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Field dependence of spin-lattice relaxation in paramagnetic salts
Solid State Communications, Vol. 24, pp. 493—497, 1977.
Pergamon Press.
Printed in Great Britain
FIELD DEPENDENCE OF SPIN-LATTICE RELAXATION IN PARAMAGNETIC SALTS M. Nogatchewsky, G. Ablart and J. Pescia Laboratoire de Magnetisme et D’Electronique Quantique, Université Paul-Sabatier, 39, AIlées Jules-Guesde, 31077 Toulouse Cédex, France (Received 13 April 1977; in revised form 17 June by P.G. de Gennes) The field dependence of the Raman SLR rate is determined for H0 of the order of the local field. Both S-state and non S-state transition salts are investigated when (i) Hd~P> Hexch and (ii) Hex~> H~p.Measurements are obtained from the response of the longitudinal magnetization to an amplitude modulated microwave field. The observed dependences agree with Orbach—Huang calculations. 1. INTRODUCTION THE EXPERIMENTS we describe hereafter have been performed at 77 K. At this temperature, spin-lattice relaxation occurs via the Raman process in which two non-resonant phonons are involved. Now we briefly review temperature and field dependences of this process. 1.1. Temperature dependence Paramagnetic diluted salts show the typical behaviour: Ti’ T~. (1) 0D (the Debye temperaFor Kramers doublets T~ ture), n = 9 [1, 2]; forand non-Kramers doublets under the same conditions, n = 7 [1] ; for multi-level Kramers states in the low temperature part of the Raman region, n = 5 [31; and at temperatures T> °D’ n = 2 always. Lastly if relaxation occurs via optical phonons: T~ ~ exp (— hw/kT) where w is the phonon’s angular frequency 1.2. Field dependence Brons [5] and van Vieck group salts:
(2)
[41.
[61have written for iron
i4 + a 14 + b
Tj’
=
i4 + ~ i4 + ~
Tj~
+ 1~H~XCh
(5) + 4H~XCh where Hexth is the exchange field. He has obtained p~and 4 < I taking into account optical phonons. Lastly if an admixture of excited doublets of the ground multiplet is present, Orbach [2, 9] has found: T~’ i4 T7. (6) Experimental verification of this equation has been given by Marchand et aL [10] in NdES, by us in CuCl 2, 2H20 [11] and by Pouw et al. in cobalt Tutton salts [18]. Many results are available for temperature dependence. On the other hand, few results exist concerning field dependence. Indeed, several spectrometers are necessary to determine this dependence because usual microwave generators exhibit a narrow frequency range. The greatest amount of data has been obtained using non-resonant techniques [12, 131 But this method -
(3)
Orbach [7] has given a more detailed expression:
is negligible. Orbach has computed p 1 and p’ 2 for rare earth ethylsulfates with g1 = 0 and p < 1 and 1’ > 2 in all the other cases. Huang [8] has found again equation (4). In the case Hexch > H~p,he has given the expression:
becomes inaccurate for T1 shorter than iO-~sec, a very usual situation in transition salts at 77K. So we have in undertaken to investigate magnetic field dependence seven paramagnetic salts, using the resonant modulation
from 75 [14—16] to 40000e. + 414~~ ~ (4) Measurements method havewell beenadapted performed to short at 77T~values. K, H0 ranging where Hb~,.,is the hyperfine field, HthP is the dipolar field, Tjj is the value of the relaxation rate for zero 2. EXPERIMENTAL ARRANGEMENT internal field strengths, H 0 isthe applied magnetic field, and The details of the modulation method and apparatus p and p’ are parameters depending on the salt. Expression have been published elsewhere [14, 161 We only recall (4) is valid quite generally so long as exchange interaction the principle. The microwave field is amplitude =
14 i4 +~
+
,
-
Vol. 24, No.7
SPIN-LATTICE RELAXATION IN PARAMAGNETIC SALTS
495
Table 2. Experimental values of 14, Hexeh, T j~deduced from equation (5’) in four salts where Hexch > H~jp.Calculated values of Hexch using equation (8) are also given Hexch comp. (Oe) Salt 14 H~x~ meas. (Oe) Tj~(sec1) K 2Cu(S04)2.6H20 Cu(N03)23H20 MnCl2 4H~O Mn(N03 )2 6H2 0
1.5 11 3 1
800 800 4400
9 x 106 l.2x 108 1.5 x 106 1.2 x 106
—
— 3400 3000
(iii) We have obtained p’ > 2 for all the samples in agreement with the Orbach calculation. (iv) All these conclusions remain valid with S-state ions. (v) According to Huang calculation, values of p’ larger than one show the optical phonons do not contribute to the relaxation of the investigated salts.
