Single-particle state

Sj Uj, and if the yth single-particle state has energy then the energy of the system in the state v is =... 

Wlren the single-particle states j are densely packed within any energy interval of k T, the sum over j can be replaced by an integral over energy such that... 

This is the classical Boltzmaim distribution m which (Uj)/(A, tire probability of finding a particle in the single-particle state j, is proportional to the classical Boltzmaim factor... 

In an ideal Bose gas, at a certain transition temperature a remarkable effect occurs a macroscopic fraction of the total number of particles condenses into the lowest-energy single-particle state. This effect, which occurs when the Bose particles have non-zero mass, is called Bose-Einstein condensation, and the key to its understanding is the chemical potential. For an ideal gas of photons or phonons, which have zero mass, this effect does not occur. This is because their total number is arbitrary and the chemical potential is effectively zero for tire photon or phonon gas. 

The continuum model with the Hamiltonian equal to the sum of Eq. (3.10) and Eq. (3.12), describing the interaction of electrons close to the Fermi surface with the optical phonons, is called the Takayama-Lin-Liu-Maki (TLM) model [5, 6], The Hamiltonian of the continuum model retains the important symmetries of the discrete Hamiltonian Eq. (3.2). In particular, the spectrum of the single-particle states of the TLM model is a symmetric function of energy. 

From equation (8.49b), we see that the wavefunction vanishes for two identical fermions in the same single-particle state... 

The number of single-particle states with energies between E and E + AE is a)(E) AE, where co(E) is the density of single-particle states and is related to... 

Here, the factor 2 accounts for spin degeneracy, and the sum is over the occupied single-particle states. In terms of y/i, the energy functional given by Eq. (6) can be written as ... 

For a two-electron system in 2m-dimensional spin-space orbital, with and denoting the fermionic annihilation and creation operators of single-particle states and 0) representing the vacuum state, a pure two-electron state ) can be written [57]... 

SAP produces a set of virtual excitations from the fully occupied to the unoccupied Kohn-Sham orbitals thus producing a fictitious statistical ensemble. A thermodynamic interpretation of SAP is presented in [50] (see [51,52] as well), where two main observations are given. First, the redistribution of single particle states in the smoothing procedure leads to... 

Figure 6.4 Energy spectrum of emitted protons from the lsO(p, 2p) reaction, showing the single-particle states. [From Tyren et al. (1958).]...

A PWC function is defined as an (N — l)-electron bound parent state (atom or ion) with well-defined spin, parity, angular momentum and energy Ia = (Sa, na, La, Ea), coupled first to the spin-angular part of a single-particle state for the Nth electron, with definite orbital angular momentum la, to form a state with definite parity n, spin S, angular momentum L, and their projections L and M (for brevity, we indicate these global quantum numbers with the collective index T)... 

A clear gap in the study of this region was a lade of information on single-particle states in deformed nuclei with N > 60. Only recently, since we began our first Y study with "r, has such information emerged. The status through the end of 1984 was recently summarized by [PETT85] ... 

As in the 208Pb region high spin single particle states are available so that simple configurations are likely to contribute to the structure of high spin states. In particular, near 6Gd the proton hn/2 orbital and the neutron 13/2 orbital should play an important role, however, there is little direct evidence about the states based on these orbitals For the Nd isotopes considered here the protons have partially filled the g7/2 d5/2 levels and the neutrons are beginning to fill orbitals above N-82. 

Shell model calculations predict a quasi-shell closure at 96Zr. Therefore, it is of interest to measure g-factors of states in 97Zr and test whether they can be described by simple shell model configurations. The 1264.4 keV level has a half-life of 102 nsec, and its g-factor was measured by the time-differential PAC method at TRISTAN [BER85a]. The result, g-0.39(4), is consistent with the Schmidt value of 0.43, which assumes no core polarization and the free value for the neutron g factor, g g free. This indicates that the 1264.4 keV level is a very pure single-particle state, thus confirming the shell model prediction of a quasi-shell closure at 96Zr. 

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