JOURNAL OF GEOPHYSICAL
RESEARCH, VOL. 89, NO. A10, PAGES 8857-8862, OCTOBER 1, 1984
A Fast Fermi Process' Energetic Electrons Accelerated
by a Nearly Perpendicular Bow Shock
C. S. Wu
Institutefor PhysicalScienceand Technology,Universityof Maryland, CollegePark
The highly localized accelerationof electronsat the foreshockreported by Andersonet al. (1979) is
explained in terms of a fast Fermi process.The basic notion is that in the solar wind frame the nearly
perpendicularbow shock at the point of tangencyof the interplanetarymagneticfield acts as a fastmoving magneticmirror which can reflect electronswith sufficientlylarge pitch angles.The reflection
processcan effectivelyenergizeelectronsand drive them upstream.If the seedelectronshave energiesof
severalhundredelectronvolts,they can attain energiesof severalkeV throughthe accelerationprocess.
1.
INTRODUCTION
Energetic electronsand ions observedin the upstream of
the earth's bow shock have attracted
much attention
and in-
terest in recent years. (See the specialissueof the Journal of
GeophysicalResearch,86(A6), 1981.) The general consensusis
that the region betweenthe so-calledforeshockand the bow
shock is very rich in particle and wave phenomena [Greenstadt and Fredricks, 1979; Russelland Hoppe, 1983]. However,
the physicsof much of the observationalresults remain obscureand controversial.For example,the generationand the
origins of the upstream ions (i.e., the so-calledreflectedand
diffuseions) are still not well understood.This topic alone has
stimulated a large number of discussionslEdralston et al.,
1982; Goslinget al., 1978, 1982; Paschmannet al., 1980, 1981;
Schwartz et al., 1983; Tanaka et al., 1983; Thomsen et al.,
classicprocessof Fermi [1949, 1954], whose interest was in
the origin of the cosmicrays. Fermi suggestedthat the individual particlescan be scatteredoff moving magnetic clouds or
irregularitiesand consequentlycan be energized.Obviously,a
shockwave can behaveas a magneticmirror, as was pointed
out previouslyby Feldmanet al. [1983]. In the presentpaper
we emphasize a fast acceleration processby a shock wave
rather than the processes
which requiremultiple collisions.
The organization of the paper is as follows.In section2 we
describethe basicphysicalpicture and explain how the Fermi
processentersinto it. The density and flux of the energized
electronsare calculated on the basis of a simple model; a
discussionof some relevant issuesis then presentedin section
3. Finally, in section4 a summary and some concludingremarks are given.
1983;Bonifaziet al., 1983; Eastmanet al., 1981].
2. BASIC CONCEPTS AND ANALYSIS
Among the prominent and outstanding problems we are
Beforegoing further we remark that in the subsequentdisparticularly interestedin the accelerationof electronsat the
point of tangency of the interplanetary magnetic field to the cussion,severalcaseswill be excluded.Theseare (1) the caseof
earth'sbow shock.The ISEE spacecraftobservedthat acceler- a perpendicularshockin which the shocknormal • is perpenated electrons originating in this region can have energies dicular to the ambientmagneticfield B, i.e., Bo ß• = 0; (2) the
case of highly energeticelectronswhose gyroradii are comfrom several keV to 100 keV. These electrons form a narrow
beam directedupstreamand parallel to the electronforeshock parable to the shockthicknessand whosemagneticmoments,
[Anderson et al., 1979; Anderson, 1979, 1981; Parks et al., consequently,are unlikely to be conserved;and (3) the caseof
1981]. Observationsreported earlier by Fan et al. [1964] and
Anderson[1969] concerningelectronswith energiesof >20
keV near the bow shock may be from the same origin. The
peculiar energization processwhich appears highly localized
and very effectiveis most intriguing.
Stimulatedby the report by Andersonet al. [1979], we have
been greatly interestedin the physicalmechanismof the observedaccelerationprocess.The purposeof the presentis to
proposean explanationof the phenomenon.The basicphysics
of suchan accelerationprocessappearsof fundamentalimportance, becauseit may be closelyrelated to some of the very
important radiation processesassociatedwith shock waves
observedin natural and laboratory plasmas.For instance,the
type II solar radio bursts[Kundu, 1965; Zheleznyakov,1970]
and the microwave emissionfrom a fast theta pinch experiment ['Chin-Fart and Griem, 1970; Chin-Fart, 1974] may be all
attributable to energeticelectronsgeneratedby similar acceleration processes.
