Mass dependence of the surface δ-interaction strength for two-particle (two-hole) multiplets

By using the surface δ -interaction, the lowest states in nuclei of atomic mass number 6≤A≤210 with a doubly magic core and two valence nucleons are investigated. A single parameter describes the strength of the multiplet splitting of the lowest configuration which includes the ground state. The dependence of the strength on the atomic mass number of the doubly magic core is determined as a power law function A-f with an exponent f = $ {\frac{{2}}{{3}}}$ for 50 ≲ A≤208 .

In this paper we investigate the simplest nuclei for which the shell model can be applied [7][8][9][10]. These are the nuclei which have one or two particles (holes) outside a doubly magic core with mass numbers A+δ, A+2δ, and A, respectively. Here the parameter δ = +1, δ = −1 refers to the systems with particles and holes, respectively.
The SDI is presented extensively in several textbooks [5][6][7][8][9][10] and known to describe the complete groundstate multiplet in a two-particle (two-hole) system in various nuclei fairly well. Of course, more complicated systems and interactions have been studied [18][19][20][21][22]. Yet in this paper we consider only the SDI interaction for systems A + 2δ.
Our goal is to find the global dependence of the multiplet splitting on the atomic mass number A, especially in medium and heavy nuclei. The global dependence described as the dependence of the effective strength of the surface δ-interaction on the atomic mass number has not been studied in such a detail before. Presumably, because at the time when the SDI was introduced few experimental a e-mail: A.Heusler@mpi-hd.mpg.de data on nuclei A +2δ near the doubly magic nuclei 90 40 Zr 50 , 100 50 Sn 50 , 132 50 Sn 82 were available. Only recently data near 78 28 Ni 50 have become available [23].
As another method to derive the global dependence of the strength of the surface δ-interaction on the atomic mass number, the differences between the masses of the nuclei with one and two valence nucleons and the mass of the doubly magic core can be investigated. However for nuclei A, A + δ, A + 2δ in the single region of interest, 50 A < 208, most mass differences are known with a precision of 10-100% only. In addition, the influence of the Coulomb force and the configuration mixing introduce systematic uncertainties up to 30%. 10 The European Physical Journal A The mass of the corresponding state depends on the same quantum numbers, A + 2δ, j, j , t z , t z , J, T . In the following, M will always refer to the mass of the system A + 2δ and we only consider the special case where the valence nucleons reside in the same orbit j. Further we assume the valence nucleons to be identical. However, in sect. 4.3 we briefly mention two odd-odd nuclei discussing the case t z = t z . Therefore the mass of the system A + 2δ is written as We compare the multiplet splitting for the configuration containing the ground state only. The mass difference for the nucleus A + 2δ with total spin J in reference to the mass with maximum spin J max is thus given by The experimental multiplet splitting is derived from the excitation energies alone, taken from ENSDF [24], as Here J max is chosen as the total spin J of the state with the highest excitation energy. Especially, we have

Surface δ-interaction
The SDI was introduced [11] as a simple but yet successful method to calculate nuclear properties in the framework of the shell model. Following Chapts. 11, 12 1 of ref. [8] the interaction between two nucleons of zero range is assumed to be localized on the nuclear surface of the core. The interaction can be written as in spherical coordinates. The radial integral evaluated as is assumed to be equal for all orbits nl with main quantum number n and orbital angular momentum l (j = l ± 1 2 ).
denotes the strength factor for each nucleus A + 2δ.
For a pure two-particle configuration one finds the multiplet splitting by (9) where ΔS SDI (j, J, T ) = S SDI (j, J, T ) − S SDI (j, J max , T ) (10) denotes a geometrical factor, a number. The factor 1 2 enters into eq. (9) since both valence nucleons occupy the same orbit j. The geometrical factor is evaluated by for T = 1 with the special value for total spin J = 0 S SDI (j, 0, 1) = −(2j + 1).
For T = 0 the geometrical factor is evaluated by The seniority is a good quantum number for the SDI. For the same isospin T and for j ≥ 5 2 , the spacings between the j 2 levels and between the j 2j−1 levels are the same (see p. 565 of ref. [8]). Hence the same formulae (eqs. (9)-(13)) apply as well to two-particle systems A + 2δ, δ = +1 as to two-hole systems A + 2δ, δ = −1. An example is shown in fig. 3 for 42 20 Ca 22 and 46 20 Ca 26 where the multiplet splitting is identical for the 1f 7/2 2 and 1f 7/2 6 configurations. In reality, because the multiplet splitting varies from one system A+2δ to another, eq. (9) will not hold exactly. But one can always exactly reproduce the experimental splitting between the states with spins J = 0 and J = J max by defining a suitable strength parameter

