sd
s,
u
ha
eu
as
Fe
er
est
transition energy between the two nuclear levels of the j th
PHYSICAL REVIEW B 1 MAY 1997-IIVOLUME 55, NUMBER 18Our interest in SF stems from the following expectation:
if a g-ray laser4 is developed, it will probably emit in a SF
mode because characteristic features of SF decay may pro-
vide ways of overcoming specific nuclear problems that pre-
vent more conventional continuous wave or pulsed-laser ac-
tion. These specific nuclear problems are discussed in Ref. 5,
where we investigated the possibility of experimentally real-
izing nuclear SF and provided several reasons why obtaining
atomic ~or molecular! SF is easier. One of these reasons,
perhaps the most important impediment to SF in long-lived
nuclear states, is the high level of inhomogeneous
broadening,6 which can destroy the resonant condition when
narrow nuclear lines are involved. Several authors have ad-
dressed the problem of inhomogeneous broadening as it per-
tains to Mo¨ssbauer experiments and g-ray laser develop-
ment, and have proposed techniques for dealing with it.7,8
nucleus divided by \, and w j is a spatial phase factor which
depends on the relative position of the nucleus in the lattice.
Inhomogeneous broadening occurs when, in general, Dv j
ÞDvk .
It is convenient to write Eq. ~2! in the form that separates
cooperative ~SF! and noncooperative emissions, so that
P
~
v!5IN1~v!1IN2~v!, ~3!
where
IN1~v!5NE
0
`
dt
8
E
0
`
dt
9
^
V ~2 !
~
t
9
!V ~1 !~ t
8
!
&
3eiv~ t82t9!2G/2~ t81t9! ~4!Recovery of superfluorescence in inhomogeneou
B. Balko an
Institute for Defense Analyse
R. V
Cornell University, It
J. W. N
University of North Tex
~Received 22
This paper shows that time-dependent hyperfine int
superfluorescence ~SF! intensity by countering the d
@S0163-1829~97!05818-9#
I. INTRODUCTION
The emission of a superfluorescence ~SF! pulse from the
ends of a thin, cylindrically shaped active medium is highly
directional. The process is characterized by the SF time de-
fined by1
tSF5
8ptn
3rl2l , ~1!
where tn is the natural lifetime of an excited level,2 r is the
density of excited atoms or nuclei, l is the emission wave-
length, and l is the length of the cylindrical inverted region.
Following a fast inversion, a pulse is emitted after a time
delay.3 The time delay tD is characteristically much shorter
than tn but several times longer than tSF . During tD , the
N individual radiators become correlated so that they can
radiate as a single dipole with a peak intensity proportional
to N2.
The presence of a dephasing mechanism, characterized by
a dephasing time t
w
, may slow down or completely inhibit
this process so that SF cannot occur. If the dephasing time is
sufficiently short, correlations between radiators do not de-
velop, and each radiator responds individually to the instan-
taneous electromagnetic field, resulting in natural spontane-
ous emission.550163-1829/97/55~18!/12079~7!/$10.00ly broadened systems through rapid relaxation
I. W. Kay
Alexandria, Virginia 22311
duc
ca, New York 14850
berger
, Denton, Texas 76203
bruary 1996!
actions of the nucleus with electrons can enhance the
ructive effect of inhomogeneous broadening on SF.
In this paper we concentrate on the effect of inhomoge-
neous broadening in SF emission. We show how inhomoge-
neous broadening produces dephasing and inhibits coopera-
tive emission, thus reducing the intensity of the SF pulse. We
also show how electronic relaxation or time-dependent hy-
perfine interactions can mitigate the effect of inhomogeneous
broadening so that SF can be recovered.
II. EMISSION FROM A SYSTEM OF IDENTICAL NUCLEI
IN DIFFERENT ENVIRONMENTS
For N atoms that are identical except for small differences
in energy caused by inhomogeneous broadening,6 the emis-
sion probability is given by9
P
~
v!5
E
0
`
dt
8
E
0
`
dt
9
^
V ~2 !
~
t
9
!V ~1 !~ t
8
!
&
eiv~ t82t9!2G/2~ t81t9!
3
(j ,k
N
ei~Dv jt82Dvkt9!1i~w j2wk!, ~2!
where V (6)(t)5eiHtV (6)e2iHt and V (6) is the interaction
term of the Hamiltonian responsible for the emission and
absorption of a photon. G is the reciprocal of the lifetime of
the excited state, H is the Hamiltonian for the total system,
including all nuclear and solid-state energy terms, Dv j is the12 079 © 1997 The American Physical Society

Dand
IN2~v!5E
0
`
dt
8
E
0
`
dt
9
^
V ~2 !
