On Th229 and time-dependent fundamental constants

0.24 MB·English

by  Elena Litvinova

On 229Th and time-dependent fundamental constants
Elena Litvinova1,2, Hans Feldmeier1
1GSI Helmholtzzentrum fu¨r Schwerionenforschung, Planckstr. 1, 64291 Darmstadt, Germany
2Institute of Physics and Power Engineering, 249033 Obninsk, Russia
Jacek Dobaczewski
Institute of Theoretical Physics, University of Warsaw, Hoz˙a 69, PL-00-681 Warsaw, Poland and
Department of Physics, P.O. Box 35 (YFL), FI-40014 University of Jyv¨askyl¨a, Finland
Victor Flambaum
School of Physics, University of New South Wales, Sydney, NSW 2052, Australia
(Dated: January 9, 2009)
9 The electromagnetic transition between the almost degenerate 5/2+ and 3/2+ states in 229Th is
0 deemed to be very sensitive to potential changes in the fine structure constant α. State of the art
0 Hartree-Fock and Hartree-Fock-Bogoliubov calculations are performed to compute the difference
2 in Coulomb energies of the two states which determines the amplification of variations in α into
variationsofthetransitionfrequency. Thekineticenergiesarealsocalculatedwhichreflectapossible
n
variation in the nucleon or quark masses. A generalized Hellmann–Feynman theorem is proved
a
J includingtheuseofdensity-matrixfunctionals. Asthetwostatesdiffermainlyintheorbitoccupied
by the last unpaired neutron the Coulomb energy difference results from a change in the nuclear
9
polarization of the proton distribution. This effect turns out to be rather small and to depend on

thenuclearmodel, theamplification variesbetween about −4×104 and+4×104. Therefore much
]
h moreeffortmustbeputintotheimprovementofthenuclearmodelsbeforeonecandrawconclusions
t fromameasureddriftinthetransitionfrequencyonatemporaldriftoffundamentalconstants. All
-
l calculations published so far do not reach thenecessary fidelity.
c
u
PACSnumbers: 06.20.Jr,21.60.Jz,27.90.+b
n
[

1 I. INTRODUCTION states are in principle experimentally accessible. Their
v measurement would reduce the uncertainty.
0
Measurementsofasuspectedtemporalvariationofthe
4
In a simplified picture the two states differ by the oc-
fine structure constant by means of atomic transitions
2
1 have reached an limit for δα/α of less than 10−16yr−1 cupation of the last neutron orbit. The change in the
Coulomb energy is thus due to a modified neutron dis-
. [1–6].
1 tribution in the excited state that will polarize via the
A tempting idea to increase the sensitivity limit is to
0
strong interaction the proton distribution in a slightly
9 use the transition between two states with very differ-
different way than in the ground state.
0 entCoulombenergies,becausethe driftintransitionfre-
: quency
v
In this paper we investigatein how far state of the art
i
X δω δα ∆V nuclear models can provide reliable answers. For that in
C
=A with A= (1)
section II we revisit first the Hellmann–Feynman theo-
r ω α ω
a rem proposed by H. Hellmann [10] and also by R. Feyn-
is given by an amplification factor A times the drift in man [11] and show that it is also valid in a wider con-
α. Using the Hellmann–Feynman theorem A is the ra- textincludingdensity-matrixfunctionaltheoryemployed
tio ofthe difference ∆V in Coulombenergiesof the two in nuclear physics. After introducing the nuclear mod-
C
statesinvolvedinthetransitiondividedbythetransition els in Sec. III we discuss in Sec. IV the amplification of
frequencyω. Apromisingcandidateforthatisthetransi- the temporal drift of fundamental constants like the fine
tionbetweenthe3/2+isomericstateofthenucleus229Th structure constantor the nucleonmass when monitoring
to its 5/2+ ground state [7, 8]. Recent measurements transition frequencies.
yield ω =7.6±0.5 eV [9] which onnuclear energyscales
isanaccidentalalmostdegeneracy. TypicalCoulomben- For the nuclear candidate 229Th we perform in Sec. V
ergiesfor this nucleus are ofthe orderof V ≈109 eV so self-consistent nuclear structure calculations and deter-
C
that even a small difference ∆V could result in a large mineviatheHellmann–Feynmantheoremthederivatives
C
amplification. ofthetransitionfrequencywithrespecttothe finestruc-
A big drawback of this idea is that the Coulomb en- ture constant and the nucleon mass. A critical assess-
ergies cannot be measured but have to be calculated ment is made on the predictive power when calculating
with sufficient accuracy in a nuclear model. Differences observables other than the energy. Finally we summa-
in charge radii and quadrupole moments of the nuclear rize.
2
II. HELLMANN–FEYNMAN THEOREM form a Slater determinant. In an energy density func-
REVISITED tional x could be the local density ρ(~r), and so on.
Steady state solutions x(n)(c), n = 0,1,2,... of the
In this section we show that the Hellmann–Feynman system are obtained by the condition
theorem holds not only for eigenstates of the Hamilto-
nian but also for all stationary solutions in approximate 0= ∂E (c,x) (5)
schemes,providedthey arevariational. Thisis ofimpor- ∂x
k
tance as the models we are using are based on density-
matrix functionals. At the stationary points the energy assumes the values
Let the Hamiltonian H(c) depend on a set of external
parameters c = {c1,c2,...}, like the strength of an in- En(c)=E(c,x(n)(c)), n=0,1,2,... . (6)
teraction or the mass of the particles. The steady state
solutionsofthetime-dependentSchr¨odingerequationare Both,the energiesandtheparametersx(n)(c) character-
given by the eigenvalue problem izingthe stationarystatesdepend onthe constantsc. In
the groundstategivenbyx(0) the energyE(c,x) is inan
H(c) Ψ ;c =E (c) Ψ ;c . (2) absolute minimum with respect to variations in x, while
n n n
the other possible solutions x(n),n6=0, represent saddle
Both,theenergies(cid:12)(cid:12)En(c)(cid:11)andtheei(cid:12)(cid:12)gensta(cid:11)tes Ψn;c de- points.
pendonc. Whendiscussingintermolecularforcesin1933 A variation of the external parameters at the station-
(cid:12) (cid:11)
H. Hellmann [10] and later in 1939 R. P. Fe(cid:12)ynman [11] ary points leads to
showed that a small variation of an external parameter
in the Hamiltonian (distance between nuclei) leads to a δE (c)= ∂ E (c)δc (7)
n n i
change in the energy given by: ∂c
i
i
X
∂E
∂ E (c)= Ψn;c ∂∂ciH(c) Ψn;c . (3) = ∂c c,x(n)(c) +
∂ci n (cid:10) Ψ(cid:12)n;c Ψn;(cid:12)c (cid:11) Xi h i(cid:0) (cid:1)
(cid:12) (cid:12) (n)
∂E ∂x
ThehrempitrioaonfHhi(ncg)e:s on Ψn;(cid:10)c bein(cid:12)(cid:12)g an e(cid:11)igenstate of the Xk ∂xk(cid:0)c,x(n)(c)(cid:1) ∂cki (c)iδci .
