this asymmetry. First, the unshifted components are centered
about zero (the S.-1 energy level) rather than the symmetry point
-id /2. Secondly, the transition probabilities for the shifted
sp
components are not equal. The shifted components are linear combinations
of the states 1200> and |210>. In the absence of any field these states
are not mixed and have relative energy levels -u> and zero. This gives
sp
rise to asymmerty because the state |200> is dipole forbidden. Only for
large electric fields is the situation symmetric, as the eigenvalues are
then (|200> ± |210>)//2.
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, 8, 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
occuring in the same experiment. This problem may be greatly reduced
through the time-resolved spectroscopy techniques presently available.

FIGURE 20
4.I 7
Helium-like a-line of Ar ' at an electron density o
1023 cm~3 and a temperature op 800 eV.

-78-

FIGURE 21
Helium-like 0-line of Ar+17 at an electron density of
10^3 cm~3 and a temperature of 800 eV.

IUO
-80-
DELTA OMEGA (RYD)

FIGURE 22
+ 17
Helium-like Y-lir.e of Ar ' at an electron density o
10cm”^ and a temperature of 800 eV.

í'") I
-82-

APPENDIX A
ZWANZIG PROJECTION OPERATOR TECHNIQUES
From Eqn. (I.A.15) we have
Km) = Jdf W(f) J(oj,4*) , (A. 1)
and
Jiu.'f) = ir”1Re / dt elültTr d•? (a ),
0 a
where = f 1(a,f) Tr p('V),
P
f(a,40 = Tr p 5(4'-$*),
P
W(4') = Tr p 5(4'-$*),
and
p(40 = p 5(4'-$*)/W(4').
Define an operator P by
PX s = f”1 (a ,4*) Tr p(f)X.
P
Note that
(A.2)
-83-

-84-
P2X - f”1(a,¥) Tr [p (f )f~ 1 (a ,¥) Tr p (y )x]
P P
.-1
f”1(a,?)(Tr p(f)) f"1(a,4') Tr p(f)X
PX
(A.3)
so P2 = P.
This property, idempoter.ee, defines a projection operator (Reed and Simon
1972). We can now write J(oj,y) as
oo
JU,4-) = u~1Re Tr d-f(a,f) / dt e1(i)tPd(-t)
a 0
=* it 1Re Tr d«f(a,¥) £). (oi), (A.4)
a
00
where o& (a>) = P^u»), and = / dt elu,td(-t) The Zwar.zig Projetion
1 0
operator technique can be used to obtain a general expression for
^(co).
Let D(t) = d(-t). We have
i D(t) = LD(t).
(A.5)
Let PD = D-| and (1 —P)D = D2 so that D-| ♦ D2 =
D. Then

-85-
i D2 = (1-P)L(D1 + D2),
and (A.6)
i D1 = PL(D1 + D2).
Therefore
Ot + i (1i-P)L)D2(t) = -*i (1-P)LD^ (t) ,
(A.7)
which has the solution
DpCt) - e‘lt(1-P)LD,(0) - i S
0
(1—P)LD1(t-s)ds. (A.8)
For our choice of P, D2(0) = 0. Therefore
i D
51 (t) = PLD] (t) - i I PL
e*i3(1*P)L(i-p)LD^(t-s)d3# (A. 9)
This can be transformed using the Convolution Theorem,
. r iuit* r itut. ... t iiiit -lit (1-P)L
i j e D.| (t )dt = PL J e D^tjdt - iPL J e e
dt
9 (1tP)L J elü)SD (s)ds,
(A.10)
or
-10,(0) + a)£,(«) = PL (a)) + PL 7¡--vr (l-PjL^, (to). (A.11)
1 1 1 (jj-(l-P)L 1
So
c0>1 (cu) = i[u-PL-PL((tf*QL)“1QL]’1D (0),
(A.12)

