Solid State Nuclear Magnetic Resonance, 1 (1992) 251-253
251
Elsevier Science Publishers B.V., Amsterdam
Nuclear spin-lattice relaxation in transition metal compounds with local tetragonal symmetry B. Nowak and O.J. AogaI W. Trzebiatowski Institute of Low Temperature and Structure Research, Polish Academy of Sciences, P.O. Box 937, 50-950 Wroclaw 2, Poland
(Received 5 June 1992; accepted 9 September 1992)
Abstract We report on the various contributions to the total spin-lattice relaxation rate in metallic materials with local tetragonal symmetry. The analytical formulae are given in the tight-binding approximation. The calculations show the relation between various partial electron densities of states and corresponding contributions to the relaxation rates. The presented formulae can be used to compare theoretically calculated electron band structure parameters with those obtained from NMR spin-lattice relaxation time measurements. Keywords: nuclear spin-lattice relaxation time; metallic substance
Introduction The important contributions to the observed relaxation rates in cubic structures have been shown [l-7] to arise from the core-polarisation, orbital and dipolar hyperfine interactions with the p and d components of the conduction-electron wave functions at the Fermi level, as well as from Fermi contact hyperfine interaction with the s component. Conduction-electron contribution associated with electron-quadrupole interaction is usually negligibly small [21 with few exceptions [5,6]. Moreover, since spin-orbit effects are generally assumed to be small, no interference terms arise in the cubic case among the various interactions.
Correspondence too: Dr. B. Nowak, W. Trzebiatowski Institute
of Low Temperature and Structure Research, Polish Academy of Sciences, P.O. Box 937, 50-950 Wrodaw 2, Poland. 09262040/92/$05.00
In the case of HCP symmetry, the relaxation rates coming from orbital, dipole and quadrupole origins are anisotropic with respect to the external field direction measured from the crystal c axis [8,9] and numerous interference terms make an important contribution of 20-30% of the total relaxation rate [9]. That is, since the A,’ representation of the D,, point-group has bases in both s and d functions and the E,’ representation in p and d functions. In contrast to the large number of theoretical and experimental studies on the cubic and hexagonal transition metals, the compounds and alloys with local tetragonal symmetry have received much less attention. Strong dependence of the various contributions of the spin-lattice relaxation rate on the point-group symmetry properties of the d component of the conduction-electron wave function in tetragonal symmetry was discussed for the case of V,X and Nb,X compounds (A-15 structure). In this structure the V (Nb) atoms
0 1992 - Elsevier Science Publishers B.V. All rights reserved
252
B. Nowak, O.J. Z!ogar/Solid State Nucl. Magn. Reson. I (1992) 251-253
have tetragonal symmetry and the point-group
is
D,, (42~2) [lo-121. Quite recently, we have mea-
sured the 91Zr and 49Ti nuclear spin-lattice relaxation rates in tetragonal phases of ZrH, and TiH, [13,14]. The corresponding point-group for Zr (Ti) sites is D4,, (4/mmm). Lately, we have realized that the expressions for the d-core polarization and d-orbital contributions derived in ref. 10 and used by us in refs. 13 and 14 are erroneous. The present calculations offer the correct expressions for the d-core polarization and dorbital contributions to the relaxation rate for the case of D4,, symmetry and they may be easily adopted for D Zd symmetry. Their application to the 91Zr relaxation ratio calculations for ZrH, gives the p and 4 coefficients (d-orbital and d-core polarization inhibition factors) equal to 2/5 and l/5, respectively, instead of p = 13/30 and 4 = 7/50 as used in ref. 13. For the sake of completeness we derived also the formulae for p-orbital and p(d) dipolar contributions, not yet reported in the literature for tetragonal symmetry.
conduction-electron wave function at the Fermi surface. Then, the total relaxation rate in the tight-binding approximation [7,9] can be expressed as: (T,‘)
= (T;l)SF+ (TC’)Porh+ +(Tl)dorb
+
(T’)P,ip
(T’)!ip
+
(T’)Zp
+.(T;‘);::,
(1)
where (T; ‘)“, represents the Fermi contact interaction with conduction electrons, (T; ‘>$$ represents the interaction with electron orbital motion. CT; ‘)j$) represents the spin dipolar interaction, CT,-‘)$ represents the core-polarization interaction and (T-‘)“,T,d, represents the interference term between Fermi contact and core-polarization interactions. Applying a calculation procedure developed by Narath [8] we have found: (T;‘); (T;l):rb
= ccrf(H;)‘nf(
EF)
= Qf(f%)2[%*“% +%,(%,
(2) ”
- %Jsin2
e]
(3)
Formulae for the spin-lattice relaxation rates (TF1)Pdip
In the Table 1 we list the irreducible representations of the s (I = O), p (I= 1) and d (1 = 2) wave functions of D4h point-group. Since both the Y: and Y: atomic wave functions belong to the Al, representations of D4h, the Fermi contact and core-polarization interactions can interfer, provided these functions are admixed in the
= ,Qa(HoPrb)2ik[4n?12u
+38&
Representations
and their bases for D4,, local symmetry Bases a
Representations A,,
A zu 6
Y,o
Yz” YP (Y: + y;‘,/@ - i(Y,’ - Y;l)JT
B 1E
(Y? + Y;?/fi
B %I
- i(Y:
4
(Yz’ + y;‘)/fi
- YT2)/Jz
- i(Yi -Y;‘)/& a Y,” denotes the usual spherical harmonics.
