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Spin-lattice relaxation mechanism for S-state ions
PHYSICS
Volume 20, number 5
SPIN-
LATTICE
RELAXATION
15 March 1966
LETTERS
MECHANISM
FOR
S-STATE
IONS
T. SHIMIZU Department
of Electronic
Engineering,
Kanazawa
University,
Kanazawa,
Japan
Received 8 February 1966 A spin-lattice relaxation mechanism is proposed for S-state ions having a tetrahedral environmentand a positive g-shift. The relaxation stems from the Van Vleck mechanism using the excited sextet state considering the covalent bonding. Thus far two mechanisms have been proposed for spin-lattice relaxation of S-state ions by Blume and Orbach [l] and Leushin [2]. Since there is no sextet excited state in the d5 configuration, the relaxation mechanism is rather complicated. In this letter we propose a relaxation mechanism which seems to be dominant for Sstate ions having a positiveg-shift and a tetrahedral environment. In order to explain a positive g-shift of S-state ions, Fidone and Stevens [3] and Watanabe [4] uses the excited sextet state which appears as a result of the covalent bonding. We use this sextet state as an excited state in the Van Vleck relaxation mechanism for the direct process [5]. For the tetrahedral environment, that is, Td symmetry, spin-orbit interaction combines 8T1-states with the ground 8AI-state. Therefore the Van Vleck mechanism is operative if the orbit-lattice interaction combines GTI-states with the ground 8A1-state. This process will be shown schematically as follows 8A -..-_!IT %30 lHoL
8AI .
Here H and HOL designate spin-orbit interaction and”orbit-lattice interaction operators respectively. I&L combines 6TI with 8A1 when HoL has a Tl-mode. The vibrational modes of the four nearest neighbour ligands are decomposed into AI + E + 2T2 and those of the 2nd neighbours into 2AI + A2 + 3E + 3Tl+ 5T2 [6], so the vibration of the 2nd neighbours combines Tl with Al. For the octahedral environment, that is, Oh symmetry, the relaxation process is as follows 8A
---JT lgHso
----6AIg
lgHo,
.
In this case the vibrational modes of the first
neighbours and the 2nd neighbours are Alg + Eg + +Eu+2Tlu+ + 2Tlu + T2u + T2g andAlg+A2u+E + 2Tzg + 2T9, respectively [6], and%ave no Tlgmode. Therefore this process is not operative unless the effect of the vibration of more distant atoms is taken into account, and is expected to be less effective than in the case of tetrahedral symmetry. If we write the matrix element of I&L(Tl) between 8TI and 8AI as c6TI 1HoL(T1) 1 Al), the order of the magnitude of the spin-lattice relaxation rate is shown to be as follows [?I T -I_ 3g2b2H2kT W2 I(% ~?-,L(T~)I 6A1)/ 2. s n&v5 (1) Here Ag designates the positive g-shift due to . Watanabe’s mechanism, and other notations are as usual, Inserting the appropriate numerical values into eq. (l),
Til M 106~2WM2~ ( 6T1( HoL(TI)(
6AI> 12sec-l. (2) Here Zf is measured in kilo auss, T in degrees . Kelvin, and (8Tll HoL(TI) 1%AI) in electron volts. T s1 calculated by Blume and Orbach for Mn2+ in MgO is as follows [l] 0.134 i?T set-’ . (3) We may expect (8TIl HoL(TI)I 8A > to be several electron volts, and our ~~1 given J y eq. (2) should be larger than the one predicted by eq. (3) for ions having positive Ag larger than 0.001. For Fe3+ in ZnTe the observed Ag amounts to about +O.l [8], and our T-S1is expected to be of the order of 104l#T set-l far larger than 0.134 H2T set-1 for Mn2+ in MgO. Since the matrix element of the angular momentum is known to be reduced due to the covalency of the central ion, ~~1 of Fe3+ in ZnTe which is expected to have strong covalency will T;l
=
441
Volume 20, number 5
PHYSICS
be smaller than the above calculated one. The effect of the eovalency on the spin-lattice relaxation is discussed by ~~~~ikov [9-j. Experimental data to ascertain whether our relaxation mechanism is operative are not available now, and it seems to be necessary to measure the spin-lattice relaxation rate of S-state ions having various large positive g-shifts whose environments are tetrahedral and octahedral.
LETTERS
15 March 1966
i. M.Blume and R.Orbach, Phys. Rev. 127(1962) 1587. 2. A.M. Leushin, Soviet Physics-Solid State 4 (1962) 1148. 3. I.Fidone and K.W.H.Stevens, Proc.Phys.Soc. 73 (1959) 116. 4. H.Watanabe, J.Phys.Chem.Soli& 25 (1964) 1471. 5. J.H.Van Vleck, Phys.Rev.57 (1940) 426. 6. H.A.JahnandE.Teller, Proc.Roy.Soc.Al61 (1937) 220. 7. R.Orbach, Pr0c.Roy.Soc.A 264 (1961)458. 8. J.C. Hensel, Bull. Am. Phys.Soc. 9 (1964) 244. 9. I.V.Ovchinnikov, Soviet Physics-Solid State 5 (1964) 1378.
