Journal of Nuclear and Radiochemical Sciences,Vol. 3, No. 1, pp. 205–208, 2002

205

Structure of Nuclei in Strong Magnetic Fields
V. N. Kondratyev∗
Advanced Science Research Center, Japan Atomic Energy Research Institute, Tokai, Ibaraki 319-1195,
Japan
Received: November 13, 2001; In Final Form: November 13, 2001
The structure of nuclei in ultrastrong magnetic fields relevant for supernovas and neutron stars is considered. The
dependence of shell-correction energy on magnetic field is analyzed and systemized by employing the shell modelwith spherical harmonic oscillator confining potential.

1. Introduction
The astrophysical environment can affect significantly properties of nuclides. In particular, the structure of nuclei canbe dramatically modified1 by ultrastrong magnetic fields leading, e.g., to a shift of nuclear magic numbers. The related
field strengths B ∼ 1015 –1017 G correspond to a flux Φ0 ∼ h̄c/πe
through an area covered by the size of nuclei and can be met,e.g., in supernovas and neutron stars. Many observations, likeshort bright outbursts of soft gamma repeaters (SGRs)2, 3 andrapid braking of relatively slowly rotating stars associated with
SGRs4, 5 and anomalous X-ray pulsars (AXPs),6, 7 provide evidences in support of such ‘magnetar’ concept8–10 implying ultramagnetized stellar media with fields of a strength ranging up to
B ∼ 1017.5 G. Consequently, the nuclear synthesis in the vicinityof supernova core can be noticeably affected by the magnetic
field of respective nascent neutron star. Such transformationsof nuclide structure can play an important role also in magnetotransport of neutron star crusts11 and contribute to some specificfeatures of magnetic emission, like soft gamma-ray bursts.12
The inhomogeneous (crusty) nuclear matter at densities D
less than the saturation density Ds has been extensively studied theoretically (cf. References 13–17 and references therein).
The outer crusts is viewed to be composed from well separated
nuclei with the largest binding energy. The shell closure in iron
region determines such a bottom of fusion and fission valleys inlaboratory.18 With increasing nuclear density large mass magic
numbers grow in importance, so that the nucleon aggregates ofmasses matching the superheavy sector might be expected tobe naturally abundant in crust inner regions.16, 17 The star activity related to a change of nuclear structure in, e.g., evolving
magnetic fields can be employed, indeed, to probe properties ofnuclei as well as crust magnetodynamics.
In this contribution we analyze general features of the field
induced modification of the shell effect for large mass nuclei.
In sect. 2 we briefly recall mean-field treatment applied for thedescription of nuclear magnetism and systemize the propertiesof spin- and orbital magnetic response of neutrons and protons.
The conclusions are in sect. 3.

wavelength λN = h̄/(mp c) ≈ 0.21 fm. These fields may affect,e.g., conditions of β-equilibrium of the neutron star bulk matter.19
Within the Hartree mean-field treatment (see, e.g., References 1, 20, 21) the nuclear structure is described in terms of
the filled up to the Fermi energy εF single-particle(sp) levels εζ ,R εFdε ρ(ε) nucleons.
which determine the properties of N = −∞Decomposing the sp level density ρ(ε) = ∑ζ δ(ε − εζ ) = ρsm +δρinto smooth ρsm and oscillating δρ components we express theenergy of a nucleus asE=dε ερ(ε) = E sm + δEn + δEp ,εn = (n + 3/2)

FAX: +81-29-282-

(1)

(2)

corresponding to principal quantum numbers n with level degeneracies (n + 1)(n + 2), and measured in units of the HO
frequency ω0 . For nuclei in vicinity of stability line ω0 ≈41/A1/3 MeV, while neutron rich nuclei of crust inner region
correspond to significantly smaller level spacing. Consequently,
such a picture of HO confinement might provide reliable description of inner crust nuclides with considerably suppressedspin-orbit (s-o) coupling.14, 15
2.1. Pauli-spin Magnetization. The interaction of magnetic field with nucleon-spin-dipole magnetic moment leads toan additional term in the energy spectraδεi = σi ∆α ,As demonstrated by Kondratyev et al.1 thermodynamic formalism within the non-relativistic mean-field treatment constitutes useful framework for an analysis of the nuclide reactivity
in magnetic fields of interest. Relativistic effects become important at considerably larger field strengths when the energyof the first Landau level ωL = µN B (with the nuclear magneton µN ) becomes comparable to the nucleon rest mass. The
magnitude of the respective limiting field for nucleons BNrel =mp c2 /2µN ≈ 1.5 ×1020 G corresponds to a flux Φ0 = h̄c/πe
through the area of a radius given by the nucleon Compton
5927.

