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Nuclear Theory of Reactor Neutrino Fluxes

This article discusses the nuclear theory behind reactor neutrino fluxes, including fission yields, beta decay branching ratios, and the conversion of electron spectra into antineutrino spectra. It also explores the corrections needed for accurate calculations and the historical background of reactor neutrino spectrum determination.

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Nuclear Theory of Reactor Neutrino Fluxes

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  1. Nuclear theory of reactor neutrino fluxes Petr Vogel Caltech Sterile Neutrino at the Crossroads Virginia Tech, Sept 25, 2011

  2. There are 6.14 ne per 235U fission. However, above the detection threshold of 1.8 MeV there are only ~30% of all ne and above the signal maximum only ~5%.

  3. Reactor spectrum: 1) Fission yields Y(Z,A,t), essentially all known with sufficient accuracy 2) b decay branching ratios bn,i(E0i) for decay branch i, with endpoint E0i and spins Ii, some known but some (particularly for the very short-lived and hence high Q-value) unknown or known only in part. 3) b decay shape, assumed allowed shape, in that case known P(En,E0i,Z) or for electrons Ee= E0 – En. Then: dN/dE = Sn Yn(Z,A,t) Sibn,i(E0i) P(En,E0i,Z) and a similar formula for electrons. 4)However, for detailed evaluation we need to include several small corrections P(En,E0i,Z) (1 + dqed + dWM + dC) as well as the effects of the forbidden decays (see below) If the electron spectrum is known, it can be `converted’ into the antineutrino spectrum, because in each branch En = E0 – Ee. Uncertainty associated with this procedure will be discussed in detail below.

  4. Contribution from b decay of fission fragments with different characteristics (information dated, more data available now, the fraction of unknown decays reduced to ~10%.) known decays unknown decays cont = summedg spectrum measured in coincidence with an electron Straightforward calculations must rely on nuclear models for the b decay of the `unknown’ fission fragments

  5. Brief history of the reactor neutrino spectrum determination: First `modern’ evaluations were done in late 1970 and early 1980 (Davis et al. 1979, Vogel et al. 1981, Klapdor & Metzinger 1982) During the 1980-1990 a series of measurements of the electron spectra associated with the fission of 235U, 239Pu and 241Pu were performed at ILL Grenoble by Schreckenbach et al. These wereconverted into the electron antineutrino spectra by the authors. This is basically what was used until now, even though some effort was made to measure the b decay of various short lived fission fragments (Tengblad et al, 1989, Rudstam et al. 1990) and new calculations were performed (see e.g. Kopeikin et al, hep-ph/0308186). New evaluation (Mueller et al. 2011) uses a combination of the ab initio approach with updated experimental data and the input from the converted electron spectra (see 2) above). This results in the upward shift by ~3% of the reactor flux (keeping the shape almost unchanged).

  6. Measured ne spectrum shape and normalization agreed with calculated predictions to ~10% and with converted electron spectra even better Calculation only Klapdor and Metzinger, 1982 Beta calibrated Schreckenbach, 1985 Hahn, 1989 Results of Bugey experiment (1996)

  7. How the conversion of the electron spectrum into the nworks? For a single b decay it is trivial: En = E0 - Ee where E0 is the decay energy For a decay with many known branches Y(Ee) = Sibi k(E0i,Z) pe Ee (E0i - Ee)2 F(Ee,Z) where k(E0i,Z) is a normalization and bi are the branching ratios Once bi and E0i are known, Y(En) can be easily calculated Now suppose that bi and E0i are unknown, but Y(Ee) is measured. One then can assumethat E0i are e.g. equidistantly distributed, and fit for bi. By varying the number of branches, one can check that the result is convergent. (30 branches were used in Schreckenbach et al.) In the actual case Z is also unknown. Some procedure for choosing Z, or Z(E) must be chosen and tested. e) The error associated with the procedure must be determined.

  8. Example of a complex spectrum. Hypothetical b decay of Z=45 nucleus, with 40 branches with random endpoints and branching ratios. The largest Q-value is 8 MeV. Allowed spectrum shape is assumed. Fit to the above electron spectrum. The spacing of slices is indicated. Deviation of the fit is shown. Same as above, but for the neutrino spectrum.

  9. Fit to the 235U spectrum assuming that all nuclei have a single Z = 47. The electron spectrum (dashed) is fitted perfectly, but the neutrino spectrum (jagged and smoothed in red) deviates from the input by as much as 10%. The average Z as a function of the endpoint energy and a quadratic polynomial fit (dashed red). With this function the 235U spectrum is fitted to better than 1%. The conversion procedure allows one to obtain the ne spectrum with < 1% error provided that the corresponding b decay shapes are all well known and described.