-86-
where Q = 1-P. Putting this in (A.4) yields
JCou.'f) = -ir'"1Im Tr d*f(a,f) [uj- - ]d. (A.13)
a
*
Let L = Ln + Lt($ ) and L_ = L(a.'j') + L where Ln is the Liouville
0 1 0 p u
operator for the isolated atom and Lp is the Liouville operator for the
plasma. Then,
= + + *) >. (A.14)
The first term is
X = f~1(a) Tr p(t)L(a )X = L(aff)Xo ,
cl cl a
p
and the second term is
Xa = f~1 (a ,y) Tr p (4») (LpXa) = 0,
because LpXa = 0, so = LCa.f) + when acting on atomic
functions. Also
QL0Xa • Va * * pCriL^
LaXa ' r'1LaXa
P
LX *- L X =
a a a a
0
(A.15)

-87-
This allows us to replace the last L by 1^(4 ) in ^(to). Now,
O
using also Q = Q, we can write
J(u),f) - -ir’1 Im Tr d-f (a ,¥) [u)-L(a )-*) >
a
(A.16)
-]"1 d.
Now we work on the first L in Pf(u). Consider
PLQX = f”1(a,f) Tr p (4»)LQX
P
= f“1(at4») Tr Lp(Y)QX
P
= f~1 (a , 4') Tr (L(a,t) + L + LjCí*) )p (4* )QX . (A.17)
P P
The first term is
f~1 (a,1?) Tr L(a,4')p(4,)QX
P
=* f~1 (a ,4* )L(a ,4*) Tr p (4» )X
P
- f *"|a ,4* )L(a ) Tr p(4-)f ~ ^a ,4») Tr pC^X
P P
- f"1 (a,¥)L(a,¥) Tr p(4»)X - f-1 (a ,4-)L(a ,4-) Tr p(4-)X = 0. (A.18)
P P
The second terra on the right hand side of (A.17) is

-88-
,”1(a,'f) Tr L p(f)QX = f~1(a,f) Tr L p(f)QX.
P P P
But Tr L M = Tr H M - MH
P P P
P P
Since Hp is an operator only on perturber
states and these states are being traced over, it is legitimate to
cyclicly permute Hp, i.e. Tr H M = Tr MH . Therefore,
P P P P
.â– *1
(a,>f) Tr L p(4)QX = 0.
P
(A.19)
This leaves
<1(10,40 = -it’1 Im Tr d *f (a ,4») [ to-L(a ,4») -
a
- f~1(a,4>) Tr LI(i*)p(4')((o-QLQ)~1QLI($*)]'1d. (A.20)
P
This is the result shown in equation (I.A.18).
Now consider
((0,4») = f ”1 (a) Tr L p (4-) (ur-QLQQ )~1QL:
P
CO
= f“’(a) Tr L.p(4) - l (— QL.Q)nQLx. (A.21)
p 1 113 r.-0 “ ° 1
From equation (A.18) we have PLqQX = 0, or QLqQX = LqQX for arbitrary X.
Using this, one may write
((0,4-) = f"1 (a) Tr L p(4f) - l [- L q]V
p 1 w n=0 “ ° 1
= f~1(a) Tr LIp(4')((o-L0)'1QLI.
P
(A.22)

-89-
Noting that (10,40 acts only on atomic functions, we can write
PLtX
I a
This yields
f'1(a) Tr p 6(4'-$*) Tr p LTXo
i e
Tr p.5(4'-4>*) Tr p
i ae
i ae
(A.23)
(io,40 = f"*1 (a) Tr LIp(4')((o-L0)”1LI ,
P
(A.24)
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 multi pole
expansion with odd values contribute a symmetric effect on the line
shape, to first order in perturbation theory.
Consider hydrogenic states |nqm> 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 o“ terms
like
M = .
lm ¿
(B. 1)
Switching to the spherical representation by
|nqm> = £ |ni,ra>
l
where
(B. 2)
gives
-90-

