Nuclear spin scissors – new type of collective motion

The coupled dynamics of the orbital and spin scissors modes is studied with the help of the Wigner Function Moments method on the basis of Time Dependent Hartree-Fock equations in the harmonic oscillator model including spin orbit potential plus quadrupole- quadrupole and spin-spin residual interactions. The relation between our results and the recent experimental data is discussed.


Introduction
The idea of the possible existence of the collective motion in deformed nuclei similar to the scissors motion continues to attract the attention of physicists who extend it on various kinds of objects, not necessary nuclei, (for example, magnetic traps, see the review by Heyde at al [1]) and invent new sorts of scissors, for example, the rotational oscillations of neutron skin against a proton-neutron core [2]. In a recent paper [3] the WFM method was applied for the first time to solve the TDHF equations including spin dynamics. As a first step, only the spin orbit interaction was included in the consideration, as the most important one among all possible spin dependent interactions because it enters into the mean field. The most remarkable result was the discovery of a new type of nuclear collective motion: rotational oscillations of "spin-up" nucleons with respect of "spin-down" nucleons (the spin scissors mode). It turns out that the experimentally observed group of peaks in the energy interval 2-4 MeV corresponds very likely to two different types of motion: the conventional (orbital) scissors mode and the spin scissors mode.
The aim of this work is to get a qualitative understanding of the influence of the spin-spin force on the new states analyzed in [3], as, for instance, the spin scissors mode. As a matter of fact we will find that the spin-spin interaction does not change the general picture of the positions of excitations described in [3]. The most interesting result concerns the B(M1) values of both scissors modes -the spin-spin interaction strongly redistributes M1 strength in the favour of the spin scissors mode, that allows us to give a tentative explanation of recent experimental findings [4,5].

TDHF equation and model Hamiltonian
We consider the TDHF equation in coordinate space keeping all spin indices: i r|ρ|r ss = s d 3 r r|ĥ|r ss r |ρ|r s s − r|ρ|r ss r |ĥ|r s s .
We do not specify the isospin indices in order to make the formulae more transparent. From the technical point of view it is more convenient to work with the Wigner function f (r, p) instead of density matrix r|ρ|r . To this end we rewrite the expression (1) with the help of the Wigner transformation [6]: where the functions h, f are the Wigner transforms ofĥ,ρ respectively, {f, h} is the Poisson bracket of the functions f (r, p) and h(r, p) and {{f, h}} is their double Poisson bracket, the dots stand for terms proportional to higher powers of . The conventional notation ↑ for s = 1 2 and ↓ for s = − 1 2 is used. The microscopic Hamiltonian of the model, harmonic oscillator with spin orbit potential plus separable quadrupole-quadrupole and spin-spin residual interactions is given by with where N and Z are the numbers of neutrons and protons andŜ µ are spin matrices [7]. The quadrupole operator q 2µ can be written as the tensor product: q 2µ (r) = 16π/5 is the Clebsch-Gordan coefficient and r 1 , r 0 , r −1 are cyclic coordinates [7].

Equations of motion
Integrating the set of equations (2) over phase space with the weights one gets dynamic equations for the following collective variables: where ς = +, −, ↑↓, ↓↑ . By analogy with isoscalar f n + f p and isovector f n − f p functions we call the functions f τ + and f τ − and the corresponding collective variables X τ + λµ (t) and X τ − λµ (t) as spin-scalar and spin-vector ones. The required expressions for h ± , h ↑↓ and h ↓↑ are where Z n+ 2µ = κR n+ 2µ +κR p+ 2µ , Z p+ 2µ = κR p+ 2µ +κR n+ 2µ and The neutron potentials V ς n are obtained by the obvious change of indices p ↔ n. We are interested in the scissors mode, the excitation with K π = 1 + , therefore, we only need the part of dynamic equations with µ = 1. These equations are nonlinear due to quadrupolequadrupole and spin-spin interactions and will be solved in a small amplitude approximation.
Let us recall that all variables and equilibrium quantities R + λ0 (eq) and Z + 20 (eq) have isospin indices τ = n, p. All the difference between neutron and proton systems is contained in the mean field quantities Z τ + 20 (eq) and V ς τ , which are different for neutrons and protons (see eq. (8)). It is convenient to rewrite the dynamical equations in terms of isovector and isoscalar variables X λµ = X n λµ + X p λµ ,X λµ = X n λµ − X p λµ and isovector and isoscalar strength constants κ 1 = (κ−κ)/2 and κ 0 = (κ+κ)/2 connected by the relation κ 1 = ακ 0 . Then the equations for the neutron and proton systems are transformed into isovector and isoscalar ones. Supposing that all equilibrium characteristics of the proton system are equal to that of the neutron system one decouples isovector and isoscalar equations. Neglecting by the fourth order moments, generated by the terms h ↑↓ f ↓↑ − f ↑↓ h ↓↑ of (2) we get the closed set of 19 equations.

Discussion and interpretation of the results
Imposing the time evolution via e iΩt for all variables one transforms dynamical equations into a set of algebraic equations. Eigenfrequencies are found as the zeros of its determinant. Excitation probabilities are calculated with the help of the linear response theory. The used spin-spin interaction is taken from the paper [8], where the notation χ = K s /A,χ = qχ was introduced. The results of calculations without spin-spin interaction (variant I) are compared with those performed with two sets of constants K s , q (variants II, III). The strength of the spin-orbit interaction is taken from [9].  I  II  III  I  II  III  I  II  III ( One can see from table 1 that the spin-spin interaction does not change the qualitative picture of the positions of excitations described in [3]. It pushes all levels up proportionally to its strength (20-30% in the case II and 40-60% in the case III) without changing their order. The most  [4] shows evident splitting into two Lorentzians. "Typically, the experimental splitting is ∆ω M 1 ∼ 0.7 MeV, and the ratio of the strengths between the lower and upper resonance components is B L /B U ∼ 2". The authors have tried to explain the splitting by a γdeformation. To describe the observed value of ∆ω M 1 the deformation γ ∼ 15 • is required, that leads to the ratio B L /B U ∼ 0.7 in an obvious contradiction with the experiment. The authors conclude that "the splitting may be due to other mechanisms". In this sense, we tentatively may argue as follows. On one side, theory [10] and experiment [11] give zero value of γ-deformation for 232 Th. On the other side, it is easy to see that our theory suggests the required mechanism. The calculations performed for 232 Th give ∆ω M 1 ∼ 0.32 MeV and B L /B U ∼ 1.6 for the first variant of the spin-spin interaction and ∆ω M 1 ∼ 0.28 MeV and B L /B U ∼ 4.1 for second one in reasonable agreement with experimental values.
Quite similar results were obtained one year earlier by A. S. Adekola et al [5]. They have studied the (γ, γ ) reaction on 232 Th and found two groups of levels with energy centroids

Concluding remarks
In this work, we continued the investigation of spin modes [3] with the Wigner Function Method. The inclusion of spin-spin interaction does not change qualitatively the described picture concerning spin modes found in [3]. It pushes all levels up without changing their order. However, it strongly redistributes M1 strength between the conventional and spin scissors mode in the favour of the last one. We mentioned the recently appeared experimental works [4,5], where for the two groups of low lying magnetic states a stronger B(M1) transition for the lower group with respect to the higher one was found. Our theory can naturally predict such a scenario with a non vanishing spin-spin force. It would indeed be very exciting, if the results of [4,5] had already discovered the isovector spin scissors mode.
In the light of the above results, the study of spin excitations with pairing included, will be the natural continuation of this work. Pairing is important for a quantitative description of the orbital scissors mode. The same is expected for the novel spin scissors mode discussed here.