Projected Shell Model in Nuclear Physics

• Projected Shell Model is a microscopic framework for nuclear structure that restores broken symmetries through angular-momentum projection on deformed quasiparticle states.
• The method employs configuration mixing and controlled truncation in a deformed quasiparticle basis to study rotational bands, chiral doublets, and weak-interaction processes.
• Advanced applications of the model include evaluating electromagnetic observables, neutrinoless double-beta decay matrix elements, and stellar weak-interaction rates.

Searching arXiv for recent and foundational Projected Shell Model papers to support the article. The Projected Shell Model (PSM) is a microscopic nuclear-structure framework in which many-body states are constructed from a deformed quasiparticle basis, symmetries broken at the mean-field level are restored by projection, and the resulting projected configurations are mixed by diagonalization of an effective Hamiltonian. In the literature summarized here, the PSM and its triaxial extension (TPSM) are used for rotational spectroscopy, $\gamma$- and $\gamma\gamma$-band structure, chiral doublets, negative-parity bands, stellar weak-interaction rates, first-forbidden $\beta$ decay, neutrinoless double-$\beta$ decay matrix elements, and electromagnetic properties of heavy deformed nuclei such as $^{229}$Th (Chen et al., 2018, Wang et al., 2021, Chen et al., 27 Aug 2025).

1. Formal structure of the model

The common PSM starting point is a deformed Nilsson single-particle basis with BCS pairing. The intrinsic states are quasiparticle configurations $|\Phi_\kappa\rangle$ built on a deformed quasiparticle vacuum, and rotational symmetry is restored by the three-dimensional angular-momentum projection operator

$$\hat P^I_{MK}=\frac{2I+1}{8\pi^2}\int d\Omega\, D^{I*}_{MK}(\Omega)\,\hat R(\Omega).$$

Projected configurations are then mixed to form laboratory-frame eigenstates with good total angular momentum:

$$|\Psi^I_M\rangle=\sum_{\kappa K} F_{\kappa K}^I \hat P^I_{MK}|\Phi_\kappa\rangle.$$

Equivalent forms of this ansatz are used throughout the axial and triaxial implementations, including odd-mass and odd-odd systems (Wang et al., 2021, Chen et al., 2018).

The shell-model Hamiltonian is typically of pairing-plus-quadrupole form,

$$\hat H = \hat H_0 - \frac{1}{2}\chi \sum_\mu \hat Q^\dagger_\mu \hat Q_\mu - G_M \hat P^\dagger \hat P - G_Q \sum_\mu \hat P^\dagger_\mu \hat P_\mu,$$

with $\hat H_0$ the spherical single-particle term and the residual interaction comprising quadrupole-quadrupole, monopole-pairing, and quadrupole-pairing terms. In heavy nuclei and octupole-deformed systems, the Hamiltonian has also been augmented to include $\gamma\gamma$0 to $\gamma\gamma$1 multipole channels,

$\gamma\gamma$2

which was used in the microscopic study of $\gamma\gamma$3Th (Chen et al., 27 Aug 2025).

Diagonalization in the nonorthogonal projected basis is carried out through a Hill–Wheeler-type generalized eigenvalue problem. The same machinery is used in axial PSM, TPSM, and number-projected variants. Some implementations enforce particle number only on average through BCS, whereas others employ explicit neutron and proton number projection before angular-momentum projection (Chen et al., 27 Aug 2025, Chen et al., 2018).

2. Configuration spaces and major extensions

A central feature of the PSM is controlled truncation in a deformed quasiparticle basis. For even-even nuclei, standard TPSM calculations use projected $\gamma\gamma$4-qp, $\gamma\gamma$5-qp, and $\gamma\gamma$6-qp states,

$\gamma\gamma$7

with the $\gamma\gamma$8 projections from the triaxial vacuum generating the ground, $\gamma\gamma$9, and $\beta$0 bands, respectively [(Jehangir et al., 2020); (Bhat et al., 2014)].

Odd-mass extensions replace the $\beta$1-qp vacuum bandhead by one-quasiparticle configurations and include higher broken-pair structures at high spin. For $\beta$2Nb, the odd-proton TPSM basis was extended to

$\beta$3

while in odd-neutron Xe isotopes the basis was enlarged further to include one-quasineutron, one-neutron-plus-two-proton, three-quasineutron, and five-quasiparticle configurations, enabling description up to and including the second band crossing [(Sheikh et al., 2010); (Jehangir et al., 2022)].