18n,42.nE
u
+ 3 sin2 8(2n21U
+ 3K4*,%, - 5&J] (TC l>:*b = ~(~o%)2(
(4)
2a, z(3Glp + nB,, + %*,)
+sin2 e 4nB,,~g,g 1
-nE,(3nA,, TABLE 1
+
+ nB,, + nB2g- ndl
> (5)
B. Nowak.
O.J. Zogal/Solid
State Nucl. Magn. Reson. I (1992) 251-253
(7) (8)
n/L, +
2n,” =n,(G-1
11.4 ,g+ nB,, + nBLg+
dEF) +n,(EF)
(10)
2nEg = nd(EF) +nd(E,)
=N(E,)
(11) (12)
In the above equations T is the temperature, k, is the Boltzmann constant, yN is the gyromagnetic ratio of the nucleus, nr is the partial density of states of the r representation, n&E,), n,(E,), n&E,) and fV(E,) are the partial s, p, d and total density of states at Fermi level for one spin direction, respectively. The Hg, Hzr’ and Hc”pare the Fermi contact, p(d) orbital and d-core polarization hyperfine fields, respectively. The quantity 1f?$,( EF) 1 in eqn. (8) is the so-called off-diagonal density of states at the Fermi energy, discussed for the first time by Asada and Terakura [93 in the case of relaxation rates in the HCP transition metals. Finally, 8 is the angle of an external magnetic field with respect to the crystal c axis of a single crystal sample. The expressions appropriate for powdered samples one obtains from eqns. (3)-(6) putting sin2 f3= 2/3.
Conclusions The detailed application of the theory to experimental data unfortunately requires a knowledge of several parameters relating to the electron band structure. These are: (1) the relevant hyperfine fields, (2) the total bare electron density of states, (3) the weights of the S, p, and d components of the conduction-electron wave
253
functions, and (4) the point symmetry properties of the p and d components. The results of our calculation we address to the NMR spectroscopist who investigates metallic compounds of transition elements. We also hope that this paper will stimulate theoretical calculations of electron band structure of transition metal hydrides. Already existing experimental NMR data [13,14] for those compounds cannot be compared with theoretical ones because of a lack of necessary details of the electron band structure. The available information on the total density of states [ME,)], although useful for such experiments as the low-temperature specific heat, magnetic susceptibility, etc., are not enough for the analysis of the NMR spin-lattice relaxation rates.
Acknowledgments The authors express their gratitude to Dr. hab. J. Mulak and Dr. Z. Gajek for many stimulating and helpful discussions.
References 1 Y. Obata, J. Phys. Sec. Jpn., 18 (1963) 1020. 2 Y. Obata, ibid, 19 (1964) 2348. 3 Y. Yafet and V. Jaccarino, Phys. Rer. A, 133 (1964) 1630. 4 A. Narath and A.T. Fromhold Jr., Phys. Rw. .4. 139 (1965) 794. 5 A. Narath and D.W. Alderman, P/zys. Rec., 143 (1966) 328. h A. Narath. Phys. Rev., 165 (1968) 506. 7 T. Asada, K. Terakura and T. Jariborg. J. Phys. F, 11 (1981) 1847. 8 A. Narath. Phys. Rev., 162 (1967) 320. 9 T. Asada and K. Terakura, J. Phys. F, 12 (1982) 1387. 10 F.Y. Fradin and D. Zamir, Phys. Rev. B, 7 (1973) 4861. 11 F.Y. Fradin and J.D. Williamson. Phys. Rel,. B, 10 (1974) 2803. 12 F.Y. Fradin and G. Cinader, Phys. Rec.. B, 16 (1977) 73. 13 O.J. iogal. B. Nowak and K. Niediwiedi. So/id Strrte Convnun.. 80 (1991) 601. 14 B. Nowak. O.J. iogal and K. Niediwiedi, J. Alloys Cotnp., 186 (1992) 53.
