Pair Production brezin

8/13/2019 Pair Production brezin

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THIRD SERIES) VOL. 2) No. 7 1 OcTQBER 1970

Pair Production in Vacuum by an Alternating Field,E. BREzIN AND C. ITzYKSQN

Service de I'hysi que Theorique, Centre d'Etudes EucLeui res de Sacluy, 91,Gif-sur-Yvette, France(Received 15 April 1970)

We discuss the creation of pairs of charged particles in an alternating electric Geld. The dependence onthe frequency is computed and found negligible. We obtain a formula for the Geld intensities required inorder to observe the effect E&nuopc/e sinh(%up/4nsc').

I. INTRODUCTIONOME experimentalists working on intense opticallasers have raised the question of testing nonlineareffects of vacuum quantum electrodynamics. In thisarticle we investigate the possibility of observing thecreation of pairs of charged particles (electrons andpositrons) in oscillating electric fields. One might thinkof the collective effects of millions ( 2mc'/Aa&0) ofphotons concentrated in a small volume and materializ-

ing their energy. On the other hand, Schwinger' longago computed the effect in a pure static field. It turnsout that the estimates based on his calculation aretotally adequate, and lead to the present inobserv-ability of that phenomenon. The only possibility wouldbe to increase the available maximal fields by fourorders of magnitude. While this rules out the observa-tion of the absorptive aspects of nonlinearities, it mightbe interesting in the future to look for dispersive effects.In order to reach the aforementioned conclusion, it isnecessary to estimate how much the production ratedepends on the frequency cop of the field. This appearsalso as a challenging theoretical exercise, since it willallow us to describe the transition between two extremedomains. The first corresponds to a vanishing frequencyand, as we shall discuss below, yields a singular expres-sion for the rate proportional to e ~~. The second isthe case of very low intensities where the field induces aweak perturbation in the vacuum sta, te. As usual insuch a, circumstance, we foresee that the response canbe expanded in powers of the perturbation. It is clearthat only very high powers will come into play, since anumber of the order of 2mc'/Ace& quanta is required tocreate a pair. Hence one expects a rate of the order of(g/g )4mc~/A(uo' J. Schvringer, Phys. Rev. 82, 664 (1951);93, 615 (1954).

The magnitude of the fields Bp and EI appearing inthe above expressions can be understood as follows. In astatic situation, the work of the field on a typical dis-tance of the problem (in our case the Compton wave-length of the electron, 5/mc) and on a unit charge eshould provide the energy 2mc'; hence cEA/mc mc' orE/Eo 1, with Zo=m'c'/ch. At the opposite end inperturbation theory, we expand in powers of the vectorpotential Z/(ao, times the coupling constant e, dividedby a momentum coming from the electron propagatormc. Thus Ei cdomc/e. Let us add a remark on thesingular behavior for the static case. In the zero-frequency limit, the e ~ behavior of the rate can beunderstood as a quantum-mechanical barrier effect.This is analogous to ionization, where for an electronwith binding energy Vp the rate is approximately givenby the square of the wave function at the exit of thepotential barrier. This can be roughly estimated as

Vp/eEexp dx 2m VpEx

Vp=exp ', 2mvp ' 'eEIn the present case the pairs might be thought as bound.in vacuum with binding energy Vp~2mc'.

We present in Sec. II a summary of the main theo-retical results of Schwinger' needed for the sequel. Theygive a formal basis but require some elaboration to beused for practical estimates. Section III is devoted tothe discussion of a number of simplifying approxima-tions leading to tractable expressions. One wants toelaborate a model which retains the main features ofthe general case but nevertheless allows one to obtain6nal expressions in closed form; Instead of estimatingi19iCopyright 1970 by The American Physical Society.


