- Email: [email protected]
27Al NMR relaxation studies of an emerald single crystal
PERGAMON
Solid State Communications 114 (2000) 311–314 www.elsevier.com/locate/ssc
27
Al NMR relaxation studies of an emerald single crystal
I.G. Kim 1,a, T.H. Yeom a,*, S.H. Choh b, K.S. Hong c, Y.M. Yu d, E.S. Choi e a
Department of Physics, Chongju University, Chongju 360-764, South Korea b Department of Physics, Korea University, Seoul 136-701, South Korea c Seoul Branch, Korea Basic Science Institute, Seoul 136-701, South Korea d Korea Research Institute of Chemical Technology, Taejeon 305-308, South Korea e Korea Institute of Ceramic Engineering and Technology, Seoul 153-023, South Korea Received 26 November 1999; received in revised form 26 January 2000; accepted 3 February 2000 by H. Akai
Abstract The nuclear magnetic resonance of the 27Al nucleus in an emerald (Be3Al2Si6O18:Cr 3⫹) single crystal grown by the flux method has been investigated in the temperature range of 120–420 K. The nuclear quadrupole coupling constant e2 qQ=h and asymmetry parameter (h ), were determined to be 3:123 ^ 0:005 MHz and 0:0076 ^ 0:0064; respectively, at room temperature. There was no appreciable temperature dependence of these parameters. The spin–lattice relaxation mechanisms were investigated in the whole temperature range, and the result revealed that the Raman process was dominant above 200 K, and the contribution from the paramagnetic impurity became much larger than the quadrupole relaxation below 200 K. This is confirmed by analyzing the activation energy of the paramagnetic Cr 3⫹ impurity. 䉷 2000 Published by Elsevier Science Ltd. All rights reserved. Keywords: C. Crystal structure and symmetry; D. Spin dynamics; E. Nuclear resonances
1. Introduction The crystal beryl (3BeO·Al2O3·6SiO2) becomes an emerald when a part of Al 3⫹ ions are substituted by Cr 3⫹ ions. The crystal has a potential for solid-state lasers and masers in high microwave frequency applications [1,2]. Therefore, the study of local site symmetry and the dynamics of the host nuclei in the crystal are very important. Magnetic resonance techniques are powerful for the microscopic information of crystals. The parameters of quadrupole Hamiltonian of 27Al nucleus in an emerald single crystal was determined for the first time by Brown and Williams [3,4]. In addition, there have been some electron paramagnetic resonance (EPR) studies on emerald, i.e. antiferromagnetic exchange coupling study [5], spin–lattice relaxation of Cr 3⫹ EPR at 22.2 GHz in the temperature range of 1.5–45 K [6], a study
* Corresponding author. Tel.: ⫹82-431-229-8555; fax: ⫹82-431229-8432. E-mail address: [email protected] (T.H. Yeom). 1 Present address: The 13th group of National Research Institute for Inorganic Materials, Tsukuba, Ibaraki 305-0044, Japan.
of the ESR line widths of the inter-doublet [7], and Cr 3⫹ and Fe 3⫹ ESR studies [8–10], and others. In this study, we have investigated the spin–lattice relaxation of 27Al nucleus in an emerald single crystal in the temperature range of 120–420 K. The effect of the paramagnetic Cr 3⫹ impurity on the 27Al relaxation is also analyzed by using a nuclear magnetic resonance (NMR) method.
2. Experimental aspects The sample was grown by the flux method using Li2O– MoO3 and Li2O–V2O5 series flux [11]. Its structure is hexagonal with the space group P6=mcc D26h ; and it has two molecules in a unit cell [12,13]. The lattice constants are and c 9:192 A at room temperature [14]. In a 9:215 A an emerald single crystal, SiO4 tetrahedra share oxygen atoms to form Si6O18 rings with each Al atom linked to six Si6O18 rings. And all aluminum sites have equivalent surroundings. The symmetry at each aluminum site is three-fold, and its symmetry axis runs parallel to the crystallographic c-axis. We measured the angular dependence of 27Al NMR
0038-1098/00/$ - see front matter 䉷 2000 Published by Elsevier Science Ltd. All rights reserved. PII: S0038-109 8(00)00061-2
312
I.G. Kim et al. / Solid State Communications 114 (2000) 311–314
Fig. 1. Rotation patterns of 27Al NMR in the ca- and aa-plane of the Be3Al2Si6O18:Cr 3⫹ single crystal at room temperature. The solid circles are experimental data and the lines are simulated ones by using the best-fit parameters obtained with the analyses by employing the EPR-NMR program.
