Effective field theory approach to electroweak transitions of nuclei far from stability

11 Pages·0.36 MB·English

by  M. A. Huertas

Effective field theory approach to electroweak transitions of nuclei far from stability
M. A. Huertas
College of William and Mary, Williamsburg, VA
(Dated: February 8, 2008)
In a previous paper, the convergence of the effective field theory approach of Furnstahl, Serot
and Tang to the nuclear many-body problem was studied by applying it to selected doubly-magic,
and neighboring single-particle and single-hole, nuclei far from stability. The success of that ap-
proach,interpretedthroughdensityfunctionaltheory,wouldimplyreliabledensities. Inthispaper,
the single-particle (Kohn-Sham) wave functions are probed using weak transitions near the Fermi
surface. The weak currents are the Noether currents derived from the effective Lagrangian. The
general single-particle transition matrix elements, from which any semi-leptonic weak rate can be
calculated,areobtainedintermsofupperandlowercomponentsoftheDiracwavefunctions. Here
beta-decaysinnucleineighboring132Snarestudiedandcomparedwithavailableexperimentaldata.
3 Calibration of the theoretical results for such decays may also have useful application in element
0 formation.
0
2
INTRODUCTION ground state density is reproduced. The wave functions
n
describing one particle outside a doubly-magic core or
a
J oneholeinadoubly-magiccoreareusedtocalculatevar-
Inarecentpaper[1]theconvergenceofcalculationsto
0 ious β-transition rates. These wave functions represent
experimental results at different levels of approximation
3 the initialor finalnuclearstates (in coordinate represen-
froma new approachto the nuclear many-body problem tation) in these transitions. Here, based on the above
2 was studied. This approach combines the principles of
arguments, an accurate description of the wave function
v Effective Field Theory (EFT) with Density Functional
is assumed to be guaranteedby the accuracywith which
6 Theory(DFT).Theresultsofthatworkshowedthatthe
7 itsbindingenergyisreproduced. Thereforetheidealcase
totalbindingenergyofeven-evenSnisotopenucleicanbe
0 of study corresponds to a ground-state to ground-state
reproducedbelowthe1%level. Inadditiontothisagree-
1 transition,sinceinthesecasesthereisacloseragreement
0 ment for the total binding energy of the doubly-magic
with the experimental energy values. Wave functions of
3 nuclei 132Sn , 100Sn , 78Ni and 48Ni , the chemical
50 82 50 50 28 50 28 20 excitedstates will be less accurateas these states lie far-
0 potential for neighboringnuclei, differing by one particle
/ therawayfromtheFermisurfaceandthustheirenergyis
h or hole from the doubly-magic ones, was also well repro-
notwellreproduced. Furthermore,itisexpectedthatthe
t duced below the 10% level. The agreement in binding
- particle-particletransitionwillgiveacleanerresultsince
l energies shows that the energy functional derived from
c the description of a particle outside the doubly-magic
u the effective lagrangian of Furnstahl, Serot and Tang [2]
core can be well approximated by a single-particle wave
n is indeed a good approximation and thus, according to
function. The description of a hole, on the other hand,
: DFT, the ground-state densities obtained in each case
v is more complicated since it represents a more complex
i are also a good approximation to the true ground-state
X many-body state and its description by a single-particle
densities. Although in [1] both proton and neutron den-
wave function might be expected to be less accurate.
r sities for 132Sn and 100Sn were presented,there has been
a Using the results obtained in [1] and the available ex-
no direct measurement of either of them and thus the
perimental data, this paper focuses on the core nucleus
comparisonwithexperimenthasnotyetbeenestablished
132Sn and its single-particle and single-hole neighbors
directly. 50 82
133Sn , 131Sn , 133Sb and 131In . Using these nu-
50 83 50 81 51 82 49 82
In the Kohn-Shamapproach,the ground-statedensity clei, the two types of transitions, particle-particle and
is constructed from single-particle wave functions, ob- hole-hole,can be investigatedand comparedwith exper-
tained by solving the Kohn-Sham equations of the sys- imental results to validate the observations made above.
tem. These equations are the energy eigenvalue equa- The first case, i.e. the particle-particle transition, corre-
tions for a system of non-interacting particles subject to spondstotheβ-decayprocess133Sn 133Sb +e−+ν
a local external potential. Except for the energy eigen- and the second one, the hole-h5o0le c8a3se→, to51the8t2ransitione
valueclosetotheFermisurface(i.e. theenergynecessary 131In 131Sn +e−+ν . Inbothcasesground-stateto
to extract one particle from the system) all other en- g4r9oun8d2→-sta5t0e tr8a1nsitions,eas well as transitions from and
ergy eigenvalues have no direct physical meaning. If the to low-lying excited states, are considered.