*****
MICROSCOPIC THEORY OF THERMAL EXCITATIONS IN DIELECTRIC CRYSTALS AND THE ANALOGY BETWEEN FIRST AND SECOND SOUND C. P. ENZ
Universit& de GenBve
Received 26 January 1966 A microscopic understanding of second sound in dielectrics is sought by deriving it in parallel with first
sound and by treating damping explicitly. The main result is a dispersion law valid for all frequencies for which local equilibrium holds.
Recently, possibilities of detecting second sound in dielectric crystals have been discussed in this journal [l, 21. The main theoretical interest of these papers is the microscopic approach to the problem which, except for an attempt in a different direction (collective excitations [3], see below), is new. Whereas in ref. 2 the damping of first sound by thermal excitations is the mechanism considered, damping of second sound is implicit in ref. 1. However, an explicit microscopic treatment of damping of thermal excitations is missing in these references. Here we adopt a point of view which is in line with the conventional phenomenological approaches [4-8] and also with the accepted concepts in solids [9], According to the latter local equilibrium excitations such as first sound in fluids are distinct from elementary excitations such as first sound in crystals. Thus we define second sound quite generally as a local equilibrium excitation in a zero-mass boson-like system. (Iu ref. 3 a different point of view is taken.) This allows for other heretical effects such as “second spin waves” 14, lo] which are not considered here. 442
The local equilibrium variation of an observable A is [e.g. 111
AAt = +@;(A-
(A))),
(1)
where H; =v-l
c (f3-fA@(f#(-q) + (I + A~&)&?)
+ ut(q)_?+q))
is the time dependent thermal Wamiltonian”, (A) the average taken with respect to overall equilibrium @$ = 0) and z 6-l t &A(-ih). For q = 0, h(O) = H, la(O)= N, j(0 $ = J are the energy, number and momentum operators, respectively. Then the macroscopic conservation laws are
x
~A~~(q~ f lq*As&z) = 0 , s
An@
+ i4 * A jt(4) = 0 ,
& Ajt(q~ f iq Ant l
=0 ,
(24 @b) m
Volume 20, number 5
SPIN-
LATTICE
RELAXATION
15 March 1966
LETTERS
MECHANISM
FOR
S-STATE
IONS
T. SHIMIZU Department
of Electronic
Engineering,
Kanazawa
University,
Kanazawa,
Japan
Received 8 February 1966 A spin-lattice relaxation mechanism is proposed for S-state ions having a tetrahedral environmentand a positive g-shift. The relaxation stems from the Van Vleck mechanism using the excited sextet state considering the covalent bonding. Thus far two mechanisms have been proposed for spin-lattice relaxation of S-state ions by Blume and Orbach [l] and Leushin [2]. Since there is no sextet excited state in the d5 configuration, the relaxation mechanism is rather complicated. In this letter we propose a relaxation mechanism which seems to be dominant for Sstate ions having a positiveg-shift and a tetrahedral environment. In order to explain a positive g-shift of S-state ions, Fidone and Stevens [3] and Watanabe [4] uses the excited sextet state which appears as a result of the covalent bonding. We use this sextet state as an excited state in the Van Vleck relaxation mechanism for the direct process [5]. For the tetrahedral environment, that is, Td symmetry, spin-orbit interaction combines 8T1-states with the ground 8AI-state. Therefore the Van Vleck mechanism is operative if the orbit-lattice interaction combines GTI-states with the ground 8A1-state. This process will be shown schematically as follows 8A -..-_!IT %30 lHoL
8AI .
Here H and HOL designate spin-orbit interaction and”orbit-lattice interaction operators respectively. I&L combines 6TI with 8A1 when HoL has a Tl-mode. The vibrational modes of the four nearest neighbour ligands are decomposed into AI + E + 2T2 and those of the 2nd neighbours into 2AI + A2 + 3E + 3Tl+ 5T2 [6], so the vibration of the 2nd neighbours combines Tl with Al. For the octahedral environment, that is, Oh symmetry, the relaxation process is as follows 8A
---JT lgHso
----6AIg
lgHo,
.