spectra on the crystallographic aa- and ca-plane in the temperature range, 120–420 K, by employing a Bruker MSL 200S FT-NMR spectrometer. The external magnetic field was 4.7 T, and the carrier frequency 52.114 MHz. The parameters of nuclear quadrupole Hamiltonian were fit with the assistance of a computer program EPR-NMR v.6.0 [15]. The spin–lattice relaxation time (T1) of 27Al was measured by using a saturation–recovery method [16], in the same temperature range. The pulse sequence was p/2–t–p/2, where t, the delay time, was varied. To excite all the transition lines, we set the orientation of the applied magnetic field to about c ⫹ 54⬚ in the ca-plane. This angle is an approximate magic angle and all the transition lines nearly collapse at this angle, as will be shown later.
3. Results and discussion The five resonance lines were recorded at each orientation of the applied magnetic field. They are due to the quadrupole interaction of the 27Al nucleus (the nuclear spin I 5=2) with the electric field gradient (EFG) under a strong magnetic field. The NMR spectra show a large angular dependence in the crystallographic ca-plane but almost independent in the aa-plane as shown in Fig. 1. From the angular dependence, we determined the nuclear quadrupole coupling constant e2 qQ=h and asymmetry parameter (h ) at room temperature, 3:123 ^ 0:005 MHz and 0:0076 ^ 0:0064; respectively. h is nearly zero, and it implies that the 27Al nucleus is in the local site with axial symmetry. The results of the local site symmetry coincide with the crystallography of the emerald. The principal Z-axis of the EFG tensor is the same as the c-axis, and the X and Y axes are in the aa-plane. These results are nearly consistent with the reports of Brown and Williams within experimental accuracy [4]. From our measurement, it was shown that the parameters hardly changed in the whole temperature range
Fig. 2. For 27Al nucleus in emerald single crystal: (a) the magnetization (represented by the signal intensity in logarithmic scale) vs. delay time, t; and (b) the magnetization recovery curves at 140 and 380 K. M0 is the saturated magnetization when the delay time t is long enough.
we covered. This implies that the local structure around the aluminum atom remains intact. The spin–lattice relaxation time (T1) was measured in the same temperature range, by employing a saturation–recovery method. When we employed the method, the full saturation was not obtained. Instead, only up to 30% of the saturation value was inverted. It was probably caused by the fast relaxation mechanism due to the paramagnetic Cr 3⫹ impurity present in emerald. The recovery curve of the saturation method shows the paramagnetic relaxation behavior in the short delay time (t) region. The relaxation
I.G. Kim et al. / Solid State Communications 114 (2000) 311–314
Fig. 3. The temperature dependence of the spin–lattice relaxation time of the 27Al nucleus in the emerald crystal. The activation energies are obtained from each slope by using Eq. (2).