The basic idea of the proposedmechanismstemsfrom the
Copyright 1984by the AmericanGeophysicalUnion.
Paper number4A0751.
0148-0227/84/004A-0751$05.00
ion acceleration
which has difficulties similar to those of case 2
unless the ion gyroradii are much greater than the shock
thickness.The caseof the accelerationof particleswith large
gyroradii by a perpendicularshock or nearly perpendicular
shock has been discussedby Chen [1975], Armstrong et al.
[1977], Sarris and Van Allen [1974], and others.
In the presentcasewe are only concernedwith the acceleration of low-energyelectronswhose energiesare of several
hundred electron volts before energizationand whose gyroradii
are much
smaller
than
the shock
thickness.
Fur-
thermore, we are particularly interested in the acceleration
process
associated
with a nearlyperpendicular
shockwith Ons,
the anglebetween• and Bo,very closeto 90ø (but OnB
• 90ø).
In order to facilitateour discussion
we first presentsomebasic
and relevantconcepts.
Effective "ShockVelocity" in the Solar Wind
Although the standingbow shock may be consideredto be
stationary (or at least quasi-stationary),as viewed from a
frame of reference fixed to earth, for an observer in the solar
wind framethe pictureis drasticallydifferent.For the purpose
of illustrationand explanation,let us considerthat locallythe
bow shockmay be approximatedby an oblique plane shock.
8857
8858
WU: FAST FERMI PROCESS
Shock surface
field
presentdiscussionthe shockvelocityVscan be far greaterthan
the upstreamvelocity V• when 0 is small. Consequently,this
leadsto energizationprocesses
much fasterand more effective
than those which require multiple interactions between the
particlesand a shock [Axford et al., 1977; Blandfordand Ostriker, 1978; Eichler, 1979a, b; Terasawa, 1981].
line
A
magnetic
Vx
omentof encounter
Trajectøry
øfan;bf:•:r
? • •d•
in the solar win
\
•-• t+dt
RelationBetweenthe deHoffman-TellerFrame
and the Solar Wind Frame
Particle accelerationdue to the reflectionby a shock wave
was discussed
by Sonnerup[1969] in terms of a simplegeometric analysis,which has been recently applied to the study
of the upstream ions by numerousauthors [Sarris and Van
I
Allen, 1974; Paschmann et al., 1980; Schwartz et al., 1983;
Fig. 1. ShockvelocityV salong a givenmagneticfield line seenby
Thomsenet al., 1983' Bonifazi et al., 1983]. The major conclusion of Sonnerup'sanalysis is that the energy gain of a
reflectedparticle dependsupon the velocityVaT, which defines
a moving frame of referencein which the motional electric
field vanishes[deHoffmanand Teller, 1950]. Here, VaT in general may be written as [Schwartzet al., 1983]
an observer in the solar wind frame of reference. The distance be-
tween the observer and the shock along B varies with time. The
geometricrelation betweenVs and deHoffman-Tellervelocity V m
definedby (2) is alsoshown.
As shown in Figure 1, an observercomovingwith the solar
wind should seethat the shockis moving along a given magnetic field line in the sensethat the relative distancealong the
field line between the shock and the observer varies with time.
Hence we may definean effectiveshockspeedVsalong the
magnetic field line in the solar wind frame. According to
Figure 1, it can be shown geometricallythat Vsmay be expressedas
Vs=Icos ½ + cot 0 sin ½1
(1)
V.T =
• x (V• x B•)
B• .•
(2)
where • is the shocknormal and V• and B• are the upstream
bulk velocity and magneticfield, respectively.It is shown by
Sonnerup[1969] that the larger the V.T, the higher the energy
gain.The geometricrelation betweenVm and V s(the effective
shockvelocity)is shownin Figure 1. It is seenthat
Vs= g.t cos0 + g•II
Furthermore, one can show that the deHoffman-Teller frame
and the solar wind frame are related by a transformation
velocitywhich is simplyVs.