single-particle (hole) energy at the Fermi surface agrees
with experiment, one canassume that the wavefunction To study these processes a general expression describ-
associated with it must also be a good approximation. ing semi-leptonic transitions has been obtained that in-
In this paper the validity of the last statement is stud- corporates the Dirac wave functions calculated by solv-
ied, and in an indirect way the accuracy to which the ing the Kohn-Sham equations derived from the effective
2
lagrangian [1, 2]. The upper and lower components of been performed [12, 13, 14] showing an agreement with
the Dirac spinors are then used to calculate the ma- experiment within a factor of two. The calculations in-
trix elements of the appropriate electroweak currents. cluded in [12] describe the β-strengthfunction by apply-
These currents are obtained from the same effective la- ing the pn-QRPA method with a Gamow-Teller residual
grangian and correspond to the leading order Noether interaction, the strength of which is fitted to experimen-
currents[3,4]. Atanylevelofapproximationintheeffec- tally known half-lives of known isotopes for a fixed mass
tive lagrangian, the axial-vector currents satisfy PCAC number A. Pairing correlations are treated in the BCS
and show a pion-pole dominance [3]. The matrix ele- model using a constant pairing force and without taking
mentsofthecurrentsthusconstructedareusedinagen- intoaccountthePauliblocking. Theprotonandneutron
eral multipole expansion from which any semi-leptonic gaps are equal and are taken from the values of global
weakinteraction canbecalculated[4]. Thispaperfocuses systematics. In addition to this, spin-isospin ground-
on β-decay rates. Applications to other semi-leptonic state correlations are included. For very neutron-rich
processes are being considered for future work. The cal- nucleionly allowedGamow-Tellertransitions areconsid-
culated decay rates have been corrected for the screen- eredandtheinfluenceoffirst-forbiddentransitionsonthe
ing effect from the valence electrons in the daughter nu- half-livesofnucleifarfromstabilityisneglected. Nuclear
cleus [5] and for the slowing down of the emitted elec- deformations also are taken into account using the Nils-
trons due to the attractive Coulomb core (Fermi func- son model from which the wave functions of the parent
tion). This last correction uses a relativistic description and daughter nuclei are calculated assuming the same
of the electron and takes into account the effects of the ground-statedeformationfor both. The main uncertain-
size of the nucleus [5]. ties in these calculations come from the mass formulas
Calculations of β-decay properties, like the half-live used as input, which in general become less accurate for
(τ ) and β-delayed neutron emission (P ), have been nuclei far from stability. The results obtained show an
1/2 n
doneextensivelyusingmainlythefollowingmodels: gross agreement with experimental data for nuclei with short
theory, quasi-particle random phase approximation (pn- half-lives (<1 s) with an averagedeviation of 1.4.
QRPA) and shell model calculations. Gross theory has Other calculations have concentrated on some spe-
been used in large-scale calculations where the discreet- cific range of nuclei important for the r-process such
ness of the final energy levels of the daughter nucleus as those close to 132Sn, as in our case. In [15] an
are smoothed out and different single-particle strength analysis of the ground-state properties and τ of nu-
1/2
functions (Gaussian, modified Lorentz) are used to cal- clei close to 132Sn is performed using the Hartree-Fock-
culate the β-decay strength [6]. This approach has been BogoliubovplusBCSpairingapproach(HFB+BCS).Al-
used to calculate both allowed and first-forbidden tran- thoughsomestudiesindicatethatthenucleiinthisregion
sitions using the Q-values from mass formulas as input. are spherical, the calculations included both spherical
Improvements to this model have been made in which and deformed nuclei. Contributions from pairing using
pairing and other shell effects are taken into account [7]. a constant strength, zero-rangeforce were also included.
Shell model calculations are one of the most elaborate Again here, only allowed GT transitions were calculated
methods used. In this type of calculation it is possi- and the GT strength function was obtained using the
ble to incorporate multi-particle transition amplitudes, method of a self-consistent treatment of the ground and
whichareuniquelydeterminedbythespecificationofthe excited states of even-even and odd-A superfluid nuclei
modelHamiltonian. Suchanapproachhastheadvantage solving QRPA-like equations in the finite Fermi-system
of predicting the state and mass dependence of observed (FFS) theory. Using this method, the Q were also de-
β
decay strengths making the results independent of any termined and no additional mass formulas were used. A
mass formulas input. The disadvantage of this method densityfunctionaldescribingthenuclearsystemwasused
liesinthe largematricesthathavetobe computedwhen with parameters fitted to stable nuclear properties ob-
the number of nucleons increases, therefore its applica- taining three different sets. One of them was special-
tions has been limited to small nuclei. A calculation for ized to reproduce not only known ground-state proper-
proton-richnucleihasbeenattemptedforthecasesofsd- ties of magic nuclei but also single particle energies of
shellnuclei andcanbe found in[8]. The thirdapproach, 132Sn. Detailsandadditionalreferencesaregivenin[15].
pn-QRPA,hasalsobeenusedinextensivecalculationsof A similar study [16], focused on nuclei in the r-process
β-decayobservables. Thisapproachcanbeconsideredto path, used a general method that combines the micro-
be between the shell model and gross theory. A descrip- scopic QRPA model for allowedGT β-decay with statis-
tionoftheformalismforeven-evenmothernuclei,aswell tic grosstheory of first-forbidden decays. In general,the
as references, canbe found in [9]. The extensions to odd results show that there is a much better agreement with
systems and odd nuclei can be found in [10] and [11]. experiment when the first-forbidden transitions are in-
Fromtheabovemodelsusedtocalculateβ-decayhalf- cluded, especially for large values of τ . When the cal-
1/2
lives, QRPA has shown to give good agreement with culated half-lives for a large range of nuclei were com-
experimental results. Some extensive calculations have pared with experimental results it was found that the
3
quantity ln(τ /τ ) lies in the range 10-0.1. and limitations as a calculational tool and will also test
calc exp
Because of new data and improved methods to cal- implications of DFT regarding the single-particle wave
culate β-decay properties, there have been efforts to functions used to construct the ground state density in
compile the existing experimental, as well as calculated the Kohn-Sham approach.