processes are shown in Fig. 2 for 140 and 380 K. In the figure, the fast decreasing part in the relaxation process which is not covered by the straight line (a) and the linearly increasing part in recovery curve (b) represents the paramagnetic relaxation mechanism described by Blumberg for the case of diffusion-limited relaxation [17]. The plot for 140 K is shifted upward for distinguishing in Fig. 2(b). To obtain the T1 value, the equation M ∞ ⫺ M t A exp ⫺t=T1
1
was used as a fitting function. As shown in Fig. 3, the relaxation time increases until about 200 K, and then decreases. At the temperature above 200 K, the relaxation rate is proportional to the square of temperature, and is
313
similar to the case when relaxation occurred by the two phonon process, the so-called Raman process, via the emission or absorption of two phonons, or absorption of one phonon followed by an emission of another. The Raman process provides the relaxation rate proportional to the square of temperature in the high temperature limit [18– 21]. There are also the direct process and the Orbach process [19]. However, the direct process is achieved via absorption or emission of one-phonon, and in that case, the spin–lattice relaxation rate 1/T1 is proportional to the square of the Larmor frequency of the nucleus, v20 ; and to the absolute temperature T for kB T=បv0 q 1 [22]. The Orbach process is another two-phonon process, but it provides an exponential behavior with temperature in relaxation rate. Since ln T1 is nearly proportional to the reciprocal temperature, if we take an approximation that the T1 is linearly dependent on the correlation time t , then we can extract out an activation energy of the nucleus by using the following Arrhenius equation:
t t0 exp Ea =kT:
2
The activation energies Ea obtained above and below 200 K are 0:053 ^ 0:005 and 0:038 ^ 0:007 eV; respectively. The smaller activation energy below 200 K implies that there is a preferable relaxation mechanism at lower temperature. Below 200 K, the paramagnetic contribution to the relaxation mechanism becomes dominant over the quadrupole relaxation induced by thermally activated phonons. In this case, the correlation time has the following relation with a constant C [23]:
t⫺1 r Cr ⫺6
3
where r is the distance between the resonant nucleus and the paramagnetic impurity. And for powder sample, " # ti 7ti 2 2 2 2 C 5 gp gn ប J J ⫹ 1 : 4 ⫹ 1 ⫹ v02 ti2 3 1 ⫹ ve2 t2i Here, the g n is the gyro-magnetic ratio of the nucleus, and J, g p, t i, and v e are the angular momentum, gyro-magnetic ratio, spin–lattice relaxation time, and Larmor frequency of the paramagnetic ion, respectively. There are three possible cases of nuclear relaxation by paramagnetic impurities [17]: the case of no spin diffusion, the case of diffusion-limited relaxation, and the case of rapid diffusion. Among them, the case of no spin diffusion has no physical meaning in practice. Nonetheless, if the delay time t is too short for the spin diffusion to occur, then the relaxation process behaves as if it is the case of no spin diffusion. That is, the magnetization obeys the following equation [17]: Mz t ⯝ 4p3=2 =3NC1=2 t1=2 :
Fig. 4. The semilog plot of the slope of the line in Fig. 2(b) vs. the reciprocal temperature in the whole temperature region.
5
Here, N is the concentration of the paramagnetic ions. We can calculate the slopes for each temperature from the recovery curve such as Fig. 2(b), by taking the straight line corresponding to the transient region, i.e. the region
314
I.G. Kim et al. / Solid State Communications 114 (2000) 311–314
of short t. As shown in Eq. (4), the constant C is closely related to the spin–lattice relaxation time of the paramagnetic ion. In general, v e is greater than v 0 by several hundred times. Since, in addition, t i has the value of ms–s—for example, the spin–lattice relaxation time of Cr 3⫹ ion in the emerald crystal is about 0.6 ms at 45 K according to the report of Martirosyan et al. [6]—we took an approximation that C is inversely proportional to t i from the first term in the parenthesis of Eq. (4). Consequently, by inspecting Eqs. (2)–(5), we could find out that the slope of the straight line in Fig. 2(b) is proportional to the square root of C and again to exp ⫺Ea =2kT: In Fig. 4, the vertical axis is the slope in natural logarithm and the horizontal axis is the reciprocal temperature in Kelvin. From the gradient of the above graph, we obtained the activation energy of 0:031 ^ 0:004 eV; which is the energy required to activate the paramagnetic impurity Cr 3⫹ and to allow it to participate in the spin–lattice relaxation mechanism of the 27Al nucleus. This value is very close to the activation energy 0:038 ^ 0:007 eV obtained from the temperature dependence of the spin–lattice relaxation time of the 27Al nucleus below 200 K, as shown in Fig. 3. It implies that those two activation energies are closely related and describe the same relaxation mechanism. As a result, the relaxation mechanism below 200 K is dominated by the paramagnetic relaxation. In other words, we could find out the activation energy of the paramagnetic impurity ions by measuring the magnetization recovery curve of the 27Al nucleus by NMR. 4. Summary From the 27Al NMR relaxation study in the emerald single crystal, the spin–lattice relaxation mechanisms of 27Al were investigated. While the quadrupolar interaction through the Raman process is dominant above 200 K, the paramagnetic relaxation is dominant below 200 K. This was confirmed by the analysis of the activation energy of the paramagnetic ions in the transient region as well as those of the 27Al nucleus. We also investigated the nuclear magnetic resonance of the 27Al nucleus in the temperature range 120–420 K. A set of 27Al NMR spectra was recorded. From the angular dependence of the spectra, the quadrupole coupling constant and the asymmetry parameter of the 27Al nucleus were determined. The local site symmetry of 27Al nucleus is also determined to be three-fold. This result is consistent with the crystal symmetry of emerald. There is no appreciable temperature dependence of the e2 qQ=h and h in the whole temperature range studied. These results show that the crystal is very stable in the investigated temperature range.
