. Research Paper .Nuclear Magnetic Moments

SCIENCE CHINAPhysics, Mechanics & Astronomy

February 2011 Vol. 54 No. 2: 198–203doi: 10.1007/s11433-010-4214-8

c© Science China Press and Springer-Verlag Berlin Heidelberg 2010 phys.scichina.com www.springerlink.com

g factors of nuclear low-lying states: A covariant descriptionYAO JiangMing1*, PENG Jing2, MENG Jie3,4,5 & RING Peter6

1School of Physical Science and Technology, Southwest University, Chongqing 400715, China;2Department of Physics, Beijing Normal University, Beijing 100875, China;

3State Key Laboratory for Nuclear Physics and Technology, School of Physics, Peking University, Beijing 100871, China;4School of Physics and Nuclear Energy Engineering, Beihang University, Beijing 100191, China;

5Department of Physics, University of Stellenbosch, Stellenbosh, South Africa;6Physik-Department der Technischen Universitat Munchen, D-85748 Garching, Germany

Received November 12, 2010; accepted December 6, 2010; published online December 30, 2010

The g factors and spectroscopic quadrupole moments of low-lying excited states 2+1 , · · · , 8+1 in 24Mg are studied in a covariantdensity functional theory. The wave functions are constructed by configuration mixing of axially deformed mean-field statesprojected on good angular momentum. The mean-field states are obtained from the constraint relativistic point-coupling modelplus BCS calculations using the PC-F1 parametrization for the particle-hole channel and a density-independent delta-force forthe particle-particle channel. The available experimental g factor and spectroscopic quadrupole moment of 2+1 state are reproducedquite well. The angular momentum dependence of g factors and spectroscopic quadrupole moments, as well as the effects of pairingcorrelations are investigated.

covariant density functional theory, magnetic moment, g factor, spectroscopic quadrupole moment

PACS: 21.10.Ky, 21.10.Re, 21.30.Fe, 21.60.Jz

1 Introduction

Magnetic moment provides substantial information on themicroscopic structure of nuclear states. It is highly sensitiveto the single particle structure and therefore serves as a strin-gent test of the nuclear models [1,2]. Advances in modernexperimental techniques and sensitive detectors have made itpossible to measure, with reasonable accuracy, magnetic mo-ments of short-lived nuclear states with lifetimes of the orderof picoseconds or less [3]. The gyromagnetic ratio (g fac-tor) of nuclear state, which is the magnetic moment dividedby the angular momentum, reflects the dedicate interplaybetween collective and single-particle degree-of-freedom inatomic nuclei. It yields valuable information on the make-upof its wave function. In particular, the variation of g fac-tors of 2+1 states across broad ranges of nuclei shows strikingfeatures that can give clues to shell structure, residual interac-tions, correlations, and collective effects. Therefore, the mea-surement of g factors for nuclear excited states has attractedmuch attention since the early days [4–8].

*Corresponding author (email: jmyao@swu.edu.cn)

On the theoretical level, the systematic behavior of g(2+1 )in both isotopes and isotones have been studied, either inthe particle-rotor model (PRM) [1], neutron-proton interact-ing Boson model [9,10], the hybrid quasi-particle randomphase approximation (QRPA) model [11], the collective Sand D nucleon pair approximation of the shell model [12], theMonte Carlo shell model [13], the spherical shell model (CD-Bonn) [14], or the projected shell model [15], most of whichare limited to several active shells with effective charges oreffective orbital and spin g factors.