-91-
M = l 1m1 |r i |n42m2>(~1)
" * Im
1/2(1+m^-q-n)
S”\l2
1/2(1+M -q-n)
9 M) /(2£ +1)(2l +1)
/ n-1 n-1
2 2
/ml ml * \
2 2 2
\
n»1—q n^+q
~r ~~t “mi,
ra2-q m2+q
-m
2/
Now,
- j R R dr- /do Y Y._Y .
4m 0 1 2 *'11 4m 2 2
The angular integral is (Edmonds 1957)
m, (24 +1)(24_+1)(24+1) / *1 * * 11 1 l2\
(-1) V 1
4ir
0 0 0 / V -m1 m m2y
Let R represent the radial integral.
*i 2
M = I n (-1)
12
(-1)
,vv
(—r—)-q-n
/
(2ü1+1)2(242+1)2(2l+1)
/ n-1
n-1
. \
/n-1
r.-1
. \
C\J
2
' 2
2
mrq
m-i -q
-J
m2-iq
m2-q
l
~m2/
V 2
2
^ 2
2
(B.3)
(B.4)
(B. 5)

-92-
'l1 l ¿2
(l1
i *2\
o
o
o
Vmi
m m 2 /
(B.6)
Now consider M' = . This is the same perturbation on
1 £m
the state which is shifted opposite to the original state, by a uniform
electric field.
m.
m1 +m2
I *0 o (-1)
12
(-1)
+ q-n {21 +1 )2 (28. +1 )2(2Z+1)
/ — 2
4ir
(B.7)
Recalling the symmetry property
('•
h i3\
, h2M
^1+^2+^3
(-1) 5
(B.8)
\m1
m2 m31
^ m2 ml m3/
and q is an integer so (-1)+<1 = one obtains

-93-
m1 +m2
M’ = I R0 0 (-D 1(-1) 2 M ‘V X~1
-q-n (2JL, -M )2(2JL2+1 )2(2£+1 )
1^2 ^ 112
4ir
/ml ml o \
2 1
m-i -Q m^q
r — 'mi/
n-1 n-1
® (-1)
♦ ——- + i +
2 2'
n-1
n-1
2
2
m2~q
m2+q
2
2
n-1
( n-31
2
2
-m.
+ l.
'^ i ^ ^ ^ 1 ^ ^ 2\
0 0 0/ \ -m^ m m2
(B. 9)
V*2
The last phase factor is (-1) . But/
J1' 1 lj
VO 0 0/
= 0 unless
- ^1+^2 l
+ ¿2 + £, is even, so (-1) = (-1) . The result is
M' = (-1)M,
If l is odd (¡1=1, dipole; ¡1=3, octapole), states with given |q| shift
symmetrically, either both out or both in towards the center.
If l is even, states with given |q| shift the same way (either all to
the blue or all to the red) and so an asymmetry results.

APPENDIX C
CALCULATION OF Qzz
In the spherical state representation, the matrix elements for Qzz,
within a given principle quantum number, can be written

*2e/ ^ 20
|n2'
m'>
r4R „R „
n2 n2
.][/*
2mY20Y2'm,di^ *
(C.1)
Consider angular integral, I
n'
:0 • M)" I
(_1 7 (22.+ 1)(22'+1)5
4ir
2 2'
4 it mm*
• 2 1
0 I
2 2’
0 0
2
0
(C .2)
Note that the result is zero unless 2 + 2' + 2 is even and (2,2',2)
satisfy the triangle inequality.
For the m = 0 case, the 3~j symbols have a simple form.
1 l' (2+2'~2)! (1—2'+2! (-2+i’+2)! 1/2
0 0
(J+1 )!
(J/2)!
TJ/2“¿1)!(j/$-22)! (!
_ Q /. _