Acknowledgements This work was supported by the Korea Science and Engineering Foundation through the RCDAMP at Pusan National University (1997–2000). One of the authors (I.G. Kim) is grateful to the support from KOSEF through Internship of Research Fellow (from May 1999). Also, the authors thank Dr J.K. Jung for his helpful discussion.
References [1] J. Buchert, A. Katz, R.R. Alfano, IEEE J. Quantum Electron. QE-19 (1983) 1477. [2] S.T. Lai, AIP Conf. Proc., USA, Vol. 160, 1986, p. 128. [3] L.C. Brown, D. Williams, Phys. Rev. 95 (1954) 1110. [4] L.C. Brown, D. Williams, J. Chem. Phys. 24 (1956) 751. [5] A. Edgar, D.R. Hutton, J. Phys. C 11 (1978) 5051. [6] R.M. Martirosyan, M.O. Manvelyan, G.A. Mnatsakanyan, V.S. Sevast’yanov, Sov. Phys.—Solid State 22 (1980) 563. [7] A.A. Akhumyah, R.M. Martirosyan, N.G. Pogosyan, Sov. Tech. Phys. Lett. 7 (1981) 371. [8] R.M. Martirosyan, M.O. Manvelyan, G.A. Mnatsakanyan, Sov. Phys.—Solid State 25 (1983) 904. [9] H. Ohkura, H. Hashimoto, Y. Mori, Y. Chiba, S. Isotarri, Jpn J. Appl. Phys. Part 1 26 (1987) 1422. [10] M.O. Manvelyan, R.M. Martirosyan, Sov, J. Contemp. Phys. 23 (1988) 279. [11] E.S. Choi, M.K. Kim, J.M. Lee, Y.P. Ahn, C.K. Seoh, C.J. Ahn, J. Kor. Assoc. Cryst. Growth 6 (1996) 44. [12] W.L. Bragg, J. West, Proc. R. Soc. Ser. A 111 (1926) 691. [13] R.W.G. Wyckoff, The Structure of Crystal, The Chemical Catalog Company, New York, 1931. [14] Powder Diffraction File, International Centre for Diffraction Data, Swarthmore, PA, 1981. Card No. 9-430. [15] M.J. Mombourquette, J.A. Weil, D.G. McGavin, Operation Instructions for Computer Program EPR-NMR, version 6.0, University of Saskachewan, Canada, 1995. [16] E. Fukushima, S.B.W. Roeder, Experimental Pulse NMR, A Nuts and Bolts Approach, Addison-Wesley, London, 1981 (chap. III). [17] W.E. Blumberg, Phys. Rev. 119 (1960) 79. [18] J. Van Kranendonk, Physica 20 (1954) 781. [19] A. Abragam, The Principles of Nuclear Magnetism, Oxford University Press, Oxford, 1961 (chaps I and IX). [20] R.L. Mieher, Phys. Rev. 125 (1962) 1537. [21] T.H. Yeom, K.S. Hong, I. Yu, H.W. Shin, S.H. Choh, J. Appl. Phys. 82 (1997) 2472. [22] T.H. Yeom, K.T. Han, S.H. Choh, K.S. Hong, J. Korean Phys. Soc. 28 (1995) 113. [23] T.T. Phua, B.J. Beaudry, D.T. Peterson, D.R. Torgeson, R.G. Barnes, M. Belhoul, G.A. Styles, E.F.W. Seymour, Phys. Rev. B 28 (1983) 6227.
