Hidden Gsymmetry in one particle and many particle spinHamiltonians
For more than 50 years radiospectroscopic investigations demonstrate both the efficiency and the fruitfulness of phenomenological spinHamiltonian (SH) conception. Nevertheless, many of the SH or generalized SH that were written earlier for noncubic centers are not practically suitable for the description of EPR, NMR and ENDOR spectra. The simplified SH has often a noncomplete basis and does not guarantee a reliability of determined parameters and an accuracy of experiment description, whereas the generalized SH contains implicitly inseparable parameter combinations, which number is less than the number of SH parameters. The simplest and obvious example of SH with inseparable parameters is H=bBgS for C_{1} symmetry (S  spin, B  magnetic field): g tensor has 9 components, however, it is well known that only 6 combinations (gg tensor) can be found from experiment. The unique determination of all coupled parameters of such SH (by hand or with the help of a computer) is impossible, since this task has Nn equations for N unknown parameters.
The critical analysis of methods of SH reduction allowed us to find the reason of the appearance of these inseparable combinations (the symmetry relative to particular gauge transformations  Gsymmetry) and the way to build the correct SH (gauge fixing for the elimination of superfluous operators and parameters). It was shown that special parts of Zeeman interaction S^{k}B (k=1,3,5,7) have to be completely eliminated from oneparticle SH. Correct expressions for S^{k}B interactions are especially important for the interpretation of spectra at high magnetic fields, since they are proportional to B. The neglecting S^{k}B terms leads to large discrepancy between calculated and observed resonance fields (hundreds of Gausses for chromium pairs in CsMgCl_{3} in Xband and much larger in high magnetic fields). SH for a cluster of three particles with the spins S_{1}=S_{2}=S_{3}=1/2 and with isotropic exchange interactions is another bright example. This SH has usually three terms H=a(S_{1}S_{2})+b(S_{1}S_{3})+c(S_{2}S_{3}). Due to hidden symmetry to Gtransformation with U=exp(ij(S_{1}S_{2}S_{3})) only two linear combinations a+b+c and 2abc can be found from the comparison of calculated and measured splittings or resonance magnetic fields.
Examples of unsuitable spinHamiltonians
SpinHamiltonian 
Shortcoming 
Symmetry C_{n }(n>2), S=I=1/2 H = bBgS  b_{n}Bg_{n}I + SAI, g, g_{n}, A Î

Inseparable parameter combinations, i.e. superfluous parameters 
Symmetry C_{1} H = bBgS  b_{n}Bg_{n}I + SAI, g, g_{n}, A Î

Inseparable parameter combinations 
Isotropic exchange or hyperfine interactions of three particles with S_{1} = S_{2} = S_{3} = 1/2 H = a(S_{1}S_{2}) + b(S_{1}S_{3}) + c(S_{2}S_{3}) 
Inseparable parameter combination 
Symmetry C_{n} (n>2), S > 1 H = b_{2}^{0}O_{2}^{0} + b[g_{^}(B_{x}S_{x}+B_{y}S_{y}) + g_{}B_{z}S_{z}] 
Noncomplete (overreduced) basis 
Reduction procedure
Correct reduction procedure was developed in:
V.G.Grachev, Gsymmetry and theory of the ENDOR frequencies for centers with lowsymmetry interactions. In: Radiospectroscopy of Solid State, Kiev: Naukova Dumka, 1993, p. 1666.
V.G.Grachev. Correct expression for generalized spinHamiltonian of noncubic paramagnetic center. JETF, 1987, v. 65(5), 10291035 (v. 92, No 5, 18341844).
Strong difference between angular dependencies calculated with and without BS^{3} terms in spinHamiltonian.
The difference increases drastically for high frequency / high magnetic field EPR. Dots  experimental lines.
1. Many spinHamiltonians have implicit (hidden) additional symmetry  Gsymmetry.
2. Taking into account of this symmetry allows to reduce the number of independent parameters in spinHamiltonian (like point group symmetry transformations) and to obtain correct maximally reduced spinHamiltonian (MRSH).
3. There is a family of equivalent MRSH. The members of the family are differ by the choice of gauge fixing only. Therefore, before comparison of theory and experiments, it is necessary to have an agreement about gauge fixing.
4. Unjustified reduction of spinHamiltonians (discarding S_{2}s_{3}, for instance) can lead to unreliability of the parameters determined and nonaccurate description of experiments.
5. In computer programs (and in analytical calculation also) the MRSH must be used only. Obtained exact solution for three particle cluster gives good check point for computer programs.
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Copyright © V.Grachev. All rights reserved.
Revised: June 01, 2013.