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E. BREZI N AND C. ITZYKSONthe effect for an arbitrary electromagnetic field varyingin space and, time, we will content ourselves with apure electric field oscillating with a frequency ~p. Inspite of the slightly unrealistic character of this assump-tion, we expect that it retains the main features of amore general situation. It is, of course, well known thatspecific anomalies can occur; for instance, there is nopair creation in a plane-wave field. ' Therefore, if therate predicted by the theory were more favorable, oneshould pay more attention to the particular geometricalcharacterization. At the present stage, however, wepresent rather an order-of-magnitude estimate.On the other hand, we are confronted with a non-trivial mathematical problem. We observe that underour assumptions we have to solve a slightly unfamiliarscattering problem in which the role of the usual con-figuration variable is played by time. This comes aboutas follows. We have to study the time evolution of asystem where a pair is created. Now an antiparticle canbe thought of as a wave packet moving backward intime. Thus we have a scattering wave which for largepositive time has only positive frequencies, while for tlarge and negative it has some negative-frequencyamplitude. As shown below, it is precisely this back-ward amplitude that we must try to Gnd. Conditionsare such that the quasiclassical approximation is valid:We have a rapidly oscillating phase with a frequency ofthe order of the energy gap as compared to a slowlyvarying amplitude, the rate of variation of which isof the order of eE/mc. The dimensionless ratio of thesetwo quantities, eEh/m'c', is assumed to be smallcompared to unity. This justifies the use of a versionof the WKB approximation which resembles some workdone in the context of the ionization problem. ' Thework is now reduced to the evaluation of some trickyintegrals by the steepest descent method which lead tothe final formula (48). This formula yields the requiredsmooth interpolation between the static and perturba-tive regimes. Finally, in Sec. IV we discuss our resultand show that the effect can only be observed for fieldintensities E&Ewith

A(pp/mc'cE,A/m'c'-sinh(A(pp/4mc')This estimate proves that, in the foreseeable future,the frequency plays a negligible role, and that the experi-

mentally obtainable fields are four orders of magnitudetoo small.II. THEORETICAL BACKGROUND

The pair creation rate will be expressed in terms ofthe solution of a one-body Klein-Gordon or Dirac equa-tion in the presence of a source. The particle current iscoupled to the vector potential A(x) of the external,c-number, electromagnetic field. At time ~, the stateof the system is the vacuum ~ 0); no particle or field ispresent. Then the electromagnetic field is turned onadiabatically and switched off at large positive times.Pair creation will have occurred if the modulus of thevacuum-to-vacuum S-matrix element is smaller thanunity. More precisely, if we can write

[(0~5~0)~'=e p( 1'x t(x)),w(x) will be interpreted as the probability of paircreation per unit volume and unit time.It is now necessary to distinguish between fermions orbosons. The final results will be indexed by Ii or 8according to whether they refer to Dirac or Klein-Gordon particles.

A. Dirac ParticlesThe S-matrix element is given in standard units,A=c=1, by

Sp(A)(OiSi0)=(0~ V'exp ie dx j(x) A(x) ~0), (2)

where the current is expressed in terms of the free Diracf'ield +(x) asand V denotes the time-ordering operator.The first step is to establish the formal expression

S,(A) = det (G 'Gp) = expLTr In(G 'Gp) j (4)where the propagators Gp and G are

Gp= G=-Pm+ip P eA(x) mipThe symbols det and Tr refer to the product of the four-dimensional Dirac space and an ~ '-dimensional spaceindexed by a space-time coordinate. The canonicaloperators I' and X satisfy the commutation rules

[X,P.7=ig.We introduce now the scattering operators satisfying

The general theory is due to Schwinger. ' In thissection we present a summary of the necessary results.The derivations are omitted except for the case of aconstant field.2A. M. Perelomov, V. S. Popov, and M. V. Terent'ev, Zh.Eksperim. i Teor. I'iz. 50, 844 (1966};51, 309 (1966} /SovietPhys. JETP 23, 924 (1966};24, 207 (1967}j.

T=eA+eA T,P m+ipE=eA+eA I',I'-es-~e (6)


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PAI R PRODUCTION I N VACUUM BY AN AI. TERNATING FIELD 1193and the state-density operators

p(~)= 22r(P+2)2) 8~(Pp) f](P2 222) . (7)Making use of unitarity and of some algebraic identities,it is shown in Ref. 1 that~Sp(A) ~2=exp[Tr ln(1Tp(+&T]&( &)g.Therefore, the pair-creation rate is

wF(*)=tr(xlln(I 'p(+)rp( )) lx),in which tr refers to the trace on Dirac indices alone.