(cid:12) (cid:11)
(cid:12)
∂ E (c)= ∂ Ψn;c H(c) Ψn;c Dthueestqouathreebstraatcikoentasrivtayncisohnedsitsioonth(a5t) twheeosbectoanindfpoarrsttain-
∂c n ∂c Ψ ;c Ψ ;c
i i(cid:10) n(cid:12) n(cid:12) (cid:11) tionary solutions
(cid:12) (cid:12)
= Ψn;c ∂∂ciH((cid:10)c) Ψn(cid:12)(cid:12);c (cid:11) (4) ∂ ∂E
(cid:10) Ψ(cid:12)(cid:12)n;c Ψn;(cid:12)(cid:12)c (cid:11) ∂ci En(c)= ∂ci c,x(n)(c) . (8)
+"(cid:10)∂∂Ψc(cid:10)inΨ;cn(cid:12);Hc(cid:12)(cid:12)(Ψc)n(cid:12);Ψc(cid:11)n;c(cid:11) + h.a.# The derivative of the energy a(cid:0)t the stati(cid:1)onary solutions
(cid:12) (cid:12) isjustthepartialderivativeoftheenergyfunctionalwith
− Ψn(cid:10);c H((cid:12)(cid:12)c) Ψn(cid:11);c ∂∂Ψcin;c Ψn;c + h.a. . respecttotheexternalparametercalculatedwiththesta-
"(cid:10) Ψn(cid:12);c Ψn(cid:12);c (cid:11)(cid:10) Ψn;c (cid:12)Ψn;c (cid:11) # tionary state. This generalizes the Hellmann–Feynman
(cid:12) (cid:12) (cid:12) theorem (3).
(cid:10) (cid:12) (cid:11) (cid:10) (cid:12) (cid:11) Taking
Inserting (2) in t(cid:12)he terms enclosed(cid:12)in square brackets
leadstoacancellationofthesetermsandweareleftwith
x H(c) x
Eq. (3), the Hellmann–Feynman theorem. The partial E(c,x)= and x = x k , (9)
rdaemriveatetrivce oisfeaqnuyaelitgoenthvaeleuxepEecnt(act)iownitvhalrueespofectthetopaartpiaa-l (cid:10) (cid:12)(cid:12) x x (cid:12)(cid:12) (cid:11) (cid:12) (cid:11) Xk k(cid:12) (cid:11)
i (cid:10) (cid:12) (cid:11) (cid:12) (cid:12)
derivative of the Hamiltonian calculated with the corre- where k denotes(cid:12)somefixedbasisanddoingavariation
sponding eigenstate Ψn;c . with respect to the parameters x = {x1,x2,...} yields
extWreemnaolwposihnotwinthaan(cid:12)(cid:12)t ethneersg(cid:11)aymfeunscttaitoenmael.ntLheotldEs(cf,oxr)anbye tohriegitnima(cid:12)(cid:12)leH-(cid:11)ienldlmepaennnd–eFnetynSmcharn¨odtihnegoerremeq(u3a)taiosna(s2p)ecainadlctahsee
theenergyofaphysicalsystemthatdepends onexternal of (8).
parameters c and on a set of variationalparameters x=
It is interesting to note that the Hellmann–Feynman
{x ,x ,···} which characterize the state of the system.
1 2 theorem even applies to every solution Ψ(t) of the
x may also represent a set of functions in which case
time-dependent Schr¨odinger equation
partialderivativesarereplacedbyfunctionalderivatives. (cid:12) (cid:11)
(cid:12)
For example, in the Hartree-Fock approximation x d
would be the set of occupied single-particle states that i Ψ(t) =H c(t) Ψ(t) . (10)
dt
(cid:12) (cid:11) (cid:0) (cid:1) (cid:12) (cid:11)
(cid:12) (cid:12)
3
For the total time derivative of the mean energy, which are represented in a working basis i = c† ∅ . Thus,
i
is the expectation value of the Hamiltonian, one obtains the expansion coefficients D represent the set x =
iν
(cid:12) (cid:11) (cid:12) (cid:11)
dE d Ψ(t) H c(t) Ψ(t) t{eDrsiν.;ν = 1,...,A,i = 1,2,...} of(cid:12)variation(cid:12)al parame-
=
dt dt Ψ(t) Ψ(t) Thegeneraldefinitionoftheone-bodydensityoperator
(cid:10) (cid:12) (cid:0) (cid:1)(cid:12) (cid:11)
(cid:12) (cid:12)
= Ψ(cid:10)(t) ∂∂c(cid:12)(cid:12)iH c((cid:11)t) Ψ(t) dci(t) ρˆ= i ρik k (16)
Xi (cid:10) (cid:12)(cid:12) Ψ(t)(cid:0)Ψ(t)(cid:1)(cid:12)(cid:12) (cid:11) dt Xi,k (cid:12) (cid:11) (cid:10) (cid:12)
Ψ(t)(cid:10)H c((cid:12)t) d (cid:11)Ψ(t) (cid:12) (cid:12)
+ (cid:12) dt + h.a. . (11) is givenin terms ofthe expectation values of c†kci, where
"(cid:10) (cid:12)Ψ((cid:0)t) Ψ(cid:1)(t)(cid:12) (cid:11) # the creation operators c† create the single-particle basis
(cid:12) (cid:12) i
Hereweusedthefa(cid:10)cttha(cid:12)taher(cid:11)mitianHamiltoniandoes i . In the HF case:
(cid:12)
nthoet tcihmaen-gdeeptehnedennotrmSchorf¨odtihnegesrtaetqeuatΨio(nt)(1.0)Iinnsteorttinhge (cid:12)(cid:12) (cid:11)ρ = ΨHF c†kci ΨHF = νDiν(Dkν)∗ . (17)
(cid:12) (cid:11) ik
expression in the square brackets let(cid:12)s this term of the (cid:10) ΨH(cid:12)F Ψ(cid:12)HF (cid:11) PΨHF ΨHF
equation vanish so that the time-dependent Hellmann– (cid:12) (cid:12)
Feynman theorem reads: The energy (cid:10)of the(cid:12)(cid:12)HF S(cid:11)later d(cid:10)etermi(cid:12)(cid:12)nant (cid:11)can be ex-
pressed in terms of the idempotent (ρˆ2 = ρˆ) one-body
dE Ψ(t) ∂ H c(t) Ψ(t) dc density as
= ∂ci i(t) . (12)
dt Xi (cid:10) (cid:12)(cid:12) Ψ(t)(cid:0)Ψ(t)(cid:1)(cid:12)(cid:12) (cid:11) dt E [c,ρˆ]= ΨHF H(c) ΨHF
Again the variation(cid:10)of the(cid:12)(cid:12)energ(cid:11)y is given by the expec- HF (cid:10) Ψ(cid:12)HF ΨH(cid:12)F (cid:11)
(cid:12) (cid:12)
tation value of the derivatives of the Hamiltonian with 1
= t (c)ρ (cid:10)+ (cid:12) v (cid:11)(c)ρ ρ
