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Curvature of paramagnetic impurity levels in Orgel—Tanabe—Sugano diagrams
.Volume
1 April 1973
CHEMICAL PHYSICS LETTERS
19, Aumber 3
CURVATURE OF PARAMAGNETIC IN ORGEL-TANABE-SUGANO
IMPURITY
LEVELS
DIAGRAMS
K.N. SHRIVASTAVA Departmellt
of Physics, Centre for Post Graduate Studies, Himachal Pradesh Simla I?IOOI, India
University,
Received 2 January 19 73
There is a new contribution to the curvature of energy levels as a function of the strength of the crystal field. Thus the levels which are flat in the Orgel-Tannbe-Sugano disgrxms, in fact, have non-zero curvature and the accidential degeneracies
are removed.
The energy levels of pammagnetic
impurities
in
crystals are usually studied using the Coulomb interactions in terms of Racah integrals [ 1, 21. Very often, a single set of parameters is not able to describe all the energy levels. Thus in diagrams [3,4] of energy
levels versus the crystal field strength parameter, one does not always find a complete matching with experiment. One of the peculiarities of these diagrams is that some of the energy levels are exactly flat, for example, the level 3G(3T,) in d4 configuration and levels 4A1(4G), 4E(4G), 4E(4D), 2A,(2D), 4F(4A2) in the configuration d5. Approximately flat levels also occur, for example, *E and IT, in d2, 2T,, 2E in d3
and 3E, 3T,, ‘A,, IT, in d4 configuration. In general, most levels have a large curvature which may or may not agree with experiment. In the present paper, we would like to suggest that there is a temperature dependent cumature in the energy levels. This curvature bends all the levels by different amounts depending on the matrix elements ofa dynamic potential. Thus any accidental degeneracy can be removed. The curvature increases with increasing temperature. As the electron-phonon interaction tends to shield the crystal field the net crystal field is reduced. Let us take thespecific example of an ion in the electronic con@uration 3d5. The ground state is t&n % (L = 0). The interaction of interest is,
where M is the mass of the crystal, uk the phonon
frequency with wavevector Ii and ak and a;,’are the phonon operators. The interaction gives a small correction to the energy levels when L = 0 [5]. However, when L # 0, the correction may be very important. The first excited state of the 3d5 configuration is 4T1. The energy of this state can be changed by the interaction /-I provided that there are other spin-quartet states. The interaction H being electric in nature cannot change the spin within the first approximation. The higher states 4A, and 4E of the configuration 3d5
may be considered for our purpose. These states have energy given by 1oA - 25B + 5C in terms of Racah parameters. The energy of the state 4T, is lOA - 25B + 6C - IO&. Following the work of a previous paper [6], the energy shift of 4T1 is given by (“T,IH’14~1)(4~11fft4~1)
SE= EC4A,M4T,)
+
$1
H’l’E)(4EI
H14T,)
E(4E)-E(4Tl)
’
which can be written as, 6E=?\(TXloDq-q-l where
,
Volume
19, number
CHEMICAL
3
PHYSICS
LETTERS
1 April 1973
existing between 4A, and 4E in the configuration 3ds are removed; (c) the system is shielded by the electron-phonon interaction so that the crystal fieId is reduced. Indeed, the reduction in the crys*A field parameter has been measured elsewhere [5]. The result of the removal of accidental
degeneracies
also
follows from the diagonalization of the matrix given previously by the present author [7]. where p is the mass density of the crystal and u the velocity of sound. The slope of SE in the OrgelTanabe-Sugano diagram is then d(GE)/d(Dq)=
-10h
,
which is negative. The effect of the interaction considered is therefore to reduce the curvature of 4T,(3ds). The calculation of the curvature is of significant importance: (a) it demonstrates a curvature which varies as T4 at low temperatures and as Tat tigh temperatures; (b) as the curvature depends on the matrix elements, it is different for different levels, that is, the accidental degeneracies, such as those
References [ 1 ] J.S. Griffith, The theory of transition metaI ions (Cambridge Univ. Press, London, 1964). (2) D.S. McClure. in: Solid state physics, Vol. 9. edr F. Seitz and D. Tumbull (Academic Prcs. New York. 1959). [31 L.E. Orgel, J. Chem. Sot. (1952) 4756; I. Chem. Phys 23 (1955) 1819. 141 Y. Tanabc and S. Sugano, J. Phys. Sac. Japan 9 (1954) 753. 151 X.N. Shrivastava, Phys. Rev. 187 (1969) 446; Phys. Letters 31A (1970) 454. ISI K.N. Shrivastava, Chem. Phys. Letters 7 (1970) 477. 171 K.N. Shrivastava, J. Phys. C (Solid State Physj 2 (1969) 2428.