−∞

where the Thomas-Fermi (i.e. semi-classical) component E sm isonly slightly affected by magnetic fields due to the Bohr-van
Leeuwen theorem.22 The leading field effect is connected withshell-correction contributions1, 23 of neutrons δEn and protons
δEp to nuclear masses which are related to the oscillating partδρ.
Great success in the understanding of many properties of stable nuclei is associated with the Nilsson model (NM) (cf. e.g.
References 20, 21) which is based on the Harmonic Oscillator(HO) confining potential approximation for the nuclear meanfield. In present study we consider the simple spherical HO
Hamiltonian yielding the equidistant sp spectrum

2. Magnetism of Nuclei within the Shell Model∗E-mail: vkondra@hadron03.tokai.jaeri.go.jp.

Z εF∆α = gα b/2(3)with the g-factor gα , and reduced field strength b = ωL /ω0 . Sucha term contributes to sp energies of both protons α = p, and neutrons α = n, and gives rise to a relative shift down and up (seeFigure 1b) of energy levels with the nucleon spin-magnetic moment directed along the field (σi=↑ = −1, majority-spin levels)and in the opposite direction (σi=↓ = 1, minority-spin levels),
respectively. The shift, eq 3, is related to the Pauli-type of themagnetic response and modifies the shell-correction energy as
δEα = δEα↑ (εF + ∆α ) + δEα↓ (εF − ∆α ) ,(4)where δEαi is determined by the unshifted sp spectrum (seeeq 1, 2).

c 2002 The Japan Society of Nuclear and Radiochemical Sciences
Published on Web 6/30/2002

206 J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002

Kondratyev

The spin magnetization represents predominant effect in themagnetic field dependence of the neutron shell-correction energy and leads to a phase-shift of shell-oscillations.1 This behavior is caused by the field dependent interference of contributions coming from the majority- and minority-spin neutrons
to the total energy. As illustrated in Figure 1a such an interference gives rise to oscillations of the shell-energy as a functionof magnetic field strength with a period bs ≈ 0.5. Figure 1b indicates that for nearly stable nuclei the respective magnetic fieldBs ∼ ω0 /µN ∼ 1016 –1017 G induces the relative shift ∆n , eq 3,of neutron majority- and minority-spin energy levels which iscomparable to the energy difference between major shells givenby the HO frequency ω0 . In particular, at field strengths corresponding to the region near b ≈ |2gn |−1 ≈ 0.26 the level spacing
and the degeneracies are decreased resulting in a suppression of
the shell effect as compared to the values related to level crossings, bcr = |k/gn | with an integer k.
At zero-field the pronounced minima are displayed at neutron
numbers, Nm0 = 8, 20, 40, 70, 112, 168, ..., associated with closed
shells, as extensively discussed in, e.g., Reference 20. In thecase of b ≈ |1/gn | the level spacing and the shell-oscillation am250

a)200

plitude are nearly the same as at b ≈ 0. However, as indicated in
Figure 1b level occupation numbers are rather different in thesetwo cases. Consequently, the positions of shell-energy minima
are replaced. At the field strength matching the first level crossing the familiar spherical HO magic numbers are turned into
anti-magic, i.e. associated with positive maxima of the shellcorrection energy, while the open shell anti-magic numbers become the closed shell magic numbers. For grand canonical ensemble the field dependence of the shell energy minima can beobtained from the condition of constant arguments in eq 4. This
yields an approximation
Nm± /Nm0 ≈ (1 ∓ gn b/2(3Nm0 )1/3 )3 ,(5)where ± corresponds to upper and lower valleys extending fromthe respective zero-field magic numbers Nm0 (see Figure 1a). As
evident from Figure 1a linear term dominates such a dependence. We note that magic number of canonical ensemble deviate from the case of grand canonical by about 1% and 5% at thefirst and second level crossings, respectively.
The effect of a sign change in the shell-correction energy remains when accounting for the spin-orbit interaction as well.1
Furthermore, the sign inversion occurs at the field strength thatis almost an order of magnitude smaller than the oscillation period Bs (see above).
2.2. Orbital Magnetism. The interaction of magnetic fieldwith moving along the confined trajectory proton charge gives
rise to an additional modification of proton spectra which atsmall field strengths b 1 readsNδεo ≈ bl3 ,where l3 denotes the projection of the proton angular momentum on the field axis. Switching off the Pauli response (i.e.
∆p = 0 in eq 4) the properties of the orbital magnetism can beeasily seen by employing the simplified expression for the shellcorrection energy at conditions of weak fields in the form (seeReferences 1, 24)100δEoi ≈ −