respect to the external parameters. See also Ref. [12]. ij ji (cid:12) ik,jl ji lk
4
The generalized theorem (8) is even valid for classical Xij Xijkl
mechanics, just take x to be the canonical variables p + 1 v (c)ρ ρ ρ +··· ,
ikm,jln ji lk nm
and q. The equations of motion are 36
ijklmn
X
dp ∂E dq ∂E (18)
=− (c,p,q) and = (c,p,q) . (13)
dt ∂q dt ∂p
where t (c) denotes the matrix elements of the kinetic
ij
At the stationary points the partial derivatives of the energy, v (c) the antisymmetrized matrix elements of
ij,kl
HamiltonfunctionE(c,p,q)vanishandoneobtainsagain the two-body interaction, v (c) of the three-body
ikm,jln
Eq. (8). interaction, and so on. The dependence on the pa-
rameter set c that includes the nucleon masses, cou-
pling strengths, interaction ranges etc. will not be in-
III. MODELS dicated againuntil required. The variationalparameters
x = {D } reside in the one-body density matrix ρˆ as
iν
InthissectionwediscussbrieflytheHartee-Fock(HF) given in Eq. (17). In order to work with familiar expres-
methodwhenaHamiltonianisused,theextensiontoHF sions in this section we will write ρˆinstead of x.
withdensity-matrixfunctionalsandtheinclusionofpair- Variation of the energy given in Eq. (18) with respect
ing correlations. We show that in all cases the general- to ρˆleads to the HF equations
ized Hellmann–Feynman relation holds. We also explain
inshortthequantitiesdiscussedinthesectioncontaining hˆHF[ρˆ] ρˆ=ρˆˆhHF[ρˆ] . (19)
calculations for 229Th.
The one-body HF Hamiltonian
hˆ [ρˆ]= i h [ρˆ] k (20)
A. Hartree-Fock with Hamiltonian HF HF ik
i,k
X(cid:12) (cid:11) (cid:10) (cid:12)
(cid:12) (cid:12)
In the HF approximation one uses a single Slater de-
is a functional of the one-body density and its matrix
terminant
elements are given by
ΨHF =a†1a†2 ···a†A ∅ (14) h [ρˆ] = ∂ E [c,ρˆ] , (21)
HF ik HF
as the many-bo(cid:12)dy tria(cid:11)l state. The cre(cid:12)at(cid:11)ion operators a† ∂ρki
(cid:12) (cid:12) ν
that create the occupied single-particle states φ ,
ν or in short notation
a†ν ∅ = φν = c†i ∅ Diν , (cid:12)(cid:12) (cid:11) (15) hˆHF[ρˆ]= δ EHF[c,ρˆ] . (22)
δρˆ
i
(cid:12) (cid:11) (cid:12) (cid:11) X (cid:12) (cid:11)
(cid:12) (cid:12) (cid:12)
4
The eigenstates φ of hˆ [ρˆ] represent the basis in the single-particle states φ
ν HF ν
which both, ˆh [ρˆ] and ρˆare diagonal:
HF(cid:12) (cid:11) (cid:12) (cid:11)
(cid:12) φ =(cid:12) i D (26)
ν iν
hˆHF[ρˆ]= φν ǫν φν (23) (cid:12) (cid:11) Xi (cid:12) (cid:11)
ν (cid:12) (cid:12)
X(cid:12) (cid:11) (cid:10) (cid:12) thatrepresenttheoccupiedstatesofasingleSlaterdeter-
ρˆ= (cid:12)φν nν φν(cid:12) , (24) minant. Astheyareexpandedintermsofaworkingbasis
Xν (cid:12) (cid:11) (cid:10) (cid:12) i theenergyEDF[c,ρˆ]is,likeintheHFcase,afunction
(cid:12) (cid:12) of the variational parameters x = {D ; i,ν = 1,2,...}
where ǫν denotes the single-particle energy and nν are (cid:12)(cid:12)or(cid:11)x=ρˆ. iν
the single-particle occupation numbers that are zero or
The difference to HF with a Hamiltonian is that the
one in the case of a single Slater determinant.
mean-field Hamiltonian hˆ [ρˆ] obtained by the func-
Eq. (19) represents the stationarity condition (5) MF
tional derivative of the density-matrix functional E
and hence the HF approximation fulfills the Hellmann– DF
Feynman theorem.
δ
In nuclear structure theory the microscopic nucleon- hˆMF[ρˆ]= EDF[c,ρˆ] (27)
δρˆ
nucleon interaction induces strong short-range correla-
tions that cannot be represented by a single Slater de- is not given by a microscopic Hamiltonian anymore, but
terminant. Therefore the HF method as explained here
by the functional form of E and the fitted parameters
DF
cannotbe used. For examplethe strongshort-rangedre- in the set c.
pulsion makes all two-body matrix elements v pos-
ik,lm The stationarity conditions (5) lead to the self-
itive and large, so that the HF Slater determinant does
consistent mean-field equations
not give bound objects. The way out is to use effec-
tive interactions that incorporate the short-range corre- hˆ [ρˆ] ρˆ=ρˆhˆ [ρˆ] (28)
MF MF
lations explicitly, see for example Ref. [13–15]. Another
approach is the density-matrix functional theory which
that have the same structure as the HF equations.
we discuss in the following section.
Because the self-consistent solution is obtained by
searching for solutions of the stationarity conditions (5).
the Hellmann–Feynman theorem (8) is fulfilled, even if
B. Hartree-Fock with density-matrix functionals one cannot refer to a microscopic Hamiltonian and a
many-body state anymore.