-95-
where J = 2 + 2' + 2 is even and triangle inequality is satisfied.
For the |m| = 1 case and |m| = 2 case (which is only necessary if
fine structure is included) the 3"*j symbols can be calculated by the
method outlined in Vidal, et al. N.B.S. Monograph 1 16 (1970). Now
consider the radial integral
00
j dr
r4R „ ,
n2 n2'
We employ a method used in Bethe and Salpeter (1977) to find the
normalization of the radial hydrogenic wavefunctions which are
/2 o 29+1
proportional to e p p^L^ '(p) where p = — r.
Using the generating function for Laguere Polynomials (Abramowitz and
Stegum 1972)
Up(p,s) =
(-s)Pe-Ps/^-s)
(1-3)P+1
I
q=p
Ln(P} n
_3 J
q!
one obtains
® s22+kt22+k’ ® 21+2 22 + 1, . 22 + 1 . .
£,=1 (22+k)! (22+k’)! 'Q P L2)l + k P L21 + k,(p)dp
22,+2 dp
/ e - p
0
p 22 + 2 0-S)2**Vp3/(1-3) (»t)24+1e’pt/(l’t)
(1-3)
22+2
(1-t)
(st)
22+1
(1-s)2i,+2(1-t)2S'+2 0
f 22+2 -p(1+s/(1-s )+t/(1-t))
7p J P e dP
(at) 211+1 (22 + 2)! (1*-s) (1-t) 2£,+3
(st)2t*1 (1 ♦ 2)!
-96-
= (3t)2i+1(1-3)(1-t)
= (St)2l+1(1-3)(1-t)
l (at)J
J-0 J-
l (3t)J .
J-0 J‘
(C.U)
Both the left and right hand side are now power series in s and t and so
must be equal term for term. Equate the coefficients of terms with
2i,+k.2i + k
st.
P 0 +1/ oo
st f _-p 2Í.+2 , 2Í. + 1 , N 2 „
Z—p J e P L t(p) dp
(2£,+k)! 0 2i,+k
(st)2»+k UI+WU +
(k-1)!
The first term on the right hand side comes from j = k-1. The
- - (22, + k)1
second term comes from j = k-2 and is 0 if k = 1, , otherwise,
J e rp
0
p 21 + 2 2)1 + 1 ,2„_ (2i,+k)
"L (p) dp
2Z + k
+k1-3 - 1 0
(2i+k+D +
(k-1)
(C .6)
k-1
Let n = k+i,. Then [ ] = 2n whether k = 1 or k > 1 .
{ e rp
0
â–  T â–  / \ (n+j.)! J
Ln+ü (p) dp (n—JL—1)!
-P-2Í. + 1 r 22,+1 , _ ^ 2^_ _ in+iui
(C .7)
This determines the normalization factor. Now consider

-97-
l
kk'
22+k.22'+k' ® „ „
3 t c -p 2 + 2
(22+k)! (22'+k* )! p
+4 22+1
L22+k
(p)
22' +1
L22'+k'
(p)
dp
f -p 2+2'+4
= J e P
0
, C.X,^ I , v
r L2fc-*-k p 2i,+k
k¿1 ( 22 + k )! S
00
l
k' = 1
22,' +1
22'+k' 22'+k'
(22'+k')! C
22+1. 22'+1
s t
d-3)21+2(l-t)2i,+2
(2+2'+4)!
(1-s)4+l,+5(1~t)1+4,+5
(1-st)
2+2'+5
22 + 1.22 1
s t
+ 1
(2+2'+4)1
(1-S)r^+3(i-t)^’+3
y (^+2 ' +4+j )!
“ j! (2+2 ' + 4)1
J
Case 1) 2=2'
Equate terms smtm.
m.m
s t
m! m!
-p 22 + 4 22+1 , w 22 + 1 , » .
p Lra (p)Lm (p)dp
m.m (m+3)1
= s t
(m-22-1 )!
+ 9
(m+2)!
(m-22-2)!
+ 9
(m+1 )!
m!
(m-22-3)! (m-22-4)!
Therefore
r°° -p 22 + 4 22+1 , >T 22+1 , , .
J e p L (p)L (p)dp
0 mm
3
= ¿~2¿rfj; t20”3 M 602m2 + (4822— 1 22 + 4)m + (-823+1 222-42) ].
(st).
(C .8)
(C .9)
(C.10)

-98-
Let m = n + l, put in the normalization, and convert to an integral over
r to get,
oo 2 a 2
J RrjiRnSlrJ4dr = F t5"2 + 1 " 31(1 + 1)] (^) . (C .11)
Case 2) Í,' = i, + 2
The same procedure yields
00 ------ Q 2
J R„oRr,o*or4dr = I n2/(n2-U + 1)2(n2-U + 2)2) (—) . (C.12)
1 q ni nü+2 2 vz
This is sufficient to give the matrix elements of Qzz in spherical
representation. If it is desirable to have Qzz in parabolic
representation, eqn. (B.2) can be used.