data. In [17] such a compilation can be found which in- The results for the ground-state particle-particle β
cludes the most recent experimental data for half-lives transition rate show an excellent agreement with exper-
as well as the results from two different models: 1) iment (to 5% in the case of the ground-state to ground-
Kratz-Herrmann formula (KHF) and 2) Macroscopic- state transition from 133Sn to 133Sb). This is in agree-
microscopic QRPA. ment with the expectation that since the binding energy
Table I summarizes the results obtained from the of the nuclei and the single-particle energy close to the
methods discussed above in the particular cases of the Fermisurface of the particle outside the coreare wellre-
nuclei133Snand131In. Here onecanappreciatethe level produced, then the single-particlewave-functionarealso
of agreement with experiment. The best agreement for wellreproduced. Transitionstoexcitedstatesshowasys-
the decay of 133Sn is given by R3, with a ratio of calcu- tematicdeviationfromexperimentaldata,againinagree-
lated to experimental half-lives, τ , equal to 0.87 or a ment with the expectation about the wave functions of
1/2
deviation of 13%. In the case of 131In, the best agree- those excited states. In the case of hole-hole transitions,
ments are obtained by KHF and R3 with a deviation of studied here, the results are less reliable.
22% and 18%, respectively. The paper is organized as follows: in section 2 a gen-
All these models aim to reproduce the known exper- eralexpressionforsemi-leptonictransitionratesinterms
imental β-decay half-lives as well as predict its value of a multipole expansionandusing Dirac wavefunctions
for other nuclei, especially those far from stability. As is presented; in section 3 this formulation is used to cal-
pointed out in [17], most of the models can be grouped culate β-transition rates for selected nuclei and results
in two categories: those which give a mathematical ex- are compared with the existing experimental data; and
pression (e.g. a polynomial) for the quantity of interest finally,section4containstheconclusionsdrawnfromthis
andthosebasedonaneffectiveinteraction. Modelsofthe study.
first type have no direct link to the underlying nucleon-
nucleon interaction and do not give additional informa-
GENERAL SEMI-LEPTONIC PROCESSES
tionregardingthe nuclearsingle-particlewavefunctions.
For models of the second type, these approaches use
different effective interactions fitted to better reproduce The calculationof β-decay ratesin this paper is based
the experimental β-decay data. In other cases, because on a generalexpressionthat can be applied to any semi-
thereisanoverparametizationoftheeffectiveinteraction, leptonic process. In this section the derivation of that
its relation with the nucleon-nucleon interaction is also expression is discussed. Much of the material presented
lost. Most of these models combine different approaches here follows [4].
and approximations to better reproduce the experimen- The starting point is the interaction hamiltonian Hw
tal data. whichforlowenergyprocesses,likeβ-decay,isdescribed
In general, it can be said that there is no consistent by the semi-leptonic weak hamiltonian of the standard
theory that by fixing its parameters once, reproduces not model. This interaction hamiltonian is written down to
only the masses but also the single-particle levels and β- firstorderintheweakconstant,G.Thisimpliesthatlep-
decay properties of nuclei. tons are treated to this order but the strong interaction
part of the hamiltonian is treated to all orders. The in-
In this paper such a consistent approach is explored
teraction hamiltonian is described by a current-current
and compared with experimental data for β-transition
rates of neighboring nuclei 132Sn . This approach corre- form [25]
spondstoanewtheorythatcombinesDensityFunctional G
H = d3xjlept(~x) (~x) (1)
TheoryandEffectiveFieldtheory,whereanenergyfunc- w −√2Z µ Jµ
tionalisconstructedwhichisconsistentwiththesymme-
tries of QCD and whose parametersarefitted to proper- where jµlept(~x) is the lepton current and Jµ(~x) the
tiesofstablenuclei[2]. Thesingle-particlewavefunctions hadronic current. By taking matrix elements of this
hamiltonian between the initial and final states (lepton
are obtained by solving the self-consistent Kohn-Sham
equations. In order to maintain a consistent approach, and hadron) the leptonic and hadronic parts factorize
and the matrix elements of the leptonic currents can be
the correctelectroweakcurrents, needed to calculate the
expressed by
β-transitionrates,arederivedfromthesameeffectivela-
grangian. These currents are the leading order Noether f jlept(~x)i =l e−iq~·~x (2)
currents. Theaxial-vectorcurrentsatisfiesPCACatany h l| µ | li µ
order in the effective lagrangian [3]. A comparison with where ~q = ~k +~k is the momentum transferred in the
e ν
experimental data will give this approach its validation process and ~k , ~k are the corresponding electron and
e ν
4
TABLE I: Calculated β-decay half-lives (in ms) of nuclei close to 132Sn.
a a a b b b b b c c d
Nuclei KHF QRPA-1 QRPA-2 DF2 DF3 R1 R2 R3 GT GT+ff exp
15303Sn83 362 9479 9479 9320 8200 823 10290 1260 - - 1450 ± 30
14391In82 216 146 146 390 350 394 1470 332 147.1 139.2 280 ± 30
aSee[17].
bSee[15].
cSee[16].
dSee[18,19].