B. Klein-Gordon ParticlesIn the case of spin-zero bosons, we note the analogousformulas:Sp(A) = [Det (G 'Gp)g '= exp[rln(G 'Gp) , (4')

whereGp= G= (5')P 22 +jp (pA)2 )22+jp

Defining nowP'=Gp 'G '=e[A( ) P+P A( )j e'A'(x)

the scattering operators are given by

C. Constant FieldLet us first use expression (4) for fermions. It can bewritten

lnSp Tr ln (P eA )Pm+ie1.=Tr ln (PA+2)2) 8+me

To obtain the last equality, use has been made of the4&4 matrix C such that CyC '= ~. Adding thetwo forms of lnSp, we get1

lnS(& 2 Tr ln [(P eA) )2') (10)P'm'+aeand from the identity

ds(peep esse)swhich incorporates the correct i e prescription, Re lnSp= deg w(g) =Re de@

T=V+V TP')2'+2pT=V+V- T,P'm' e

and the density of states

(6')

~tr(~ ~ 4 is(P2s) &i [(P seA)2s] ~ g) (12)Hence the rate wF(x) readswF(x) =Re tr

p S)((Z~ ' [(ePseA)2 Wme/2&e F]&is(P2-ms) ~ +) (13)r

p(p) = 2&r[]p(Pp) (&(P2222) . (&')The square modulus of the amplitude is

where O'F=o-& F,with the usual conventionsP =2i[y sy 7 ) F,= BAs BsA

~Sp(A) ~ 2=exp[+Tr ln(1 Tp(+)Tp( &)j. (8') In the case of bosons, (4') yields directlyAnd, 6nally,

w~(~) =(~l(I') (+)2') (-)) I*). (9')Expressions (9) and (9') are interesting since theyinvolve on-shell matrix elements. All the singularitiesof the perturbation expansion are explicitly extracted

and, in particular, the thresholds are in evidence.Expressions (4) and (4'), which at 6rst sight might seemmore appropriate, do not have these properties. Never-theless, for the case of a constant field, which is anexactly soluble model, ' we shall start from (4) or (4').A unitary transformation will reduce the problem to aone-dimensional harmonic oscillator. For the case of anoscillating 6eld, studied in Sec.III,we shall be interestedin recovering the constant 6eld result at a zero-frequencylimit. That is why we present this calculation in somedetail.

lnSp Tr in( [(PA)'22'7(P' )22+2p) '), (10')and, after some manipulations similar to the ones above,

dSws(2:) =Rep S

)((g~ pi(e[PeA) ] pis(P2s) ~g) ( 3s)In fact, if one wants to compute the imaginary parts,instead of the real ones, of the integrals (13) and (13'),one has to remove a logarithmic divergence at s=0.This is done by a subtraction at s=0, the logarithmicdivergence being absorbed, as shown by Schwinger, intoa renormalization of the fields and charges. But for thereal part the calculations are straightforward.Pair creation does not occur in a pure magnetic 6eld.This is clear since a constant fjLeld cannot transfer energy


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ii94 E. 8 REZ I N AN D C. I TZ YKSONto a charged particle. It is the acceleration due to the Henceelectric 6eld that enables particles to leak through the2m potential barrier. Hence the physically relevant casecan be taken to be the one of a pure electric field. Wethen choose the gauge as

2(2o-) 'ds- eE 1- sin(sm') . (19b)s' sinh(eEs) s

A (x)= (0, 0, 0, go) .The following identities hold:

tr exp(-,iescr F)=4 cosh(seE)and(P eA)'= (P') ' P,' (P+eE') '

oE2 ~ n7rm2res exp-7r' eE (20a)For this constant-field case, z J; and x~ are, as expected,x independent. The integrals (19a) and (19b) are easilyperformed by contour integration, and the final for-mulas read(15)