Solid State Communications 114 (2000) 311–314 www.elsevier.com/locate/ssc
27
Al NMR relaxation studies of an emerald single crystal
I.G. Kim 1,a, T.H. Yeom a,*, S.H. Choh b, K.S. Hong c, Y.M. Yu d, E.S. Choi e a
Department of Physics, Chongju University, Chongju 360-764, South Korea b Department of Physics, Korea University, Seoul 136-701, South Korea c Seoul Branch, Korea Basic Science Institute, Seoul 136-701, South Korea d Korea Research Institute of Chemical Technology, Taejeon 305-308, South Korea e Korea Institute of Ceramic Engineering and Technology, Seoul 153-023, South Korea Received 26 November 1999; received in revised form 26 January 2000; accepted 3 February 2000 by H. Akai
Abstract The nuclear magnetic resonance of the 27Al nucleus in an emerald (Be3Al2Si6O18:Cr 3⫹) single crystal grown by the flux method has been investigated in the temperature range of 120–420 K. The nuclear quadrupole coupling constant e2 qQ=h and asymmetry parameter (h ), were determined to be 3:123 ^ 0:005 MHz and 0:0076 ^ 0:0064; respectively, at room temperature. There was no appreciable temperature dependence of these parameters. The spin–lattice relaxation mechanisms were investigated in the whole temperature range, and the result revealed that the Raman process was dominant above 200 K, and the contribution from the paramagnetic impurity became much larger than the quadrupole relaxation below 200 K. This is confirmed by analyzing the activation energy of the paramagnetic Cr 3⫹ impurity. 䉷 2000 Published by Elsevier Science Ltd. All rights reserved. Keywords: C. Crystal structure and symmetry; D. Spin dynamics; E. Nuclear resonances
1. Introduction The crystal beryl (3BeO·Al2O3·6SiO2) becomes an emerald when a part of Al 3⫹ ions are substituted by Cr 3⫹ ions. The crystal has a potential for solid-state lasers and masers in high microwave frequency applications [1,2]. Therefore, the study of local site symmetry and the dynamics of the host nuclei in the crystal are very important. Magnetic resonance techniques are powerful for the microscopic information of crystals. The parameters of quadrupole Hamiltonian of 27Al nucleus in an emerald single crystal was determined for the first time by Brown and Williams [3,4]. In addition, there have been some electron paramagnetic resonance (EPR) studies on emerald, i.e. antiferromagnetic exchange coupling study [5], spin–lattice relaxation of Cr 3⫹ EPR at 22.2 GHz in the temperature range of 1.5–45 K [6], a study
* Corresponding author. Tel.: ⫹82-431-229-8555; fax: ⫹82-431229-8432. E-mail address: [email protected] (T.H. Yeom). 1 Present address: The 13th group of National Research Institute for Inorganic Materials, Tsukuba, Ibaraki 305-0044, Japan.
of the ESR line widths of the inter-doublet [7], and Cr 3⫹ and Fe 3⫹ ESR studies [8–10], and others. In this study, we have investigated the spin–lattice relaxation of 27Al nucleus in an emerald single crystal in the temperature range of 120–420 K. The effect of the paramagnetic Cr 3⫹ impurity on the 27Al relaxation is also analyzed by using a nuclear magnetic resonance (NMR) method.