neutrino momenta. f and i represent initial and final expansion of the hadronic current to project irreducible
l l
lepton states. In addition, l = (~l,il ) and l = ~l qˆ, tensor operators, one arrives at the following expression
µ 0 3
·
where qˆis a unit vector in the direction of ~q, i.e. ~q/~q . for the matrix elements [26]. In this and the following
| |
UsingtheexpressioninEq.(1)andmakingamultipole equations the magnitude of ~q is defined by k = ~q .
| |
G
f H i = − f 2π(2J +1)( i)J l λ mag(k)+ el (k)
h | w| i √2h |{− − λ TJ−λ TJ−λ
JX≥1p λX=±1 (cid:2) (cid:3)
+ 4π(2J +1)( i)J[l (k) l (k)] i (3)
3 J0 0 J0
− L − M }| i
JX≥0p
The four multipole operators occurring in the previous where λ and λ are the anomalous magnetic moments
p n
expression are defined in the following way [4, 21]: of the nucleon and ∂ ∂ stands for the D’alambertian
α α
operator. F is taken as a constant, i.e. we consider
A
(k)= d3x[j (kx)Y (Ω )] (~x) (4) here small momentum transfers, and its numerical value
JM J JM x 0
M Z J
is -1.257. These currents are the leading currents as de-
scribedin[2,3]. Therearenoadditionalcontributionsto
(k)= i d3x ~ [j (kx)Y (Ω )] ~(~x) (5) the one-body current coming from higher order terms in
JM J JM x
L k Z {∇ }·J the effectivelagrangian[3]. The nextcontributionto the
currentcomesfromtwo-bodyeffectswhichareestimated
TJeMl (k)= k1 Z d3x{∇~ ×hjJ(kx)Y~JMJ1(Ωx)i}·J~(~x) (6) ttoranbsefeorf. tShinecoertdheermofaxOim(qu/mmπv)a,luweiothfqqisthoeftmheomorednetrumof
afewMeV,thiscontributioncorrespondstoafewpercent
correction. Additionalcontributions,fromstillhigheror-
mag(k)= d3x j (kx)Y~M (Ω ) ~(~x) (7) der terms, are reduced by a factor of q/M, with M the
TJM Z h J JJ1 x i·J nucleon mass. These estimates do not take into account
The multipole operators have the vector–axial-vector the spin structure of these two-body currents, and the
(V-A) structure. The general form of the hadronic cur- actual correction will depend on a detailed calculation,
rent is given by which lies beyond the scope of the present work [27].
Inordertoevaluatethenuclearmatrixelementsofthe
=J +J (8)
Jµ µ µ5 four multipole operators it is necessary to define the ba-
sis wave functions that are going to be used. For this we
and the different parts of the total current (vector
µ
J employthewavefunctionsobtainedbysolvingtheKohn-
and axial) used in this paper are given by
Shamequationscorrespondingtotheeffectivelagrangian
that describes the nuclear many-body system. This la-
(λ λ ) ∂
J =iψ†γ γ ψ+ p− n (ψ†γ σ ψ) (9) grangian and the corresponding Kohn-Sham equations
µ 4 µ 4 µν
2m ∂xν can be found in [1, 2].
At this point, the important aspect of the wave func-
1 ∂ ∂ tionsusedisthattheyaredescribedbyDiracspinorsand
J = δ + F iψ†γ γ γ ψ
µ5 (cid:18) µν m2 ∂ ∂ ∂x ∂x (cid:19) A 4 5 ν thus the contributions of both upper and lower compo-
π− α α µ ν
(10) nents have to be taken into account to all orders. Both
5
currents and wave functions are used without a non- + n′(l′1/2)j′m′ dJM n(l1/2)jm
h j| 2 | ji
relativistic reduction. The form of the Dirac spinors is +i n′(l′1/2)j′m′ bJM n(l1/2)jm
given by h j| | ji
i n′(l′1/2)j′m′ bJM n(l1/2)jm
− h j| | ji
1 iG(r) Φ (Ω )
ψ (~x)= nκ κm x η (11) (17)
nκm r (cid:18) F(r)nκΦ−κm(Ωx)(cid:19) t
−
whereO representsanyofthemultipoleoperatorsoc-
JM
where Φκm(Ωx) are spin spherical harmonics defined by curring in Eq.(3). By applying the Wigner-Eckard the-
orem to the above matrix elements a reduced formed is
1 1
Φ (Ω )= lm m l jm Y (θ,φ)χ (12) obtained.
κm x h l2 s| 2 i lml ms
mXlms ( 1)j−mj
j′m′ O jm = − j′m′j m j′jJM
and χ are two-component spinors. η is a two com- h j| JM| ji √2J +1 h j − j| i
ms t
ponent isospinor describing either a proton or a neu- j′ O j
J
× h k k i
tron [28].
(18)
The currents as well as the multipole operators have
a matrix structure (two by two block matrix) due to the wherethe reducedmatrixelements ofOJM aregivenby:
Dirac matrices included in them, mixing the upper and
j′ O j = n′(l′1/2)j′ dJ n(l1/2)j
lower components of the initial and final nucleon wave h k Jk i {h k 1k i
functions. In what follows, the elements of the matrix +hn′(l′1/2)j′kdJ2kn(l1/2)ji
form of these multipoles play an important role and to +i( n′(l′1/2)j′ bJ n(l1/2)j
h k k i
keep track of them they will be labeled as follows. i n′(l′1/2)j′ bJ n(l1/2)j )
− h k k i }
dJM bJM (19)
O = 1 (13)
JM (cid:18) bJM dJM (cid:19)
2 Each of these reduced matrix elements is composed of
a coefficient corresponding to the angular momentum
Using Eqs.(9) and (10) and the following matrix form
structure of the matrix element and a radial integral in-
for the Dirac γ matrices (here the conventions of [4] are
volving the radial wave functions, G(r) and F(r) and
used)
sphericalBesselfunctions. Theangularmomentumcoef-
0 i~σ ficients are tabulated and can be found in [21, 22]. The
~γ =(cid:18)i~σ −0 (cid:19) (14) radial integrals were performed numerically.