It has turned out that bypassing the construction One should note that it is not mandatory that the
of an effective microscopic Hamiltonian by postulating single-particlestates φν withlowestsingle-particleen-
an ansatz for the energy as functional of the one-body ergies are occupied. Any combination of occupied states
(cid:12) (cid:11)
density-matrix ρˆ, as originally proposed by Skyrme for leads to a stationary(cid:12)solution fulfilling Eq. (28).
the non-relativistic and by J. Boguta and A.R. Bodmer Another interesting case is a one-body density with
[16] and D. Walecka [17] for relativistic nuclear physics fractionaloccupation numbers 0≤n ≤1 that may also
ν
(or by Kohn and Sham [18] for the atomic case), is very commutewiththemean-fieldHamiltonianandhenceful-
successfulin describinggroundstate properties. The en- fillsthestationaritycondition(28)sothattheHellmann–
ergy functional E [c,ρˆ] contains a finite number of pa- Feynman theorem is applicable. In such a situation one
DF
rameters, c, that are adjusted by fitting observables to cannot attribute a single Slater determinant to the one-
nuclear data. The shape of the functional is subject of bodydensityanymore,becauseρˆ6=ρˆ2isnotidempotent.
past and present research [19–21] and is being improved In any case the occupation numbers play the role of
toalsoapplyfornucleifaroffstability. Differentfromthe externalparametersandshould be regardedas members
HF case with Hamiltonian, Eq. (18), the densities may of the set c and not as variational parameters.
appear also with non-integer powers and the exchange
terms are not calculated explicitly but absorbed in the
form of the energy functional. C. Hartree-Fock-Bogoliubov
Not all of the information residing in the one-body
density-matrix ρˆ is used. Usually one uses the proton Thesolutionofthe eigenvalueproblemofhˆ [ρˆ]pro-
MF
andneutrondensityρ (~r),ρ (~r),kineticenergydensities
p n videsnotonlythe coefficientsD oftheoccupiedsingle-
τp(~r),τn(~r), current densities~j(~r), etc. particle states but also a represeinνtationfor empty states
so that one has a complete representation of the one-
EDF[c,ρˆ]=EDF(c,ρp(~r),ρn(~r),τp(~r),τn(~r),~j(~r),...) body Hilbert space. With that one can define creation
(25) operators for fermions for occupied and empty states
In order to keep densities and currents consistent and a† = c†D , (29)
ν i iν
correspondingto fermionsthey areexpressedin termsof
i
X
5
with Both can be expressedin terms of D and v . The gen-
iν ν
eralized density matrix is idempotent and hermitian
φ =a† ∅ and i =c† ∅ . (30)
ν ν i
R2 =R and R† =R (37)
andtheirc(cid:12)(cid:12)orre(cid:11)spondin(cid:12)(cid:12)g(cid:11)annihila(cid:12)(cid:12)tio(cid:11)noper(cid:12)(cid:12)ato(cid:11)rsaν andci.
Pairingcorrelationsinthe many-bodystate canbe in- and the stationaribty conbdition lebads tob
corporatedbyBogoliubovquasi-particlesthatarecreated
by H R R=RH R (38)
MF MF
α†ν =uνa†ν −vνaν¯ (31) quiteinanalogybtoth(cid:2)ebm(cid:3)eban-fibeldbequ(cid:2)abti(cid:3)ons(28)without
α† =u a† +v a pairing correlations. The pseudo-Hamiltonian
ν¯ ν ν¯ ν ν
as linear combinations of the creation and annihilation δ
H R = E c,R
operators,a†ν,aν¯,ofthe eigenstatesofthe one-body den- MF δR HFB
seitteyrsmuatraixnd(tvhecsaon-cablelecdhcoasnenonriecaallasntadtetsh)e. rTehqeuipreamraemnt- b (cid:2)b(cid:3)= ˆhMF−(cid:2)λ b(cid:3) ∆ˆ (39)
ν ν b −∆ˆ∗ λ−hˆ∗
thatᆠandᆠarefermionicquasi-particleoperatorsim- (cid:18) MF (cid:19)
ν ν¯
plies u2+v2 =1. The pairingpartnerstates ν andν¯are
ν ν containsthemean-fieldHamiltonianandthepairingpart
usually mutually time-reversed states. ∆ˆ. The chemical potentials λ = (λ ,λ ) determine the
p n
The many-body trial state is expressed as
mean proton and neutron number. For details and fur-
ther reading see Refs. [22–24].
ΨHFB = a†µ 1−vν2+vν a†νa†ν¯ ∅ , (32) Let us write down only the one-body density because
(cid:12) (cid:11) Yµ Yν (cid:16)p (cid:17) (cid:12) (cid:11) it is referred to in the results for 229Th. The one-body
(cid:12) (cid:12) density matrix is given by
where the product overµ runsoverthe socalledblocked
statesorunpairedstatesandν runsoverallotherpaired
ρˆ= φ v2 φ + φ n φ , (40)
states. Besides the variational parameters residing in ν ν ν µ µ µ
the operators a†ν that create eigenstates of the mean- ν Xpaired(cid:12)(cid:12) (cid:11) (cid:10) (cid:12)(cid:12) µ bXlocked(cid:12)(cid:12) (cid:11) (cid:10) (cid:12)(cid:12)
field Hamiltonian the energy depends now also on the
wherethe φ denotecanonicalbasisstates. The occu-
variational parameters v , hence x = {D ,v ; i,ν = ν
ν iν ν pationnumbersaregivenbyn =n =v2 forthepaired
1,2,...}. (cid:12) (cid:11) ν ν¯ ν
states and(cid:12)are the same for ν and ν¯. For the unpaired
As the trial state (32) has no sharp particle number
or blocked states n = 0,1. In the application we will
thestationarityconditionsEq.(5)havetobeaugmented µ
consider only one blocked neutron state for 229Th.
by a constraint on mean proton number Z and mean
neutron number N to obtain the selfconsistent Hartree-
Fock-Bogoliubov (HFB) equations:
IV. AMPLIFICATION
E =E −λ Z−λ N . (33)
HFB p n
To be more specific let us consider as external param-
The proton and neutron chemical potentials, λ and λ , eters the fine structure constant α and the proton and
p n
have to be regarded as members of the set c of external neutronmassm ,m ,thusc={α,m ,m }. InthisSec-
p n p n
parameters. The additional constraints do not alter the tion, we write down expressions pertaining to a Hamil-
argumentsleadingtheHellmann–Feynmantheorem,thus tonian; those characteristic to a density functional are
it is also valid in the HFB case. entirely analogous, cf. Sec. II. The partial derivatives of
InHFBitisconvenienttointroduceageneralizedden- a non-relativistic Hamiltonian are
sity matrix
∂ 1
H(α,m ,m )= V (41)
p n C
ρˆ κˆ ∂α α
R= (34)
−κˆ∗ 1−ρˆ∗ ∂ 1
(cid:18) (cid:19) H(α,mp,mn)=Z− Tp (42)
∂m m
p p
b
where ρˆis the normal one-body density
∂ 1
H(α,m ,m )=N − T , (43)
p n n
Ψ c†c Ψ ∂mn mn
ρ = HFB k i HFB (35)
ik
Ψ Ψ where V denotes the operator for the Coulomb energy,
(cid:10) HF(cid:12)B (cid:12)HFB (cid:11) C
(cid:12) (cid:12) Z,N the proton and neutron number, respectively, and
and κ the so called abn(cid:10)ormal d(cid:12)(cid:12)ensity (cid:11) Tp,Tn stand for the proton and neutron kinetic energy
operator, respectively.
