Analytic solution of fourth degree secular equations: Zeeman-quadrupole interactions and pure quadrupole interaction

Creel, R. B.

doi:10.1016/0022-2364(83)90179-8

Cited by 7 publications

(8 citation statements)

References 8 publications

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“…Higher-order corrections to the wavefunctions have been published [14][15][16][17][18][19], as have a number of exact solutions [10,[20][21][22][23][24][25]. In practice, we find that second-order perturbation theory works remarkably well.…”

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confidence: 85%

A simple proof that third-order quadrupole perturbations of the NMR central transition of half-integral spin nuclei are zero

Bain

2006

Journal of Magnetic Resonance

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confidence: 85%

A simple proof that third-order quadrupole perturbations of the NMR central transition of half-integral spin nuclei are zero

Bain

2006

Journal of Magnetic Resonance

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“…For spins I = 2 − 9 the energy relations as functions of η are illustrated in Figs. 3,4,5,6,7,8,9,10, respectively. For each Figure, the energy states are labeled using the m-state notation.…”

Section: Appendixmentioning

confidence: 99%

“…Several sources [2] 1 [3] 2 [4] - [5] 3 [6] 4 and theses include all cases up to quadratics and some cubics. Furthermore, except for the spin 7/2 case, the other quartic solutions are absent from the literature.…”

Section: Introductionmentioning

confidence: 99%

All 36 exactly solvable solutions of eigenvalues for nuclear electric quadrupole interaction Hamiltonian and equivalent rigid asymmetric rotor with expanded characteristic equation listing

Menke

2012

Hyperfine Interact

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This paper derives all 36 analytical solutions of the energy eigenvalues for nuclear electric quadrupole interaction Hamiltonian and equivalent rigid asymmetric rotor for polynomial degrees 1 through 4 using classical algebraic theory. By the use of double-parameterization the full general solution sets are illustrated in a compact, symmetric, structural, and usable form that is valid for asymmetry parameter η ∈ (−∞, +∞). These results are useful for code developers in the area of Perturbed Angular Correlation (PAC), Nuclear Quadrupole Resonance (NQR) and rotational spectroscopy who want to offer exact solutions whenever possible, rather that resorting to numerical solutions. In addition, by using standard linear algebra methods, the characteristic equations of all integer and half-integer spins I from 0 to 15, inclusive are represented in a compact and naturally parameterized form that illustrates structure and symmetries. This extends Nielson's [1] listing of characteristic equations for integer spins out to I = 15, inclusive.

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“…[11] and [12] are not needed, only their superspin and its z component. For this reason, we deliberately do not use the tensor notation (52-54) of T 31 and T 20 for [11] and [12], respectively. We denote them as kets, just with the quantum numbers ͉3 1 ) and ͉2 0 ).…”

Section: The Full Solutionmentioning

confidence: 99%

From NQR to NMR: The complete range of quadrupole interactions

Bain

Khasawneh

2004

Concepts Magnetic Resonance

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ABSTRACT:The theoretical transition energies of a spin 3/2 under a combination of quadrupolar and Zeeman interactions are mapped out as a function of magnetic field. This allows us to follow the spectroscopy continuously from zero magnetic field (NQR, nuclear quadrupole resonance) through to high magnetic field (quadrupole-perturbed NMR). This calculation is made possible by a theoretical approach that makes maximum use of the angular momentum properties of the spin system. In this approach, the detailed form of the spin operators are not required, only their angular momentum quantum numbers. No commutators need be evaluated, and the method is easily coded for computer calculation. Application of the Wigner-Eckart theorem and selection rules means that there are only two nonzero reduced matrix elements that are needed for a spin 3/2, and there are explicit formulae for them. In looking at the transition energies, the crucial parameter is the orientation of the quadrupole tensor to the magnetic field. If the angle is nonzero, then the quantization of the spins must shift. At zero field, the quantization is defined by the electric field gradient tensor, which is molecule-fixed at some arbitrary angle to the magnetic field. This direction of quantization must change as the magnetic field increases, to lie close to the z axis at high field. In the region where the two interactions are comparable, all the familiar rules break down and the response is highly nonlinear.

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Analytic solution of fourth degree secular equations: Zeeman-quadrupole interactions and pure quadrupole interaction

Cited by 7 publications

References 8 publications

A simple proof that third-order quadrupole perturbations of the NMR central transition of half-integral spin nuclei are zero

A simple proof that third-order quadrupole perturbations of the NMR central transition of half-integral spin nuclei are zero

All 36 exactly solvable solutions of eigenvalues for nuclear electric quadrupole interaction Hamiltonian and equivalent rigid asymmetric rotor with expanded characteristic equation listing

From NQR to NMR: The complete range of quadrupole interactions

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