Calculationof ReflectedElectrons
First of all, let us ignore the weak thermal anisotropyand
where ½ denotesthe angle betweenB• and V•, which is the
assumefor simplicitythat the electrondistribution function in
upstreamsolar wind velocity.Whether the senseof the velocishownin Figure 3.
ty Vs is positive or negativedependson (1) the relative posi- the solarwind framemay be schematically
tions of the observerand the shock wave and (2) definition.
For simplicity,in Figure 1 we assumeimplicitlythat the shock
40
normal •, the solar wind velocity V•, and the magneticfield
B• are coplanar. Equation (1) can be easily extended to situationsin which the three vectorsare not coplanar.
Obviously,for a givenV• when0 decreases,
Vsprogressively
8I
3O
increases.For illustration,considering½ = 45ø, which is typical at 1 AU, we presentFigure 2 in which Vs/V•is plotted as a
functionof 0. The importanceof Vscan be appreciatedeasily.
Let us consider that the guiding center of an electron is
moving with velocityV (definedin the solar wind frame)along
2O
the interplanetaryfield B•, and let us assumethat V and Vs
are opposite.In this case,if the electron can have an "elastic
encounter" with the shock and be reflected (by whatever
mechanism),the speed of the particle after the encounter is
IV + 2Vslin the solar wind frame. Thus the electroncan gain
energy through the reflectionprocess.In the subsequentdiscussionwe are particularly interestedin the reflectionby a
magnetic-mirroreffectassociatedwith the magneticfieldjump
~••.Shock
at the shock front.
The physicalprocessjust describedis in essencethe mechanism suggestedby Fermi [1949, 1954] for the explanationof
the origin of cosmic rays, although Fermi's original work
stressesthe concept of accelerationthrough statistical processeswhich are usually slow. In the model consideredin the
Io
2o
5o
4ø
5ø 6 ø
7 ø 8ø 9 ø
I0 o
o
Fig. 2. EffectivevelocityV s normalizedby V• as a functionof 0.
Here n/2 - 0 denotesthe anglebetweenthe upstreammagneticfield
B• and the shocknormal ti. The calculationis for the case½ --45 ø,
which is typical at 1 AU.
WU:
FAST FERMI PROCESS
8859
y
This distribution, which consistsof two components,the core
and the halo, may be representedin the deHoffman-Teller
frameby a simpletranslation
of Vsalongthe V•iaxis.
It is convenientto discussthe mirror reflection processin
the deHoffman-Teller frame. In Figure 3 the portion of the
distribution
function
associated
with
those
electrons
which
may be reflectedby a magnetic-mirroreffectis indicatedby
the shadedarea.The lossconeangle0cis givenby
Oc
sin
l(BB-•ax)
1/2
=.
-
Motional
Ey
I •'• •
:l:l•t
;inc
a •/
shock
rest
(3)
we need to consider
a model
distribution
profile
fram•.•,
Ex
where B•naxdenotesthe maximum value of the magneticfield
across the shock transition. Equation (3) is valid when the
electronmagneticmoment is conserved.
Evidently, whether the accelerationprocessis significantor
not dependsupon the seedpopulation of the solar wind electrons in the shaded area of the velocity space indicated in
Figure 3. In order to calculate the density of the reflected
electrons
•agn•tic
field
•
•BI
Shocksurface
x
z
function
which may be suggestedon the basis of observationswhich
have been reported by a number of authors [Montgomery et
al., 1968; Ogilvie et al., 1971; Montgomery, 1972; Serbu, 1972;
Feldmanet al., 1973; Scudderet al., 1973; Feldman et al., 1975,
1982, 1983]. For the present discussionwe choose a simple
model which is described as follows. If the solar wind electron
Fig. 4. Coordinate system and shock geometry consideredin the
presentanalysis.
solar wind frame. In the deHoffman-Tellerframe, V 2 should
bereplaced
byv•_
2 + (vii- Vs)
2,asshown
in Figure3.