APPENDIX D
THE INDEPENDENT PERTURBER MICROFIELD
The probability distribution function (microfield) for finding an
electric field of strength e in a plasma is given by
Q(e) = <5(e~E)>,
(D. 1)
“► r» .
where <...> denotes a plasma average and E = ¿ E. is the sum of ion
i
fields (possibly screened).
Q(e)
lim j / — Jd3Nr e'6V6(e-E)
N,V+«
(D. 2)
where Z is the configurational partition function. We will assume that
the thermodynamic limit exists and drop its explicit mention hereafter.
Inserting the integral definition of a delta function,
(2n)3
yields
Q(e)
—!—=• Jd3i, e’U*eT(£)
(2ir)á
(D. 3)
where
-99-

-100-
T(i)
(D.4)
It is easy to see that T(H) is a function only of the magnitude
of Í, and not its direction. This allows the angular integrations in
(D.3) to be done readily, yielding
CO
(D .5)
2tt e 0
The Independent Perturber approximation can now be used to find T(H).
This approximation contends that interactions between the radiator and
perturbing particles are more important for determining the microfield
than are interactions between different perturbers. Therefore all
pert urber-per turber interactions are neglected, leaving only central
interations.
I -8v.+U*E.
u l A
N
0
r
0

-101-
where R is the radius of the system (assumed to be large). For large r,
v(r) + 0 so the denominator is essentially the volume of the system.
TU) = lim [1 - / dr r2 e~Bv(r)(j (8. E (r ))-1) ]N
N+® 0
00
= expj^im J dr r2 e Bv^r ^ (jQ(8.E(r ) )-1)}.
This is TU) in the Independent Perturber Model. It is interesting to
note that TIpU) is exactly the same as for the first term of Baranger-
Mozer formalism except that for the I.P. model v(r) is the bare potential
and for B.M. it is screened.

APPENDIX E
CALCULATION OF AN IMPORTANT INTEGRAL
From (I.B.9), we have
2 2
sin k/e +e -2ee y
I0 = Jda(1-3y ) â–  . (E.1)
2 2
k/e + e^ -2ee^y
From Abramowitz and Stegun (1972),
I (2r.+ 1)j (ke)j (ke )P (y). (E.2)
n=0
2
Also (1-3y ) = -2P2(y). Therefore,
1 “
In = ~4ir / dy P2(m) l (2n+1)jn(ke)jn(ke1)Pn(y). (E.3)
Since
one obtains
2 2
sin k /e +e1 -2ee^y
k /e^+e1^-2ee1y
Ijj = -8irj2(ke )J2(ke1).
(E. 5)
-102-

APPENDIX F
NUMERICAL PROCEDURES
In this appendix, the numerical procedures used in calculating
(I.B.12) will be discussed.

e
- nh (k)
dk k e J.(kE) J
0
dx
-ax 2 2
g(x) ^— (1+ax+ ~Y~ j2(ke:, (x))
~nh (k )
J dk k e sin(ke)
0
(F. 1)
Let W(k) be defined as
“> -ax 2 2
W(k) * j dx g(x) (1 + ax + j (ke(x)).
0 X J
(F. 2)
This integral is done using trap rule integration. The outer integral is
broken into small intervals,
J* dk * £ Jki+1 dk.
0 i k.
i
On the interval (kitki + 1) let W(k) be approximated by
2
W(k) - AW^k + BVLk where the constants AV^ and BWi are determined by
2
matching the endpoints of the interval. (Note that W(k) - k for small k
so the quadratic first term makes sense.) Similarly let nh.j(k) = AH^k +
BH^ on (kifki+1). This yields
-103-

-104-
AH.k+BH.