ΨHFB ckci ΨHFB According to the Hellmann–Feynman theorema small
κ = . (36)
ik
Ψ Ψ variation δα of the fine structure constant results in a
(cid:10) HF(cid:12)B (cid:12)HFB (cid:11)
(cid:12) (cid:12)
(cid:10) (cid:12) (cid:11)
(cid:12)
6
variation of an energy eigenvalue given by with the abbreviation
dm dm
δEn = Ψn VC Ψn +Zα dαp +Nα dαn (44) ∆X = Ψ1;α X Ψ1;α − Ψ0;α X Ψ0;α (50)
(cid:18) Ψ ;α Ψ ;α Ψ ;α Ψ ;α
(cid:10) (cid:12)(cid:12) (cid:12)(cid:12)αdm(cid:11)p αdmn δα (cid:10) 1 (cid:12)(cid:12) (cid:12)(cid:12)1 (cid:11) (cid:10) 0 (cid:12)(cid:12) (cid:12)(cid:12)0 (cid:11)
− Ψ T Ψ dα − Ψ T Ψ dα . (cid:10) (cid:12) (cid:11) (cid:10) (cid:12) (cid:11)
n p n m n n n m α for the difference o(cid:12)f the expectation val(cid:12)ues of the oper-
p n (cid:19) ators X = {V ,T ,T } calculated with the stationary
(cid:10) (cid:12) (cid:12) (cid:11) (cid:10) (cid:12) (cid:12) (cid:11) C p n
In princip(cid:12)le th(cid:12)ere could also be a(cid:12)dep(cid:12)endence of the nu- states.
clear interaction V on α, e.q., through meson masses,
N TheresultsdiscussedinSec.Vshowthat∆T and∆T
n p
which we neglect here. The dependence of the nucleon
areof the same order as ∆V so that the terms with the
C
massesonαcanbeestimatedfromthepaperbyMeißner
proton and neutron mass variations can be neglected.
et al. [25] (Eq. (36)). Their estimate of the neutron-
If there is a system, where the energy difference ω is
proton mass difference due to the electromagnetic inter-
much smaller than the difference between the Coulomb
action is ∆m(nEpM) =−0.68 MeV which yields energiesofthetwostatesonegetsanamplificationfactor
αdmp ∆m(EM) inthemeasurementofδα/α. Flambaum[8]hasproposed
dα =− np ≈+0.36·10−3 (45) thenucleus229Thasagoodcandidate[7]becauseitpos-
m 2m
p p sesses two almost degenerate eigenstates with ω ≈ 8 eV
αdmn ∆m(EM) according to a recent measurement [9].
dα = np ≈−0.36·10−3 . (46)
The theoreticaltaskisto investigatethese states care-
m 2m
n n
fullyinordertogetareliableestimatefortheirCoulomb
A possible variation of some QCD constant c would
and kinetic energies. For the calculation of these quan-
lead to
tities we employ in the following section state-of-the-art
d dmp dmn mean-field models and also include the effects of pairing
δE = Ψ V (c) Ψ +Z +N (47)
n n N n
dc dc dc correlations.
(cid:18)
(cid:10) (cid:12)(cid:12) dm(cid:12)(cid:12)p (cid:11) dmn
− Ψ T Ψ dc − Ψ T Ψ dc δc ,
n p n n n n
mp mn (cid:19) V. RESULTS FOR 229Th
(cid:10) (cid:12) (cid:12) (cid:11) (cid:10) (cid:12) (cid:12) (cid:11)
where VN is(cid:12)the(cid:12)nuclear part of t(cid:12)he i(cid:12)nteraction. This
variationisactuallylinkedtothedimensionlessratioc= The nucleus 229Th with 90 protons and 139 neutrons
mq/ΛQCD,wheremq denotesthecurrentquarkmassand occurs in nature as the daughter of the α-decaying 233U
Λ the strong interaction scale [8, 26, 27].
QCD and decays itself with a half life of 7880 years, again by
While the dependence of the nucleonmass on the cur-
α emission. This nucleus has attracted lot of interest as
rent quark mass has been calculated [28, 29], the QCD
ithas the lowestlying excitedstate known. InFig.1 the
constants enter the effective interaction V (c) in a very
N spectrum is arranged in terms of rotational bands [30].
complicated and yet unknown way. In this paper we do Two low lying rotational bands with Kπ = 5/2+ and
not consider such variations of QCD constants explic- 3/2+ can be identified, with band heads that according
itly but calculate besides the total energies the kinetic
to recent measurements differ in energy by only about
energies and the Coulomb energies which then can be 8 eV. The first negative parity band with Kπ = 5/2−
combined according to Eqs. (44) or (47) to obtain the
occurs at 146.36 keV.
variationswithrespecttovariationsofαorQCDparam-
BecauseoftheamplificationeffectdiscussedinSec.IV
eters. thetransition3/2+ →5/2+isregardedasapossiblecan-
As the temporal variation of the fundamental con-
didatetomeasurethetimevariationofthefinestructure
stants α,m and m are tiny, if they exist at all, it has
p n constant. Asimple estimate ofthe momentofinertia for
been proposed to consider transition frequencies
the two bands assuming a J(J +1) energy dependence
ω =E (α)−E (α) (48) shows that the two rotational bands have very similar
1 0
intrinsic deformation. Therefore, one does not expect
that can be measured with high precision.
a large difference between the Coulomb energies of the
Ifthe twoenergylevelsbelongtothesamenucleusthe
5/2+ and 3/2+ band heads that would be due to differ-
termswiththeprotonandneutronnumberdropoutand
ences in shapes. Instead, one has to consider and work
weobtainfortherelativevariationδω/ωofthetransition
outdetailedeffectsduetodifferentconfigurationsofthese
frequency
states.