U,
~—;~-~50 F
~ 20 0
10
L C
400
6. RESULTS FOR THE SALTS OF GROUP II (Hexch > H~jp) 20
50
100
Though the case of paramagnets with predominant exchange is less clear, we have tried again to fit experimental data by equation (5). But taking into account the inequality Hexch > ~ we have used the simplified equation: 14 + (5’) Tj~ = 10 14+~ll~XCh
300 Tempercii~ureT( K)
Fig. 3. log—log plot of the Raman relaxation rate as a function of temperature Tat H0 = 3000 1 ~ Oe T2.for MnC12 4H20. The solid curve fits Ti 4. RESULTS FOR THE SALTS OF GROUP I (Hdip > Hexch) .
The obtained values of 1.L’i Hexch and T~O’ are listed in Table 2 and also the values Of Hexch computed from ,
The observed dependences T 1(Ho) are seen in Figs. I and 2. We have tried to fit our experimental data to equation (4), but taking into account the inequality Hdlp ~‘Hh~P,we have used the simplified equation: 1 = Tj~ i4 + 4H~~ ~14 + T~ (4’)
the second Anderson formula [17] 1
3kTN 4ZS(S + 1) (8) number of nearest neighbours. where TN is the Néel temperature [19] and Z is the
Hexch
=
8
3
The so-obtained values of p’, Hdjp and Ti 0’ are listed in Table 1. We have also given the values of HdjP computed by the first Anderson formula [17] : Hd~P = 1 .5gf3ps../~~J)
(7)
where p is the density, S is the spin, j3 is the Bohr magneton and g is the Landé g-factor. 5. DISCUSSION (i) The experimental dependences are in fair agreement with equation (3). (ii) Comparing the observed variations to equation (4), we have found values of Hd~Pwhich fit well with the values computed from equation (7). This can be considered as a verification of equation (4’).
7. DISCUSSION (i) Here again the experimental data are in good agreement with equation (3). (ii) Comparing these data to equation (5’), we have found values of HexCh not very different from the values computed from equation (8). Indeed the equation gives a value of the mean exchange field which is always too low. Hexch values listed in Table 2 are of the order of Hd~Pvalues
Hexth
~‘
listed in Table 1. However, condition
Hd1P is well obeyed with group II salts, as it can
be clearly seen from the typical Lorentzian line shape: Anderson [20] suggested the resonance line is essentially a Lorentz curve when Hexth ~ H,jjp. Condition Hdjp cannot be verified computing Hd1P by equation (7) because it is impossible to use
Hexth
~‘
SPIN-LATTICE RELAXATION IN PARAMAGNETIC SALTS Vol. 24, No.7 7sec at 230K in manganese simultaneously equations (7)and (8). So the verification 77K and T1 = 1.5 x lO of equation (5’) can be considered as reasonable. nitrate. A strong exchange interaction would prevent (iii) The values of 14 are found to be higher than the variation of T 1 with temperature, as it is seen for one. According to Huang theory, this means the optical instance in the unidimensional paramagnet TMMC. So phonons play no role in the relaxation. Confirmation of the first explanation is ruled out. On the other hand, the this has been looked for in the temperature dependence second explanation see. s to be more reasonable. Only of T, which we have measured in MnC12, 4H20. Over the calculation of i4 woL d allow to conclude.Unfortunthe whole range 45 to 300 K we have observed: ately, the lack of informati ~nconcerning the microscopic data makes such a calculation impossible. 1 T2 (Fig. 3). Ti The variation is clearly slower than an exponential 8. CONCLUSION law and the optical phonons seem to play no role here. At 77 K the field dependence T, (H 0) has been The observed temperature dependence is normal for determined in seven paramagnetic salts using a resoT< o~ lOOK. On the0Drange 45
—
REFERENCES 1.