~
t
9
!V ~1 !
3
~
t
8
!
&
eiv~ t82t9!2G/2~ t81t9!
3
(j ,kÞ j
N
ei~Dv jt82Dvkt9!1i~w j2wk!. ~5!
The first term on the right-hand side of Eq. ~3! as defined
by Eq. ~4! represents the total single nucleus uncooperative
emission. The second term on the right side of Eq. ~3! as
defined by Eq. ~5! represents the SF cooperative emission
with the characteristic narrow angular distribution and short-
ened emission time following an initial delay. This double
sum exhibits the inhomogeneous broadening effect. Assum-
ing that the nuclei are identical except for the different posi-
tions in the solid lattice and different amounts of detuning,
the expectations in Eq. ~3! are the same for any two atoms,
and, therefore, the double sum involves only the exponential
terms.
Since our primary interest is the total intensity as a func-
tion of time and not as a function of the line shape or photon
energy distribution, we can integrate IN2(v) over v, to get
1
2p E
2`
`
IN2~v!dv5E
2`
`
IˆN2~ t8!dt8,
where
IˆN2~ t8!5^V ~2 !~ t8!V ~1 !~ t8!&
3
(j ,kÞ j
N
ei~Dv j2Dvk!t82Gt81~w j2wk!i. ~6!
The exponential on the right-hand side of Eq. ~6! exhibits
phase differences in the contributions of the different radia-
tors to the SF emission term. This effect of inhomogeneous
broadening reduces the intensity of the emitted SF pulse.
Assuming that the inhomogeneously broadened line shape
can be modeled by a Lorentzian distribution, Eberly10
showed that the SF intensity given by Eq. ~6! after the sum-
mation is performed takes the form11
IN2~ t !5N2u^V&u2e22G*t, ~7!
where G* is the inhomogeneously broadened linewidth and
^
V
&
is the product of the matrix elements in Eq. ~6!. For fast
pumping, essentially completed near t50 ~pumping time
tpump!tSF!1/G*,1/G!, inhomogeneous broadening has
little effect on the SF emission because the nuclei do not
have time to dephase. An examination of the exponential
factor on the right-hand side of Eq. ~7! shows that only after
a relaxation time of the order of T2*51/G* do the phase
differences caused by the energy differences between differ-
ent nuclei
@
i , j ,(Dv i2Dv j)t# become large enough to affect
12 080 B. BALKO, I. W. KAY, R. VUthe emission significantly or prevent the SF pulse from oc-
curring.III. TIME-DEPENDENT PERTURBATIONS
A. In absorption measurements
Inhomogeneous broadening in samples used in a reso-
nance experiment can reduce the resonance effect observed.
However, for fluctuations that are fast enough, if the inho-
mogeneous broadening is caused by resonances that are
coupled by fluctuating fields, the resonance effect often can
be recovered.
For example, assume that the field at the nucleus is jump-
ing between two values at a particular site in the source or
absorber, so that the resonance energy correspondingly
jumps between 6d . Blume9 has shown how to calculate the
emission and absorption line shapes for resonators acting
independently when the system under consideration is sub-
jected to time-dependent perturbing fields. We follow his
analysis, assuming a time-dependent field acting on the
nucleus, and write
H
~
t !5H01d i f ~ t !, ~8!
where H0 is the unperturbed Hamiltonian, f (t) is a random
function of t , d i is the perturbation energy with i50 for the
ground state, and i51 for the excited state.
For the equilibrium case, Blume derives the emission
probability
P
~
v!5
2
G
Re
E
0
`
dteivt2~G/2!t
^
V ~2 !V ~1 !
~
t !
&
, ~9!
where the bar indicates an average with respect to all func-
tions f (t), so that
^
V ~2 !V ~1 !
~
t !
&
5ei~E02E1!t
^
V ~2 !V ~1 !
&^
ei~d02d1!*0
t f
~
t
8
!dt
8
&av .
~10!
For convenience, we have selected units for which \51.
Blume9 also shows that the following identity holds for any
value of a5d02d1 :
^
eia*0
t f
~
t
8
!dt
8
&av5S cosxVt1
1
x
sinxVt
D
e2Vt5G
~
V ,a ,t !,
~11!
where
x5
F
a
2
V
221 G
1/2
. ~12!