TableIIshowstheformofeachmatrixelementofevery
multipole operator defined by Eqs.(4)-(7). Here we have
β =(cid:18)10 −01(cid:19) (15) dq2e)fi,nwedheµr(ekq)2==kk(λ2p−−ωλ02na)n/2dmka=nd|~q|η.(kT)h=e q1u−ankt2it/y(mω2π0+is
identified as the total decay energy.
then the elements of the multipole matrices can be ob- Severalnewdefinitionstakenfrom[21]havebeenused
tained. inthistable. Theyarereproducedhereforcompleteness.
The matrixelements ofthe multipole operatorsshown
ΣM(k~x)=M~M(k~x) ~σ
in Eq.(3) will mix the upper and lower components of J JJ ·
the Dirac wave functions as shown in the following ex- Σ′M(k~x)= i 1~ M~M(k~x)
pression. Hereallquantumnumbersreferringtothefinal J − nk∇× JJ o (20)
Σ′′M = 1~MM(k~x) ~σ
state are indicated by a prime. Those corresponding to J nk∇ j o·
the lower components of the wave functions, either from MJM(k~x)
initial or final states, are denoted by an underline. Thus
and where
the Dirac wave function given in Eq.(11) is symbolically
written as MJM(k~x) = jJ(kx)YJM(Ωx) (21)
M~M(k~x) = j (kx)Y~M (Ω )
JL J JL1 x
i ψ
ψ (~x)= n(l1/2)jmj (16)
nκm (cid:18)− ψn(l1/2)jmj (cid:19) Allreducedmatrixelementsofthemultipoleoperators
are now given by combining Eq.(19) and the expressions
Combining this with the matrix given in Eq.(13) one given in Table II. From the derivation given in [4], one
obtainsanexpressionforthematrixelementsofthemul- arrives at the following expression which is general for
tipole operators between initial and final nuclear states any semi-leptonic process.
hf|OJM|ii ≡ hj′m′j|OJM|jmji 1 f Hw i 2 = G2 4π
2j+1 |h | | i| 2 2j+1
= hn′(l′1/2)j′m′j|dJ1M|n(l1/2)jmji Xm′j Xmj
6
TABLE II: This table shows the matrix elements for each of the multipole operators. Here we have used the notation
µ(k)=k(λp−λn)/2m and η(k)=1−k2/(m2π+q2), where q2 =k2−ω02 and k=|~q|. Hereω0 is thetotal decay energy.
Multipole Operator dJM dJM bJM
1 2
′′
M MM dJM −iµ(k)Σ M
JM J 1 J
MJ5M −i(ω0/k)[1−η(k)]FAΣ′′JM dJ1M FAMJM
L 0 dJM iΣ′′M
JM 1 J
L5 iη(k)F Σ′′M dJM 0
JM A J 1
Tel µ(k)ΣM −dJM iΣ′M
JM J 1 J
Tel,5 iF Σ′M dJM 0
TJmMag iµ(Ak)ΣJ′M −d1JM ΣM
JM J 1 J
Tmag,5 η(k)F ΣM dJM 0
JM A J 1
[( j′ el j 2 + [(1 νˆ β~+2(qˆ β~)(qˆ νˆ)) j′ j 2
×{ |h kTJ k i| − · · · |h kLJk i|
JX≥1 JX≥0
+ j′ mag j 2)1(~l ~l∗ l l∗) +(1+β~ νˆ) j′ J j 2
|h kTJ k i| 2 · − 3· 3 · |h kM k i|
qˆ (νˆ+β~)2Re( j′ j j′ j ∗)]
1i(~l ~l∗) 2Re( j′ el j j′ mag j ∗)] − · h kLJk ih kMJk i }
−2 × 3 h kTJ k ih kTJ k i (24)
+ [l3l3∗|hj′kLJkji|2+l0l0∗|hj′kMJkji|2 where qˆ= ~q/~q, νˆ =~ν/ν and β~ =~k/ǫ, ~q is the momen-
JX≥0 tum transfer,|~ν|is the momentum of the neutrino and ν
−2Re(l3l0∗hj′kLJkjihj′kMJkji∗)]} its energy, ~k is the momentum of the electron and ǫ its
(22) energy.