2. Experimental aspects The sample was grown by the flux method using Li2O– MoO3 and Li2O–V2O5 series flux [11]. Its structure is hexagonal with the space group P6=mcc D26h ; and it has two molecules in a unit cell [12,13]. The lattice constants are and c 9:192 A at room temperature [14]. In a 9:215 A an emerald single crystal, SiO4 tetrahedra share oxygen atoms to form Si6O18 rings with each Al atom linked to six Si6O18 rings. And all aluminum sites have equivalent surroundings. The symmetry at each aluminum site is three-fold, and its symmetry axis runs parallel to the crystallographic c-axis. We measured the angular dependence of 27Al NMR
0038-1098/00/$ - see front matter 䉷 2000 Published by Elsevier Science Ltd. All rights reserved. PII: S0038-109 8(00)00061-2
312
I.G. Kim et al. / Solid State Communications 114 (2000) 311–314
Fig. 1. Rotation patterns of 27Al NMR in the ca- and aa-plane of the Be3Al2Si6O18:Cr 3⫹ single crystal at room temperature. The solid circles are experimental data and the lines are simulated ones by using the best-fit parameters obtained with the analyses by employing the EPR-NMR program.
spectra on the crystallographic aa- and ca-plane in the temperature range, 120–420 K, by employing a Bruker MSL 200S FT-NMR spectrometer. The external magnetic field was 4.7 T, and the carrier frequency 52.114 MHz. The parameters of nuclear quadrupole Hamiltonian were fit with the assistance of a computer program EPR-NMR v.6.0 [15]. The spin–lattice relaxation time (T1) of 27Al was measured by using a saturation–recovery method [16], in the same temperature range. The pulse sequence was p/2–t–p/2, where t, the delay time, was varied. To excite all the transition lines, we set the orientation of the applied magnetic field to about c ⫹ 54⬚ in the ca-plane. This angle is an approximate magic angle and all the transition lines nearly collapse at this angle, as will be shown later.
3. Results and discussion The five resonance lines were recorded at each orientation of the applied magnetic field. They are due to the quadrupole interaction of the 27Al nucleus (the nuclear spin I 5=2) with the electric field gradient (EFG) under a strong magnetic field. The NMR spectra show a large angular dependence in the crystallographic ca-plane but almost independent in the aa-plane as shown in Fig. 1. From the angular dependence, we determined the nuclear quadrupole coupling constant e2 qQ=h and asymmetry parameter (h ) at room temperature, 3:123 ^ 0:005 MHz and 0:0076 ^ 0:0064; respectively. h is nearly zero, and it implies that the 27Al nucleus is in the local site with axial symmetry. The results of the local site symmetry coincide with the crystallography of the emerald. The principal Z-axis of the EFG tensor is the same as the c-axis, and the X and Y axes are in the aa-plane. These results are nearly consistent with the reports of Brown and Williams within experimental accuracy [4]. From our measurement, it was shown that the parameters hardly changed in the whole temperature range
Fig. 2. For 27Al nucleus in emerald single crystal: (a) the magnetization (represented by the signal intensity in logarithmic scale) vs. delay time, t; and (b) the magnetization recovery curves at 140 and 380 K. M0 is the saturated magnetization when the delay time t is long enough.
we covered. This implies that the local structure around the aluminum atom remains intact. The spin–lattice relaxation time (T1) was measured in the same temperature range, by employing a saturation–recovery method. When we employed the method, the full saturation was not obtained. Instead, only up to 30% of the saturation value was inverted. It was probably caused by the fast relaxation mechanism due to the paramagnetic Cr 3⫹ impurity present in emerald. The recovery curve of the saturation method shows the paramagnetic relaxation behavior in the short delay time (t) region. The relaxation
I.G. Kim et al. / Solid State Communications 114 (2000) 311–314
Fig. 3. The temperature dependence of the spin–lattice relaxation time of the 27Al nucleus in the emerald crystal. The activation energies are obtained from each slope by using Eq. (2).