Here a remark is necessary.Strictly speaking, the model
distributionfunctionis denotedby F e, then it may be written distribution function describedby (5) is not consistentwith the
as the sum of two parts: the core F• and the halo FH, say,
physical picture which we intend to discuss,because once
insidethe electronforeshockall electronsmoving away from
F e = F• + F H
(4)
the shockcan only be thosewhich either have originated from
where both F• and F H may be fitted by slightly displaced the downstreamregion or have been reflectedby the shock.
bi-Maxwellian distributions.Since the displacementsin both However, the calculation mainly involves those which are
F• and FH are small (in comparisonwith the thermal speedof moving toward the shock. The distribution function of those
the core electrons)and the thermal anisotropiesare also weak, electronsmoving away from the shock does not matter, as can
thesefeaturesare ignored for the present purpose.Hereafter, be seenin the followingdiscussion.
we assume
To calculate the density of the reflectedelectrons,nr, we
write
c /t3/2-•
3exp -
(5)
•3/2--•H
3exp -[Feldman et al., 1975]. So far, F• and FH are defined in the
Solar wind
frame
IVx
IVx
© d%_v•_Fe
(6)
IItan 0½
•••.................
.......
Evalu•tting
theintegral,
weobtain
n•=•cos0•
l+erf
•cos0•
exp(-sin
20•
Pc2J
+• cos
0•1+erf cos
Ocexp
_sin
20c
•2•
OH2]
•
Mirror-reflected
••••••":'••
•'•'"'•'"'"'
"••'
""
'"'"
''' '''
•
electrons •"" '"•'•'•••••
dVll
where the integral is expressedin the deHoffman-Teller frame.
and considerthat mVc2/2
_• 10 eV and typically1)H2/l)c
2• 6
de Hoffman-Teller
frame
nr= 2re
%
XX
(7)
Hereerf(x) • 2/n1/2•0• doe-• istheusualerrorfunction.
In order to obtain numerical results we consider the follow-
ing parameters:
m•c2/2= 10 eV, mVH2/2
= 60 eV, nc = 9.9
cm-3, and nH = 0.06 cm-3. Here we haveconsidered
an effectivenH which is smallerthan the usually measuredvalue.
The reasonis that in the presentdiscussiona plane shockis
considered, whereas in the real situation the bow shock is two
dimensional. In a two-dimensional case the finite excursion
I re•tive
velocity
I'
Z;
time of an electronduring the reflectionprocesscan significantly reducethe effectiveness
of the accelerationprocess.A
more satisfactoryanalysiswill be presentedin a forthcoming
article.The parameter0• is definedto be an effectivelosscone
Fig. 3. Model electrondistributionexpressedin the solar wind
•/2, whereB2' denotesan
frame and the deHoffman-Tellerframe. The shadedarea represents anglesuchthat 0• • sin-1 (B1/B2')
thoseelectronswhich may be mirror reflected.
effectivemaximum magneticfield.
8860
Wu'
FAST FERMI PROCESS
perpendicularvelocitycomponentv•_satisfiesthe condition
10-a
107
2
nr
v•_2
> vii
+2le[Aq>aT/m
(10)
[(Bmax/B
•) -- 1]
10-3
106
whereA(I)HT> 0 indicatesan increaseof the potential.In pass-
ing,weremindthereaders
herethatvii2 denotes
theupstream
105
10-4
104
10-5
i0•
2
4
6
8
I0
parallel velocity componentdefined in the deHoffman-Teller
frame. Here it is implicitly assumedthat A•H, and AB -- Bm,x
-B• occur in the same region. It is seenthat the potential
jump broadens the loss cone and consequentlyit tends to
reduce the reflectedelectrons.Obviously, the crucial point is
the magnitude of lelA%x in comparison with the kinetic
o
energy
mvl12/2
of theelectron.
If eA•m is small,its effectis
expectedto be insignificant.The effectof A•m on the shockrelated mirror reflection processwas first discussedby Feldman et al. [1983].
Here it is important to reiteratethat (10) is applicableto the
shock wave case only if we work in the deHoffman-Teller
frame. Thus we cannot use a A(I)HTif it is calculatedor mea-
10
-6
grll(keY)
sured in a frame other than that.
Considering the coordinate system defined in Figure 4,
Fig. 5. Calculated density and flux of the reflected electrons
which is a shock rest frame with normal incidence,i.e., V•
versus
the energys,H,whichis definedin section2. For given0c,•,
andV•,therelationbetween
%1andtheangle0 canbereadilydeter- we note that there are two electric field components.One is
mined.
Ey,whichis dueto themotionof theplasma,andtheotheris
We can also compute the flux of the reflectedelectrons,J,.
