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Version1.0 Radiative orbital electron capture by the atomic nucleus K. Pachucki,1 U.D. Jentschura,2 and M. Pfu¨tzner3 1 Institute of Theoretical Physics, Warsaw University, Hoz˙a 69, 00-681 Warsaw, Poland 2Max-Planck-Institut fu¨r Kernphysik, Saupfercheckweg 1, 69117 Heidelberg 3 Institute of Experimental Physics, Warsaw University, Hoz˙a 69, 00-681 Warsaw, Poland Abstract 7 0 The rate for the photon emission accompanying orbital 1S electron capture by the atomic nucleus is 0 2 recalculated. While a photon can be emitted by the electron or by the nucleus, the use of the length gauge n a significantly suppresses thenuclearcontribution. Ourcalculations resolvethelongstandingdiscrepancy of J 0 theoretical predictions with experimental data for ∆J = 2 forbidden transitions. We illustrate the results 3 by comparison with the data established experimentally for the first forbidden unique decays of 41Ca and 1 v 204Tl. 1 9 0 PACSnumbers:23.20.-g,23.40.-s,32.80.Ys 1 0 7 0 / h t - l c u n : v i X r a 1 I. INTRODUCTION Orbital electron capture by the atomic nucleus (EC) is one of nuclear decay modes governed by weak interactions, a common and well known type of radioactivity [1]. In this process the released energy Q (equal to the transition energy Q minus the binding energy of the captured EC electron in the daughter atom) is shared between an emitted electron neutrino and the final atom. With a small probability, of the order of 10−4 with respect to the normal EC decay, a photon can beemittedtogetherwiththeneutrino. Insuchradiativeelectroncapturedecay(REC)theenergyis sharedstatisticallybetweenthreebodies,thustheenergyspectrumofthesephotonsiscontinuous. ThetheoreticaldescriptionofRECwasinitiatedbyMorrisonandSchiff[2]whoneglectedthe Coulomb field of the nucleus and took into account just 1S electrons and only in non-relativistic approximation. Anadvancedandmoreaccuratetheoryofradiativeelectroncapturewasdeveloped byGlauberandMartininRefs. [3,4]. TheyincludedexactlytheCoulombfieldinthepropagation of the electron, took into account relativistic effects as well as screening and considered captures from higher shells. However, they limited themselves to the allowed decays where the nuclear spinchanges by∆J = 0,1withnoparitychange. Predictionsofthismodelweretestedinalarge numberofexperiments,and satisfactoryagreement was found[1]. AmoregeneraltheoryofREC,extendedtoanyorderofforbiddenness(i.e. forarbitrarychange ofnuclearangularmomentumandparity),wasdevelopedbyZonandRapoportinRef. [5]andZon in Ref. [6]. For verification of their results, first forbidden unique transitions (∆J = 2, π π = i f 1)areofspecialimportancebecauseofacancellationofthenuclearmatrixelementsintheratio − of radiativeto the nonradiativecapture rate. Measurements of radiation accompanyingthe 1S EC decayinthecaseof41Ca[7],whichbelongstothiscategory,revealedaseriousdisagreementwith results of Refs. [5, 6]. The shape of the photon spectrum differed from the prediction, and the total probability of the REC process per ordinary nonradiativedecay was found to be larger by a factorof 6than thepredicted one. To resolvethisdiscrepancy apossibilityof photonemissionby the nucleus, in so