δω 1 ∂E ∂E
1 0
= − δα (49)
ω ω ∂α ∂α
(cid:18) (cid:19)
A. Hartree-Fock with density-matrix functional
1 αdmp αdmn δα
= ∆V −∆T dα −∆T dα theory
C p n
ω m m α
(cid:18) p n (cid:19)
δα To come to more quantitative statements we calculate
=A
α the energies of these two states with the best methods
7
TABLEI:Total,Coulomb,neutronandprotonkineticenergiesofthe229Th5/2+ groundstatecalculatedwithdifferentenergy
functionals. Differences of these energies between 3/2+ first excited state and 5/2+ ground state.
∗
Exp. SkM SIII NL3
5/2+ Ref. [30] HF HFB HF HFB RH
Etot [MeV] -1748.334 -1739.454 -1747.546 -1741.885 -1748.016 -1745.775
VC [MeV] 923.927 924.854 912.204 912.216 948.203
Tn [MeV] 2785.404 2800.225 2783.593 2794.909 2059.640
Tp [MeV] 1458.103 1512.705 1442.018 1477.485 1106.697
3/2+−5/2+ Ref. [9]
∆Etot[MeV] 0.000 008 0.619 -0.046 0.141 -0.074 2.407
∆VC [MeV] 0.451 -0.307 -0.098 0.001 1.011
∆Tn [MeV] 2.570 0.954 -0.728 0.087 -2.181
∆Tp [MeV] 0.688 0.233 -0.163 -0.022 -1.996
HO frequency of ~ω = 8.05MeV in all three Cartesian
directions.
In the discussion, the resulting basis φ will be as-
ν
5/2- 146.36 signed Nilsson quantum numbers Ωπ[N,Nz,Λ] (for de-
+ - (cid:12) (cid:11)
9/2 125.44 K = 5/2 tails see Ref. [22, 37]) by looking for t(cid:12)he largest over-
9/2+ 97.14 lap Ωπ[N,Nz,Λ] φν with a Nilsson state. N =
N +N +N =N +N +Λdenotesthetotalnumberof
7/2+ 71.83 x (cid:12)(cid:10) y z ρ(cid:12) z(cid:11)(cid:12)
oscil(cid:12)lator quanta, N(cid:12)z the(cid:12)number of quanta in the direc-
7/2+ 42.43 + tionofthe symmetryaxis. Λ=Nρ,Nρ−2,...0or1and
5/2 29.19 Ω = Λ± 1 are the absolute values the projection of or-
2
5/2+ 0 3/2+ 0.008 bitalangularmomentumandtotalspinonthesymmetry
+ + axis, respectively. π =(−1)N is the parity.
K = 5/2 K = 3/2
229 It turns out that for both functionals the lowest HF
Th state has Ωπ =5/2− and thus corresponds to the intrin-
sic state of the Kπ = 5/2− band that is experimentally
locatedat146.36keV.Thetotalbindingenergyamounts
FIG. 1: Measured low lying states of 229Th with spin and to−1739.454MeVforSkM∗ and−1741.885MeVforSIII
whichshouldbe comparedtothe experimentalenergyof
parity assignments [30]. Energies are in keV.
−1748.334 MeV [30]. On this absolute scale the calcu-
lated HF energies are already amazingly good.
In order to get the experimentally observed parity we
available: density-matrix functional theory without and perform the variation procedure in the subspace with
with pairing, and a spherical relativistic mean-field the- positive parity. For both energy functionals the Slater
ory [31]. In the non-relativistic case we use the com- determinant with the lowest energy has Kπ =5/2+. All
puter programHFODD(v2.33j)[32,33]andemploytwo energies are summarized in Table I. The last neutron
successful energy functionals, SIII [34] and SkM∗ [35]. occupies the level labeled with 5/2+[622] in the SkM∗
Because of large uncertainties related to polarization ef- and with 5/2+[633] in the SIII case. One should keep in
fects due time-odd mean fields [36], in the present study mind that the single-particlestates are superpositions of
these terms in the energy functionals are neglected. The Nilsson states and the labeling refers only to the largest
standard Slater approximation is used to calculate the component. The states aredetailedin Eqs.(51)-(54)be-
Coulomb exchange energies. The code works with a low.
Cartesian harmonic oscillator (HO) eigenstates as work- The neutron single-particle energies ǫ and the occu-
ν
ing basis i and allows for triaxial and parity-breaking pation numbers are displayed in Fig. 2(a) for SkM∗ and
deformations, but the states determined in 229Th turn inFig.3(a)forSIII.The5/2−[752]statewhichwouldbe
(cid:12) (cid:11)
out to ha(cid:12)ve axial shapes with conserved parity. In our occupied in the negative parity case is very close to the
calculations we have used the basis of HO states up to 5/2+[633] state – for SIII they are almost degenerate.
the principal quantum number of N =18 and the same As can be seen from Table I, the total HF binding
0
8

+ + + +
-5 5/2 3/2 -5 5/2 3/2
+ + + +
1/2 [631] 1/2 [631] 1/2 [631] 1/2 [631]
-6 5/2+ [622] 5/2+ [622] -6 5/2+ [622] 5/2+ [622]
V] 35//22+- [[765422]] 35//22+- [[674522]] V] 35//22+- [[674522]] 35//22+- [[765422]]
e -7 e -7
M M
[ 3/2- [741] 3/2- [741] [ 3/2- [741] 3/2- [741]
-8 3/2+ [651] 3/2+ [651] -8 3/2+ [651] 1/2- [501]
neutrons neutrons
-9 229Th (a) SKM* - HF -9 229Th (b) SKM* - HFB

FIG. 2: (Color online) Neutron single-particle energies and occupation numbers labeled by the asymptotic Nilsson quantum
numbers ν = Ωπ[N,Nz,Λ] for the SkM∗ energy functional. (a) HF mean-field energies ǫν and occupation numbers nν. Full
bars denote nν +nν¯ =2, i.e. two particles in degenerate pair of states with mj =±Ω. Half full gray (pink) bars denote one
particle, nµ = 1, nµ¯ = 0. (b) HFB mean-field energies ǫν (eigenvalues of hˆMF) and occupation numbers vν2. Length of bars
indicates nν +nν¯. Gray (pink) bars stand for blocked states with nµ =1, nµ¯ =0.

