Hidden G-symmetry in one particle and many particle spin-Hamiltonians

Hidden symmetry 

For more than 50 years radiospectroscopic investigations demonstrate both the efficiency and the fruitfulness of phenomenological spin-Hamiltonian (SH) conception. Nevertheless, many of the SH or generalized SH that were written earlier for non-cubic centers are not practically suitable for the description of EPR, NMR and ENDOR spectra. The simplified SH has often a non-complete 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 C1 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 N-n 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 - G-symmetry) 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 SkB (k=1,3,5,7) have to be completely eliminated from one-particle SH. Correct expressions for SkB interactions are especially important for the interpretation of spectra at high magnetic fields, since they are proportional to B. The neglecting SkB terms leads to large discrepancy between calculated and observed resonance fields (hundreds of Gausses for chromium pairs in CsMgCl3 in X-band and much larger in high magnetic fields). SH for a cluster of three particles with the spins S1=S2=S3=1/2 and with isotropic exchange interactions is another bright example. This SH has usually three terms H=a(S1S2)+b(S1S3)+c(S2S3). Due to hidden symmetry to G-transformation with U=exp(ij(S1S2S3)) only two linear combinations  a+b+c  and  2a-b-c  can be found from the comparison of calculated and measured splittings or resonance magnetic fields.

Examples of unsuitable spin-Hamiltonians



Symmetry  Cn (n>2), S=I=1/2

 H = bBgS - bnBgnI + SAI,

                                        g, gn, A



Inseparable parameter combinations,

i.e. superfluous parameters

Symmetry  C1

 H = bBgS - bnBgnI + SAI,

                                        g, gn, A 



Inseparable parameter combinations

Isotropic exchange or hyperfine interactions

of three particles with  S1 = S2 = S3 = 1/2

H = a(S1S2) + b(S1S3) + c(S2S3)


Inseparable parameter combination

Symmetry  Cn (n>2), S > 1

H = b20O20 + b[g^(BxSx+BySy) + g||BzSz]



(overreduced) basis


Reduction procedure


Correct reduction procedure was developed in:       

V.G.Grachev, G-symmetry and theory of the ENDOR frequencies for centers with low-symmetry interactions.- In: Radiospectroscopy of Solid State, Kiev: Naukova Dumka, 1993, p. 16-66.

V.G.Grachev. Correct expression for generalized spin-Hamiltonian of non-cubic paramagnetic center.- JETF, 1987, v. 65(5), 1029-1035 (v. 92, No 5, 1834-1844).


Strong difference between angular dependencies calculated with and without BS3 terms in spin-Hamiltonian.

 The difference increases drastically for high frequency / high magnetic field EPR. Dots - experimental lines.


 1.    Many spin-Hamiltonians have implicit (hidden) additional symmetry - G-symmetry.

2.    Taking into account of this symmetry allows to reduce the number of independent parameters in spin-Hamiltonian (like point group symmetry transformations) and to obtain correct maximally reduced spin-Hamiltonian (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 spin-Hamiltonians (discarding S2s3, for instance) can lead to unreliability of the parameters determined and non-accurate 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.


Copyright V.Grachev. All rights reserved.
Revised: June 01, 2013.