I'OI'3=exp P' ', ''E' Xo 2eE~Eo ~ ( 1)n'Mg=2vr & n

N7rm'exp . 20b

joOp'3Xexp i . (16)Denoting by co and p the eigenvalues of I'o and I',respectively, we thus have(*I exp(isL(P eA)' m'j'rI x)= (2~) dcodco'

3X d'pexp i (coo') t+(pp+m')eEX (co ~ expLis(Po' 'E'Xo')] ) co') . (1l)

The integrals over po and pc are readily done, and wefind(z ~ exp(isL(P A)' m']) ~ x)

m-eE esm2'LS

dM(2m-)'

X (co f expLis(Po' 'E'Xo') 7 [ co) (18)The last integral over ~ is the trace of the evolutionoperator of a harmonic oscillator with a pure imag-inary frequency. We compute this trace by summirigover the discrete levels:

dco(oo I expLcs(P oo'EoXo )]00=P expL2n+1)seEj=0 2 sinh(seE)

Using this result in (13),we get

(2o-)'~ ds-eE coth(eEs) sin(sm'), (19a)s S

where the 1/s term is the zero-Geld subtraction whichappears in (13). For bosons, a similar expression holdswith the spin factor 4 cosh(eEs) of (15) replaced by .

We conclude this section with two remarks aboutEqs. (20a) and (20b).(i) Even with the most intense laser beams presentlyavailable, m'/eE is a large number (see Sec. IV), andthe terms with n& j. are exponentially small as com-pared with the first term. Therefore, the ratio betweenthe rates for charged fermions or bosons is essentiallya factor of 2 due to spin.(ii) The oscillation in sign for the bosons shows thatthe successive terms cannot be interpreted as the prob-ability for the creation of I, 2, . . . , n pairs.

III. ALTERNATING FIELDThe aim of this section is to find practical estimatesfor the rate of pair creation in an oscillating field. Thesmallness of two typical parameters enables one toapproximate in a manageable way the previous theo-retical formulas valid in an arbitrary field. We require,of course, that the final expression agrees, in the zero-frequency limit, with the constant-field result (20).On the other hand, in weak fields we should recover thepredictions of perturbation theory. This means thatone has to find. an interpolation between a power lawin E and an exponential in E '. I.et us then introduceour simplifying assumptions.First we limit ourselves to frequencies Mo&(m. Further-more, eE is assumed to be small compared to m'. Thusterms in the expansion of the logarithm in (9) and (9')with thresholds at energies corresponding to the creationof 2, 3, . . . pairs can be safely ignored. This is also sup-

ported by remark (i) at the end of Sec. II, where weobserved that in the static case under the above as-sumption, only the leading term in the probabilitycould be retained. Consequently, we replace the loga-rithm in the expression for the probability by its 6rstterm. Then, as we learned from Sec. II, the spins of thefinal pairs contribute essentially to a counting factor.Therefore, we study the creation of charged bosons.Furthermore, we limit ourselves to an oscillating fieldconstant throughout space. We expect that the spacevariation of the field might produce similar effects.


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PAIR PRODUCTION IN VACUUM BY AN ALTERNATING FIELD ~~9~They are neglected for the sake of simplicity in thisorder-of-magnitude calculation.The vector potential is then chosen as

&(x)=(0,0,0,A(t)).We assume A(t) to be periodic with frequency &ao andadiabatically damped for large times. A straightforwardapplication of (9') gives for the probability per unitvolumedW I dtp I( I& I )I' =(p'+~')' (2&)dV 2tr (2(d)'The Tt, matrix occurring in (21) is the solution of theintegral equation:

and is bounded by8E eE

(0 Bt +Pi tttin which 8 is the amplitude of the oscillating 6eld.Therefore, the calculation wiH assume eB/ttt'((I; allthe existing laser beams satisfy this condition. The%KB method suggests that we look for a wave func-tion of the form

p(t}= (t)e-' )+-p(t)e' &'),withx(t) = dt' )(t') .

T=V+t/' T,Po' )'+ieV=eptA (X())+e'A '(X())

[Xp,PO]= i. & (e)j),, = (+ )ep(In these expressions, y appears as a parameter. Thespace-translational invariance has reduced the problemto a one-dimensional I ippmann-Schwinger equation.Time plays the role of the usual conhguration variableand E'0 of its conjugate momentum. The matrix element(Ol 2'pI)) then corresponds to the backward scatter-ing amplitude. In turn, this amplitude is related to theasymptotic behavior of the solutions of the difterentialequation: [+d'/dt2+(dt+ V(t)]lt (t)=0.

+0& (e)3)t( )xp(=-'

In (28), P(t) has been replaced by two unknown func-tions; we require that they fu16ll the conditionne '~+pe')(=0.

Equation (24) then gives-'-' -p' =-I (t)/ (t)]( -' -p' ).his equation is simply the Klein-Gordon equation forour one-dimensional problem. Ke look now for thesolution which behaves asymptotically as The following system is then completely equivalent tothe original equation:)ft(t)~e eet+ eteet-

t~+ tx) tf,(t) &te t'eet n=(d(t)/2'(t) ](nPe '),p=- [ (t)/2 (t)](pe ') (32)The T-matrix elements are then related to u and b bynormalization factors prom the adiabatic switching assumption, M(t) visheswhen I tI~; therefore, n and p tend to constants rlarge times. Furthermore, according to the previoushypothesis n and p vary slowly in time. The phase e )(oscillates very rapidly Rs compared to the variation ofn and p, since x=tp(t)((l )(t)/)(t) I. As a first step, wecan neglect these oscillating terms in (32). Taking intoaccount the boundary conditions, one 6nds

( I 2 I~)= (t'rv/~)b,(M I 2 .I ~)= (f~/~) (o&). (26)

We notice now that ~ is a large energy, greater than m.This fact suggests that we look at the classical approxi-mation. More precisely, if we deke the variablefrequency

(22) The boundary conditions aren(~)= I, P(+~)=0. (30)

The identification with the asymptotic behavior (25)yields

~(t)= [~'+V.(t)] '[ttt'+ p,+[pt eA(t)]']' (27)the classical approximation can be applied if )(t) variesslowly, i.e., if the dimensionless ratio ~(t)/(d (t) is muchsmaller than unity. This ratio is given by

aZ(t) [P()A (t)]&02 (tm'+p2+Qg A(t)]'}'('

(~l &.I~)M cxp (& '+V(e)1 ))t,

(34)

'(t)=L / (t)]'p&')(t) =0.

In tcIIQS of the T matrix, this means


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1196 r Tz YKSON. BREZIN AND C.

~(t )p(t) = dt'=e ~x i&'i2p~(t') (35)1 we have /fiJ = /P( ) J and, from

~P(~)~. Thi c bin or the robability [Eq. (21)7:erted in the expression for the pro a i i2 pi(t)dti')

2pp(t)d p(2m)' (36)

rrect we cano convince oneself t a et th factors are correest order in the field:xtract from (36) the lowest or er4~2 3/2

dV 3

nize the familiar result of the classical

k d tt i hi hair creation There-erms means no pairt f th F th'to o one sep. . /32' her -1-dlve the second Eq. w1 '(t), ' ' (33. Ty its approximate value esolution readslimit in Eq. (37):

Tip ~ t) iC (t)eiOter, p 2pi(t)11Ql-Q~QO T=2m- g B(mip 0) (cp('.

d p P B(nppp 0)jc j 2(27r)' p(39 )(2pr)'

= c, with c given bye have set (,eE)cpa/ c,+ dg sing [pp eE/&pp) coaxcosx+p '+[pp eE/ppp) cosx' SEeE 1/2Ig' mp+p '+ pp osxexp dS fs

Q)p

&m cop))1, the discrete summation1 d h ob-37bility per unit time and unit vo umreads

m a direct calcu-th the exact result from ahich agrees wit ede6neapro a ii ' eb'lity per unit time,ation. ' In order to d its vector potentialow that the field an i s vessentially van ssh outsi e eThe pair creation raate+ is t en

dW 1 d'p

(39b)