Thisexpressioncanbeintegratedoverthecorrespond-
The rest of this derivation, and the calculations done, ing phase-space to obtain the desired β-transition rate.
concentrateonβ-transitionprocesses. Forthisparticular All the nuclear structure input to this formula is em-
semi-leptonic process the transition rate is given by bedded in the reduced matrix elements of the multipole
operators. These in turn are generally composed of four
V2 1 termsasgivenbyEq.(19)andtheexpressionsofTableII.
dω = dΩ dΩ kǫ(ω ǫ)2dǫ
(2π)5 ǫ ν 0− 2j+1 Forthelastpartofthissectionasummaryoftheexpres-
sions needed to evaluate these reduced matrix elements
f H i 2
|h | w| i| is included.
leXptonXm′j Xmj As mentioned above, all reduced matrix elements of
spins
themultipoleoperatorsarecomposedoftwofactors: one
(23)
correspondingtotheangularmomentumstructureofthe
matrix elements and the other corresponding to an inte-
where the ǫ and ν subscripts identify quantities related
graloverthe initialandfinalwavefunctions weightedby
to the electron and the neutrino, respectively, and ω
0
spherical Bessel functions. Their expressions are given
is the total decay energy. In addition, V corresponds
in [22].
to the volume of quantization for the lepton wave func-
tions, Ω is the solid angle in the direction of emission of n′(l′1/2)j′ M (k~x) n(l1/2)j =
J
the lepton, either electron or neutrino, 2j+1 is the sta- h k k i
(4π)−1/2A (l′j′;lj) n′l′j′ j (ρ)nlj
tistical factor corresponding to the initial nuclear state J h | J | i
and m corresponds to the projections of the total an- (25)
j
gular momentum of the initial and final nuclear states.
Using Eq.(22) and evaluating the spin sums (i.e. lepton n′(l′1/2)j′ Σ (k~x) n(l1/2)j =
J
h k k i
traces) one arrives at the final expression that describes (4π)−1/2D (l′j′;lj) n′l′j′ j (ρ)nlj
J J
β-transition rates [4]. h | | i
(26)
V2 4πG2
dω = (2π)5dΩǫdΩνkǫ(ω0−ǫ)2dǫ2j+1 n′(l′1/2)j′ Σ′ (k~x) n(l1/2)j =
h k J k i
{ [(|hj′kTJelkji|2+|hj′kTJmagkji|2)(1−(qˆ·β~)(qˆ·νˆ)) (4π)−1/2{−J1/2DJ+(l′j′;lj)hn′l′j′|jJ+1(ρ)|nlji
JX≥1 +(J +1)1/2DJ−(l′j′;lj)hn′l′j′|jJ−1(ρ)|nlji}
+qˆ (νˆ β~)2Re( j′ el j j′ mag j ∗)] (27)
· − h kTJ k ih kTJ k i
7
DFT with regard to excited states. On the other hand,
based on the agreement obtained for the ground-state
hn′(l′1/2)j′kΣ′J′(k~x)kn(l1/2)ji= binding energies of the parent and daughter nuclei [1],
(4π)−1/2 (J +1)1/2D+(l′j′;lj) n′l′j′ j (ρ)nlj it is expected that the wave functions corresponding to
{ J h | J+1 | i
these states, i.e. 2f and 1g respectively, are a good
+J1/2D−(l′j′;lj) n′l′j′ j (ρ)nlj 7/2 7/2
J h | J−1 | i} approximation. These particles are outside a filled core
(28)
and in this approach they only interact with it through
themean-fields. Anassumptionhasbeenmadeherethat
Inalltheseexpressionsj (ρ)correspondstoaspherical
J thesenuclei,withoneparticleoutsideitscore,canbede-
Bessel function of order J and argument ρ = kx. The
scribed by spherically symmetric wave functions.
coefficients A , D , D+ and D− are tabulated in [21,
J J J J In the case of the nuclei considered here for hole-hole
22] for a large, but limited number of transitions. For
transitions,severalcomplicationsemerge. First,thelevel
coefficients of transitions not included in these tables,
ordering of the daughter nucleus, 131Sn is reproduced
explicit expressions of the above coefficients in terms of 50 81
except for the (2d )−1 state, which is the measured
3-jand6-jsymbolscanbefoundin [21][29]. Theintegral 3/2
ground-state. The calculated ground-state corresponds
overtheinitialandfinalradialwavefunctionsisgivenby
to the 1h level, which lays approximately 200 keV
11/2
∞ off the experimental value. Another complication arises
n′l′j′ θ(ρ)nlj = dr u(r)n′κ′θ(ρ)v(r)nκ (29) from the fact that there are no ground-state to ground-
h | | i Z
0
statetransitionsinthehole-holecase,soadirectcompar-
Here θ(ρ) stands for the appropriate spherical Bessel ison with the particle-particle situation cannot be made.
function and u(r) and v(r) for either the upper or lower Here all transitions go from or to excited states. Yet
component of the Dirac wave function, i.e. either G(r) another complication comes from the assumption that
or F(r) of Eq.(11), of the initial and final nucleon state. it is possible to approximate a hole state wave function
by a single-particle wave function. The hole state cor-
responds to an unfilled core, with one particle missing,
RESULTS and no interactions among the particles in the unfilled
corehavebeentakenintoaccountbeyondthe mean-field
The calculations of β-transition rates are done by nu- one. These three situations make it difficult to compare
mericalintegrationofEq.(24) overthe electronand neu- the hole-hole type of transitions with experiment. Still
trino phase-space. These transition rates are very sensi- the results, as will be shown below, are not so different
tive to the exact decay energy ω . Although the results than other calculationsandin generalfollowthe pattern
0
obtained for total binding energies of the nuclei involved of behavior of the experimental transition rates.