50
0

0.1

0.2

0.3

0.4

0.5

b

168
112
70
40
20

45

285

36

204

36

(7)

249

28

176
28

140

21

91

15

55

10

30

21

200

119

15

76

10

150

45

6

24
6

0

ω0 X(X + 1)
∑ k−2 cos(kX) j0 (bkX) ,
8π4
k=1

250

Z

b) 240
Energy Levels

(6)

150

100

3

0.5

1

Magnetic Field
Figure 1. (a) Spherical HO model prediction of the neutron shellcorrection energy versus the neutron number N and the parameter b.
Smaller energies are indicated by white regions, while the dark regions
denote larger energies. The contours are plotted with the step 0.5 starting from −1.5 for the quantity δEn · 8π4 /(ω0 X 2 ). The thick solid lines
are related to the relationship eq 5. (b) The dependence of the majority(solid lines) and minority-spin (dashed lines) energy level on the magnetic field. The energy and the field are measured in the units of the HO
level spacing ω0 and [ω0 /gn µN ], respectively. The figures at the levelcrossings (indicated by triangles) show the total nucleon number when
the levels are filled starting from the bottom. The level degeneracies are
displayed by the numbers attached to respective lines.

50
0.05

0.1

0.15

0.2

0.25

0.3

b
Figure 2. Effect of orbital magnetism in the proton shell-correctionenergy versus the proton number Z and the parameter b for the spherical
HO potential. The white regions display to the energy minima (i.e. thewells), while the dark regions indicate the maxima (i.e. the hills) in theshell-correction energy landscape. The contours are plotted with thestep 0.15 starting from −1 for the quantity δEp · (1 + b/b0 )8π4 /(ω0 X 2 ),where b0 = (4 · 901/3 )−1 ≈ 0.056. Thick solid lines represent eq 9, while
dashed lines correspond to eq 8.

Structure of Nuclei in Strong Magnetic Fields

J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002

where X = 2πεF /ω0 ≈ 2π(3N)1/3 counts the number of filledshells in zero-field limit. The effect of orbital magnetism consists, therefore, in a modulation of components for a nonperturbed spherical HO by the spherical Bessel function with thefield dependent argument. As indicated by eq 7 and illustrated
in Figure 2 the weak field leads to regular oscillations of theshell-correction energy as a function of the field strength as well
as overall suppression the shell effect with increasing b at zeros-o coupling. As seen in Figure 2 local maxima of the shellenergy amplitude arise along the lines
Z p ≈ [(p + 1/2)/2b]3 /3(8)with an integer p. The shell-correction component vanishes
when in eq 8 p = k + 1/2 at integer k. Therefore, orbital magnetism yields the oscillation period bo ≈ X −1 ∼ A−1/3 corresponding to the field strength Bo = ω0 /µN X ∼ A−2/3 . This estimate (cf. also Figure 2) yields a weaker field for heavier nuclei
in order to invert the sign of the proton shell-correction energy
as well as to wash out the shell-structure. The minima valleys inshell-correction energy landscape follow the lines
Zm± /Zm0 ≈ [(1 ∓ bm )/(1 ∓ b)]3 , bm = 1/(4(3Zm0 )1/3 ) ,(9)with a good accuracy up to the third oscillation.
2.3. Paramagnetism versus Orbital Magnetism of Protons
in Nuclei. The magnetic field dependence of the proton shellcorrection energy is given as a combination of the discussed insect. 2.1, Figure 1 and eq 4, Pauli-magnetism and the Landautype of orbital magnetism related to proton ballistic dynamics(see Figure 2 and eq 6, 7). Switching on the spin-magnetic
response we obtain the proton shell-correction energy whichis shown in Figure 3 as a function of the proton number Z
and magnetic field strength b. As seen the damped oscillations of the shell energy correspond to the period bso decreasing with the mass number similarly to a pure orbital magnetism.
The Pauli-type magnetization reduces, however, additionally thefield strength that is required to change the sign in the protonshell-correction energy. The phase-shift on π (i.e. the reversedsign) occurs at the parameter b which is considerably smaller
than the value 1/gp . The interplay between the Landau-type
and spin-magnetism gives rise to an extra-decrease of the periodof a sign oscillation. The relevant field strength is, therefore,
250

Z

200

150

100

50
0.05

0.1

0.15

0.2

0.25

0.3

b
Figure 3. Spherical HO model prediction for the magnetic field dependent proton shell-correction energy when including spin and orbital
magnetization. The contours are the same as in Figure 2, while thicksolid lines are associated with eq 10.