-6 el Jki + 1dk k2e 1 1 {(^-r - ^)sin(ke)- }} (AW k2+BH. k)
i i k^e"5 k e
k. AH.k+BH.
£ J 1+ dkke 1 1 sin(ke)
i i
Let
k. AH.k+BH.
S.(N) = J 1+ dk e 1 1 k sin(ke) and
1 k.
i
(F.3)
k AH.k+BH.
C.(N) = J 1 dk e 1 1 k cos(ke).
1 k.
i
(F.4)
Then
AW. BW. 3AW. 3BW. 3AW.
= -6e I S. (4) is. (3) + —==■ S. (2) + —^ S. (1) =*■ C. (3)
£ £1 £1 3 1 3 1 2l
Í £ £ £
3BW.
—Y c.(2) /l S.(2)
E Í
(F. 5)
From Gradshteyn and Ryzhik (1980), we have
S. (N)
AH.x+BH. N , ,%K+1, N-K , k. .
e 1 1 l (-—jr—— sin(£x + Kt) 1 1 + 1
K=1 (N-K)!(AH.2+e2)K/2
k.
x
and
AH.x+BH. N ,,,xK+1,M N+K k. ,
, „ 1 1 r (-1) (N~1)!x , . v.s 1+1
C . (N) = e I 0-3-V7Ó 003 (eX Kt)
K=1 (N-K)!(AH. +e ) k.
(F. 6)
where

-105-
AH.
. . e i
t = arcsm = arccos
/AH.2+c2
i
/AH.2+e2
Let
SW<»> - ¿
K+1 k.
i
K)!
N-K
-—2T72 sin(eki + Kt)'
(AH. + e )
N , ,K+1 k.N'K
SUi(N) â–  1 WT7 1 t t
1 K,, in iW. (AH, + e )
2-075 sln (F.7)
and similarly for CL^N) and CU^N) except use cos instead of sin. Then
AH.k. +BH. AH.k.+BH.
S. (N) = (N~ 1 )! (e 1 1+ ^U.tN) - e 1 1 ^L. (N))
l i i
and
AH.k. +BH. AH.k.+BH.
C.(N) = (N— 1 )! (e 1 1+1 ^U.tN) «- e 1 1 uL.(N))
(F. 8)
SL, SU, CL, and CU are calculated as above; then and are formed.
These are used in (F.5) to get ^. If ke << 1, it is appropriate to
use an expansion for j2(ke) and sin(ke). In this case

k. . AH.k+BH. .22 .
-6e l I dk k2e (AW.k2+BW.k)
i Ki 1b i i
k~ AH.k+BH.
I Í 1+1 dk k e 1 1 ke
i k.
i
(F. 9)

-106-
Integrals of the following form are needed
x. ,
i + 1 n r.-r
r . ax+b n ax+b r , n!x
I dx e x = e ¿ (-1)
r=0
(n-r )!a
r+1
x. ,
i + 1
X.
1
(F.10)
These can be easily coded, but, for small ax, the following numerical
problem still exists;
I = e
ax+b
n! I (-1)
r x
n-r
r=0
r+1 , .,
a (n-r)!
x. ,
i + 1
X.
1
ax+b ,
e n!
n
I
j=0
(-1)
n-j
J
j! a
n-j +1
(-1)nn!
n+1
a
ax+b
e
00
I
J-0
(-ax)1
i!
. . ax+b
(-1) n! e
n+1
-ax
00
I
j=n+1
(-ax)^
j!
(-1)nn!
n+1
a
b
e
00
J-n+1
(-ax)^
j!
(F.11)
For small ax, the 1 dominates the summation; but 1 is just a constant
and so drops out when taking the difference between upper and lower
limits, leaving very little accuracy. One may approximate