Example for the multiplet splitting
As an example, in fig. 1 (left side) the level scheme is displayed for the nucleus 210 Po with two protons in the orbit 1h 9/2 and 208 Pb as the doubly magic core. The multiplet splitting (eq. (4)) is shown on a keV scale, too. The lowest states have spins 0 + , 2 + , 4 + , 6 + , 8 + . They consist mainly of the configuration h 2 9/2 while the next multiplet is described by the configuration 1h 9/2 2f 7/2 , see table 3. The energy difference between each pair of states with spins 2 + , 4 + , 6 + , 8 + in these two multiplets is at least three times the mean splitting in each multiplet [24]. Therefore the admixture of other configurations to the ground-state multiplet can be assumed to be small. and total spins J π = 0 + , 2 + , 4 + , 6 + , 8 + derived from experimental excitation energies. The vertical lines show the multiplet splitting ΔM expt (eq. (4)) between the states with spins J = 0, 2, 4, 6, 8 and J = J max = 8. Right: multiplet splitting ΔM expt transformed to ΔS expt (eq. (16), right scale) for J π = 0 + , 2 + , 4 + , 6 + , 8 + (diamonds) plotted versus the classical angle θ (eq. (15)). The cross and the dashed line mark the reference value ΔS expt = 0 from which the multiplet splitting is calculated (eqs. (5), (16)). The drawn curve connects the calculated values ΔS SDI (eq. (10)) in order to guide the eye. It is continued until We introduce the classical angle between the two valence nucleons as a convenient method to present the multiplet splitting. It is defined as (see eq. (II.2) in ref. [21]) where j, j are the spins of the two interacting nucleons in a state with total spin J. In this paper we deal with j = j only, see eq. (3). The central part (diamonds in fig. 1) shows the experimental multiplet splitting ΔM expt (eq. (4)) as function of the angle θ (eq. (15)). The reference energy for the state with spin J max (eq. (5)) is marked by a cross. A parabolic dependence is found.
Similar to the calculated multiplet splitting (eq. (9)), the corresponding experimental values are given by (16) by using eqs. (10), (12).  5)) on the atomic mass number A of the core (eq. (1)). The shaded areas cover the values used to derive the mean strengthv 0 and its uncertainty (eq. (21)). For protons it is about 10% larger than for neutrons. The exponent f = 2 3 for the power law function A −f (eqs. (18), (19)) is valid only for heavier nuclei A > A min (eq. (20), drawn line). The dotted curve shows the fit by interpolating (eq. (22)) between the experimental data point at A = 4 and the flat region with an exponent f = 2 3 . Double diamonds mark values for multiplets where the main quantum number is n = 2 ( 134 52 Sn82 and 210 82 Pb128); otherwise it is n = 1.

SDI strength parameter
In order to find the strength V 0 (eq. (8)) we look at fig. 2, see also table 1. Here we plot the quantity V expt 0 A 2/3 for the two-proton and two-neutron multiplets. One finds The European Physical Journal A For heavier nuclei, this suggests the fit formula by a power law function with an index For lighter nuclei the fit formula is more complicated. By interpolating linearly between A = 4 and A min on a doubly logarithmic scale one obtains for 4 ≤ A ≤ A min (t z ) The origin for the deviation of the fit function from A −2/3 for lighter nuclei is not easy to explain.  (14)) for heavy nuclei A + 2δ with a core of atomic mass number A ≥ 90. The quantum numbers nlj of the valence nucleons are given. By averaging the values for n = 1 only one finds V fit 0 A 2/3 = 11.3 ± 0.5.

Multiplet splitting for the ground state
Equation (14) is valid only for pure configuration states.
In deriving the strength factor V fit 0 (eq. (18)) we select states which can be assumed to be rather pure.
The multiplet splitting is largest for the 0 + ground state (eq. (12)). Therefore we consider only configurations where the valence nucleons occupy orbits with spins j ≥ 5 2 . In most nuclei the excitation energy of the lowest 2 + state is lower than predicted by the SDI because several configurations admix. As seen in fig. 3, only for 134 Te and 210 Po the multiplet splitting is very well described by the SDI. Similarly, the excitation energies for states with higher spins 2 < J < J max are often lowered by admixing configurations.
Besides the ground state, only the multiplet member with the highest spin J max is rather pure in most nuclei with a doubly magic core. For this reason we discuss only the multiplet splitting between the states with spins J = 0 and J = J max (eq. (5)).
For 148 Dy the ground state is lower than calculated whereas the other multiplet members are fairly well described. A large admixture of another configuration to the ground state may be the reason, see sect. 4 Corresponding to eq. (9) we define The ratio between the experimental and the calculated multiplet splitting for the ground state (eqs. (4), (23)) R(A, j, t z ) = ΔM expt (A, j, t z , 0, 1) ΔM fit (A, j, t z , 0, 1) is unity if the SDI describes the experimental multiplet splitting.

Multiplet splitting for all spins
It is convenient to transform the experimental multiplet splitting into a number similar to ΔS SDI (eq.  25)) within a few percent for all spins J, for nuclei A +2δ with valence nucleons of a certain spin j, the maximal experimental multiplet splitting is generally different from the fit value, ΔS expt (A, j, t z , 0, 1) = ΔS fit (A, j, t z , 0, 1).
Some higher excited states are indicated by asterisks (as far as spins are known), arbitrarily using the angle θ(j, j , J) (eq. (15)) with the same spins j = j and J as for the ground-state multiplet, since often the dominant configuration is not known (see table 3).   We note that for 50 22 Ti 28 the low isospin of the core (T core z = 4) may add another component to the wave function (eq. (1)).