207

smaller than the corresponding field for neutrons by a factor exceeding the ratio |gp /gn |. At the Fermi energy the respective
magnetic field shifts the proton majority- and minority-spin levels on an energy (δεi + δεo ) comparable to the HO level spacing
ω0 . The energy of the first Landau level ωL is, however, by thefactor 6–7 smaller than the energy difference between the major shells, while the radius of the proton cyclotron orbit is larger
than the radii of nuclei on approximately the same factor. Consequently, Landau levels give almost no contribution to the magnetism of nuclei. The orbital magnetism in such a case is mainly
given by an interaction of the magnetic field with the magnetic
dipole arising from the quantum orbital motion of protons inside the nucleus. The orbital magnetic response of such an inhomogeneous system is considerably amplified as compared tothe magnetism of a homogeneous liquid which originates fromthe quantization of proton orbital motion in magnetic field. The
Landau levels are expected to contribute noticeably to the magnetic response when their and nuclear radii become comparable.
Corresponding field strength BL ∼ 1019 G.
As seen from the comparison of Figures 1, 2 and 3 the interplay between spin and orbital magnetic reactivities yields more
complicated proton shell-correction energy landscape as compared to their independent responses in a field. In this case themagnetic field evolution of the proton shell-energy valleys canbe guided by the lines3
Zm± /Zm0 ≈ [(1 ∓ bm )/(1 ∓ b)] ± gpb/2(3Zm0 )1/3
(10)
which represent a combination of eq 5, but with proton g-factor
gp , and eq 9. As illustrated in Figure 3 the valleys display irregular shapes passing from upper to lower lines.
It is worthy to recall here that the presence of s-o interaction leads, in addition, to an anomalous field dependence of the
proton shell-correction energy amplitude.1 The proton orbital
magnetism yields rather pronounced enhancement of the shelleffect. Especially, when the values of the parameter b are close
to the spin-orbit strength, the shell-oscillation amplitude can beconsiderably amplified.
3. Conclusion
In summary, we have discussed an effect of ultrastrong magnetic fields on the structure of nuclei. As shown within the employed mean-field treatment such a field influence can be viewed
as a shift of nucleon energy levels which is governed by the projection of respective magnetic moments on the field axis. Making use of the shell model with spherical HO confining potential we have analyzed common features of magnetic field dependence of shell-correction energy and magic numbers. For
neutrons such effects entirely originate from the Pauli-type paramagnetic response. As a consequence, for equidistant HO spectrum the neutron shell-correction energy displays almost periodic behavior as a function of the magnetic field strength. As
seen within the considered model the field evolution of valleys
associated neutron magic numbers follow nearly straight lines
corresponding to eq 5.
The interplay between spin- and orbital-magnetism determines the proton magnetic response. As demonstrated at zerospin-orbit interaction the Landau-type of orbital magnetic response results in a dumped oscillations of the shell energy as afunction of magnetic field. The overall amplitude of shell oscillations is inversely proportional to the magnetic field strength,
while the amplitude maxima arise at the fields given by eq 8.
The energy minima valleys associated with magic numbers follow the lines associated with eq 9. The combination of spinand orbital magnetic response brings complicated shell correction energy landscape. The valleys of nuclear magics extend
attaching upper and lower lines of eq 10, repeatedly.
As a matter of fact, the nuclear level density represents animportant ingredient to the prediction of nuclear reaction cross208 J.Nucl.Radiochem.Sci.,Vol. 3, No. 1, 2002sections.25 Therefore, our analysis indicates possible magnetic
field dependence of the nuclear reaction (e.g. s- and r-processes)
rates and, consequently, abundance of elements in, e.g., supernova remnants.
We note finally that similar shift of magic numbers canshow up in, e.g., atomic clusters. The scale of the respective
field strength for e.g. an alkaline cluster of Nac atoms Bac ∼εFac /(3Nac )1/3 µB ∼ 109 /(3Nac )1/3 G (with the Bohr magneton µB )suggests a possibility for laboratory tests of the predicted phaseshift of shell-oscillations.
Acknowledgments. The author is indebted to A. Iwamoto,Y. Abe, T. Tatsumi, S. Chiba, T. Maruyama, K. Niita, K. Oyamatsu, and M. Matsuzaki for valuable discussions and thanks
the Research Group for Hadron Science at JAERI for the warm
hospitality.
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