In this case the vibrational modes of the first
neighbours and the 2nd neighbours are Alg + Eg + +Eu+2Tlu+ + 2Tlu + T2u + T2g andAlg+A2u+E + 2Tzg + 2T9, respectively [6], and%ave no Tlgmode. Therefore this process is not operative unless the effect of the vibration of more distant atoms is taken into account, and is expected to be less effective than in the case of tetrahedral symmetry. If we write the matrix element of I&L(Tl) between 8TI and 8AI as c6TI 1HoL(T1) 1 Al), the order of the magnitude of the spin-lattice relaxation rate is shown to be as follows [?I T -I_ 3g2b2H2kT W2 I(% ~?-,L(T~)I 6A1)/ 2. s n&v5 (1) Here Ag designates the positive g-shift due to . Watanabe’s mechanism, and other notations are as usual, Inserting the appropriate numerical values into eq. (l),
Til M 106~2WM2~ ( 6T1( HoL(TI)(
6AI> 12sec-l. (2) Here Zf is measured in kilo auss, T in degrees . Kelvin, and (8Tll HoL(TI) 1%AI) in electron volts. T s1 calculated by Blume and Orbach for Mn2+ in MgO is as follows [l] 0.134 i?T set-’ . (3) We may expect (8TIl HoL(TI)I 8A > to be several electron volts, and our ~~1 given J y eq. (2) should be larger than the one predicted by eq. (3) for ions having positive Ag larger than 0.001. For Fe3+ in ZnTe the observed Ag amounts to about +O.l [8], and our T-S1is expected to be of the order of 104l#T set-l far larger than 0.134 H2T set-1 for Mn2+ in MgO. Since the matrix element of the angular momentum is known to be reduced due to the covalency of the central ion, ~~1 of Fe3+ in ZnTe which is expected to have strong covalency will T;l
=
441
Volume 20, number 5
PHYSICS
be smaller than the above calculated one. The effect of the eovalency on the spin-lattice relaxation is discussed by ~~~~ikov [9-j. Experimental data to ascertain whether our relaxation mechanism is operative are not available now, and it seems to be necessary to measure the spin-lattice relaxation rate of S-state ions having various large positive g-shifts whose environments are tetrahedral and octahedral.
LETTERS
15 March 1966
i. M.Blume and R.Orbach, Phys. Rev. 127(1962) 1587. 2. A.M. Leushin, Soviet Physics-Solid State 4 (1962) 1148. 3. I.Fidone and K.W.H.Stevens, Proc.Phys.Soc. 73 (1959) 116. 4. H.Watanabe, J.Phys.Chem.Soli& 25 (1964) 1471. 5. J.H.Van Vleck, Phys.Rev.57 (1940) 426. 6. H.A.JahnandE.Teller, Proc.Roy.Soc.Al61 (1937) 220. 7. R.Orbach, Pr0c.Roy.Soc.A 264 (1961)458. 8. J.C. Hensel, Bull. Am. Phys.Soc. 9 (1964) 244. 9. I.V.Ovchinnikov, Soviet Physics-Solid State 5 (1964) 1378.
*****
MICROSCOPIC THEORY OF THERMAL EXCITATIONS IN DIELECTRIC CRYSTALS AND THE ANALOGY BETWEEN FIRST AND SECOND SOUND C. P. ENZ
Universit& de GenBve
Received 26 January 1966 A microscopic understanding of second sound in dielectrics is sought by deriving it in parallel with first
sound and by treating damping explicitly. The main result is a dispersion law valid for all frequencies for which local equilibrium holds.
Recently, possibilities of detecting second sound in dielectric crystals have been discussed in this journal [l, 21. The main theoretical interest of these papers is the microscopic approach to the problem which, except for an attempt in a different direction (collective excitations [3], see below), is new. Whereas in ref. 2 the damping of first sound by thermal excitations is the mechanism considered, damping of second sound is implicit in ref. 1. However, an explicit microscopic treatment of damping of thermal excitations is missing in these references. Here we adopt a point of view which is in line with the conventional phenomenological approaches [4-8] and also with the accepted concepts in solids [9], According to the latter local equilibrium excitations such as first sound in fluids are distinct from elementary excitations such as first sound in crystals. Thus we define second sound quite generally as a local equilibrium excitation in a zero-mass boson-like system. (Iu ref. 3 a different point of view is taken.) This allows for other heretical effects such as “second spin waves” 14, lo] which are not considered here. 442
The local equilibrium variation of an observable A is [e.g. 111
AAt = +@;(A-
(A))),
(1)
where H; =v-l
c (f3-fA@(f#(-q) + (I + A~&)&?)
+ ut(q)_?+q))
is the time dependent thermal Wamiltonian”, (A) the average taken with respect to overall equilibrium @$ = 0) and z 6-l t &A(-ih). For q = 0, h(O) = H, la(O)= N, j(0 $ = J are the energy, number and momentum operators, respectively. Then the macroscopic conservation laws are
x
~A~~(q~ f lq*As&z) = 0 , s
An@
+ i4 * A jt(4) = 0 ,
& Ajt(q~ f iq Ant l
=0 ,
(24 @b) m