In the past decades, the framework of nuclear energy den-sity functionals has been used extensively in the analysis ofstructure properties for nuclei over the whole nuclear chart[16]. In particular, the covariant density functional (CDF)theory, incorporating many important relativistic effects, hasbeen successful in describing many nuclear phenomena forboth stable and exotic nuclei with a few universal parame-ters [17–21]. However, a straightforward application of thesingle-particle relativistic mean-field model with only time-even fields fails to reproduce the experimental magnetic mo-ments of ground states for even near LS double-closed shellnuclei [22–26], where the traditional Schmidt values are in

Yao J M, et al. Sci China Phys Mech Astron February (2011) Vol. 54 No. 2 199

agreement with the data. This failure is attributed to thesmall Dirac effective mass (M∗ ∼ 0.6M) which results inthe enhancement of the relativistic effect on the Dirac cur-rent [27]. The solution of this problem is performing theself-consistent RMF calculation for (A ± 1) nucleons, ratherthan for A nucleons. Before much involved numerical calcu-lations for (A ± 1) nucleons, one can consider the effect ofextra unpaired nucleons on the core in an approximate wayby allowing for excitations from the core creating particle-antiparticle vibration. The coupling single-particle state ina nuclear medium to such a vibration state through mesonexchange in the framework of relativistic RPA could restorethe single-particle electromagnetic current to its free-nucleonvalue [26,28,29]. A more general discussion starting froma Ward identity in which the coupling to a vibration staterepresents a vertex correction arrived at the same conclusion[30]. In terms of Landau-Migdal quasiparticle approach or inthe language of quantum liquids, the effective single-particlecurrents in nuclei or the “back-flow” effect were also intro-duced to resolve this problem [31]. In the fully self-consistentRMF calculations of A±1 nucleons, there are time-odd fieldsgenerated by the unpaired valence nucleon that should betreated properly. To consider these time-odd components inmean-field theory, one has to break spherical symmetry atthe mean-field level. After taking into account the time-oddnuclear magnetic potential in axial [32–34] or triaxial [35]deformed relativistic mean-field models, one can reproducethe iso-scalar magnetic moments of light LS double-closedshell ± 1 nucleon systems. Moreover, it has been shown re-cently that the cranking calculation based on the CDF theorycan describe nuclear magnetic rotation successfully after tak-ing into account the nuclear currents self-consistently [36].In view of these facts, it is interesting to study the g factor ofnuclear excited states starting from the CDF theory.

Since the angular momentum is not a good quantum num-ber for non-spherical mean-field state, the extension of CDFtheory to the description of g factors for nuclear excited statesrequires the restoration of rotational symmetry breaking inthe mean-field approximation. Recently, we have projectedthe triaxial states of the relativistic point coupling model cal-culations to the good angular momentum [37,38]. Later, thismodel has been further extended to include fluctuations fortriaxial deformations within the framework of the generatorcoordinate method (GCM) [39]. The success of this modelhas been illustrated in the description of excitation energiesand B(E2) transition strengths of low-lying states in both car-bon isotopes [40] and magnesium isotopes [41].

In this work, we apply the angular-momentum projectedgenerator coordinate method (AMP+GCM) based on theCDF theory with axial symmetry to study the g factorsand spectroscopic quadrupole moments of low-lying excitedstates in 24Mg. The paper is arranged as follows. The modelis introduced briefly in sec. 1. The results and correspond-ing discussions are given in sec. 2. A brief summary of thepresent investigation is presented in sec. 3.

2 The model

In the AMP+GCM model, the trial nuclear wave function|ΨJ

α〉 for an axially deformed nucleus is a superposition ofintrinsic states with a different deformation,

|ΨJα〉 =

∫dq20 f J

α (q20)PJ00|Φ(q20)〉, (1)

where α = 1, 2, · · · labels collective eigenstates for a givenangular momentum J, and q20 =

√5/16π〈Φ(q20)|2z2 − x2 −

y2|Φ(q20)〉 is the mass quadrupole moment. PJ00 denotes the

angular-momentum projection operator:

PJ00 =

2J + 18π2

∫dφ sin θdθdψdJ

00(θ)eiφJz eiθ JyeiψJz , (2)

where (φ, θ, ψ) denotes the set of three Euler angles.The intrinsic state |Φ(q20)〉 is a set of axially symmetric in-

trinsic wave functions generated from the relativistic mean-field plus BCS calculations, which is based on the covariantdensity functional [42,43]:

ERMF=Tr[(α · p+ βm)ρv]

+

∫dr(αS

2ρ2

S +βS

3ρ3

S +γS

4ρ4

S +δS

2ρS�ρS

+αV

2jμ jμ +

γV

4( jμ jμ)2 +

δV

2jμ� jμ

+αTV

2jμTV ( jTV )μ +

δTV

2jμTV�( jTV )μ

+14

FμνFμν − F0μ∂0Aμ + e

1 − τ3

2ρV A0

),

where ψk(r) denotes a Dirac spinor, and A0 is the Coulombfield. The local isoscalar and isovector densities and currentsρS , ρTS , jμ, jμTV are calculated in the no-sea approximation.

The weight functions f Jα (q20) in the collective wave func-

tion eq. (1) are determined by the Hill-Wheeler-Griffin(HWG) integral equation:∫

dq′20

[H J

00(q20, q′20) − EJ

αNJ

00(q20, q′20)]

f Jα (q′20) = 0, (3)

where H and N are the angular-momentum projected GCMkernel matrices of the Hamiltonian and the norm, respectively[39,44]. The solution to HWG equation determines both theenergies EJ

α and the amplitudes f Jα (q20) for nuclear low-lying

states Jπα with good angular momentum.Since the weight functions f J

α (q20) are not orthogonal andcannot be interpreted as collective wave functions for the de-formation variable, the wave functions gJ

α(qi) are usually cal-culated from the eigenstates of norm overlap kernel,

gJα(qi) =

∑k

gJαk uJ

k (qi), (4)

which are orthogonal and their module squared can, there-fore, be interpreted as a probability. The gJα

k is the solutionto the following standard eigenvalue equation:∑

l

H Jklg

Jαl = EJ

αgJαk , (5)

200 Yao J M, et al. Sci China Phys Mech Astron February (2011) Vol. 54 No. 2

which is equivalent to eq. (3). The matrix element H Jkl is

determined by the GCM kernel matrix of the Hamiltonian,

H Jkl =

1√nJ

k

1√nJ

l

∑qi ,qj

uJk (qi)H J

00(qi, q j)uJl (q j), (6)

where nJk,l and uJ

k,l are the non-zero eigenvalue and eigenstatesof norm overlap kernel N J

00(qi, q j).Since the restoration of rotational symmetry provides wave

functions in the laboratory frame of reference, one can calcu-late the expectation values of electromagnetic operators. Thespectroscopic quadrupole moment of excited state Jπα is givenby

Qspec(Jπα)= e

√16π

5〈J, M = J, α|Q20|J, M = J, α〉

= e

√16π

5(2J + 1)2

2(−1)J

(J 2 JJ 0 −J

)

×∫

dqi

∫dq j f ∗Jα (q j) f J

α (qi)∑μ

(J 2 J0 μ −μ

)

×∫

dθ sin θdJ−μ0(θ)〈Φ(q j)|Q2μeiθ Jy |Φ(qi)〉. (7)

The g factor of excited state is defined by g(Jπα) =μ(Jπα)μN J

,

where the magnetic moment μ(Jπα) of excited state Jπα can becalculated with the angular momentum projected wave func-tion,

μ(Jπα)= 〈J, M = J, α|μ10|J, M = J, α〉=

(J 1 JJ 0 −J

)∑qi ,qj

f ∗Jα (q j) f Jα (qi)

× (2J + 1)2

2(−1)J

∑ν

(J 1 J−ν ν 0

)

×∫

dθ sin θdJ−μ0(θ)〈Φ(q j)|μ1νeiθ Jy |Φ(qi)〉 (8)

with the one-body magnetic dipole moment operator definedby

μ1ν = (μ−1, μ0, μ+1) =

(1√2

(μx − iμy), μz,− 1√2

(μx + iμy)

).