V is the relaxation rate, or rate of jumping between values of
f (t)511 and 21, with equal probabilities of finding the
system in either 11 or 21. The identity given in Eq. ~11!
can be used to calculate the average on the right side of Eq.
~10! and the emission probability ~line shape! given by Eq.
~9!.
To apply Eq. ~11! to inhomogeneous broadening, we fur-
ther assume that a is a random variable with a probability
density g. In this paper, we consider both Gaussian and
Lorentzian distributions as examples. For a Gaussian distri-
bution, we characterize the broadening by the standard de-
viation s. For a Lorentzian distribution, the broadening is
55UC, AND J. W. NEUBERGERcharacterized by the inhomogeneous broadening parameter12
a . Thus, for a Gaussian distribution

FLg
~
a ,s!5
1
s
A2p
e2a
2/2s2
.
The inhomogeneous broadening factor is given by
G
~
V ,s ,t !5
E
2`
`
g
~
a ,s!G~V ,a ,t !da , ~13!
which can be used with Eq. ~9! to calculate the inhomoge-
neously broadened line.
We can now examine two special cases: V/a@1 and
V/a!1. From Eq. ~11!, for V@a , G(t)'1 and the inho-
mogeneous broadening effect disappears. For V!a , G(t)
'1/2(eiat1e2iat) and with a as a random variable distrib-
uted so that it represents the appropriate inhomogeneous
broadening in Eq. ~13!, the maximum broadening effect oc-
curs.
For values of V/a between these two extremes, we do not
get such simple results, and a complete calculation using
Eqs. ~9!, ~10!, ~11!, and ~13! is required to obtain the line
shape of an inhomogeneously broadened system in the pres-
ence of relaxation. Figure 1 depicts the effect of such fluc-
tuations in the fields causing relaxation between resonances.
The first column shows the effect of fluctuations on the ab-
sorption cross section at a particular site. The second column
shows the effect on the inhomogeneously broadened system
of nuclei with a Gaussian distribution of s ~the positions of
the resonances shown in the first column indicated by ar-
rows!. The third column shows the observed line shape in a
Mo¨ssbauer experiment when both the source and absorber
have the same amount of inhomogeneous broadening. In row
~a!, s50, implying no inhomogeneous broadening. The
maximum effect is observed. For rows ~b!–~f!, the inhomo-
geneous broadening is assumed constant at s520, and the
relaxation rate, V, varies from 0.001 to 500, in units of the
natural lifetime. At high relaxation rates @Fig. 1~f!#, the in-
homogeneous broadening is essentially wiped out and the
line shape approaches the result obtained when no broaden-
ing is present @Fig. 1~a!#. For arbitrary s, the relaxation rate
has to be much greater than the inhomogeneous broadening
(V@s) for the broadening effects to be wiped out.
B. Effect on the SF intensity
To illustrate the effect of time-dependent perturbations on
the SF pulse, we generalize Eq. ~10! by introducing the ran-
dom functions f i(t), which represent mutually independent
random processes that are otherwise identical with f (t), and
by replacing Dv i , for all i , with a f i(t). To simplify the
calculation we assume in addition f j5fk , thus ignoring
spatial phase variations. We then get from Eq. ~6!
IˆN2~ t !5^V ~2 !V ~1 !&e2Gt(jÞk
^
eia*0
t f j~ t8!dt82ia*0
t f k~ t8!dt8
&av .
~14!
Since the exponential factors on the right-hand side of Eq.
~14! are statistically independent, the average of the product
is equal to the product of the averages; therefore,
55 RECOVERY OF SUPERIˆN2~ t !5~N22N !^V ~2 !V ~1 !&e2Gt@^eia*0
t f
~
t
8
!dt
8
&av#
2
.
~15!The average can then be calculated as in Eq. ~13!. After
approximating N22N by N2, the result is
IˆN2~ t !5N2^V ~2 !V ~1 !&e2GtG~V ,a ,t !2, ~16!
where G(V ,a ,t) is given by Eq. ~11!.
As expected from the previous emission results based on
Eq. ~11!, for V@a , the inhomogeneous broadening effect
that destroys the coherence disappears, and a maximum SF
pulse is obtained. For V!a , the maximum phase destruc-
tion occurs and the SF pulse disappears.
We now examine the dependence of the reduction factor
G(V ,s ,t)2 on the inhomogeneous broadening that produces
dephasing ~characterized by s! and the relaxation that re-
duces the dephasing ~characterized by the relaxation rate V!.