in this study agree within 1% of its experimental values, Thecalculationofβ-transitionratesusingEq.(24)em-
herethecalculationsofthedecayratesaredoneusingthe ploys the initial and final nuclear wave functions. These
experimentaldecayenergiessoastotakeintoaccountthe wave functions enter into the calculation in a different
fullphase-spaceavailabletotheprocess. Thisway,noad- form depending on the type of transition, i.e. particle-
ditional uncertainties are introduced and the direct con- particleorhole-hole. Intheparticle-particlecasetheini-
tributionscomingfromthecalculatednuclearwavefunc- tial and final wave functions correspond with the initial
tionscanbeaccountedfor. Thedecayenergiesusedhere and final nuclear states. For example, for the ground-
aretakenfrom[18,19]. TheDiracwavefunctionsusedin state to ground-state transition 15303Sn83 →15313 Sb82 the
these calculations are solutions to the Kohn-Sham equa- initial and final wave function are 2f and 1g re-
7/2 7/2
tions derived from the effective lagrangian given in [2] spectively, see Fig. 1. Hole-hole transitions, on the con-
using the G1 parameter set. trary, can be interpreted as a particle transitions going
Two types of transitions are investigated in this pa- in the opposite direction [24]. Thus, for example, in the
per: 1) particle-particle and 2) hole-hole. The diagram- transition 14391In82 →15301 Sn81 going from ground-state to
matic representation of the transitions studied here are the (1h11/2)−1 state, the initial and final wave functions
showninFig.1fortransitionsoftheparticle-particletype would be (1h ) and (1g ) respectively [30].
11/2 9/2
and in Fig. 2 for the hole-hole type. The level structure Examples of the single-particle wave functions used in
indicated on the left side of each figure corresponds to the calculation of β-transition rates are given in Fig. 3
the results obtained using the EFT/DFT approach. On through 6. In these figures the upper and lower compo-
the right are the measured levels. In both cases the en- nentsoftheDiracwavefunctions,i.e. theG(r) andF(r)
ergylevelsaremeasuredwithrespecttotheground-state functions,areplottedsimultaneouslyasafunctionofthe
level, either the calculated (left) or experimental (right). radial distance r, in Fermis.
Theresultsobtainedgive,forthenucleiconsideredinthe The results of the calculation of β-transition rates for
particle-particletransitions,thecorrectlevelorderingal- the particle-particle type are shown in Fig. 7. These are
though not the right splitting. This is in accord with compared with experimental values for each of the tran-
8
10
133Sn →133Sb 2f7/2 - G(r)
8 2f 7/2- 133Sn 0.2 50 83 51 82 21fg7/2 -- FG(r()r)
7/2 7/2
) 50 83 1g - F(r)
V ) 7/2
r
e 6 F( 0.1
M
Particle-particle
d
( β--transitions n
2d a
y 4 3/2
g ) 0
r
r (
e 2d G
n
5/2
E 2
-0.1
3/2+
133
1g 5/2+ Sb
0 7/2 7/2+ 51 82
-0.2
0 5 10
Calculated Experimental r (fm)
FIG. 1: Single-particle level spectrum of parent and daugh- FIG. 3: Upper and lower components of the initial and fi-
ter nuclei in the case of particle-particle transitions. All en- nal nucleon wave functions. This corresponds to a particle-
ergy levels aremeasured from thecalculated orexperimental particle transition proceeding from the ground-state of
ground-state level, accordingly. 15303Sn83 to theground-state of 15313Sb82.
10 (2p )-1 1/2- 131
In
1/2 9/2+
-1 49 82
(1g )
8 9/2 133Sn →133Sb 22ff7/2 -- GF((rr))
V) 0.2 50 83 51 82 2d7/2 - G(r)
6 5/2
e 2d - F(r)
M Hole-hole ) 5/2
r
( β--transitions (
y 4 (2d )-1 F 0.1
g 5/2 d
r 5/2+ n
e -1 a
n 2 (2d )
E 3/2 1/2+ 131 r) 0
-1 Sn (
0 (3s1/2) 11/2- 50 81 G
(1h )-1 3/2+ -0.1
11/2
-2
Calculated Experimental
-0.2
0 5 10
r (fm)
FIG. 2: Single-hole level spectrum of parent and daughter
nuclei in the case of hole-hole transitions. All energy levels
are measured from the calculated or experimental ground-
FIG. 4: Upper and lower components of the initial and fi-
state level, accordingly.
nal nucleon wave functions. This corresponds to a particle-
particle transition proceeding from the ground-state of
15303Sn83 to an excited state of 15313Sb82.
9
0.06 1g - G(r)
9/2
1g - F(r)
9/2
1h11/2 - G(r) 1
1h - F(r)
r) 0.04 11/2 )
( 1
F 131 →131 -s
d 49In82 50Sn81 ω( 0.1
n
a
0.02 s
) e
r t
( a
G r 0.01

y
0 a
c
De 0.001 133Sn →133Sb
50 83 51 82
-0.02
0 5 10
r (fm)
0.0001
2f →1g 2f →2d* 2f →2d*
FIG. 5: Upper and lower components of the initial and fi-
7/2 7/2 7/2 5/2 7/2 3/2
nal nucleon wave functions. This corresponds to a hole-hole
transition proceeding from theground-stateof 14391In82 to the FIG. 7: This figure shows the calculated results of the β-
calculated ground-stateof 15301Sn81. transitionratesinthecasesofparticle-particletransitionsand
theexperimental values.