It is apparent that for Vs>>vn, J• can be approximatelyexpressedas J•-• 2Vsn•.Since the reflectedelectronsare antici-
E,,, which is inherentlyassociatedwith the shock transition.
By following a velocity transformation,the correspondingE,,'
definedin the deHoffman-Teller frame may be written as
patedto possess
a distribution
functionwhichpeaksat vii=
Vs(1+ cos2 0c)in thesolarwindframe,we definea character- It has been shown recently by Goodrichand Scudder[1984]
isticparallel
energy
s•l
that
It is interestingto point out that the densityn, as shownin
(7),is proportionalto thefactorexp(- Vs2 sin20c/vn2).Physically,the quantitymV•2 sin2 0c/2 may be conceived
as the
characteristicenergy of the seed electrons before reflection.
This can be appreciated from Figure 3, as the distribution
functionof the seedelectrons
peaksat V2 = Vs2 sin2 0c.Thus
onecan showthat an energygain is approximately
(1 + cos2
0c)2/sin
• 0c,whichis veryimpressive.
For example,if we take
0c = 30ø,the gain is slightlyover 12.
3.
Two points relevant to the calculation presentedin the precedingsectionneed discussion.
by (2). SinceV• and V: are smallin comparison
with the upstreamelectronthermalspeedand mVnV:<<mV•2/2 (which
can be shown on the basisof the Rankine-Hugoniot condition
eA• m = AT•
where Teis in the energyunit. SinceATe,accordingto observations [see Feldman et al., 1983], is typically a few tens of
Effect of the ElectrostaticPotential at
the Shockon the ReflectionProcess
The discussionand calculation presentedin section 2 have
ignored the effect of the electrostaticpotential which existsat
the shock front. Such an approximation needsjustification. It
is well known that in general, the effect of the electrostatic
potential jump can affect a mirror reflection process.For example, if we work in the &Hoffman-Teller frame and consider
the relationsof energyconservation
(8)
and the conservationof the magneticmomentof an electron
# = mv.•2/B= const
(12)
In (11) and (12) the deHoffman-Teller velocityVm is defined
of V0, it is seenthat the potentialjump A(I)HT(where(I)HTis
definedin (12)) is approximatelyequal to the electrontemperature increase[Goodrichand Scudder,1984], i.e.,
DISCUSSION
«my2 -- eOHT(X)
= const
]
eEx'= Ox (Vy2
+ K2)+ mVHTKnOx- e 0---•-
(9)
electronvoltsto 102eV, the effectA• m on the lossconeis
apparently unimportant for those electrons with energies
greaterthan 102eV.
If a finite A•.x is includedin the calculationof n, onefinds
that in (7), nc and nn shouldbe replacedby nc exp (--eAq>m
tan20c/vc2)andnn exp(--eA(I).xtan20c/VH2),
respectively.
In
practicalcomputationthe coreelectronsare insignificanteven
if A(I)m= 0. Thusonly the factorexp (-eA(I)m tan20c/VH2)
entersthe computation.If we assumeeAq)HT/VH2=
2 and
Oc= 30ø, it is foundthat exp (--eA(I>.xtan20c/l)H2)•'•0.5.
Although A•.T doesnot give rise to a dramatic effect,it can
be easilyincludedin our calculation.
Comparisonof the Theory With Observations
Numerical valuesof n, and J, are plotted in Figure 5 versus
it is readily seenthat the electron is reflectedif its upstream
the energy%1definedin section2. In obtainingtheseresults
WU: FAST FERMI PROCESS
8861
we have consideredthree valuesof the effectivelosscone angle
man et al. [1982], which may lead to a better quantitative
0½,say,30ø, 35ø,and 40ø.In an attemptto comparethe calcu-
result.
Despitethe unsatisfactory
quantitativecomparison,
two primary conclusionsof the presentmodel should be stressed.
dJr'/derll,
whereJr'is Jr dividedby a solidangle2•. Theresult First, the accelerationis highly effectivenear the point of
tangency of the interplanetary magnetic field (IMF) to the
is presentedin Figure 6, in which a case of the ISEE 3measureddifferentialflux reportedby Anderson[1981] is also bow shock.Second,the energizedelectrons(or the reflected
shownby thedashed
curve.It is seenthat for •rll• 2 keV the electrons)possessa loss cone-type distribution function becalculateddifferentialflux is more than 1 order of magnitude causeof the nature of the mirror reflectionprocess.This prehigherthan the observedvalue.This discrepancymay be at- diction is consistentwith the measurementtaken deep inside
tributable to severalfactors:(1) the calculateddifferential flux the foreshockreportedby Feldmanet al. [1983], who observed
includes electrons with all pitch angles greater than 0½, the lossconefeatureof electronswith energiesof severaltens
whereasthe actual measurementmay cover only a limited of electron volts.
rangeof pitch anglesdependingupon the nature of the detec4.