SCOTT P.L & JEFFRIES C.D.,Phys. Rev. 127, 32 (1962).
2. 3.
ORBACH R.,F~oc.Roy. Soc. A264, 458 (1961). ORBACH R. & BLUME M.,Phys. Rev. Lett. 8,478 (1962).
4.
HUANG C.Y.,Phys. Rev. 154,2215 (1967).
5. 6.
BRONS F., Thesis, Groningen (1938). V~NVLECK J.H., Phys. Rev. 57, 426 (1940).
7. 8.
ORBACH R.,F~oc.Roy. Soc. A264, 485 (1961). HUANG C.Y.,Phys. Rev. 161,272(1967).
9. 10. 11.
ORBACH R.,F~oc.Phys. Soc. 77, 821 (1961). MARCHAND R.L. & STAPLETON H.J., Phys. Rev. B9, 14(1974). ABLART G., BOUJOL P., LOPEZ P. & PESCIA J., Solid State Commun. 17, 1085 (1975).
12. 13.
SING A. & SAPP R.C.,Phys. Rev. 5, 1688 (1972). GORTER C.J. & SMITS L.J., Tables de constantes et données numériques, Masson et Cie (1957).
14.
GOURDON J.C., VIGOUROUIX B. & PESCIA J.,Phys. Lett. 45A, 69(1973).
15. 16.
NOGATCHEWSKY M., These 3éme cycle, Toulouse (1977). PESCIA J., Ann. Phys. 10, 389 (1965).
Vol. 24, No.7
SPIN-LATTICE RELAXATION IN PARAMAGNETIC SALTS
497
17. 18.
ANDERSON P.W. &WEISS P.R., Rev. Mod. Phys. 25, 269 (1953). POUW C.LM. & VAN DUYNEVELDT AJ., Physics 83B, 267 (1976).
19.
KONIG E., Landolt Bornstein in Group 11, VoL 2 Magnetic Properties. Springer Verlag, New York (1966); AMAYA K., TOKUNAGA Y., KURAMITSU Y. & HASEDA T.,J. Phy& Soc. Japan 33,49(1972). PAKE G.E., Paramagnetic Resonance. Benjamin, New York (1962).
20.
Pergamon Press.
Printed in Great Britain
FIELD DEPENDENCE OF SPIN-LATTICE RELAXATION IN PARAMAGNETIC SALTS M. Nogatchewsky, G. Ablart and J. Pescia Laboratoire de Magnetisme et D’Electronique Quantique, Université Paul-Sabatier, 39, AIlées Jules-Guesde, 31077 Toulouse Cédex, France (Received 13 April 1977; in revised form 17 June by P.G. de Gennes) The field dependence of the Raman SLR rate is determined for H0 of the order of the local field. Both S-state and non S-state transition salts are investigated when (i) Hd~P> Hexch and (ii) Hex~> H~p.Measurements are obtained from the response of the longitudinal magnetization to an amplitude modulated microwave field. The observed dependences agree with Orbach—Huang calculations. 1. INTRODUCTION THE EXPERIMENTS we describe hereafter have been performed at 77 K. At this temperature, spin-lattice relaxation occurs via the Raman process in which two non-resonant phonons are involved. Now we briefly review temperature and field dependences of this process. 1.1. Temperature dependence Paramagnetic diluted salts show the typical behaviour: Ti’ T~. (1) 0D (the Debye temperaFor Kramers doublets T~ ture), n = 9 [1, 2]; forand non-Kramers doublets under the same conditions, n = 7 [1] ; for multi-level Kramers states in the low temperature part of the Raman region, n = 5 [31; and at temperatures T> °D’ n = 2 always. Lastly if relaxation occurs via optical phonons: T~ ~ exp (— hw/kT) where w is the phonon’s angular frequency 1.2. Field dependence Brons [5] and van Vieck group salts:
(2)
[41.