0.03 10
131 →131 1g - G(r)
In Sn 9/2
49 82 50 81 1g9/2 - F(r) 1
2d -G(r)
0.02 5/2 )
2d - F(r) 1
) 5/2 -
r s
( ( 0.1
F ω
d s
n 0.01 e
a t 0.01
a
) r
r
( y
G a 131In →131Sn
0.001
c 49 82 50 81
0 e
D
0.0001
-0.01
0 5 10
1e-05
r (fm) (1g )-→1 (1h *)-1 (1g )-→1 (2d*)-1 (2p* )-→1 (2d )-1
9/2 11/2 9/2 5/2 1/2 3/2
FIG. 6: Upper and lower components of the initial and fi- FIG. 8: This figure shows the calculated results of the β-
nal nucleon wave functions. This corresponds to a hole-hole transition rates in the cases of hole-hole transitions and the
transition proceeding from the ground-state of 14391In82 to an experimental values.
excited state of 15301Sn81.
10
0.292
0.291 imentalresultsareshownhereasarrowsindicatingupper
limitstothedecayrates. Fromthethreecasescalculated
s 0.29 |M (k)|2
it 0 here, only one lies within the experimentally determined
n 0.289 range. This transition corresponds to a ground-state to
u
excited state hole-hole transition. The two other cases
s 0.288
s shown occur from ground-state to excited state and ex-
e
l 0.287 cited to ground-state respectively. Their numerical re-
n
o sults are very similar. In these cases, there is no direct
mag 2
si 10-2 |Tel(k)|2 |T (k)| ground-state to ground-state transition which makes it
n 1 1 2 difficult to compare with the particle-particle case.
e |L (k)|
m 10-3 0
i
D 2
10-4 |L1(k)| |M (k)|2 CONCLUSIONS
1
-5
10 0 0.01 0.02 0.03 0.04 This paper shows the results of calculations of the β-
k(fm-1) transition rates of nuclei close to 15302Sn182. These cal-
culations were inspired from the success of the results
obtained in a previous paper [1] dealing with the appli-
FIG. 9: Magnitude of the multipoles defined in Eqs. (4)-(7) cation of the EFT/DFT approach to the nuclear many-
for J=0 and J=1. These multipoles are the dominant terms body systemto nucleifar fromstability. In reachingthis
contributingtotheparticle-particletransitionundergoingbe- goal several important results have been obtained.
tweentheground-statesof15303Sn83 and15313Sb82. Inthisfigure
MJ ≡MJ, LJ ≡LJ, TeJl ≡TJel and TmJag ≡TJmag. 1. Ageneralexpressiontocalculateanysemi-leptonic
process has been derived. The main equations and
definitionsneededaregivenbyEqs. (19),(20)-(21),
sitions. Here the calculated results are indicated with a
(22), (25)-(28) and Table II. The nucleon wave
cross. Itcanbeseenthattheground-statetoground-state
functions are the Dirac wave functions obtained
transition is well reproduced. The ratio of calculated to
from solving the Kohn-Sham equations derived in
experimental values is 1.05, or a deviation of 5%. The
the EFT/DFT approach and considers relativis-
other transitions go from the ground-state of the parent
tic corrections to all orders in the nucleon wave
nucleus to an excited state in the daughter nucleus.
functions. Additionally, the electroweak currents
The results for these transitions showsystematically a
usedcorrespondtotheleadingNoethercurrentsob-
largervalue ofthe decay rates comparedwith the exper-
tained directly from the same effective lagrangian.
imental ones. This agrees well with what is expected of
Inthis sense,the calculationis self-consistentsince
DFT. Since the calculations have been performed using
all relevant elements have been obtained directly
the experimentaldecayenergies,the onlyinput fromthe
from a single theory.
EFT/DFT approach at this point has been the single-
particle Diracwavefunctions. These,onthe otherhand,
2. The results of the β-transition rates calculated us-
are used to construct the nuclear ground-state densities
ingthisformalismagreewithin5%withtheexperi-
and since this quantity is in principle well reproduced,
mentalvaluesforthetransitiongoingfromground-
the result of the ground-state to ground-state transition
state to the ground-state in the particle-particle
serves as an indirect way of checking it.
case; see Fig. 7. DFT can reproduce ground-state
Returning to the results for the ground-state to observables and since the results of the total bind-
ground-state transition in the particle-particle case, ingenergyofthe parentanddaughternucleiessen-
Fig. 9 shows the dominant multipoles as a function of tially agree with experiment, it would be expected
the k = ~q . The values ploted correspond to the square thatthelastparticleinthesystemiswelldescribed.
| |
ofthemagnitudofthemultipolesdefinedinEqs. (4)-(7). Thefactthattheβ-transitionratesdeviatesystem-
The multipoles plotedcorrespondto values ofJ =0 and atically when the transitions go to excited states
J = 1, with 0(k)2 being the most dominant term. gives additional support to this view.
|M |
From this figure it is also clear that at low momentum
transferthe only non-vanishing multipoles are 0(k)2, 3. The results of the β-transition rates in the case
|M |
|T1el(k)|2 and |L1(k)|2, in accordance with the analysis of hole-hole transitions are less accurate than the
made in [4]. particle-particle case; see Fig. 8. Here the absence
The results obtained in the hole-hole type of transi- of a pure ground-state to ground-state transition
tionsareshowninFig.8. Inthiscase,theresultsareless doesnotallowabetterassessmentoftheargument
satisfactorythanintheparticle-particlecase. Theexper- given above for the particle-particle case. The fact