Ttp .~(t)dt e '& . (37)p tp 2M(t)

1 'T, ,T, we takethe time intervauring2 (t)= (E/Mp) cosMpt . (38)

Thuscos~p t = [m'+ p, '+(pp eE/p~p)cosp~pt)'

tx(t) = dt'~(t') =at+ C (t)4~~ ' p ' ' the field itself with fre-~t~ is periodic as e1' dfd 0 is a renorma izeuency ~p, an

eE~ dxMp7r

2pp(t) 7e~&'id the periodic function [pi t pp eet us expand t e periAS a Fourier series:

+i+ (t) C eincept2pp(t)

we can roceed to the largesing this expansion, we can proc

' s re uired to extract' s the mathematics req 'Let us now discussuseeful informatio' n from (39).or c contains a very rrapidly oscilla-/ o.re uencies oevaluation requiresh d

to e inte ated is pe ioth t2O/, i&+ th f bore& in the strip-&Rex

2 2

d gp, h that both its real andp, where xp is such t a oimaginary parts are posi ive2 2 1/2,eE/cop) cosxp~(m +pi0(Rexp( ,p. , 0(Imgp.

n even functionsince c is ane have assumed pp), ' ' nof pp. 'cit condition we can computetec bppo yy k 1 I dle cuts are ta en a onh h lo ydex onential in suc a wa idl .ion the function ecr2 other iree ion


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M BY AN ALTERNATING FIELDAIR PRODUCTION IN VACUUM BY Ap ~In the vlclnl y of the singularities thc cxponentlal 18stationary. ere or,Th fore the main contrlbutlons o

1Dtcgl Rl come f1oIIl this 1cglon.o 6nd the best contour, ere must study lndetail the topology of the surface of e expthe nei hborhood of xo thehis ls sketched ln Flg. 1.In the n g ) ' ex t (xp) 't'g. entegrand behaves hke (xo) pnd end in the valleys o owingoDtoUr must avoid so RIld cD kh two steepest descent lines.. These lines ma e antvlo arts], I'~ The contour ls then made o. vr pDg C 3Z'. Cu d F around o. Furthermor, 'e lf ci and1OUnu Xo, RD c =y RIld2 are their respective contributions, c~= ~,therefore

c=2& Imcy,E I GPo0c . p d* wP+p '+ p :osx)o o

exp x ' 2i(p, '+m') ' inxocoo2x'x 3(doThrough a change of scale, the la,st Integral can bervritten

dgxp(y'&2) ., 2myT econ ourt I is depicted in Fig. 2. Through the map-circuit' is transformed into a positiveing N=y, z is

I 2 C ntour on integration I'q around theIG. . on lane.ranch point in the complex g paround the origin with end points at ~.Hence

We de6ne Doer the real qua, ntities A and 8 through

Q)o

2- 1j20 eldx m'+p, '+ p8osx ~G)0 p (41)@&here A is posltlve. i'th this notation and the evalua-tion of c) 'we obtRlll thc anal cxpI'cssloIl

(27r)'

e ual-modulus lines of the functioneE/ p) cosx') 1' ) needed inEq. (39a).The integration contour consists o q an

se ofal oint in 42) is the exponential decreasee ' , vrhich indicates, as expected, t at aparessential threshold factors the pairs tend to be emittededith small momenta.Th' allows us to estimate (42) as follows.by the classical equation of motion.

(ii) cos'8 is replaced by its average value, ~~.


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u'g(u) is rnonotonically increasing:

FZG. 3. The CurVe g(y}= (4/'7i-) JP' dyr (1')/(1+y'y')g' .

p

X III'+pip e2+2 (slnhz)Alp

Making the change of integration. variable y= (eE/Ipp)X (III'/PI')'~' sinhx we obtain2(m'+p ') ' 1'

cE p 1+y'IPp'(m'+Pis)/c'E'I,et us de6ne the function

. (44)