Odd-odd implementations use proton-neutron two-quasiparticle configurations,

$\beta$4

which form the basis for chiral doublet studies in $\beta$5Cs and $\beta$6Cs, as well as the multi-configuration PSM treatment of $\beta$7Cs [(Bhat et al., 2011); (Bhat et al., 2013); (Chen et al., 2018)].

Several formal enlargements have broadened the scope of the method. A negative-parity TPSM for even-even nuclei allowed quasiparticles to occupy two major oscillator shells of opposite parity, thereby making negative-parity rotational bands accessible microscopically (Nazir et al., 2023). In stellar weak-interaction work, one extended PSM included up to six-quasiparticle configurations for rates in $\beta$8-process waiting-point nuclei (Chen et al., 2023), while an electron-capture study of $\beta$9Nb employed explicit multi-quasiparticle configurations up to seven quasiparticles (Wang et al., 2021). In heavy odd-mass $\beta$0Th, the configuration space was truncated at the 3-quasiparticle level, with the authors stating that 5-qp contributions are small (Chen et al., 27 Aug 2025).

3. Collective spectroscopy, $\beta$1 bands, and high-spin structure

In the TPSM, $\beta$2-band structure emerges from projection rather than from a phonon degree of freedom introduced by hand. For even-even nuclei, the staggering quantity

$\beta$3

is used as a diagnostic of the $\beta$4-band phase. A systematic study of twenty-three nuclei found that most exhibit even-$\beta$5-below-odd-$\beta$6 staggering after quasiparticle configuration mixing, while only $\beta$7Ge, $\beta$8Ru, $\beta$9Er, and $^{229}$0Th retain odd-$^{229}$1-below-even-$^{229}$2 staggering (Jehangir et al., 2020).

The Ge/Se region provides a particularly explicit TPSM test of rigid triaxiality against $^{229}$3-softness. For $^{229}$4Ge, the best description of the yrast and $^{229}$5-vibrational bands required $^{229}$6 and $^{229}$7, corresponding to $^{229}$8, and the calculation predicted a $^{229}$9 band around $|\Phi_\kappa\rangle$0 MeV. In neighboring Ge and Se isotopes, the opposite staggering phase emerged after configuration mixing, and the paper argued that the resulting $|\Phi_\kappa\rangle$1-soft feature comes from mixing of the ground-state configuration with multi-quasiparticle states (Bhat et al., 2014).

In odd-mass nuclei, the same projected-$|\Phi_\kappa\rangle$2 logic yields multi-$|\Phi_\kappa\rangle$3 bands built on a quasiparticle configuration. In $|\Phi_\kappa\rangle$4Nb, the ground, one-$|\Phi_\kappa\rangle$5, and two-$|\Phi_\kappa\rangle$6 bands were identified primarily with the $|\Phi_\kappa\rangle$7, $|\Phi_\kappa\rangle$8, and $|\Phi_\kappa\rangle$9 projections of the same lowest one-quasiparticle configuration. The calculated bandhead ratio $$\hat P^I_{MK}=\frac{2I+1}{8\pi^2}\int d\Omega\, D^{I*}_{MK}(\Omega)\,\hat R(\Omega).$$0, compared with the experimental value $$\hat P^I_{MK}=\frac{2I+1}{8\pi^2}\int d\Omega\, D^{I*}_{MK}(\Omega)\,\hat R(\Omega).$$1, supported the assignment of simultaneous one- and two-$$\hat P^I_{MK}=\frac{2I+1}{8\pi^2}\int d\Omega\, D^{I*}_{MK}(\Omega)\,\hat R(\Omega).$$2-phonon bands while also indicating residual anharmonicity (Sheikh et al., 2010).