251
Elsevier Science Publishers B.V., Amsterdam
Nuclear spin-lattice relaxation in transition metal compounds with local tetragonal symmetry B. Nowak and O.J. AogaI W. Trzebiatowski Institute of Low Temperature and Structure Research, Polish Academy of Sciences, P.O. Box 937, 50-950 Wroclaw 2, Poland
(Received 5 June 1992; accepted 9 September 1992)
Abstract We report on the various contributions to the total spin-lattice relaxation rate in metallic materials with local tetragonal symmetry. The analytical formulae are given in the tight-binding approximation. The calculations show the relation between various partial electron densities of states and corresponding contributions to the relaxation rates. The presented formulae can be used to compare theoretically calculated electron band structure parameters with those obtained from NMR spin-lattice relaxation time measurements. Keywords: nuclear spin-lattice relaxation time; metallic substance
Introduction The important contributions to the observed relaxation rates in cubic structures have been shown [l-7] to arise from the core-polarisation, orbital and dipolar hyperfine interactions with the p and d components of the conduction-electron wave functions at the Fermi level, as well as from Fermi contact hyperfine interaction with the s component. Conduction-electron contribution associated with electron-quadrupole interaction is usually negligibly small [21 with few exceptions [5,6]. Moreover, since spin-orbit effects are generally assumed to be small, no interference terms arise in the cubic case among the various interactions.
Correspondence too: Dr. B. Nowak, W. Trzebiatowski Institute
of Low Temperature and Structure Research, Polish Academy of Sciences, P.O. Box 937, 50-950 Wrodaw 2, Poland. 09262040/92/$05.00
In the case of HCP symmetry, the relaxation rates coming from orbital, dipole and quadrupole origins are anisotropic with respect to the external field direction measured from the crystal c axis [8,9] and numerous interference terms make an important contribution of 20-30% of the total relaxation rate [9]. That is, since the A,’ representation of the D,, point-group has bases in both s and d functions and the E,’ representation in p and d functions. In contrast to the large number of theoretical and experimental studies on the cubic and hexagonal transition metals, the compounds and alloys with local tetragonal symmetry have received much less attention. Strong dependence of the various contributions of the spin-lattice relaxation rate on the point-group symmetry properties of the d component of the conduction-electron wave function in tetragonal symmetry was discussed for the case of V,X and Nb,X compounds (A-15 structure). In this structure the V (Nb) atoms
0 1992 - Elsevier Science Publishers B.V. All rights reserved
252
B. Nowak, O.J. Z!ogar/Solid State Nucl. Magn. Reson. I (1992) 251-253
have tetragonal symmetry and the point-group
is
D,, (42~2) [lo-121. Quite recently, we have mea-
sured the 91Zr and 49Ti nuclear spin-lattice relaxation rates in tetragonal phases of ZrH, and TiH, [13,14]. The corresponding point-group for Zr (Ti) sites is D4,, (4/mmm). Lately, we have realized that the expressions for the d-core polarization and d-orbital contributions derived in ref. 10 and used by us in refs. 13 and 14 are erroneous. The present calculations offer the correct expressions for the d-core polarization and dorbital contributions to the relaxation rate for the case of D4,, symmetry and they may be easily adopted for D Zd symmetry. Their application to the 91Zr relaxation ratio calculations for ZrH, gives the p and 4 coefficients (d-orbital and d-core polarization inhibition factors) equal to 2/5 and l/5, respectively, instead of p = 13/30 and 4 = 7/50 as used in ref. 13. For the sake of completeness we derived also the formulae for p-orbital and p(d) dipolar contributions, not yet reported in the literature for tetragonal symmetry.
conduction-electron wave function at the Fermi surface. Then, the total relaxation rate in the tight-binding approximation [7,9] can be expressed as: (T,‘)
= (T;l)SF+ (TC’)Porh+ +(Tl)dorb
+
(T’)P,ip
(T’)!ip
+
(T’)Zp
+.(T;‘);::,
(1)
where (T; ‘)“, represents the Fermi contact interaction with conduction electrons, (T; ‘>$$ represents the interaction with electron orbital motion. CT; ‘)j$) represents the spin dipolar interaction, CT,-‘)$ represents the core-polarization interaction and (T-‘)“,T,d, represents the interference term between Fermi contact and core-polarization interactions. Applying a calculation procedure developed by Narath [8] we have found: (T;‘); (T;l):rb
= ccrf(H;)‘nf(
EF)
= Qf(f%)2[%*“% +%,(%,
(2) ”
- %Jsin2
e]
(3)
Formulae for the spin-lattice relaxation rates (TF1)Pdip
In the Table 1 we list the irreducible representations of the s (I = O), p (I= 1) and d (1 = 2) wave functions of D4h point-group. Since both the Y: and Y: atomic wave functions belong to the Al, representations of D4h, the Fermi contact and core-polarization interactions can interfer, provided these functions are admixed in the
= ,Qa(HoPrb)2ik[4n?12u
+38&
Representations
and their bases for D4,, local symmetry Bases a
Representations A,,
A zu 6
Y,o
Yz” YP (Y: + y;‘,/@ - i(Y,’ - Y;l)JT
B 1E
(Y? + Y;?/fi
B %I
- i(Y:
4
(Yz’ + y;‘)/fi
- YT2)/Jz
- i(Yi -Y;‘)/& a Y,” denotes the usual spherical harmonics.