(9)The magnetic moment vector μ entering the above expressionis related to the space-like components of effective electro-magnetic current operator,

μk =12

∫d3r[r × j]k, k = x, y, z (10)

that is defined microscopically by ref. [35],

j = eψ†αψ +κ

2M∇ × [ψ†βΣψ], (11)

where κ is the free anomalous gyromagnetic ratio of the nu-cleon: κp = 1.793 and κn = −1.913. The first term is the

Dirac charge current jD = eψ†αψ, and the second term isthe so-called anomalous current. In the ground state of even-even nucleus, the expectation value of current operator (11)vanishes because of time-reversal invariance. However, thereare non-zero currents in rotating excited states, which willgive rise to the magnetic moment.

Since the full configuration space is considered, no effec-tive charges and effective g factors for neutron and proton areneeded. Further details on the computational procedure canbe found in refs. [38,39].

3 Results and discussion

The intrinsic wave functions that are used in the configura-tion mixing calculation have been obtained as solutions tothe self-consistent relativistic mean-field equations with theconstraint on the axial mass quadrupole moments. The inter-action in the particle-hole channel is determined by the rela-tivistic density functional of PC-F1 parametrization [42], anda density-independent δ-force is used as the effective interac-tion in the particle-particle channel. Pairing correlations aretreated within the BCS approximation. The pairing strengthparameters Vτ are determined in the following two ways:

Type I: taking from the PC-F1 set, i.e., Vn = 308.00,Vp = 321.00 fm3 MeV for neutron and proton respectively.

Type II: fitting the average single-particle pairing gaps ofmean-field ground state for 24Mg weighted with the occupa-tion probability v2

k

〈Δ〉 ≡∑

k fkv2kΔk∑

k fkv2k

(12)

to the experimental odd-even mass difference obtained with afive-point formula, i.e., Δ(5)

n = 3.193 MeV, Δ(5)p = 3.123 MeV.

The resulted pairing strengths are Vn = 511.30,Vp = 518.35fm3 MeV for neutron and proton respectively.

For axially deformed intrinsic wave functions, the inte-grals over two of the three Euler angles φ, ψ in the normand hamiltonian kernels can be calculated analytically andone is left with a one-dimensional integration. The Gaussian-Legendre quadrature is used for integrals over the Euler an-gle θ. The number of mesh points in the interval [0,π] isNθ = 18. The generator coordinates are chosen as β =−1.0, 0.9, · · · , 1.2, 1.3, where the β = (4π/3AR2

0)q20 withR0 = 1.2A1/3. Eigenstates of the norm overlap kernel N J

00with eigenvalues nJ

k/nJmax < ζ are removed from the GCM

basis (nJmax is the largest eigenvalue of the norm kernel for a

given angular momentum), where ζ = 5 × 10−3 is used as inref. [39].

In Figure 1 we plot the excitation energies of low-lyingstates in 24Mg as functions of angular momentum up to J = 8.It shows that the calculations with pairing strengths of “TypeI” give consistent results of excitation energies with the data.However, it is noted that the excitation energy of 2+1 is a littlelower than the data because of weak pairing strengths that re-sult in the relative larger moment of inertia. With the adjusted

Yao J M, et al. Sci China Phys Mech Astron February (2011) Vol. 54 No. 2 201

pairing strengths (Type II), the calculated excitation energiesare systematically higher than the corresponding data. Thereare two reasons: 1) Stronger pairing correlations always giverise to smaller moment of inertia; 2) AMP is carried out onlyafter variation. Time-odd components in the wave functions,and alignment effects are, therefore, neglected, which cancompress the spectrum.

In Figure 2, we show the probability distributions |gJα|2 as

functions of the deformation parameter β for the low-lyingstates in both calculations. Obviously, the calculation withthe pairing strengths of “Type I” gives the well prolate de-formed shapes for both ground and excited states of 24Mg.However, a prolate-oblate coexistence in the ground state of24Mg is obtained in the calculation with the pairing strengthsof “Type II”. With the increase of angular momentum, theshape becomes more and more prolate deformed.