High-spin rotational behavior is likewise treated through band diagrams and configuration mixing. In the superdeformed band of $$\hat P^I_{MK}=\frac{2I+1}{8\pi^2}\int d\Omega\, D^{I*}_{MK}(\Omega)\,\hat R(\Omega).$$3Ar, an axial PSM with $$\hat P^I_{MK}=\frac{2I+1}{8\pi^2}\int d\Omega\, D^{I*}_{MK}(\Omega)\,\hat R(\Omega).$$4-, $$\hat P^I_{MK}=\frac{2I+1}{8\pi^2}\int d\Omega\, D^{I*}_{MK}(\Omega)\,\hat R(\Omega).$$5-, and $$\hat P^I_{MK}=\frac{2I+1}{8\pi^2}\int d\Omega\, D^{I*}_{MK}(\Omega)\,\hat R(\Omega).$$6-qp configurations reproduced the nearly linear $$\hat P^I_{MK}=\frac{2I+1}{8\pi^2}\int d\Omega\, D^{I*}_{MK}(\Omega)\,\hat R(\Omega).$$7-ray energy sequence and showed that the high-spin yrast structure is dominated by mixed $$\hat P^I_{MK}=\frac{2I+1}{8\pi^2}\int d\Omega\, D^{I*}_{MK}(\Omega)\,\hat R(\Omega).$$8-, $$\hat P^I_{MK}=\frac{2I+1}{8\pi^2}\int d\Omega\, D^{I*}_{MK}(\Omega)\,\hat R(\Omega).$$9-, and $$|\Psi^I_M\rangle=\sum_{\kappa K} F_{\kappa K}^I \hat P^I_{MK}|\Phi_\kappa\rangle.$$0-qp configurations. The triaxial deformation extracted from the calculated observables was small, roughly $$|\Psi^I_M\rangle=\sum_{\kappa K} F_{\kappa K}^I \hat P^I_{MK}|\Phi_\kappa\rangle.$$1–$$|\Psi^I_M\rangle=\sum_{\kappa K} F_{\kappa K}^I \hat P^I_{MK}|\Phi_\kappa\rangle.$$2, leading to the conclusion that triaxiality is not very important for this superdeformed band (Yang et al., 2015).

A recurring result in comparative studies is that energies alone do not exhaust the model discrimination. When a collective Bohr Hamiltonian was fitted to TPSM energies for $$|\Psi^I_M\rangle=\sum_{\kappa K} F_{\kappa K}^I \hat P^I_{MK}|\Phi_\kappa\rangle.$$3-bands, the staggering pattern could be reproduced reasonably well, but interband $$|\Psi^I_M\rangle=\sum_{\kappa K} F_{\kappa K}^I \hat P^I_{MK}|\Phi_\kappa\rangle.$$4 values differed strongly. This suggests that transition strengths, not only level energies, are required to distinguish fixed-triaxial configuration mixing from collective $$|\Psi^I_M\rangle=\sum_{\kappa K} F_{\kappa K}^I \hat P^I_{MK}|\Phi_\kappa\rangle.$$5-soft dynamics (Jehangir et al., 2020).

4. Chiral rotation and triaxial angular-momentum geometry

The PSM and TPSM provide a microscopic description of nuclear chirality by projecting good angular momentum from triaxial proton-neutron quasiparticle configurations. In $$|\Psi^I_M\rangle=\sum_{\kappa K} F_{\kappa K}^I \hat P^I_{MK}|\Phi_\kappa\rangle.$$6Cs, a TPSM calculation with $$|\Psi^I_M\rangle=\sum_{\kappa K} F_{\kappa K}^I \hat P^I_{MK}|\Phi_\kappa\rangle.$$7 and $$|\Psi^I_M\rangle=\sum_{\kappa K} F_{\kappa K}^I \hat P^I_{MK}|\Phi_\kappa\rangle.$$8, corresponding to $$|\Psi^I_M\rangle=\sum_{\kappa K} F_{\kappa K}^I \hat P^I_{MK}|\Phi_\kappa\rangle.$$9, reproduced the observed energies and electromagnetic transition probabilities of the nearly degenerate chiral dipole bands. The splitting between the two bands was shown to depend strongly on $$\hat H = \hat H_0 - \frac{1}{2}\chi \sum_\mu \hat Q^\dagger_\mu \hat Q_\mu - G_M \hat P^\dagger \hat P - G_Q \sum_\mu \hat P^\dagger_\mu \hat P_\mu,$$0, becoming small for $$\hat H = \hat H_0 - \frac{1}{2}\chi \sum_\mu \hat Q^\dagger_\mu \hat Q_\mu - G_M \hat P^\dagger \hat P - G_Q \sum_\mu \hat P^\dagger_\mu \hat P_\mu,$$1, which confirmed that triaxiality is essential for chirality in this case (Bhat et al., 2011).