  10. Corrections to the b spectrum shape: Note that all of them have been worked out in some detail for the allowed decays. It is not clear how to apply them to the forbidden decays. These energy dependent corrections are: dqed (vertex and bremsstrahlung corrections of order a/2p), dWM (weak magnetism correction, linear in Ee and thus also En) dC (Coulomb correction to the axial vector decays due to nuclear size, also primarily linear in Ee)

  11. QED corrections of the order a/2p are different for the electrons and antineutrinos. It is not enough to take the known correction to the electron spectrum and substitute En= E0 – Ee. The antineutrino correction was evaluated by Batkin and Sundaresan (1995) And a simpler analytic formula was derived and published by Sirlin (2011). Remarkably the expression by Sirlin was used already earlier in Mueller et al. (2011).

  12. Coulomb corrections: In zeroth order F0(Z,Ee) = relativistic amplitude in point Coulomb field evaluated at r=R Including nuclear finite size (possibly also electron screening) F(Z,Ee) = F0(Z,Ee)L0(Z,Ee) = (g-12(r=0) + f+12(r=0)/2pe2 (this is extensively tabulated, e.g. by Behrens and Janecke)

  13. Correction term AC, convolution of the lepton and nucleon wavefunctions. Correction is generally in the form const*ZaREe = ACEe A simple analytic formula has been worked out for the j=1/2 electron waves, i.e. for the allowed transitions. The simple expression, used previously is AC = - 9/10 ZaR/hc <sr2>/R2, with <sr2> = k<s>R2 , k = 1(3/5) for surface(volume) spin distribution (Vogel 1984). (Mueller et al use -9/10) Analogous formula was obtain earlier by Holstein and Calaprice (1976) AC = -11/15ZaR/hc (close to average of the two possibilities above) On the other hand, Wilkinson in 1990 has a formula that contains also a quadratic term in energy AC = -3/5ZaR/hc + 4/9R2En/(hc)2 (close to the volume distribution above) Thus, while the basic form of AC is the same in all cases, there is a rather wide variation of the numerical coefficients and it is not clear how the formula should be modified for forbidden decays

  14. Weak magnetism correction AWM, in spectrum as AWMEe Weak charged current has a simple form for quarks, but need be generalized for the nucleons. The vector part is <p|Va|n> = cosqC exp(iqx) u’[f1(q2) ga + if2(q2) sanqn]t+u In the case of b decay we need the form factors just for q2 = 0. The CVC requires that f1(0) = 1 and f2(0) = [mp – mn]/2mp Where mp – mn = 3.7 is the anomalous isovector magnetic moment. After some algebra one finds (for the allowed GT transitions) Amplitude ~ gA<s> + i[(mp – mn + 1)<s> + <L>] x q/2mp In the decays rate the two terms interfere, Often the orbital momentum <L> is neglected. I used instead the for ji = jf ± 1 and li = lf <L> = -<s>/2. Thus finally AWM = 4/3 x [(mp + 1) – mn – ½] / gA mp≅ 0.47%/MeV For the allowed GT decays AWM is independent of the GT matrix element.

  15. Table I of computation of the weak magnetism slope parameters (P. Huber, Phys.Rev.C84, 024617(2011)) . To test the AWM one can relate it to the rate of the M1 decay from the analog state to the final state of beta decay (parameter b) Note, the relation between bg and slope AWM requires transition from the isobar analog state. Egfrom analog state 8638 829 ~10400 wrong T, 0->1 ~9300 ~11300 wrong T, 10->11 wrong T, 9->10 tiny c 5072 7002

  16. Given a set of endpoints and branching ratios, the resulting relative change in the neutrino spectrum when these corrections are included.

  17. Forbidden b decays: Until now we considered only the allowedb decays, in which the nuclear spin is changed by no more than one unit, |DI|≤ 1, and the parity is not changed. Such decays have the shape dN/dE = pE(E0-E)2F(Z,E)(1 + dqed + dWM + dC). However, when these selection rules are not fulfilled the decay proceeds anyway, but the decay rate is reduced usually by ~(pR)2 << 1 for each order. Note that for a typical fission fragment A~120, pR~ E0(MeV)/30 << 1. For our application we need to consider essentially only the first forbidden decays, with |DI|≤ 2 and Dp = yes. The transitions with |DI|= 0,1 and Dp = yes are governed by several nuclear matrix elements each, and might have complicated spectrum shape (the decay of RaE = Bi210 is a textbook example). However, most of such decays have approximately allowed shape due to a combination of the Coulomb and relativistic effects. This quasi-allowed(x) approximation is valid if x = Za/E0R >> 1, for Z=46 this means (11/E0(MeV)) >> 1 which is barely fulfilled. Thus, for the first-forbidden b decays we can use the allowed shape with caution, and the corrections dWM and dC are not applicable. It is difficult to estimate the error this causes.