[61have written for iron
i4 + a 14 + b
Tj’
=
i4 + ~ i4 + ~
Tj~
+ 1~H~XCh
(5) + 4H~XCh where Hexth is the exchange field. He has obtained p~and 4 < I taking into account optical phonons. Lastly if an admixture of excited doublets of the ground multiplet is present, Orbach [2, 9] has found: T~’ i4 T7. (6) Experimental verification of this equation has been given by Marchand et aL [10] in NdES, by us in CuCl 2, 2H20 [11] and by Pouw et al. in cobalt Tutton salts [18]. Many results are available for temperature dependence. On the other hand, few results exist concerning field dependence. Indeed, several spectrometers are necessary to determine this dependence because usual microwave generators exhibit a narrow frequency range. The greatest amount of data has been obtained using non-resonant techniques [12, 131 But this method -
(3)
Orbach [7] has given a more detailed expression:
is negligible. Orbach has computed p 1 and p’ 2 for rare earth ethylsulfates with g1 = 0 and p < 1 and 1’ > 2 in all the other cases. Huang [8] has found again equation (4). In the case Hexch > H~p,he has given the expression:
becomes inaccurate for T1 shorter than iO-~sec, a very usual situation in transition salts at 77K. So we have in undertaken to investigate magnetic field dependence seven paramagnetic salts, using the resonant modulation
from 75 [14—16] to 40000e. + 414~~ ~ (4) Measurements method havewell beenadapted performed to short at 77T~values. K, H0 ranging where Hb~,.,is the hyperfine field, HthP is the dipolar field, Tjj is the value of the relaxation rate for zero 2. EXPERIMENTAL ARRANGEMENT internal field strengths, H 0 isthe applied magnetic field, and The details of the modulation method and apparatus p and p’ are parameters depending on the salt. Expression have been published elsewhere [14, 161 We only recall (4) is valid quite generally so long as exchange interaction the principle. The microwave field is amplitude =
14 i4 +~
+
,
-
Vol. 24, No.7
SPIN-LATTICE RELAXATION IN PARAMAGNETIC SALTS
495
Table 2. Experimental values of 14, Hexeh, T j~deduced from equation (5’) in four salts where Hexch > H~jp.Calculated values of Hexch using equation (8) are also given Hexch comp. (Oe) Salt 14 H~x~ meas. (Oe) Tj~(sec1) K 2Cu(S04)2.6H20 Cu(N03)23H20 MnCl2 4H~O Mn(N03 )2 6H2 0
1.5 11 3 1
800 800 4400
9 x 106 l.2x 108 1.5 x 106 1.2 x 106
—
— 3400 3000
(iii) We have obtained p’ > 2 for all the samples in agreement with the Orbach calculation. (iv) All these conclusions remain valid with S-state ions. (v) According to Huang calculation, values of p’ larger than one show the optical phonons do not contribute to the relaxation of the investigated salts.
U,
~—;~-~50 F
~ 20 0
10
L C
400
6. RESULTS FOR THE SALTS OF GROUP II (Hexch > H~jp) 20
50
100
Though the case of paramagnets with predominant exchange is less clear, we have tried again to fit experimental data by equation (5). But taking into account the inequality Hexch > ~ we have used the simplified equation: 14 + (5’) Tj~ = 10 14+~ll~XCh
300 Tempercii~ureT( K)
Fig. 3. log—log plot of the Raman relaxation rate as a function of temperature Tat H0 = 3000 1 ~ Oe T2.for MnC12 4H20. The solid curve fits Ti 4. RESULTS FOR THE SALTS OF GROUP I (Hdip > Hexch) .
The obtained values of 1.L’i Hexch and T~O’ are listed in Table 2 and also the values Of Hexch computed from ,
The observed dependences T 1(Ho) are seen in Figs. I and 2. We have tried to fit our experimental data to equation (4), but taking into account the inequality Hdlp ~‘Hh~P,we have used the simplified equation: 1 = Tj~ i4 + 4H~~ ~14 + T~ (4’)
the second Anderson formula [17] 1
3kTN 4ZS(S + 1) (8) number of nearest neighbours. where TN is the Néel temperature [19] and Z is the
Hexch
=
8
3
The so-obtained values of p’, Hdjp and Ti 0’ are listed in Table 1. We have also given the values of HdjP computed by the first Anderson formula [17] : Hd~P = 1 .5gf3ps../~~J)
(7)
where p is the density, S is the spin, j3 is the Bohr magneton and g is the Landé g-factor. 5. DISCUSSION (i) The experimental dependences are in fair agreement with equation (3). (ii) Comparing the observed variations to equation (4), we have found values of Hd~Pwhich fit well with the values computed from equation (7). This can be considered as a verification of equation (4’).