-4 -4
+ + + +
5/2 3/2 5/2 3/2
- -
7/2 [743] 7/2 [743] - -
-5 -5 7/2 [743] 7/2 [743]
eV] -6 355///222++- [[[667335132]]] 355///222++- [[[676353123]]] eV] -6 355///222++- [[[676353123]]] 355///222++- [[[766533213]]]
M M
[ [
-7 3/2- [741] 3/2- [741] -7 3/2-- [741] 3/2-- [741]
- - 1/2 [501] 1/2 [501]
1/2 [501] 1/2 [501]
-8 -8
neutrons neutrons
229 229
Th (a) SIII - HF Th (b) SIII - HFB
-9 -9

FIG. 3: (Color online) Same as Fig. 2, butfor theSIII energy functional.
energyagreeswiththe measuredoneuptoabout9MeV the excited states occur at 0.619 MeV for the SkM∗ and
for the SkM∗ andup to about6 MeVfor the SIII energy at 0.141 MeV for the SIII density functional. The dif-
functional. Keeping in mind that no parameters have ferenceinCoulombenergies∆V amountsto0.451MeV
C
beenadjustedtothespecificnucleusconsideredhereitis forSkM∗ andto−0.098MeVforSIII.Thekineticenergy
surprising that these mean-field models can predict the differences ∆T and ∆T for protons and neutrons, re-
p n
energy with an uncertainty of only about 0.5 %. spectively, dissent even more. These deviations between
the two energy functionals reflect the differences in the
Putting two neutrons in the 5/2+[622] (for SkM∗) or
structure of the intrinsic states as also seenfrom the dif-
5/2+[633] (for SIII) level and one in the 3/2+[642] (for
ference in the single-particle states discussed above.
SkM∗) or 3/2+[631] (for SIII) level and minimizing the
total energy yields an excited HF state that is to be re-
garded as the intrinsic state of the experimentally ob-
served Kπ = 3/2+ band. As can be seen in Table I,
9
B. Relativistic mean-field Its value decreases from 619 keV down to −46 keV for
SkM∗andfrom141keVdownto−74keVforSIII.Onthe
We alsoperforma sphericalrelativisticmean-fieldcal- accuracy level one can expect from this model the 5/2+
culationwiththeNL3parameterset[38]andfindthatin and the 3/2+ states are degenerate, like in experiment.
the ground state the last neutron occupies a 2g orbit.
9/2
For vanishing deformation the Nilsson state 5/2+[633] By looking at the occupation numbers displayed in
belongs to the subspace spanned by the spherical 2g Figs. 2(b) and 3(b) one sees that about 5 single particle
9/2
orbits so that this result is not unreasonable when com- levels near the Fermi edge assume fractional occupation
paring with the deformed SIII calculation. The leading numbers significantly different than 0 or 2. In the SIII
component for the 3/2+ state is for SIII the 3/2+[631] case(Fig.3(b))theyarealmostidenticalforthe5/2+and
orbit (cf. Fig. 3) which for deformation zero belongs to 3/2+ statesexceptforthe5/2+[633]and3/2+[631]levels
the1i subshell. Thereforewecreatetheexcitedstate that switch their role. For the SkM∗ case (Fig. 2(b)) the
11/2
with Ωπ = 3/2+ by a particle-hole excitation from the blocked states are energetically further away from each
other which causes more deviations in the occupation
1i to the 2g shell so that there are 11 neutrons in
11/2 9/2
numbers.
the 1i and 2 neutrons in the 2g shell.
11/2 9/2
The resulting energies are listed in the last column
of Table I. The total binding energy is similar to the Thischaracteristicpatternofoccupationnumbersren-
non-relativistic one, but the particle-hole excited state ders the HFB results qualitatively different than the HF
is 2.41 MeV higher. Different from the deformed mean- ones. Indeed,intheHFcaseeitherthe 5/2+ orthe3/2+
fieldthesingle-particlestatesofasphericalpotentialhave orbitalhastheoccupationnumberequalto1. Therefore,
good total spin j and are (2j+1)-fold degenerated with polarizationeffectsexertedbythesetwoorbitalsareable
large gaps between them. The single-particle energy dif- to render different values of observables calculated for
ferenceǫ −ǫ =2.74MeVexplainsthelargeexcita- the 5/2+ and 3/2+ states. In the HFB case, occupation
9/2 11/2
tion energy of the particle-hole pair of 2.41 MeV, which numbers of the 5/2+ and 3/2+ orbitals are close to 1 for
includes the rearrangementenergy. both 5/2+ and 3/2+ configurations. Differences in ob-
We conclude that a spherical calculation is not appro- servables may here only occur due to the fact that the
priateforthisparticularquestion. Inarecentpublication occupation number of the blocked state is exactly equal
[39]asimilarsphericalrelativisticmean-fieldcalculations to1,while thatofthe other stateis approximatelyequal
comes to results comparable to ours with a Coulomb en- to 1, depending on its closeness to the Fermi level.
ergy difference of 0.7 MeV. Because of the unphysical
properties of a spherical 229Th we will not commit our- In the paired case (HFB), we are faced with the sit-
selves to the spherical case any longer but proceed to uation, where the zero-order approximation renders ob-
consider the effects of pairing. servablescalculatedin the 5/2+ and3/2+ exactly equal.
Indeed, such equality would be the case for the 5/2+
and 3/2+ orbitals located exactly at the Fermi surface
C. Hartree-Fock-Bogoliubov and having exactly the same occupation numbers. Note
that values of all observables calculated with the HFB
We include the pairing correlations with the Bogoli- approach depend only on the canonical states and oc-
ubov ansatz (32) and perform a self-consistent HFB cal- cupation numbers, irrespective of which state has been
culationbasedontheSkM∗andSIIIdensity-matrixfunc- blocked in obtaining them. Of course, one can obtain
tional. For the SIII case proton and neutron pairing different occupation numbers for both orbitals in ques-
strengthsofV =−260and−180MeVfm3(foravolume- tionwhentheyareenergeticallysplit. However,thismay
0
type contact force) are adjusted to reproduce the total contradict the experimental energetic degeneracy of the
binding energies and the odd-even staggering with the corresponding configurations. All in all, within a zero-
neighboring nuclei. Proton and neutron density ma- order paired approach, polarization effects of the 5/2+
trices and pairing tensors are calculated by including and 3/2+ orbitals become exactly averaged out and the
contributions from quasiparticle states up to the cutoff anticipateddifferences in observablescanoccur only due
energy of 60MeV. Calculations are performed by self- to first-order corrections.
consistently blocking the 5/2+ and the 3/2+ quasipar-
ticle states. In the 5/2+ and 3/2+ configurations, this This fact is perfectly well visible in our results for the
yields the averageHFB protonand neutronpairinggaps Coulomb energy differences shown in Table I. For the
of∆ =2.4MeVand∆ =0.65MeV,and∆ =2.4MeV SIIIenergyfunctional,only1keVremainsfor∆V . This
p n p C
and ∆ = 0.68MeV, respectively, for the SIII case and reducestheamplificationfactorofEq.(49)toabout100.
n
slightly larger values for the SkM∗ case. For SkM∗ a larger value of ∆V of about 300 keV is
C
The results for the energies aresummarized in Table I obtained due to a larger splitting of the corresponding
and for the mean-field single-particle energies and the single-particleorbitals. Fromthisonemustconcludethat
occupation probabilities in Fig. 2(b) and Fig. 3(b). The pairing correlations result in two states with even more
first to note is that the excitation energy is improving. similar charge distributions than in the HF calculation.