1 April 1973
CHEMICAL PHYSICS LETTERS
19, Aumber 3
CURVATURE OF PARAMAGNETIC IN ORGEL-TANABE-SUGANO
IMPURITY
LEVELS
DIAGRAMS
K.N. SHRIVASTAVA Departmellt
of Physics, Centre for Post Graduate Studies, Himachal Pradesh Simla I?IOOI, India
University,
Received 2 January 19 73
There is a new contribution to the curvature of energy levels as a function of the strength of the crystal field. Thus the levels which are flat in the Orgel-Tannbe-Sugano disgrxms, in fact, have non-zero curvature and the accidential degeneracies
are removed.
The energy levels of pammagnetic
impurities
in
crystals are usually studied using the Coulomb interactions in terms of Racah integrals [ 1, 21. Very often, a single set of parameters is not able to describe all the energy levels. Thus in diagrams [3,4] of energy
levels versus the crystal field strength parameter, one does not always find a complete matching with experiment. One of the peculiarities of these diagrams is that some of the energy levels are exactly flat, for example, the level 3G(3T,) in d4 configuration and levels 4A1(4G), 4E(4G), 4E(4D), 2A,(2D), 4F(4A2) in the configuration d5. Approximately flat levels also occur, for example, *E and IT, in d2, 2T,, 2E in d3
and 3E, 3T,, ‘A,, IT, in d4 configuration. In general, most levels have a large curvature which may or may not agree with experiment. In the present paper, we would like to suggest that there is a temperature dependent cumature in the energy levels. This curvature bends all the levels by different amounts depending on the matrix elements ofa dynamic potential. Thus any accidental degeneracy can be removed. The curvature increases with increasing temperature. As the electron-phonon interaction tends to shield the crystal field the net crystal field is reduced. Let us take thespecific example of an ion in the electronic con@uration 3d5. The ground state is t&n % (L = 0). The interaction of interest is,
where M is the mass of the crystal, uk the phonon
frequency with wavevector Ii and ak and a;,’are the phonon operators. The interaction gives a small correction to the energy levels when L = 0 [5]. However, when L # 0, the correction may be very important. The first excited state of the 3d5 configuration is 4T1. The energy of this state can be changed by the interaction /-I provided that there are other spin-quartet states. The interaction H being electric in nature cannot change the spin within the first approximation. The higher states 4A, and 4E of the configuration 3d5
may be considered for our purpose. These states have energy given by 1oA - 25B + 5C in terms of Racah parameters. The energy of the state 4T, is lOA - 25B + 6C - IO&. Following the work of a previous paper [6], the energy shift of 4T1 is given by (“T,IH’14~1)(4~11fft4~1)
SE= EC4A,M4T,)
+
$1
H’l’E)(4EI
H14T,)
E(4E)-E(4Tl)
’
which can be written as, 6E=?\(TXloDq-q-l where
,
Volume
19, number
CHEMICAL
3
PHYSICS
LETTERS
1 April 1973
existing between 4A, and 4E in the configuration 3ds are removed; (c) the system is shielded by the electron-phonon interaction so that the crystal fieId is reduced. Indeed, the reduction in the crys*A field parameter has been measured elsewhere [5]. The result of the removal of accidental
degeneracies
also
follows from the diagonalization of the matrix given previously by the present author [7]. where p is the mass density of the crystal and u the velocity of sound. The slope of SE in the OrgelTanabe-Sugano diagram is then d(GE)/d(Dq)=
-10h
,
which is negative. The effect of the interaction considered is therefore to reduce the curvature of 4T,(3ds). The calculation of the curvature is of significant importance: (a) it demonstrates a curvature which varies as T4 at low temperatures and as Tat tigh temperatures; (b) as the curvature depends on the matrix elements, it is different for different levels, that is, the accidental degeneracies, such as those
References [ 1 ] J.S. Griffith, The theory of transition metaI ions (Cambridge Univ. Press, London, 1964). (2) D.S. McClure. in: Solid state physics, Vol. 9. edr F. Seitz and D. Tumbull (Academic Prcs. New York. 1959). [31 L.E. Orgel, J. Chem. Sot. (1952) 4756; I. Chem. Phys 23 (1955) 1819. 141 Y. Tanabc and S. Sugano, J. Phys. Sac. Japan 9 (1954) 753. 151 X.N. Shrivastava, Phys. Rev. 187 (1969) 446; Phys. Letters 31A (1970) 454. ISI K.N. Shrivastava, Chem. Phys. Letters 7 (1970) 477. 171 K.N. Shrivastava, J. Phys. C (Solid State Physj 2 (1969) 2428.