7. DISCUSSION (i) Here again the experimental data are in good agreement with equation (3). (ii) Comparing these data to equation (5’), we have found values of HexCh not very different from the values computed from equation (8). Indeed the equation gives a value of the mean exchange field which is always too low. Hexch values listed in Table 2 are of the order of Hd~Pvalues
Hexth
~‘
listed in Table 1. However, condition
Hd1P is well obeyed with group II salts, as it can
be clearly seen from the typical Lorentzian line shape: Anderson [20] suggested the resonance line is essentially a Lorentz curve when Hexth ~ H,jjp. Condition Hdjp cannot be verified computing Hd1P by equation (7) because it is impossible to use
Hexth
~‘
SPIN-LATTICE RELAXATION IN PARAMAGNETIC SALTS Vol. 24, No.7 7sec at 230K in manganese simultaneously equations (7)and (8). So the verification 77K and T1 = 1.5 x lO of equation (5’) can be considered as reasonable. nitrate. A strong exchange interaction would prevent (iii) The values of 14 are found to be higher than the variation of T 1 with temperature, as it is seen for one. According to Huang theory, this means the optical instance in the unidimensional paramagnet TMMC. So phonons play no role in the relaxation. Confirmation of the first explanation is ruled out. On the other hand, the this has been looked for in the temperature dependence second explanation see. s to be more reasonable. Only of T, which we have measured in MnC12, 4H20. Over the calculation of i4 woL d allow to conclude.Unfortunthe whole range 45 to 300 K we have observed: ately, the lack of informati ~nconcerning the microscopic data makes such a calculation impossible. 1 T2 (Fig. 3). Ti The variation is clearly slower than an exponential 8. CONCLUSION law and the optical phonons seem to play no role here. At 77 K the field dependence T, (H 0) has been The observed temperature dependence is normal for determined in seven paramagnetic salts using a resoT< o~ lOOK. On the0Drange 45
—
REFERENCES 1.
SCOTT P.L & JEFFRIES C.D.,Phys. Rev. 127, 32 (1962).
2. 3.
ORBACH R.,F~oc.Roy. Soc. A264, 458 (1961). ORBACH R. & BLUME M.,Phys. Rev. Lett. 8,478 (1962).
4.
HUANG C.Y.,Phys. Rev. 154,2215 (1967).
5. 6.
BRONS F., Thesis, Groningen (1938). V~NVLECK J.H., Phys. Rev. 57, 426 (1940).
7. 8.
ORBACH R.,F~oc.Roy. Soc. A264, 485 (1961). HUANG C.Y.,Phys. Rev. 161,272(1967).
9. 10. 11.
ORBACH R.,F~oc.Phys. Soc. 77, 821 (1961). MARCHAND R.L. & STAPLETON H.J., Phys. Rev. B9, 14(1974). ABLART G., BOUJOL P., LOPEZ P. & PESCIA J., Solid State Commun. 17, 1085 (1975).
12. 13.
SING A. & SAPP R.C.,Phys. Rev. 5, 1688 (1972). GORTER C.J. & SMITS L.J., Tables de constantes et données numériques, Masson et Cie (1957).
14.
GOURDON J.C., VIGOUROUIX B. & PESCIA J.,Phys. Lett. 45A, 69(1973).
15. 16.
NOGATCHEWSKY M., These 3éme cycle, Toulouse (1977). PESCIA J., Ann. Phys. 10, 389 (1965).
Vol. 24, No.7
SPIN-LATTICE RELAXATION IN PARAMAGNETIC SALTS
497
17. 18.
ANDERSON P.W. &WEISS P.R., Rev. Mod. Phys. 25, 269 (1953). POUW C.LM. & VAN DUYNEVELDT AJ., Physics 83B, 267 (1976).
19.
KONIG E., Landolt Bornstein in Group 11, VoL 2 Magnetic Properties. Springer Verlag, New York (1966); AMAYA K., TOKUNAGA Y., KURAMITSU Y. & HASEDA T.,J. Phy& Soc. Japan 33,49(1972). PAKE G.E., Paramagnetic Resonance. Benjamin, New York (1962).
20.