called detour transitions, was examined by Kalinowski et al. [8, 9] following ideas developed by Ford and Martin [10]. According to Refs. [8, 9, 10], the nuclear contribution to the REC process accompanying forbidden transitions can be substantial. In particular, for the caseof41Caitwasclaimed[8,9]thatthedetourtransitionsfullyaccountforthemissingintensity established by the experiment. However, for another first forbidden unique transition — the 1S EC decay of 204Tl — a different situation was encountered. The measured intensity of the REC 2 spectrum [11] was found to be smaller by a factor of 4 than the value predicted by the model of Zonleaving noroomforthenuclearcontribution,incontradictionwithresultsofRef. [9]. Although we agree in general with Refs. [9, 10] that nuclear radiation takes place, we point out that the separation of radiation emitted by the electron and the nucleus, respectively, is not physicalbecauseitdepends on theparticulargaugeusedin thedescriptionoftheelectromagnetic field. We argue, that although physical results do not depend on the selected gauge, the so called length gauge is preferred for the actual calculations. First, it suppresses the nuclear contribution, and second, it makes possible important simplifications in the calculations. We note that some formulae of Zon in [6] for ∆J = 2 transition are divergent for the point nucleus. It means that approximationswhichleadtotheseformulaemaynotbecorrect. Forexample,anassumptionthat the radius of the region where the photon emission occurs is much larger than the dimension of the nucleus where the capture takes place, is not valid in the Coulomb (velocity) gauge used by theauthorsofRefs. [5, 6, 9]. The failure of the theory of Zon and Rapoport to describe the experimental data for forbidden ECtransitions,andthecontroversyconcerningthenuclearcontributiontotheRECprocess,moti- vatedustoaddressthesequestionstheoreticallyinan independentanddifferentway. Inthiswork we recalculate the radiative electron capture process in the length gauge. We restrict ourselves to the 1S electron (K-capture) only, but the extension to other states is straightforward. In the following section we describe our calculations in detail. The results, in terms of a dimensionless shape factor, for the case of the first forbidden unique transitions are presented in Section III and are compared to theexperimental data for 41Ca and 204Tl. Comparison to previouscalculations is madeinSection IV followedby asummaryinSection V. II. THEORY A. Preliminaries In the following we use natural units h¯ = c = 1, e2 = 4πα and set the electron mass m = 1. TheHamiltonianfortheEC-decay is G H = F [ψ¯ γ (1 λγ5)ψ ] [ψ¯ γµ(1 γ5)ψ ], (1) EC n µ p ν e √2 − − where ψ , ψ , ψ and ψ denote neutron, proton, neutrino and electron bispinors respectively, n p ν e with Fermi constant G = 1.16639(1)10−5 GeV−2 and λ = 1.26992(69). We use conventionof F 3 Bjo¨rken and Drell[12]forDiracmatrices withγ5 = γ = iγ0γ1γ2γ3, namely 5 I 0 0 ~σ 0 I γ0 = , ~γ = , γ5 = , (2) 0 I ~σ 0 I 0 (cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19) − − and α~ = γ0~γ. It is convenient to use states with a definite angular momentum. For this reason we introduce spinspherical harmonicsχm [14] κ −κ κ+1/2−m Ym−1/2 χm(θ,φ) = |κ| 2κ+1 |κ+1/2|−1/2 , (3) κ  qκ+1/2+m Ym+1/2  2κ+1 |κ+1/2|−1/2  q  where j +1/2 forj = l 1/2, κ = − (4)  (j +1/2) forj = l+1/2. − Theyhavethefollowingproperties  ~σ rˆχm = χm , (5) · κ − −κ ~σ L~χm = (κ+1)χm. (6) · κ − κ whichare usedto solvetheDiracequation. We considerat first theleft handed neutrino, 1 1 γ5 ψ = ψ . (7) ν ν 2 − (cid:0) (cid:1) Its wavefunctionψ withtheangularmomentumj andtheenergy q is ν ν ψ (~r) = , (8) ν   ν − ν = q[j (qr)χm (θ,φ)+ij (qr)χm (θ,φ)], (9) j−1/2 −(j+1/2) j+1/2 j+1/2 withj (x)beingthesphericalBesselfunctionandν solutionof~σ p~ν = qν. Theaboveneutrino l · − wavefunctionisnormalizedwithrespect to theenergy, namely d3rψ+(~r)ψ (~r) = 2πδ(q q′). (10) ν ν − Z The electron wave function is the solutionof the Dirac equation in the Coulomb field Zα/r − [13]. It isoftheform G(r)χm(θ,φ) ψ (~r) = κ , (11) e iF(r)χm (θ,φ)  −κ   4 withtheenergy E = E n,κ Z2α2 −1/2 E = 1+ , (12) n,κ (n+1+γ)2 (cid:18) (cid:19) whereγ = 1+√κ2 Z2α2. Theexplicitform ofradial wavefunctionsG andF is[13] − − G (r) = C (2λr)γe−λr 1+E [g(2)(r)+g(1)(r)], (13) n,κ n,κ n,κ n,κ n,κ F (r) = C (2λr)γe−λrp1 E [g(2)(r) g(1)(r)], (14) n,κ n,κ − n,κ n,κ − n,κ g(1)(r) = √Zαλ−1 κ L(2p+2γ)(2λr), (15) n,κ − n g(2)(r) = (n+2+2γ)√Zαλ−1 κ L(2+2γ)(2λr), (16) n,κ − − n−1 2λ4n! C = , (17) n,κ sZαΓ(n+3+2γ) whereL(α) areLaguerre polynomials n n Γ(α+n+1) L(α)(z) = ( z)k, (18) n k!(n k)!Γ(α+k +1) − k=0 − X with L 0 and λ = √1 E2. In particular, the energy and the wave function of the ground −1 ≡ − staten = l = 0,κ = 1, and j = 1/2 aregivenby − E = √1 Z2α2 , (19) 0,−1 − ≡ E 4Z3α3(2+γ) G = (2Zαr)γe−Zαr , (20) 0,−1 s Γ(3+2γ) ≡ G 4Z3α3( γ) F = − (2Zαr)γe−Zαr . (21) 0,−1 −s Γ(3+2γ) ≡ F ForthecalculationofradiativetransitionratesoneneedstheDirac-CoulombGreen’sfunctionGD. Its explicitformis [13] 1 GD(~r,~r′,E) ~r ~r′ = GD (~r,~r′,E), (22) ≡ h |H E| i κ,m − κm X GD (~r,~r′,E) = Θ(r′ r)ψ< (~r) ψ> (~r′)† +Θ(r r′)ψ> (~r) ψ< (~r′)†, (23) κ,m − κm ⊗ κm − κm ⊗ κm whereΘ(x) is aHeavisidestep function,and bothψ> and ψ< areoftheform (11)with G<(r) = (2λr)γe−λr√1+E(f +f ), (24) κ 2 1 F<(r) = (2λr)γe−λr√1 E(f f ), (25) κ − 2 − 1 2λΓ(a) G>(r) = (2λr)γe−λr√1+E(f +f ) , (26) κ 4 3 Γ(c) 2λΓ(a) F>(r) = (2λr)γe−λr√1 E(f f ) , (27) κ − 4 − 3 Γ(c) 5 where f = (Zαλ−1 κ) F (a,c,2λr), (28) 1 1 1 − f = a F (a+1,c,2λr), (29) 2 1 1 f = U(a,c,2λr), (30) 3 f = (Zαλ−1 +κ)U(a+1,c,2λr), (31) 4 anda = 1+γ EZαλ−1,c = 3+2γ,while F andU areconfluenthypergeometricfunctions 1 1 − regularattheoriginand at theinfinity,respectively. Todescribephotonwavefunctionweintroducevectorsphericalharmonics[14], Y~M(θ,φ) = Ym(θ,φ)~e Lm;1q L1;JM , (32) JL L qh | i mq X where~e = 1/√2(~e +i~e ),~e =~e ,and~e = 1/√2(~e i~e ). ThesolutionsoftheMaxwell 1 x y 0 z −1 x y − − equationswithdefiniteangularmomentumandparityarerepresentedbyboth,themagneticphoton A~(M)(~r) = √2kj (kr)Y~M(rˆ) (33) JM J JJ A0(M)(~r) = 0 (34) JM and theelectric photonin thelengthgauge[15] 2J +1 A~(E)(~r) = √2k j (kr)Y~M (rˆ), (35) JM J J+1 JJ+1 r J +1 A0(E)(~r) = i√2k j (kr)YM(rˆ), (36) JM − J J J r whiletheelectricphotonintheCoulomb(velocity)gaugeis J J +1 A~(E)(~r) = √2k j (kr)Y~M (rˆ) j (kr)Y~M (rˆ) , (37) JM 2J +1 J+1 JJ+1 − 2J +1 J−1 JJ−1 "r r # A0(E)(~r) = 0. (38) JM Thesesolutionsarenormalizedwithrespect to energy,so that 2k d3r[A~ (~r)∗ A~′ (~r) A0 (~r)∗ A′0 (~r)] = 2πδ(k k′), (39) JM JM − JM JM − Z The use of the length gauge is essential for performing several simplifications in the calculations oftransitionrates, as itis explainedin thenextsections. 