The odd-odd Cs isotopes $$\hat H = \hat H_0 - \frac{1}{2}\chi \sum_\mu \hat Q^\dagger_\mu \hat Q_\mu - G_M \hat P^\dagger \hat P - G_Q \sum_\mu \hat P^\dagger_\mu \hat P_\mu,$$2Cs were studied with a multi-quasiparticle TPSM in terms of energies, staggering, $$\hat H = \hat H_0 - \frac{1}{2}\chi \sum_\mu \hat Q^\dagger_\mu \hat Q_\mu - G_M \hat P^\dagger \hat P - G_Q \sum_\mu \hat P^\dagger_\mu \hat P_\mu,$$3, $$\hat H = \hat H_0 - \frac{1}{2}\chi \sum_\mu \hat Q^\dagger_\mu \hat Q_\mu - G_M \hat P^\dagger \hat P - G_Q \sum_\mu \hat P^\dagger_\mu \hat P_\mu,$$4, and $$\hat H = \hat H_0 - \frac{1}{2}\chi \sum_\mu \hat Q^\dagger_\mu \hat Q_\mu - G_M \hat P^\dagger \hat P - G_Q \sum_\mu \hat P^\dagger_\mu \hat P_\mu,$$5 ratios. An explicit methodological point in that work is that near degeneracy alone is not sufficient to establish chirality; the electromagnetic transition patterns are also required. For $$\hat H = \hat H_0 - \frac{1}{2}\chi \sum_\mu \hat Q^\dagger_\mu \hat Q_\mu - G_M \hat P^\dagger \hat P - G_Q \sum_\mu \hat P^\dagger_\mu \hat P_\mu,$$6Cs in particular, the agreement with extensive transition-probability data was taken to support the chiral interpretation, while $$\hat H = \hat H_0 - \frac{1}{2}\chi \sum_\mu \hat Q^\dagger_\mu \hat Q_\mu - G_M \hat P^\dagger \hat P - G_Q \sum_\mu \hat P^\dagger_\mu \hat P_\mu,$$7Cs were used to generate predictions for future tests (Bhat et al., 2013).

The most detailed intrinsic-geometry analysis in the material summarized here is the PSM study of $$\hat H = \hat H_0 - \frac{1}{2}\chi \sum_\mu \hat Q^\dagger_\mu \hat Q_\mu - G_M \hat P^\dagger \hat P - G_Q \sum_\mu \hat P^\dagger_\mu \hat P_\mu,$$8Cs with configuration mixing. In that work, chiral geometry was analyzed through the $$\hat H = \hat H_0 - \frac{1}{2}\chi \sum_\mu \hat Q^\dagger_\mu \hat Q_\mu - G_M \hat P^\dagger \hat P - G_Q \sum_\mu \hat P^\dagger_\mu \hat P_\mu,$$9 plot, which displays distributions of angular-momentum components along the intrinsic axes, and the azimuthal plot, which gives distributions of orientation angles $\hat H_0$0 of the total angular momentum. The resulting picture evolved with spin from chiral vibration at $\hat H_0$1, to static chirality at $\hat H_0$2, and then to principal-axis rotation at $\hat H_0$3. The authors concluded that the chiral geometry is stable against configuration mixing (Chen et al., 2018).

These studies jointly establish that, within the projected-shell-model framework, chiral doublets are not inferred from energy quasi-degeneracy alone. Rather, they are identified through a conjunction of projected triaxial geometry, band mixing, and characteristic $\hat H_0$4 and $\hat H_0$5 systematics [(Bhat et al., 2011); (Bhat et al., 2013); (Chen et al., 2018)].

5. Weak interactions, $\hat H_0$6 decay, and stellar rates

The PSM has been extended beyond spectroscopy to weak-interaction observables in thermally populated nuclei. For stellar electron capture, the rate was written as

$\hat H_0$7

with the partition function

$\hat H_0$8

In the $\hat H_0$9Nb $\gamma\gamma$00 $\gamma\gamma$01Zr example, the PSM was used precisely because excited configurations are explicitly included as multi-quasiparticle states. That study concluded that the Brink-Axel approximation is not reliable for odd-mass nuclei such as $\gamma\gamma$02Nb, and that at $\gamma\gamma$03 the inclusion of excited parent states can increase the electron-capture rate by more than an order of magnitude (Wang et al., 2021).

A broader stellar weak-interaction application proposed a PSM for even-even and odd-odd nuclei with an extended configuration space including up to six-quasiparticle configurations, and calculated stellar weak-interaction rates for eight $\gamma\gamma$04-process waiting-point nuclei: $\gamma\gamma$05Ge, $\gamma\gamma$06Se, $\gamma\gamma$07Kr, $\gamma\gamma$08Sr, $\gamma\gamma$09Zr, $\gamma\gamma$10Mo, $\gamma\gamma$11Ru, and $\gamma\gamma$12Pd. In that work, higher-order quasiparticle configurations were found to affect the underlying Gamow-Teller strength distributions and the corresponding stellar weak-interaction rates. Under $\gamma\gamma$13-process environments with high temperatures and densities, thermal population of excited parent states tends to decrease the stellar $\gamma\gamma$14 decay rates, whereas electron capture tends to contribute increasingly with temperature and density. The effective half-lives under peak $\gamma\gamma$15-process conditions were predicted to be reduced relative to the terrestrial case, especially for $\gamma\gamma$16Ge and $\gamma\gamma$17Se (Chen et al., 2023).