SUMMARY
tor; (2) in the calculationwe have ignored the effectof finite
In this paper we have discussed
a fast Fermi processwhich
excursiontime during reflection;(3) the calculationsassume
that all electronsdo not loseenergyduring the reflectionpro- may explain how solar wind electronsare energizedin the
cess;and (4) the magneticmoments of all reflectedelectrons vicinityof the point of tangencyof the IMF to the earth'sbow
are assumedto be adiabatic invariants. Becauseof the pres- shock.The key point is that in the solarwind framethe nearly
enceof plasmaturbulenceassociated
with a varietyof instabil- perpendicularbow shock behavesas a rapidly moving magities [Wu et al., 1984], assumptions(3) and (4) may not hold netic mirror in such a manner that electronswith sufficiently
for all electrons.Moreover, the ISEE 3 spacecraftwas in the large pitch anglescan be reflectedand accelerated.Recently
far upstream of the bow shock (approximately 200 Re from Holmanand Passes[-1983-]have pointed out that this accelerthe earth). The flux of the reflectedelectronscan be somewhat ation processmay be relevantto the type II solar radio emisreducedbecauseof the excitationof plasmawavesand wave- sion.We sharethe sameview, althoughwe tend to think that
emission
mechanism
maybedifferent.
A quantitaparticle scatteringprocesses
while they travel from the bow thedetailed
shockto the spacecraftsite.
tive comparisonof the calculateddifferential flux with the
The calculated flux and differential flux decreaseexponen- observationsreported by Anderson[1981] and Parks et al.
tiallywithœrll
or Vs
2. Thisis attributed
to themodeldistri- [1981] indicates that the model distribution function of the
bution function (equation (5)) used in the calculation.When halo electrons used in the present calculation may not be
•rll increases,
the energies
of the seedelectrons
becomepro- pertinent when the energies of the seed electrons become
gressivelyhigher. The halo Maxwellian distribution may not greaterthan 3 x 102eV.
A final remark may be appropriate. After the presentpaper
be pertinentfor electrons
with energies
greaterthan 4 x 102
eV. As is shownby Feldmanet al. [1975], the fitting beginsto had beensubmittedto the Journalof GeophysicalResearchfor
showsomesignificant
deviationwhenv > 10'• km/s, which publication,M. M. Leroy and A. Mangeney at Meudon inapproximately
corresponds
to energies
greaterthan 3 x 102 formed the author that they had developed a similar idea
eV. Thus in order to improve the calculation, perhaps we independentlyand were writing a paper on the subject.Their
shoulduse the more sophisticatedmodel suggestedby Feld- paper will be publishedin the AnnalesGeophysicae
[Leroy and
Mangeney, 1984]. According to the preprint of their paper,
which we received in March 1984, the major difference be106
tween the two papersis that Leroy and Mangeneyconsidera
single Maxwellian distribution in describingthe solar wind
electrons,whereasthe presentpaper usesa bi-Maxwellian dislation with the observational result reported by Anderson
[1981], we have further computed the differential flux
•
Calculated
tribution
function.
T._.
I0
5••• Observed
Acknowledgments.The present work was supported by the National Aeronautics and Space Administration under grants NAGW81 and NGL 21-002-005. The author is indebted to J. D. Scudder, C.
I \
C. Goodrich, K. A. Anderson, R. P. Lin, and B. Tsurutani for valu-
able discussionsand helpful commentsduring the preparation of this
paper.
E
The Editor thanksthe two refereesfor their assistance
in evaluating
this paper.
x
• IO3
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4
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I0
•r II (keV}
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Wu: FAST FERMI PROCESS
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(ReceivedNovember 2, 1983;
revisedApril 30, 1984;
acceptedMay 17, 1984.)