Nuclei with a 90
40 Zr 50 , 100 50 Sn 50 , 132 50 Sn 82 or 146 64 Gd 82 core In the nucleus 92 42 Mo 50 the ground-state multiplet mainly consists of the configuration g 2 9/2 . The multiplet splitting is well described by the SDI (fig. 3). We note, however, that the low isospin of the core (T core z = 5) may add another component to the wave function (eq. (1)).
By using the tentative spin assignments [24] and by assuming the configuration g 2 9/2 , a calculation reproduces the multiplet splitting for 102 50 Sn 52 (fig. 3). Yet the energy of a second 0 + state is uncertain [24].
The ground-state multiplet of 148 66 Dy 82 is expected to have a dominant configuration h 2 9/2 . In 147 65 Tb 82 however, the configuration h 11/2 is just 50 keV above the configuration s 1/2 followed by d 3/2 lying 253 keV higher in energy; in 149 67 Ho 82 the configuration h 11/2 is just 49 keV below the configuration s 1/2 followed by d 3/2 which is 171 keV apart. In 148 66 Dy 82 therefore, the lowest 0 + and 2 + states are expected to contain considerable admixtures of the configurations s 2 1/2 , s 1/2 d 3/2 , d 2 3/2 . The excitation energies of these two states being lower than predicted may thus be explained. The next states with spins 0 + and 2 + (and 1 + ) expected to have low excitation energies are not yet known. Hence, the data are omitted in fig. 2 but shown in fig. 3.

Nuclei with a 208 Pb core
In the nuclei 210 Pb and 210 Po the configurations g 9/2 and h 9/2 , respectively, produce states with spins 0 + , 2 + , 4 + , 6 + , 8 + , all of which can be expected to be rather pure. In the shell model the next configurations are at a large distance relative to the mean matrix element of the residual interaction of about 100 keV.

Radial dependence
A basic assumption of the SDI is the neglect of any dependence of the radial wave function on the details of the orbit (eq. (7)). In effect, the region where the valence nucleons interact is assumed to be similar for all nuclei (see figs. 3-4 of ref. [4]).
In tables 1, 3 the quantum numbers nlj of the orbits for the valence nucleons are given. In fig. 2, for protons as valence nucleons no significant dependence on the quantum numbers nlj is visible; the main quantum number is n = 1 throughout.
However, some dependence on the radial part of the wave function C 0 (nl) (eq. (7)) is indicated for nuclei with a doubly magic core 132 50 Sn 82 and 208 82 Pb 126 where the valence neutrons have main quantum number n = 2. Both values are lower than the mean value. By using only experimental values with main quantum number n = 1 (single diamonds in fig. 2) the mean strength factorv 0 (eq. (18)) is found to be the same for neutrons and protons for nuclei with a core of mass number A ≥ 90 (table 1).

Odd-odd nuclei
Only few odd-odd nuclei with a doubly magic core and two valence nucleons are known where the configuration mixing is assumed to be small, namely 42 21 Sc 21 [25], 54 27 Co 27 . Here, because the isospin of the valence nucleons differs, states with spins J π = 1 + -7 + are present (eq. (13)), see also table 2.
For the states with isospin T = 0 the values ΔS expt for the multiplet splitting are small and range near unity. For the states with isospin T = 1, fig. 5  Since the exponent f in the power law function (eq. (18)) starts to deviate from f = 2 3 for lighter nuclei with mass number A 50 (see fig. 2), we do not discuss the data from these odd-odd nuclei. Table 3. The lowest states in nuclei with a doubly magic core and two valence nucleons assumed to have a negligible admixture to some main configuration. The next configurations expected from the shell model to admix considerably are indicated. The order of the single-particle levels in dependence on the atomic mass number is taken from ref. [4]. The multiplet splitting between the state with the highest known spin J max (eq. (5), underlined in the table) and the lowest spin (the ground state with spin 0 + ) is given by ΔM expt (eq. (4)). By using the mean valuev0 (eq. (21)) and the function A −f (eqs. (18)- (22)) the multiplet splitting is calculated as ΔM fit (eq. (23)). The ratio R (eq. (24)) should be unity if the SDI describes the experimental multiplet splitting.

Nucleus
Range The experimental values for the nuclei with a 4 He core are used for the fit of the function A −f (eq. (22)).

16
The European Physical Journal A

Summary
For nuclear systems A +2δ with two valence nucleons outside a doubly magic core A (δ = +1, δ = −1 referring to particles and holes, respectively) the lowest multiplet built with the single particle (hole) in the ground state of the system A + δ is investigated. From the multiplet splitting of nearly 20 such systems, the strength of the residual interaction is found to be proportional to the atomic mass number by a power law function A −f with an exponent f = 2 3 for atomic mass number 50 A ≤ 208. For light and medium heavy nuclei (A < 40) a departure of the power law function with an exponent f = 2 3 is determined.