6 B. Electroncapturerate The electron capture rate W is equal to the square of the the matrix element M, summed over final statesand averaged overinitialstates 1 1 W = M 2, (40) 2J +1 2J +1 | | e i MeMXiMνMf where G M = f H i = d3r F [ψ¯ γ (1 λγ5)ψ ] [ψ¯ γµ(1 γ5)ψ ]. (41) EC f µ i ν e h | | i √2 − − Z Althoughweusesinglenucleonmatrixelements,resultscaneasilybetransformedforthenuclear matrixelements,byassumingthatψ andψ arefieldoperators,andinsteadofψ¯ γ (1 λγ5)ψ n p f µ i − one considers f ψ¯ γ (1 λγ5)ψ i . The tensor decomposition of the matrix element in Eq. n µ p h | − | i (41)leads to G M = F r2dr ( 1)J+M f TM (1 λγ5) i ψ T−M (1 γ5) ψ , (42) √2 − | JLS − | ν| JLS − | e Z JLSM X (cid:0) (cid:1)(cid:0) (cid:1) where TM = iLδ YM , (43) JL0 JL L TM = ( 1)J+L+1iLY~M α~ , (44) JL1 − JL · and (. .) denotes the integral over angular coordinates. Each state f,i,ψ and φ has definite ν e | angularmomentumJ,M numbers,so onecan usereduced matrixelement j k j′ j,m Tq j′,m′ = ( 1)j−m j T j′ , (45) h | k| i −  m q m′  h || k|| i −   and orthogonality properties of 3j symbol [14] to obtain simple formula for the electron capture rate G2 1 1 1 W = F 2 2J +1 2J +1 2J +1 e i J X 2 r2dr J T (1 λγ5) J J T (1 γ5) J . (46) f JLS i ν JLS e × || − || || − || (cid:12)Z LS (cid:12) (cid:12) X(cid:0) (cid:1)(cid:0) (cid:1)(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 7 Thereduced matrixelementsofspherical harmonicsaregivenby 2l+1 κ Y κ = C (κ ,κ ), (47) f l i l f i h || || i 4π r j l j C (κ ,κ ) = ( 1)jf+1/2 (2j +1)(2j +1) i f Π(l ,l ,l), (48) l f i f i f i −   1/2 0 1/2 q − 1   Π(l ,l ,l) = 1+( 1)lf+li+l , (49) f i 2 − (cid:2) (cid:3) and spinsphericalharmonicsby 2J +1 κ Y~ ~σ κ = S (κ ,κ ), (50) f JL i JL f i h || · || i 4π r J +1 κ +κ S (κ ,κ ) = 1+ f i C ( κ ,κ ), (51) J,J+1 f i J f i 2J +1 J +1 − r (cid:18) (cid:19) κ κ S (κ ,κ ) = i − f C (κ ,κ ), (52) J,J f i J f i J (J +1) J κ +κ S (κ ,κ ) = p 1+ f i C ( κ ,κ ). (53) J,J−1 f i J f i 2J +1 − J − r (cid:18) (cid:19) With the use of the above formulae for reduced matrix elements, the capture rate W of the 1S electron bythenucleusis W = W , (54) J J X 2G2 Q2 1 W = F J T (R) T (R) (1 λγ ) J 2, (55) 0 f 000 011 5 i 2J +1 4π h || G − F − || i i (cid:12) (cid:2) (cid:3) (cid:12) (cid:12) (cid:12) 2G2 Q2 1 W = F J T (R)+ T /√3+T 2/3 (R) (1 λγ ) J 2, 1 f 101 110 111 5 i 2J +1 4π h || G F − || i i (cid:12) (cid:2) (cid:0) p (cid:1) (cid:3) (cid:12) (56) (cid:12) (cid:12) 2 G2 Q4 1 W = F J r T (R)+ T 2/5+T 3/5 (R) (1 λγ ) J 2, 2 f 211 220 221 5 i 9 2J +1 4π h || G F − || i i (cid:12) (cid:2) (cid:0) p p (cid:1) (cid:3) (cid:12)(57) (cid:12) (cid:12) whereQis theenergy released inthedecay and R isthenuclearradius. C. Radiativeelectron capturerate Theprobabilityamplitudefortheelectroncapture withthesimultaneousphotonemissionis G M = d3r F [ψ¯ γ (1 λγ5)ψ ] [ψ¯ γµ(1 γ5)ψ′], (58) R √2 f µ − i ν − e Z 8 where 1 ψ′ e(A0 α~ A~) ψ ψ′(~r) = ~r e(A0 α~ A~) ψ = ψ′(r) h n| − · | ei . (59) e h | H k − · | ei − n E +k n E − − n −E X Weusethelatterform toperform tensordecompositionofM and obtain R G e M = F r2dr ( 1)J+M+1 J ,M TM (1 λγ5) J ,M (60) R √2 − f f| JLS − | i i Z JLSMM′ n X e (cid:0) (cid:1) X 1 J ,M T−M (1 γ5) n,J′,M′ n,J′,M′ A0 α~ A~ J ,M ν ν| JLS − | e e E +k h e e| JAMA − · JAMA| e ei n −E (cid:0) (cid:1) Theratefortheradiativeelectron captureW is R Q W = dk W (k), (61) R R Z0 where 1 1 1 W (k) = M 2, (62) R R 2J +1 2J +1 2π | | e i MeMiMXνMfMA with the sum over final state and the average over initial states. The W (k) can be expressed in R terms of the reduced matrix elements and the summation over magnetic states can be carried out withtheaidoftheorthogonalityofthe3j symbols. Theresultsis 1 1 1 1 1 W (k) = G2 α R F 2J +1 2J +1 2J +1 2J′ +1 E +k e i XJ XJe′ e (cid:12)Xn n −E (cid:12) r2dr J T (1 λγ5) J J T (1(cid:12) γ5) J′,n × f|| JLS − || i ν|| JLS − || e Z LS X(cid:0) (cid:1)(cid:0) (cid:1) 2 n,J′ A0 α~ A~ J . (63) ×h e|| JA − · JA|| ei (cid:12) (cid:12) WecanusenowtheexplicitformoftheDirac-Co(cid:12)ulombGreen’sfunctioninEq. (22-31)toreplace the sum over intermediate electron states n in Eq. (63) by GD. Since the electron capture takes placewithinthenucleus,andthephotonradiationinaregionoftheelectronwavefunctionwhich isseveralordersofmagnitudelarger, weapplyan identityΘ(r′ r) = 1 Θ(r r′) andneglect − − − Θ(r r′) inEq. (23)completely,sotheGreen’s functionbecomes − GD (~r,~r′,E) ψ< (~r) ψ> (~r′)†. (64) κ,m ≈ κm ⊗ κm 9 In otherwords,thisapproximationisallowed,becausetheintegralwithΘ(r r′) givescontribu- − tion,whichis higherorderin thesmallparameterξ = QR. Afterthisassumptiononeobtains G2 1 1 1 W (k) = F R 2 2J +1 2J +1 2J′ +1 i J J′ e X Xe 2 r2dr J T (1 λγ5) J J T (1 γ5) J′ψ< × f|| JLS − || i ν|| JLS − || e (cid:12)Z XLS (cid:0) (cid:1)(cid:0) (cid:1)(cid:12) (cid:12) 2α (cid:12) (cid:12) J′,ψ> A0 α~ A~ J 2 . (cid:12) (65) × 2J +1 h e || JA − · JA|| ei (cid:26) e (cid:27) (cid:12) (cid:12) (cid:12) (cid:12) The approximation in Eq. (64) would not be valid in the Coulomb gauge, as the integral with Θ(r r′) isofthesameorderintheparameterξ,what wediscussinmoredetailsinSec. IV. − Weconsidertransitionswith∆J = 0,1,2 W (k) = W (k)+W (k)+W (k), (66) R R0 R1 R2 since no electron radiative capture has been observed for higher multipolarities. Due to the as- sumption in Eq. (64), transition rate for each value of J can be decomposed into a product of a term corresponding to the photon emission and a term corresponding to the nuclear transition, namely W (k) = W (k,S S> )W (Q k,S< ν ) R0 M1 1/2 → 1/2 0 − 1/2 → 1/2 +W (k,S P> )W (Q k,P< ν ), (67) E1 1/2 → 1/2 0 − 1/2 → 1/2 W (k) = W (k,S S> )W (Q k,S< ν ) R1 M1 1/2 → 1/2 1 − 1/2 → 1/2 +W (k,S P> )W (Q k,P< ν ), (68) E1 1/2 → 1/2 1 − 1/2 → 1/2 W (k) = W (k,S S> )W (Q k,S< ν ) R2 M1 1/2 → 1/2 2 − 1/2 → 3/2 +W (k,S P> )W (Q k,P< ν ) (69) E1 1/2 → 1/2 2 − 1/2 → 3/2 + W (k,S P> )+W (k,S P> ) W (Q k,P< ν ) E1 1/2 → 3/2 M2 1/2 → 3/2 2 − 3/2 → 1/2 +(cid:2)W (k,S D> )+W (k,S D> )(cid:3) W (Q k,D< ν ). M1 1/2 → 3/2 E2 1/2 → 3/2 2 − 3/2 → 1/2 (cid:2) (cid:3) Theexplicitformulaeforthephotonemissionrate 2α α W = J′,ψ> A0 α~ A~ J 2 k (70) EM 2J +1 h e || JA − · JA|| ei ≡ π REM e (cid:12) (cid:12) (cid:12) (cid:12) 10