I ] y2 I/Qg(s) = dy , (45)1+SPyPwhere the normalization is chosen in such a way thatg(0)= 1. Using formula (44) for A and replacing cos'8by ~ as discussed above, we obtain

dp' exp ~(Iu'+p') Ippg III'-+P') 'eE eEor, with III'+P'= (eE/Ipp)'u',

(eE)'187lM p

xeEdu u exp u'g(u) . (46)G)p

In this last expression appears the essential parametery=mppp/eE,

which ls tile 1R'tlo of tile two small quantltles (pip/84)/(eE/III'). Up to now we have not made any assumptionon its size. Under the integral sign in (46) the function

These simplifications amount to multiplying theresult by a numerical factor. From Eq. (40) we getxp ,x+4s-inh 'L(ppp/cE)(IR'+ pp)1~'7. (43)

Then, the signi6cant quantity 2 is expressed as,l -I ~ &~0(.Z) (~2+2,2) I/2] gh) =1v'+Oh ),g(y) = (4/Iry) 1n(2y)+0(1/y) . (49)

In general g(7) ~(4/aery) sinh Iy is an almost uniformapproximation except for a factor of order ~sr for small y.Hence GE ~rg2y(&1, m --- exp2m eE

The value for small y agrees indeed with the calcula-tion of Sec. II apart from an inessential factor x, whilefor large y (or cE/2mcpp((1) we find an amplitude ofprobability proportional to the potential raised to apower equal to the minimum number of photons neces-sary to produce a pair, namely, 2414/Ipp. The generalformula (48) interpolates between these two extremesituations.The most optimistic data using presently availableoptical lasers (frequency Ipp 3X10 sec ') correspondto peak-fields of the order of 7X10 V/cm. These ex-tremely intense fields can only be obtained in pulsesof order 10 sec and al e concentI'ated ln voluQles10 cm . Our two small parameters are, under thesecll cumstances~

IIIpp/Iuc' 4X 10 ', eEA/III'cp 3X10 4,and, not too surprisingly, the parameter y is also small:

'r= 5 QJpc/8E~ 10indicating that the frequency plays no significant role.Kith these numerical values the probability of paircieatloll ls zei'0 to R fRIltRstlc Rcclll'Rcy t ~exp(0 )7.Clearly the observation of the effect would require

u'a(u) & y'g(v) =(v)1up2 mp2 e+This allows us to perform the integration and leads tothe Anal result

o.E' ~msw = exp g(y), (48)2~ g(v)+2vg'(v)where the smooth function g has been given in (45).

IV. DISCUSSION OF RESULTIn this section we discuss our final formula (48) andmake some numerical estimates using existing valuesfor the Geld and frequency.The parameter y describes the transition from thehigh-field, low-frequency limit (y1: constant-field

case), to the low-field, perturbative regime (y))1). Thecurve g(y) is represented in Fig. 3. One readily derivesfrom (45) that


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2 PA I R P RODUCTION I N VACUUM 8 Y AN ALTERNATING FIELD 1199either an increase in the Geld, or in the frequency, orboth.The condition for observing pairs in vacuum can thusbe summarized as eE&m.m'g(y),or, using the very good approximate form for g(y)(4/n-y) sinh 'y,

5ZcdpC

sinh(A(so/4mc2)

Hence even if x-ray lasers would become feasible,A&so/mc' would still remain very smal1 and the effectcould only be observed through a huge increase of theintensity of four orders of magnitude.

ACKNOWLEDGMENTSWe have bene6ted from helpful discussions with ourcolleagues from the DPh-T, Saclay, and with J.Robieux from the C.G.E.Marcoussis Laboratory.

P Il YS I CAL REVIEW 0 VOLUME 2, NUMBER 7 1 OCTOBER 1970Neutron Polarization in ~-p Charge-Exchange Scattering at 310 Mev*

R. E. HILI,f N. E. BooTH) R. J. EsTERLING, ) D. A, JENKINs, ~~ N. H. LIPMAN, ** H. R. RUGGK, f't AND O. T. VIED)Lawrence Radiation Laboratory, University of Calij'ornia, Berkeley, California P47ZO(Received 21 May '1970)