In Fig. 2, we show G(V ,s ,t)2 with parameters V50 and
s50, 1, 10, or 100, plotted as a function of time normalized
to the natural lifetime tn . The horizontal line labeled s50
shows no reduction in the SF pulse intensity because no
inhomogeneous broadening exists. The other curves show
the decrease in SF pulse intensity when different dephasing
rates caused by different amounts of inhomogeneous broad-
ening are present.
Figure 3 shows the effect of relaxation on the reduction
factor when s5103 for different values of the relaxation
rate, V. The curves decay rapidly for small values of V and
less rapidly as V increases, approaching the limit of 1.0 as V
approaches infinity. The results shown in Fig. 3 indicate that
time-dependent hyperfine interactions can reduce the effect
of inhomogeneous broadening. As the relaxation rate, V, in-
creases, the intensity factor also increases. In the limit of
V@s , the SF pulse intensity approaches unity, which is the
value it would have all the time if no inhomogeneous broad-
ening (s50) were present.
C. Effect on the SF pulse shape
In Ref. 5, we showed that inhomogeneous broadening de-
grades the SF pulse shape and that this effect is more pro-
nounced than the degradation resulting from homogeneous
broadening. In the previous sections, we also showed how
this effect of inhomogeneous broadening can be mitigated
through time-dependent hyperfine interactions.
In this section, we employ the Maxwell-Bloch equations
developed in Ref. 5 and summarized in the Appendix here to
show how the time-dependent effects alter the total SF pulse
shape. With this approach, we can study the effect of time-
dependent interactions in inhomogeneously broadened sys-
tems on the complete SF pulse shape, i.e., time delay, mul-
tiplicity of peaks, and the peak intensities. In this formalism,
the effect of both the inhomogeneous broadening and time-
dependent hyperfine interactions is contained in what was
originally a ‘‘coupling constant’’ ~in the Appendix! and is
now a time-dependent function g
8
(t), as discussed in Ref. 5.
We use Eqs. ~11! and ~13! to calculate g
8
(t)5G(V ,s ,t)
which replaces the constant g in Eq. ~1! of Ref. 5. The SF
pulse shape is calculated using the resulting equation.
Figure 4 shows some typical results. The solid curves give
12 081UORESCENCE IN . . .the result obtained when a Lorentzian distribution13 is as-
sumed for the inhomogeneous broadening, and the dashed

Dcurves give the result when a Gaussian distribution is as-
sumed. The curve labeled ~a! is obtained assuming no inho-
mogeneous broadening and no relaxation ~a50, V50!. The
curves labeled ~b!–~e! are obtained by increasing inhomoge-
Figure 5 shows the effect of relaxation on the system with
an inhomogeneous broadening parameter a5103. The curve
labeled ~a! is obtained with some relaxation present (V
50.001). The curves labeled ~b!–~e! are obtained with in-
FIG. 1. Effect of relaxation on inhomogeneously broadened lines. Column I shows two resonances that contribute to the inhomoge-
neously broadened line ~absorption cross section! shown in column II. Column III shows the Mo¨ssbauer line shape, assuming both the source
and absorber are broadened as in column II. Rows ~a!–~e! give the results for different values of broadening and relaxation. Row ~a! shows
the result when there is no broadening (s50) and no relaxation (V50). Rows ~b!–~f! show the result when s520 and the relaxation
increases (V50.01–500). For all these results, the natural linewidth is assumed to be G51.12 082 B. BALKO, I. W. KAY, R. VUneous broadening and exhibit the degradation of the SF
pulse. We next consider the recovery of the SF pulse.55UC, AND J. W. NEUBERGERcreasing relaxation. At V5104 @curve ~e!#, the optimal SF
@curve 4~a!# is almost fully recovered.

FLIV. CONCLUSIONS
The number of cooperating nuclei and the inherent
nuclear properties determine the SF pulse intensity. Our re-
sults show that if a candidate exists that, when inverted,
would produce SF were it not for inhomogeneous broaden-
ing, it is still possible to get an SF emission if a fast enough
relaxation of the levels producing the inhomogeneous broad-
ening can be realized.
Our calculations show that inhomogeneous broadening af-
fects an SF emission by causing a dephasing of the different
nuclear dipoles that are forming correlations. This dephasing
is time dependent and has a rate proportional to the inhomo-
geneous broadening. Immediately after inversion, SF emis-
sion is less affected by the broadening than later. Thus, for a
strong SF pulse, the SF delay time should be much shorter
than the dephasing time tSF!T2* .