The operator content has also been expanded. A PSM development for first-forbidden nuclear $\gamma\gamma$18 decay was presented as a first application of the model to that transition class, with 35 dominant first-forbidden transitions calculated and compared systematically with data. The corresponding experimental Log$\gamma\gamma$19 values were described reasonably, and quenching factors of nuclear matrix elements were reported to affect both the Log$\gamma\gamma$20 values and the related shape factors (Wang et al., 2023).

Within this body of work, the PSM serves not merely as a spectroscopic model but as a state-by-state framework for weak processes in nuclei where thermal population, deformation, and explicit excited configurations are all structurally important (Wang et al., 2021, Chen et al., 2023).

6. Double-$\gamma\gamma$21 decay, electromagnetic observables, and computational techniques

The triaxial projected shell model has also been used for neutrinoless double-$\gamma\gamma$22 decay. In that application, mother and daughter states were built from triaxially deformed quasiparticle vacua with angular-momentum projection and quasiparticle configuration mixing, and odd-odd intermediate nuclei were treated explicitly as two-quasiparticle proton-neutron configurations. The decay matrix element was decomposed as

$\gamma\gamma$23

and the numerical analysis found that the total NME is dominated by the Gamow–Teller channel at about $\gamma\gamma$24, with the Fermi term contributing about $\gamma\gamma$25 and the tensor term about $\gamma\gamma$26. Configuration mixing reduced the NME by $\gamma\gamma$27–$\gamma\gamma$28 depending on nucleus, explicit odd-odd intermediate states enhanced it by $\gamma\gamma$29–$\gamma\gamma$30, and varying the triaxial deformation $\gamma\gamma$31 from $\gamma\gamma$32 to $\gamma\gamma$33 changed the NME by $\gamma\gamma$34 to $\gamma\gamma$35, depending on the decay candidate (Wang et al., 2021).

Electromagnetic observables are evaluated directly from projected wavefunctions. Standard reduced transition probabilities,

$\gamma\gamma$36

and

$\gamma\gamma$37

are used repeatedly to diagnose collectivity, alignment, and band mixing. In $\gamma\gamma$38Th, a modern PSM treatment with quadrupole, octupole, and hexadecapole deformation reproduced the low-energy spectrum and obtained

$\gamma\gamma$39

which was stated to agree well with the recently measured radiative lifetime of $\gamma\gamma$40Th. The same calculation found that the $\gamma\gamma$41 ground state is dominated by the $\gamma\gamma$42 one-quasiparticle component at about $\gamma\gamma$43, while the $\gamma\gamma$44 isomer is dominated by the $\gamma\gamma$45 component at about $\gamma\gamma$46, thereby explaining large in-band $\gamma\gamma$47 strengths and weak cross-band $\gamma\gamma$48 transitions in terms of small wave-function overlap (Chen et al., 27 Aug 2025).

A significant computational ingredient in modern implementations is the Pfaffian algorithm. It is used to compute rotational overlaps in the TPSM treatment of neutrinoless double-$\gamma\gamma$49 decay and to evaluate reduced one-body transition densities in the $\gamma\gamma$50Th calculation (Wang et al., 2021, Chen et al., 27 Aug 2025). This suggests that efficient overlap technology is integral to extending the projected-shell-model program to odd-mass systems, large configuration spaces, and transition operators beyond standard spectroscopy.

The literature also records explicit limitations. In $\gamma\gamma$51Nb, the remaining discrepancy in the two-$\gamma\gamma$52 band was attributed to $\gamma\gamma$53-softness and to the use of a single fixed $\gamma\gamma$54, with projection over multiple $\gamma\gamma$55 values and generator-coordinate mixing proposed as a better treatment (Sheikh et al., 2010). In $\gamma\gamma$56Cs, one observed band was argued to lie beyond the present model space because it likely has larger deformation (Chen et al., 2018). Such cases indicate that PSM accuracy depends not only on projection and configuration mixing, but also on the adequacy of the chosen intrinsic manifold.