We report the measurement of the polarization in the reaction m +p x'+n at an incident-pion kineticenergy (lab) of 310 MeV and at an angle of 30 in the c.m. system. The polarization was obtained frommeasurements of the left-right asymmetry in the scattering of the neutrons from liquid helium at lab-scatter-ing angles of 75' and 125'. The measured polarization is 0.24~0.07.I. INTRODUCTION

HE experiment reported here was performed sometime ago as part of a program to obtain sufhcientexperimental data on pion-nucleon scattering atT =310MeV so that a unique set of x-X phase shiftscould be determined. Although this goal was onlypartly realized, subsequent experiments and detailedphase-shift analyses have established the m-E phaseshifts rather uniquely up to 1 GeV, and possibly up to2 GeV.

Although the result reported here has been used insome of the detailed phase-shift analyses performedover the last few years, ' ' it has apparently been*Work supported by the U. S. Atomic Energy Commission.t Present address: Geonuclear Nobel Paso, 1 Chantepoulet,Geneva, Switzerland. Present address: Nuclear Physics Laboratory, Oxford Univer-sity, Oxford, England.) Present address: Physics Department, Rutgers, The StateUniversity, New Brunswick, N. J. 08903ii Present address: Virginia Polytechnic Institute, Blacksburg,Va. 24061.~* Present address: Rutherford High Energy Laboratory,Chilton, Didcot, Berkshire, England.ff Present address: The Aerospace Corporation, El Segundo,Calif. 90245.t Present address: Lawrence Radiation Laboratory, Liver-more, Calif. 94551.' L. D. Roper, R. M. Wright, and B.T. Feld, Phys. Rev. 138,B190 (1965).' L. D. Roper and R. M. Wright, Phys. Rev. 138, 8921 (1965).' P. Bareyre, C. Brickman, and G. Villet, Phys. Rev. 165, 1730(1968).4 J. F. Arens, O. Chamberlain, H. E. Dost, M. J. Hansroul,L. E. Holloway, C. H. Johnson, C. H. Schultz, G. Shapiro, H. M.Steiner, and D. M. Weldon, Phys. Rev. 167, 1261 (1968).

omitted in some of the others. ' ' Because of this, andbecause our result has been omitted from a recentcompilation of pion-nucleon scattering data, ' we feelthat it should be properly published rather than onlybe available in its present obscure form. ' 'Apart from the result of our polarization measure-ment, the experimental technique of using liquidhelium as a polarization analyzer continues to be ofinterest

II. MOTIVATIONIn 1959 an extensive set of measurements was begunon pion-proton scattering at an incident lab kinetic

energy of 310 MeV. Measurements were erst made of' B.H. Bransden, P. J. O'Donnell, and R. G. Moorhouse, Phys.Letters 11, 339 (1964); 19, 420 (1.965); Phys. Rev. 139, 81566(1965).' R. J. Cence, Phys. Letters 29, 306 (1966).7 A. Donnachie, R. G. Kirsopp, and C. Lovelace, Phys. Letters268, 161 (1968}.G. Giacomelli, P. Pini, and S. Stagni, CERN Report No.CERN/HERA 69-1 (unpublished).9 R. E. Hill, N. E. Booth, R. J. Esterling, D. A. Jenkins, N. H.Lipman, H. R. Rugge, and O. T. Uik, Bull. Am. Phys. Soc. 9,410 (1964).Note that the result quoted here was a preliminary oneand is superseded by the result given in the present paper.R.E. Hill, Ph.D. thesis, University of California, LRL ReportNo. UCRL-11140, 1964 (unpubliShed, available from UniversityMicrofilms, Ann Arbor, Mich.T.G. Miller, Nucl, Instr. Methods 40, 93 (1966); 48, 154(1967). S. T. Lam, D. A. Gedche, G. M. Stinson, S. M. Tang, andJ.T. Sample, Nucl. Instr. Methods 62, 1 (1968).'3 J. PiQaretti, J. Rossel, and J. Weber, in Proceedings of theSecond International Symposium on Polarization, Phenomena ofEucleorrs, edited by P.Huber and H. Schopper (Birkhauser Verlag,Stuttgart, 1966), p. 152.

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