  18. First forbidden b decays are common in fission fragments In the lighter peak of the fission yields, A ~ 90-100, take as an example 37Rb94, near its maximum. Protons and neutrons near the Fermi level are in states of opposite parity. N=57, gds shell, p=+ Z=37, fp shell, p=-

  19. First forbidden decays with |DI| = 2 are governed by only single matrix element and thus have again a simple shape (at least for small Z)

  20. Conclusions: • All evaluations of the reactor neutrino flux depend on two • basic assumptions: • 1a) That the electron spectra measured at ILL by Schreckenbach • et al. are correct (there is little doubt of that, even though • the results were not independently confirmed). • 1b) That the spectrum shape of the individual b decays is well • understood. • For the allowed b decays (DI = 0,1, Dp = no) the assumption 1b) • is reasonably well fulfilled, even though the slope parameters • AWM and AC have considerable uncertainty. • However, many transitions involve forbidden b decays since • Dp=yes is often the case, there is an additional uncertainty involved. • It is at present difficult to estimate how large it is, but simple • examples suggest that it could be comparable to the stated • uncertainty 2-3%.

  21. Spares

  22. Figure from the memo by T. Mueller and D. Lhuillier (graciously sent to me) Corrections to the spectrum of a Z=46 nucleus with 20 b decay branches and Q value of 10 MeV. The blue line is the weak magnetism plus coulomb slope factor (Note,it is less than unity everywhere). The dotted line is the corrections used in my old notes, 0.65(En– 4)% The red line is the sum of all corrections, including dqed.

  23. Estimate of the fraction of forbidden b decays: • The GT strength typically increases with excitation energy, thus • the transition to higher excited states will be dominantly GT. • (the cross sections below are proportional to the GT strength, i.e. to • the square of the corresponding GT matrix element) • But the transitions to states near the ground state of the final • nucleus are determined by the structure of these nuclei, and will • be mostly first forbidden. These might account for 30-50% of • the antineutrino flux at En ≥ 4-5 MeV. In the paper by Mueller • et al. all the known forbidden decays were assumed to have the • unique shape, in reality only ~1/3 will be unique.

  24. First forbidden b decays are common in fission fragments In the heavier peak of the fission yields, A ~ 132-145, take as an Example 55Cs140, near its maximum. Protons and neutrons near the Fermi level are in states of opposite parity. N=85, hfp shell, p=- Z=55, gds shell, p=+

  25. Important example of the DI = 2, Dp= yes 2- -> 0+ transition is the 90Y decay, Following the T1/2 = 29 y decay of 90Sr, this decay has Q=2.28 MeV

  26. History of neutron lifetime measurement. Latest result differs from the previous ones by ~6.5 s. Needs verification 885.7 +-0.8 878 +-0.7

  27. Ezhov (2009) unpublished

  28. Electron and ne spectra do not saturate immediately. The low energy part reaches equilibrium only after a long time, while for E > 2 MeV the equilibrium is reached in about a day. Calculated for 235U. At 1 MeV the spectrum changes by ~20% between 1 day and 2 years. At 2 MeV it changes by ~3%, and at 3 MeV by ~0.3%.

  29. Neutrino industry

  30. Cross section is simply related to the neutron lifetime: s (Ee) = (2p2h3)/(me5c8ftn) peEe(1 + small corrections)

  31. The measured spectra at several distances in Goesgen (1986) also agreed well in shape and normalization.

  32. QED corrections of the order a/2p are different for the electrons and antineutrinos. It is not enough to take the known correction to the electron spectrum and substitute En= E0 – Ee.

  33. and precision tests of ne disappearance slide of K. Heeger

  34. Electron antineutrinos are produced bythe b decay of fission fragments

  35. Transforming thermal power into fission rate (all energies in MeV/fission) 235U 202.7 +-0.1 192.9 +- 0.5 201.7 +- 0.6 201.9+-0.5 238U 205.9 +-0.3 193.9 +- 0.8 205.0 +- 0.9 205.5+-1.0 239Pu 207.2 +-0.3 198.5 +- 0.8 210.0 +- 0.9 210.0+-0.6 241Pu 210.6 +-0.3 200.3 +- 0.8212.4 +- 1.0213.6+-0.6 Energy per fission from the mass excesses Energy per fission without neutrinos and long lived fragments but including the energy associated with the neutron captures, (See M.F. James 1969) Energy per fission without neutrinos and long lived fragments (En ~ 9 MeV) From Kopeikin et al. 2004

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