— -L\J / —
I
n+1
ax+b
e
00
I
j-n+1
(~ax)^
j!
x
i + l
X.
1
. n+1 n+2 x
ax+bfx ax i
1 n+1 " (n+1) (n+2) +
. n+1 2 2
ax+b x r _ ax + a x
n+1 ^ n+2 + (n+2)(n+3)
(F.12)
This is a better expression to use for computation when ax << 1.
For large values of e, the calculation of ^ detailed above has
non-physical oscillations. This is a common feature of numerical Fourier
transforms. To avoid this problem, we use an asymptotic form. For
large e values, the nearest neighbor limit ( = -2z3l^2//z) is
eNN
valid. Divide the expression for ^ by the nearest neighbor form to get
1 /2
a function that goes to 1 for large z and goes as z for small e. This
can be fit by a Pade approximate of the form

e

eNN
h/z + 3e + Ce3/2
1 + d/F + Ee + Cz3/2
(F.13)
where A, B, C, D, and E are fitting parameters. These parameters are
determined by minimizing the function

z

eNN
A/e + Be + Ce3/2 \ 2
1 + D/e + Ee + Ce3/2/
for small values of e. A code "Simplex" provided by Dr. Robert Coldwell
does the minimization.

APPENDIX G
ELECTRIC FIELD BEHAVIOR OF THE ENERGY LEVELS
OF HELIUM-LIKE IONS
From equation (IV.A.1) we have
J(w,e) â–  - - Im Tr d{w-(H -w.) - p-e - V (u>) }_1 . (G.1)
it r u
a
Using (IV.A.2), we can rewrite this, for a Ly-a line, as
J((o,e)=- — ImTrD(Aw + w 5(i.,0)-p*e-ir} (G.2)
it sp
a K
where
Aw = o)-(Z-1) 2( 1 - '—) - w( 2,1, Z)
n
u> = w(2,1,Z) - w(2,0,Z)
sp
ir = pU).
We have expressed ^/(w) as ir to show that it is pure imaginary and that
since it is essentially constant out to the electron plasma frequency, we
will treat it as a constant here. We have also chosen the isolated p
state (£=1) as our energy reference point. We will use the spherical
representation, |ni,m>, and chose the states, for our matrix, in the
-108-

-109-
following order |200>, | 21 0>, | 211 >, and |21 — 1 >. In this representation,
the resolvent R, {aw + oj 5(£,0) - p*e - ir} , has the following form
sp
(Smith 1966)
where
X = (Aoj+nj -iD (Aco-iD-E2 and
sp
E - 3eaQe/h.

-110-
The matrix D has the form
This yields
D =
0
0
0
\ 0
0
1
0
0
0
0
1
0
0 \
0
0
’/
J(w,e) = - - Im{
Au)+iü -ir
sp
Aw
^Trl
(G.4)
(G .5)
The second term corresponds to the unshifted components. Since we
are interested in the field dependence of the shifted components, consider
only the first term. The peaks in J(w,e) corresponding to the energy
levels of the shifted components occur at points for which the real part
of X is equal to zero. Neglecting r, these are given by
2 2
Aw +Aw w - E = 0
sp
Aw
-w ±/w 2+4E2
sp sp
2
(G.6)
In order to better understand this expression, we shall consider two
limits. First, for w >> E,
sp

-III-
Aco
(G.7)
For oí << E, we have
sp
0) 0)
r - _SP + SP
h 2 8E
Aid
(G .8)
-E -
to a)
sp sp
2 8E
The first limit is the small ion field limit. Here we see the states
2
shift as e . The second limit is for large ion fields and shows linear
dependence. The general expression, (G.6), goes smoothly from the
quadratic Stark effect for small fields to the linear Stark effect for
large fields.