10
TABLE II: Rms-radius and intrinsic quadrupole moments of neutron and proton densities of the 229Th 5/2+ ground state
calculated with different energy functionals. Differences of these moments between 3/2+ first excited state and 5/2+ ground
state.
∗
SkM SIII
5/2+ HF HFB HF HFB
Rrms(neutron)[fm] 5.8789 5.8716 5.8971 5.8923
Rrms(proton) [fm] 5.7180 5.7078 5.7817 5.7769
Q20 (neutron)[fm2] 9.4407 9.2608 9.1990 9.0711
Q20 (proton) [fm2] 9.5461 9.3717 9.3542 9.1643
3/2+−5/2+
∆Rrms(neutron)[fm] -0.0040 0.0036 -0.0008 -0.0005
∆Rrms(proton) [fm] -0.0038 0.0039 0.0000 -0.0005
∆Q20 (neutron)[fm2] -0.2427 0.2647 -0.0767 -0.0516
∆Q20 (proton) [fm2] -0.1824 0.2756 -0.0339 -0.0495
D. Radii and quadrupole moments and for SkM∗-HFB including pairing as
5+ = +.504 622 +.487 642 +.248 862
2
In Table II the rms-radii and quadrupole moments +.418 633 −.383 613 +···
(normalized with proton and neutron number, respec- (cid:12)(cid:12) (cid:11) (cid:12)(cid:12) (cid:11) (cid:12)(cid:12) (cid:11) (cid:12)(cid:12) (cid:11) (52)
tively) of the neutron and proton point-densities are 23+ = +.015(cid:12)(cid:12)622(cid:11) +.642(cid:12)(cid:12)642(cid:11) +.235 862
given for the ground state and as differences for the ex- +.305 611 −.582 631 +··· .
(cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11)
cited state. The quadrupole moments of the protons are (cid:12) (cid:12) (cid:12) (cid:12)
The SIII-HF calcu(cid:12)latio(cid:11)n gives (cid:12) (cid:11)
somewhat larger than for the neutrons. When reoccu- (cid:12) (cid:12)
pying the last neutron the neutron quadrupole moment 5+ = +.066 622 +.418 642 +.180 862
decreases substantially in the SkM∗-HF calculation and 2
+.755 633 −.398 613 +···
drags along via the nuclear interaction the quadrupole (cid:12)(cid:12) (cid:11) (cid:12)(cid:12) (cid:11) (cid:12)(cid:12) (cid:11) (cid:12)(cid:12) (cid:11) (53)
moment of the protons. At the same time both rms- 3+ = +.134(cid:12)622(cid:11) +.360(cid:12)642(cid:11) +.165 862
2 (cid:12) (cid:12)
radii are decreased. A smaller charge radius and smaller
+.428 611 −.642 631 +···
charge quadrupole moment are consistent with the in- (cid:12)(cid:12) (cid:11) (cid:12)(cid:12) (cid:11) (cid:12)(cid:12) (cid:11) (cid:12)(cid:12) (cid:11)
crease of the Coulomb energy which explains the large and the SIII-HFB(cid:12)with(cid:11)pairing(cid:12) (cid:11)
(cid:12) (cid:12)
positive ∆V in the SkM∗-HF case. When including
pairingtheeCffectgoesintheoppositedirectionforSkM∗- 52+ = +.024 622 +.423 642 +.159 862
HFB. +.775 633 −.367 613 +···
(cid:12)(cid:12) (cid:11) (cid:12)(cid:12) (cid:11) (cid:12)(cid:12) (cid:11) (cid:12)(cid:12) (cid:11) (54)
For the SIII functional HF and HFB calculations do 3+ = +.156(cid:12)622(cid:11) +.342(cid:12)642(cid:11) +.149 862
2 (cid:12) (cid:12)
not lead to noteworthy changes in the moments when +.412 611 −.636 631 +··· .
(cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11)
reoccupying the last neutron and thus the Coulomb dif- (cid:12) (cid:12) (cid:12) (cid:12)
InSIII-HFandSI(cid:12)II-HF(cid:11)Bthebl(cid:12)ocked(cid:11)statesexhibitmore
ferencesremainalsosmall. Thiscanbeanticipatedwhen (cid:12) (cid:12)
concentration on the dominant Nilsson orbits [633] and
lookingatthesingle-particleenergiesoftheinvolvedneu-
[631]. As both have the same nodal structure in z-
tron states and the occupation numbers (Fig. 3).
direction(N =3)oneexpects lessdifference inthe den-
z
The mean-field single-particle state that corresponds sitydistributionthanintheSkM∗ casewherebothstates
to the blockedHFB state occupied by the unpaired neu- are superpositions of several Nilsson orbits with similar
tron is represented in Nilsson orbits for SkM∗-HF as amplitudes. In the SkM∗ case the leading component of
5+ is[622]andof 3+ is[642],whichimpliesachange
2 2
in nodal structure in z-direction. In summary the polar-
5+ = +.509 622 +.467 642 +.266 862 (cid:12)(cid:12)izatio(cid:11)noftheproton(cid:12)(cid:12)distr(cid:11)ibutionduetothereoccupation
2
of the level with the unpaired neutron has less effect in
+.402 633 −.397 613 +···
(cid:12)(cid:12) (cid:11) (cid:12)(cid:12) (cid:11) (cid:12)(cid:12) (cid:11) (cid:12)(cid:12) (cid:11) (51) the SIII than in the SkM∗ case.
3+ = −.010(cid:12)622(cid:11) +.662(cid:12)642(cid:11) +.249 862 In Ref. [40] the finite-range microscopic-macroscopic
2 (cid:12) (cid:12)
+.305 611 −.562 631 +··· model has been used to study the problem. The authors
(cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11) (cid:12) (cid:11)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:11) (cid:12) (cid:11)
(cid:12) (cid:12)