We also show that time-dependent hyperfine interactions
of the nucleus with electrons can induce a reduction of the
dephasing and an increase in the probability of SF emission
FIG. 2. SF pulse emission intensity reduction as a function of
inhomogeneous broadening and time. For each curve shown, V
50 and s50, 1, 10, or 100.
FIG. 3. SF pulse emission intensity reduction as a function of
55 RECOVERY OF SUPERinhomogeneous broadening and relaxation. For each curve shown,
s5103 and the relaxation rate V510, 103, 104, or 108.FIG. 4. SF pulse shape in the presence of inhomogeneous broad-
ening. ~a!–~e! show the results for increasing broadening character-
ized by the full width at half maximum in units of the natural
unbroadened linewidth G or the inhomogeneous broadening param-
12 083UORESCENCE IN . . .eter a . The solid lines give the results for a Lorentzian distribution,
and the dashed lines give the results for a Gaussian distribution.

FIG. 5. SF line shape in the presence of inhomogeneous broad-
ening and relaxation. ~a! shows the SF pulses when inhomogeneous
broadening with a51000 is present. ~b!–~e! show the pulses for
increasing values of the relaxation rate V given in units of G. The
12 084 B. BALKO, I. W. KAY, R. VUDsolid lines give the results for a Lorentzian distribution, and the
dashed lines give the results for a Gaussian distribution.over a large time range. These electronic relaxation effects
are well known for their line-shape modification as observed
in Mo¨ssbauer and nuclear magnetic resonance
experiments.9,14–18 For SF, relaxation effects provide a
means of reducing the dephasing caused by inhomogeneous
broadening and, thus, may be useful in overcoming one of
the more difficult obstacles to nuclear SF.
ACKNOWLEDGMENT
This research was supported by the Innovative Science
and Technology Directorate of the Ballistic Missile Defense
Organization ~BMDO!.
APPENDIX: THE NUCLEAR SUPERFLUORESCENCE
MODEL
In this appendix, we present a summary of the modified
Haake-Reibold Model ~Maxwell-Bloch equations! of nuclear
SF for reference. The model is discussed in detail in Ref. 5.
The nuclear transitions are shown in the energy-level dia-
gram given in Fig. 6, where N0 , N1 , N2 , and N3 represent
the populations of the levels and g, G1 , G2 , and the transi-
tion rates between levels as shown. The pumping mechanism
is modeled by the transition from level 3 to level 2. The SF
transition occurs between levels 2 and 1. The modified
Haake-Reibold equations in dimensionless units are
]N3
]t
5gN3 , ~A1!
]N2
]t
52
~
E1R11E2R2!2G2N21gN3 , ~A2!
]N1
]t
51
~
E1R11E2R2!1G2N22G1N1 , ~A3!
]N0
]t
5G1N1 , ~A4!
]R6
]t
5
~
N22N1!E72
1
2 ~G21Gf!R
6
1j
6
, ~A5!
FIG. 6. Energy-level structure assumed in the calculation dis-
cussed in this paper.
55UC, AND J. W. NEUBERGER]E6
]x
5g
8
~
t !R72
1
2 mE
6
. ~A6!

The first set of equations, ~A1!–~A4!, govern the time rate
of change of the level populations. Equation ~A5! determines
the rate of buildup of the system polarizations R6, resulting
from the electric fields E7 and a noise source j6. A repre-
sentation of the noise source used in this model has been
derived from quantum electrodynamic considerations by
Polder, Schuurmanns, and Vrehen.3 The last equation, Eq.
~A6!, governs the spatial transport of the fields and connects
the system polarization with the electric-field gradient
through a coupling parameter g (t). This coupling parameter
Haake-Reibold theory. When the line broadening statistical
distribution is Lorentzian, the time-dependent parameter is
given by an exponential function as discussed in Ref. 5. In
the present paper we also deal with Gaussian distributions
and relaxation effects which also modify g
8
(t). The second
term in Eq. ~A6! models the attenuation of the beam as it
propagates through the system. In the equation, m is the lin-
ear attenuation coefficient, and G
f
is the homogeneous
broadening dephasing rate. Inhomogeneous and homoge-
neous broadening play quite different and distinct roles in
wiping out the effect of inhomogeneous broadening. In general,
we believe that this approach will not work to wipe out inhomo-
geneous broadening substantially. While it provides overlap be-
tween lines from different nuclei by broadening them, it does
15G. R. Hoy, M. Corson, and B. Balko, Phys. Rev. B 27, 2652
~1983!.