REFERENCES
Abramowitz, M. and Stegon, I. A., Handbook of Mathematical Functions
(Dover, New York, 1972).
Bashkin, S. and Stoner, J. 0., Jr., Atomic Energy Levels and Grotian
Diagrams (North-Holiand, Amsterdam, 1975).
Bethe, H. A. and Sal peter, E. E., Quantum Mechanics of One- and Two-
electron Atoms (Plenum, New York, 1977).
Delamater, N. D., Ph.D Dissertation, University of Florida (1984).
Demura, A. V. and Sholin, G. V., J. Quant. Spectrosc. Radiat. Transfer J_5,
881 (1975).
Dufty, J. W. and Boercker, D. B., J. Quant. Spectrosc. Radiat. Transfer
J_6, 1065 (1976).
Dufty, J. W. and Iglesias, C. A., Spectral Line Shapes, K. Burnett, Ed.
(Walter de Gruyter, New York, 1 983).
Edmonds, A. R., Angular Momentum In Quantum Mechanics (Princeton
University Press, Princeton, New Jersey, 1957).
Gradshteyn, S. and Ryzik, I. M., Table of Integrals, Series, and Products
(Academic, New York, 1980).
Griem, H. R., Spectral Line Broadening by Plasmas (Academic, New York,
1974).
Hooper, C. F., Jr., Phys. Rev. 165, 215 (1968).
Iglesias, C. A., Phys. Rev. A29, 1 366 (1984).
Iglesias, C. A., Hooper, C. F., Jr. and H. E. DeWitt, Phys. Rev. A28, 361
(1983).
Jackson, J. D., Classical Electrodynamics (Wiley, New York, 1972).
Joyce, R. F., Woltz, L. A. and Hooper, C. F., Jr., Laser Interaction and
Related Plasma Phenomena, Vol. 15 M. Lubin, B. Yakkobi, Ed. (Plenun,
New York, 1981).
Knight, R. E. and Scherr, C. W., Rev. Mod. Phys. 35_, 431 (1963).
Lílders, G., Ann. Phys. [6] 8, 301 (1951 ).
-112-

-113-
Reed, M. and Simon, B., Functional Analysis (Academic, New York, 1972).
Sholin, G. V., Optics and Spectroscopy, _26, H19 (1969).
Smith, E. W., Ph.D. Dissertation, University of Florida (1966).
Smith, E. W., J. Cooper and C. R. Vidal, Phys. Rev. 185, 1*10 (1969).
Tighe, R. J., Ph.D. Dissertation, University of Florida (1977).
Vidal, C. R., Cooper, J. and Smith, E. W., National Bureau of Standards
Monograph 116 (Boulder, Colorado, 1970).
Woltz, L. A., Ph.D. Dissertation, University of Florida (1982).
Woltz, L. A. and Hooper, C. F., Jr., Proceedings of the Second
International Conference on Radiative Properties of Hot Dense Matter
(World Scientific, Singapore, 1985).
Woltz, L. A., Iglesias, C. A. and Hooper, C. F., Jr., J. Quant. Spectrosc.
Radiat. Transfer 27. 233 (1982).

BIOGRAPHICAL SKETCH
Robert F. Joyce was born on April 10, 1953» in Lawton, Oklahoma. He
graduated from Jesuit High School of Tampa, Florida, in June 1971. He
attended Rice University, in Houston, Texas, graduating with a Bachelor of
Arts degree in 1975. From September 1975 until the present he has been a
graduate student in the Department of Physics at the University of
Florida.
-114-

I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for/the degree of Doctor
of Philosophy.
Chairman
fifes
loo
Professor of p:
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of Doctor
of Philosophy.
James W. Duftr T\
Professor of Phy^tcs
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of Doctor
of Philosophy.
Jean R. Buchler
Professor of Physics
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertatiojp^for the degr<
of Philosophy.
of Doctor
Professor of Physics
I certify that I have read this study and that in my opinion it
conforms to acceptable standards of scholarly presentation and is fully
adequate, in scope and quality, as a dissertation for the degree of Doctor
of Philosophy.
X. Ritter
fsociate Professor of Mathematics

This dissertation was submitted to the Graduate Faculty of the Department
of Physics in the College of Liberal Arts and Sciences and to the Graduate
School and was accepted as partial fulfillment of the requirements for the
degree of Doctor of Philosophy.
May 1986
Dean, Graduate School

UNIVERSITY OF FLORIDA
3 1262 08554 1497




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