Using the configuration mixed wave functions projectedon good angular momentum, one can calculate the spectro-scopic quadrupole moments Qspec and g factors of low-lyingstates. In Figures 3 and 4 we plot the spectroscopic qua-

Figure 1 Excitation energies of the low-lying states as functions of angular

momentum in 24Mg. The experimental data are taken from ref. [45].

Figure 2 Probability distributions |gJα |2 as functions of deformation param-

eter β for the low-lying states in 24Mg.

drupole moments Qspec and g factors of low-lying states Jπαas functions of angular momentum J in 24Mg. The nega-tive Qspec corresponds to the prolate deformed shape. It isshown that the available experimental g factor and spectro-scopic quadrupole moment Qspec of 2+1 state are reproducedquite well in both calculations. The Qspec(J+1 ) as a functionof J in the calculations with pairing strengths of type “I” is inbetter agreement with values of rotor (scaled to Qspec(2+1 ))than that in the calculations with type “II”. Furthermore,the calculated g factors are found to be almost the same forthe low-lying excited states with different angular momen-tum and close to the empirical value gR = Z/A = 0.5 of rigidrotor [1]. The effects of pairing correlation strengths on theg factors are negligible, but much evident for spectroscopicquadrupole moments. It can be understood from the changesof probability distributions along deformation parameter β inthe calculations with different pairing strengthes. Contrary tothe spectroscopic quadrupole moments that are proportionalto the deformation, the g factors are not sensitive to the de-formation but to the configuration of nuclear wave functions.

Figure 3 The spectroscopic quadrupole moment Qspec of low-lying states

as functions of angular momentum in 24Mg. The empirical values of spectro-

scopic quadrupole moment Qspec(J+1 ) = 7J/[2(2J + 3)]Qspec(2+1 ) of a rigid

rotor are indicated as dashed lines. The experimental data are taken from

refs. [46].

Figure 4 The g factors of low-lying states as functions of angular momen-

tum in 24Mg. The g factor Z/A of axial rotor (K = 0) is indicated as dashed

line. The experimental data are taken from ref. [46].

202 Yao J M, et al. Sci China Phys Mech Astron February (2011) Vol. 54 No. 2

4 Summary

The g factors and spectroscopic quadrupole moments of low-lying excited states in 24Mg have been studied microscopi-cally without introducing effective charge or effective orbitaland spin g factors for neutron and proton, but using the wavefunctions constructed by configuration mixing of relativisticmean-field states projected on good angular momentum. Thepairing correlations have been treated using the BCS approx-imation with density-independent delta-pairing force. Twotypes of pairing strengths: 1) taking from the PC-F1 set; 2)fitting the average single-particle pairing gaps to the experi-mental odd-even mass difference, have been adopted.

The available experimental g factor and spectroscopicquadrupole moment of 2+1 state have been reproduced quitewell. Furthermore, the calculated g factors have been foundto be almost the same for the low-lying excited states withdifferent angular momenta and close to the empirical valuegR = Z/A of rigid rotor. The effects of pairing correlationstrengths on the g factors of low-lying states are negligible,though much evident for excitation energies (moment of in-ertia) and spectroscopic quadrupole moments.

This work was partly supported by the National Basic Research Program

of China (Grant No. 2007CB815000) and the National Natural Science

Foundation of China (Grant Nos. 10947013, 10975008, 10705004 and

10775004), the Fundamental Research Funds for the Center Universities

(Grant No. XDJK2010B007), the Southwest University Initial Research

Foundation Grant to Doctor (Grant No. SWU109011), the Bundesmin-

isterium fur Bildung und Forschung, Germany (Grant No. 06 MT 246),

and the DFG cluster of excellence “Origin and Structure of the Universe”

(www.universe-cluster.de).

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