16B. Balko, Phys. Rev. B 33, 7421 ~1986!.
17M. Blume and J. A. Tjon, Phys. Rev. 165, 446 ~1968!.
55 12 085RECOVERY OF SUPERFLUORESCENCE IN . . .not increase the resonance effect substantially because, at the
same time, it also reduces the maximum cross section on reso-18A. Abragam, The Principles of Nuclear Magnetism ~Oxford Uni-
versity Press, London, 1961!, pp. 448–450.8
is time dependent because we assume inhomogeneous broad-
ening; otherwise, it would be a constant as in the original
1F. Haake and R. Reibold, Phys. Rev. A 29, 3208 ~1984!.
2 In this paper, we are ignoring internal conversion and assuming
that the natural lifetime is equal to the radiative lifetime.
3The time delay tD is a function of the number of cooperating
radiators and has been estimated by several authors, notably, R.
Bonifacio and L. A. Lugiato Phys. Rev. A 11, 1507 ~1975!; M.
Gross and S. Haroch Phys. Rep. 93, 301 ~1982!; and D. Polder,
M. F. H. Schuurmans, and Q. H. F. Vrehen Phys. Rev. A 19,
1192 ~1979!.
4
~a! G. C. Baldwin and R. V. Khokhlov, Phys. Today, 28 ~2!, 32
~1975!; ~b! G. C. Baldwin, J. C. Solem, and V. I. Gol’danskii,
Rev. Mod. Phys. 53, 687 ~1981!.
5B. Balko, I. W. Kay, and J. W. Neuberger, Phys. Rev. B 52, 858
~1995!.
6The sources of inhomogeneous line broadening are discussed in
Ref. 4 and generally include isomer shifts, quadruple interac-
tions, magnetic hyperfine interactions, magnetic nuclear dipole-
dipole interactions, and gravitational shifts.
7Yu. A. Ilinskii and R. V. Khoklov, Sov. Phys. JETP 38, 809
~1974!, advanced a technique for dealing with magnetic dipole-
dipole broadening and V. I. Gol’danskii, Yu Kagan, and V. A.
Namiot, Sov. Phys. Solid State 16, 1640 ~1975!, discussed a
technique for dealing with broadening due to inhomogeneous
chemical shifts.
8More recently, J. Odeurs, Phys. Rev. 52, 6166 ~1995! and 53,
9095 ~1996! proposed homogeneous broadening as a way ofthis model, unlike some of the earlier models where they
were assumed to have essentially indistinguishable effects on
the emission of SF.
nance of each nucleus. What is needed is a mechanism to col-
lapse the lines to some average value like the mechanisms pro-
posed in Ref. 7. This is discussed by B. Balko, I. W. Kay, J. F.
Nicoll, G. M. Herling, J. D. Silk, and D. A. Sparrow, Phys. Rev.
B. 48, 27 1993, and in Hyperfine Interact. ~to be published!. In
the approach described in the present paper, relaxation does not
just homogeneously broaden lines but collapses the spectrum to
an average value that approaches the unbroadened limit. This is
what is required to recover SF in inhomogeneously broadened
systems.
9M. Blume, in Hyperfine Structure and Nuclear Radiations, edited
by E. Matthias and D. A. Shirley ~North-Holland, Amsterdam,
1968!, pp. 911–927.
10J. H. Eberly, Acta Phys. Polonica A 39, 633 ~1971!.
11The assumption of a Lorentzian line shape is a mathematical con-
venience and provides us with an exponential time dependence.
It was also used in Ref. 3 to obtain the exponentially varying
coupling factor g(t). Other line shapes may be more appropriate
in specific cases. In this paper, we compare the effects of
Lorentzian and Gaussian distributions.
12The inhomogeneous broadening parameter characterizes the
broadening in units of the natural linewidth so that the full width
at half maximum of the broadened line ~Lorentzian distribution!
is (11a)G . For a Gaussian distribution, it is related to the stan-
dard deviation, s, by a5A2 ln2 s/G.
13For a Lorentzian distribution, g(a , a)5aG/2p/
@
a
2
1(aG/2)2
#
.
14H. H. Wickman et al., Phys. Rev. 152, 345 ~1966!.