The FTBCS1+SCQRPA theory proposed in the present chapter includes the effect of quasiparticle-number fluctuation as well as dynamic coupling of quasiparticles to pairing vibrations. This theory also incorporates the corrections caused by the particle- number projection within the LN method. We have carried out a thorough test of the developed approach within the Richardson model as well as two realistic nuclei, 56Fe
and120Sn. The analysis of the obtained pairing gaps, total energies, and heat capacities
shows that in the region of moderate and strong couplings, the QNF within the FTBCS1 (with or without SCQRPA corrections) smoothes out the sharp SN phase transition. As a result, the pairing gap does not collapse at critical temperature T =Tc, but has
a tail, which extends to high temperatureT.
The correction due to the particle-number projection within the LN method to the pairing gap is significant atT ¿Tc, which leads to a steeper temperature dependence
of the pairing gap in the region aroundTc. At the same time, the SCQRPA correction
smears out the signature of a sharp SN phase transition even in heavy realistic nuclei such as 120Sn.
The dynamic coupling to SCQRPA vibrations causes the deviation of the quasipar- ticle occupation number from the Fermi-Dirac distribution for non-interacting fermions. However, for a realistic heavy nucleus such as 120Sn, this deviation is negligible. Con-
sequently, in these nuclei, the FTBCS1 and FTBCS1+SCQRPA predict similar results for the pairing gap and total energy. At the same time, for light systems, this deviation is stronger, therefore, the FTBCS1+SCQRPA offers a better approximation than the FTBCS1 in the study of thermal pairing properties of these nuclei.
The fact that the total energies and heat capacities obtained within the FTBCS1 (FTLN1) + SCQRPA predictions agree reasonably well with the exact results for
N = 10 as well as those obtained within the finite-temperature quantum Monte Carlo method for56Fe shows that the FTBCS1(FTLN1)+SCQRPA can be applied in further
study of thermal properties of finite systems such as nuclei, where pairing plays an im- portant role. The best agreement is see between the FTLN1+SCQRPA and the GCE results.
Compared to existing methods, the merit of the present approach lies in its fully microscopic derivation and simplicity when it is applied to heavy nuclei with strong pairing, where the effect of coupling to SCQRPA is negligible so that the solution of the SCQRPA can be avoided. In this case, thermal pairing can be determined solely by solving the FTBCS1 gap equation, which is technically as simple as the FTBCS one, whereas the exact diagonalization is impracticable (at T 6= 0).
Chapter 4
SCQRPA at finite temperature and
angular momentum
4.1
Pairing Hamiltonian for rotating system
For hot rotating nuclei, we consider the Hamiltonian in rotating frame, which describes a system ofN particles interacting via a pairing force with the parameterG, and rotat- ing about the symmetry axis (noncollective rotation) at an angular velocity (rotational frequency)γ with a fixed projectionM (orK) of the total angular momentum operator along this axis. For a spherically symmetric system, it is always possible to make the laboratory-framez axis, taken as the axis of quantization, coincide with the body-fixed one, which is aligned within the quantum mechanical uncertainty with the direction of the total angular momentum, so that the latter is completely determined by its z-axis projectionM alone. As for deformed systems, where the symmetry axis is the principal (body-fixed) axis, this noncollective motion is known as the “single-particle” rotation, which takes place when the angular momenta of individual nucleons are aligned parallel to the symmetry axis, resulting in an axially symmetric oblate shape rotating about this axis. Such noncollective motion is also possible in high-K isomers [6], which have many single-particle orbitals near the Fermi surface with a large and approximately conserved projection K of individual nucleonic angular momenta along the symmetry axis. Therefore, without losing generality, further derivations are carried out below for the pairing Hamiltonian of a spherical system rotating about the z axis [40, 50, 51],
namely
H=HP −λNˆ −γM ,ˆ (4.1)
where HP is the well-known pairing Hamiltonian
HP = X k ²k(Nk+N−k)−G X k,k0 Pk†Pk0 , N±k =a†±ka±k , Pk†=a†ka†−k , (4.2)
witha†±k (a±k) denoting the operator that creates (annihilates) a particle with angular
momentum k, spin projection±mk, and energy ²k. For simplicity, the subscriptsk are
used to label the single-particle states |k, mki in the deformed basis with the positive
single-particle spin projectionsmk, whereas the subscripts−k denote the time-reversal
states |k,−mki (mk > 0). The particle number operator ˆN and angular momentum
ˆ
M can be expressed in terms of a summation over the single-particle levels:
ˆ N =X k (Nk+N−k) , Mˆ = X k mk(Nk−N−k), (4.3)
whereas the chemical potentialλand angular velocityγ are two Lagrangian multipliers to be determined.
By using the Bogoliubov transformation the Hamiltonian (4.1) is transformed into the quasiparticle Hamiltonian in the rotating frame as
H =a+X k b+ kNk++ X −k b− kNk−+ X k ck(A†k+Ak) + X kk0 dkk0A† kAk0 +X kk0 gk(k0)(A†k0Nk+NkAk0) + X kk0 hkk0(A†kAk†0 +Ak0Ak) + X kk0 qkk0NkNk0 , (4.4) where N+ k =α † kαk , Nk− =α † −kα−k , Nk =Nk++Nk− , (4.5) A†k =α†kα†−k , Ak = (A†k)† . (4.6)
They obey the following commutation relations
[Ak , A†k0] =δkk0Dk , where Dk= 1− Nk , (4.7)
The coefficients b±
k in Eq. (4.4) are given as
b±k ≡bk∓γmk= (²k−λ)(u2k−vk2) + 2Gukvk
X
k0
uk0vk0 +Gv4k∓γmk , (4.9)
whereas the expressions for the other coefficientsa, bk,ck,dkk0,gk(k0),hkk0, andqkk0 in
Eqs. (4.4) and (4.9) are the same as those in Eqs. (1.8) – (1.12).