18n,42.nE
u
+ 3 sin2 8(2n21U
+ 3K4*,%, - 5&J] (TC l>:*b = ~(~o%)2(
(4)
2a, z(3Glp + nB,, + %*,)
+sin2 e 4nB,,~g,g 1
-nE,(3nA,, TABLE 1
+
+ nB,, + nB2g- ndl
> (5)
B. Nowak.
O.J. Zogal/Solid
State Nucl. Magn. Reson. I (1992) 251-253
(7) (8)
n/L, +
2n,” =n,(G-1
11.4 ,g+ nB,, + nBLg+
dEF) +n,(EF)
(10)
2nEg = nd(EF) +nd(E,)
=N(E,)
(11) (12)
In the above equations T is the temperature, k, is the Boltzmann constant, yN is the gyromagnetic ratio of the nucleus, nr is the partial density of states of the r representation, n&E,), n,(E,), n&E,) and fV(E,) are the partial s, p, d and total density of states at Fermi level for one spin direction, respectively. The Hg, Hzr’ and Hc”pare the Fermi contact, p(d) orbital and d-core polarization hyperfine fields, respectively. The quantity 1f?$,( EF) 1 in eqn. (8) is the so-called off-diagonal density of states at the Fermi energy, discussed for the first time by Asada and Terakura [93 in the case of relaxation rates in the HCP transition metals. Finally, 8 is the angle of an external magnetic field with respect to the crystal c axis of a single crystal sample. The expressions appropriate for powdered samples one obtains from eqns. (3)-(6) putting sin2 f3= 2/3.
Conclusions The detailed application of the theory to experimental data unfortunately requires a knowledge of several parameters relating to the electron band structure. These are: (1) the relevant hyperfine fields, (2) the total bare electron density of states, (3) the weights of the S, p, and d components of the conduction-electron wave
253
functions, and (4) the point symmetry properties of the p and d components. The results of our calculation we address to the NMR spectroscopist who investigates metallic compounds of transition elements. We also hope that this paper will stimulate theoretical calculations of electron band structure of transition metal hydrides. Already existing experimental NMR data [13,14] for those compounds cannot be compared with theoretical ones because of a lack of necessary details of the electron band structure. The available information on the total density of states [ME,)], although useful for such experiments as the low-temperature specific heat, magnetic susceptibility, etc., are not enough for the analysis of the NMR spin-lattice relaxation rates.
Acknowledgments The authors express their gratitude to Dr. hab. J. Mulak and Dr. Z. Gajek for many stimulating and helpful discussions.
References 1 Y. Obata, J. Phys. Sec. Jpn., 18 (1963) 1020. 2 Y. Obata, ibid, 19 (1964) 2348. 3 Y. Yafet and V. Jaccarino, Phys. Rer. A, 133 (1964) 1630. 4 A. Narath and A.T. Fromhold Jr., Phys. Rw. .4. 139 (1965) 794. 5 A. Narath and D.W. Alderman, P/zys. Rec., 143 (1966) 328. h A. Narath. Phys. Rev., 165 (1968) 506. 7 T. Asada, K. Terakura and T. Jariborg. J. Phys. F, 11 (1981) 1847. 8 A. Narath. Phys. Rev., 162 (1967) 320. 9 T. Asada and K. Terakura, J. Phys. F, 12 (1982) 1387. 10 F.Y. Fradin and D. Zamir, Phys. Rev. B, 7 (1973) 4861. 11 F.Y. Fradin and J.D. Williamson. Phys. Rel,. B, 10 (1974) 2803. 12 F.Y. Fradin and G. Cinader, Phys. Rec.. B, 16 (1977) 73. 13 O.J. iogal. B. Nowak and K. Niediwiedi. So/id Strrte Convnun.. 80 (1991) 601. 14 B. Nowak. O.J. iogal and K. Niediwiedi, J. Alloys Cotnp., 186 (1992) 53.
