The nuclear ?Twist?

Abstract

The nuclear fluid-dynamical Hamiltonian which takes the distortion of the local Fermi surface into account predicts the Twist mode (T = O, J = 2 ) at a_finite frequency (about 7.5 MeV for 20~pb).The validity of hydrodynamics in its usual Newtonian form for the nuclear fluid is highly questionable because the mean free nucleon path (for energies less than about 50 MeV/particle) is comparable to nuclear dimensions and therefore the concept of local equilibrium seems doubtful. As we have discussed in detail elsewhere /i/, the non-equilibrium effect leads to a distortion of the local Fermi surface throughout the whole nucleus, which, however, can be incorporated into a fluid dynamical formulation, i.e. a formalism involving only local densities and currents. The essential difference to hydrodynamics is a non-local contribution to the local kinetic energy density. In order to calculate this contribution for a twisting (T=O) mode, we define the "Twist" through the operator = e -i~zs = e ~'~, ~ = (yz,-xz,o) , i.e., a rotation around the body-fixed z-axis with rotation angle proportional to z. The operator zl z has spin-parity 2because the scalar ~art of the tensor product ~ ~ ~, namely r-~, vanishes identically. Although for axially symmetric nuclei there evidently is no change in the local density, the Twist still creates a distortion of the local Fermi surface which in lowest order is given by PF(r,p) = PF(r)N(e) (i-~2~/15" (Y(Y21 (~)-Y2-t (P))+ix(Y21 (P) +Y2-I (P)) ) ) with N(~) = 1 -a2(x2+y2)/15. In Thomas-Fermi approximation we then obtain for the kinetic energy P~<PF (~'P) /T(p)d~ = fat I p~d~ = /to(p) (l+a 2(x2+y2)/3)d~ z~ 3h2 3~2. s/s with T o = lOm (-2-) P All contributions to the restoring force for the Twist from volume terms in the total ~nergy vanish because the ~elocity field u is free of divergences: V.u ~ o. For axially symmetric nuclei there are no surface-energy contributions, therefore the total restoring force is ~2 C = --IT d~ ~2 (3~2. 2/3 S13 ~e2 = ~,-~--) Ip (x2+y2)d~ The inertia parameter B for the Twist follows from the kinetic part of the fluid-dynamical Hamiltonian: B = mfpu2d~ = m/pz2(x2+y 2) d~ . Frequency and B(M2)+ value are then obtained as = (C/B) I/2 B(M2)+ ~ .5Z. 2 1 tel% 2 <0,2> where <~2> is the mean square twisting amplitude in the ground state <e2> = I%/(2(BC) I/2)Of course, this model is not able to predict how much of the M2-strength is actually concentrated into one single collective 2 state. This question must be

Shell effects and the fluid dynamical model for nuclear giant resonances

“…Exactly the same relations between the semiclassical kinetic energies Ta and TO were obtained by HE [3].…”

“…which is the result of HE [3]. We have thus shown for the twist mode, too, that the approach of HE using the distortion of the Fermi surface is no more and no less than a clever way of taking into account the shell effects m the intrinsic kmetic energy (which are just the static ones!…”

“…For notatmnal simplimty, we shall use the mdependent particle (Hartree-Fock) picture; the arguments are unchanged if one starts from many-body wave functions as done in ref. [3].The essential assumption of the imposed collective scaling modes is that during the oscillations of the nucleus, all static wave functions are scaled m phase and the total intrinsic kinetic energy is then given by r~ = f r (~, r) d 3r. …”

“…[1] for a recent review) have raised much interest in the theoretical description of these exmtation modes within nuclear fluid dynamical models [1][2][3][4]. Studying imposed collective oscillations of the nucleus defined through generalized scaling transformahons [5 ] of the wave functions, Holzwarth and Eckart [3] (in the following abbreviated as HE) stressed the importance of what they called dynamical corrections to the ThomasFermi method due to distortions of the local Fermi surface m momentum space.…”

“…For notatmnal simplimty, we shall use the mdependent particle (Hartree-Fock) picture; the arguments are unchanged if one starts from many-body wave functions as done in ref. [3].…”