processes are shown in Fig. 2 for 140 and 380 K. In the figure, the fast decreasing part in the relaxation process which is not covered by the straight line (a) and the linearly increasing part in recovery curve (b) represents the paramagnetic relaxation mechanism described by Blumberg for the case of diffusion-limited relaxation [17]. The plot for 140 K is shifted upward for distinguishing in Fig. 2(b). To obtain the T1 value, the equation M ∞ ⫺ M t A exp ⫺t=T1
1
was used as a fitting function. As shown in Fig. 3, the relaxation time increases until about 200 K, and then decreases. At the temperature above 200 K, the relaxation rate is proportional to the square of temperature, and is
313
similar to the case when relaxation occurred by the two phonon process, the so-called Raman process, via the emission or absorption of two phonons, or absorption of one phonon followed by an emission of another. The Raman process provides the relaxation rate proportional to the square of temperature in the high temperature limit [18– 21]. There are also the direct process and the Orbach process [19]. However, the direct process is achieved via absorption or emission of one-phonon, and in that case, the spin–lattice relaxation rate 1/T1 is proportional to the square of the Larmor frequency of the nucleus, v20 ; and to the absolute temperature T for kB T=បv0 q 1 [22]. The Orbach process is another two-phonon process, but it provides an exponential behavior with temperature in relaxation rate. Since ln T1 is nearly proportional to the reciprocal temperature, if we take an approximation that the T1 is linearly dependent on the correlation time t , then we can extract out an activation energy of the nucleus by using the following Arrhenius equation:
t t0 exp Ea =kT:
2
The activation energies Ea obtained above and below 200 K are 0:053 ^ 0:005 and 0:038 ^ 0:007 eV; respectively. The smaller activation energy below 200 K implies that there is a preferable relaxation mechanism at lower temperature. Below 200 K, the paramagnetic contribution to the relaxation mechanism becomes dominant over the quadrupole relaxation induced by thermally activated phonons. In this case, the correlation time has the following relation with a constant C [23]:
t⫺1 r Cr ⫺6
3
where r is the distance between the resonant nucleus and the paramagnetic impurity. And for powder sample, " # ti 7ti 2 2 2 2 C 5 gp gn ប J J ⫹ 1 : 4 ⫹ 1 ⫹ v02 ti2 3 1 ⫹ ve2 t2i Here, the g n is the gyro-magnetic ratio of the nucleus, and J, g p, t i, and v e are the angular momentum, gyro-magnetic ratio, spin–lattice relaxation time, and Larmor frequency of the paramagnetic ion, respectively. There are three possible cases of nuclear relaxation by paramagnetic impurities [17]: the case of no spin diffusion, the case of diffusion-limited relaxation, and the case of rapid diffusion. Among them, the case of no spin diffusion has no physical meaning in practice. Nonetheless, if the delay time t is too short for the spin diffusion to occur, then the relaxation process behaves as if it is the case of no spin diffusion. That is, the magnetization obeys the following equation [17]: Mz t ⯝ 4p3=2 =3NC1=2 t1=2 :
Fig. 4. The semilog plot of the slope of the line in Fig. 2(b) vs. the reciprocal temperature in the whole temperature region.
5
Here, N is the concentration of the paramagnetic ions. We can calculate the slopes for each temperature from the recovery curve such as Fig. 2(b), by taking the straight line corresponding to the transient region, i.e. the region
314
I.G. Kim et al. / Solid State Communications 114 (2000) 311–314
of short t. As shown in Eq. (4), the constant C is closely related to the spin–lattice relaxation time of the paramagnetic ion. In general, v e is greater than v 0 by several hundred times. Since, in addition, t i has the value of ms–s—for example, the spin–lattice relaxation time of Cr 3⫹ ion in the emerald crystal is about 0.6 ms at 45 K according to the report of Martirosyan et al. [6]—we took an approximation that C is inversely proportional to t i from the first term in the parenthesis of Eq. (4). Consequently, by inspecting Eqs. (2)–(5), we could find out that the slope of the straight line in Fig. 2(b) is proportional to the square root of C and again to exp ⫺Ea =2kT: In Fig. 4, the vertical axis is the slope in natural logarithm and the horizontal axis is the reciprocal temperature in Kelvin. From the gradient of the above graph, we obtained the activation energy of 0:031 ^ 0:004 eV; which is the energy required to activate the paramagnetic impurity Cr 3⫹ and to allow it to participate in the spin–lattice relaxation mechanism of the 27Al nucleus. This value is very close to the activation energy 0:038 ^ 0:007 eV obtained from the temperature dependence of the spin–lattice relaxation time of the 27Al nucleus below 200 K, as shown in Fig. 3. It implies that those two activation energies are closely related and describe the same relaxation mechanism. As a result, the relaxation mechanism below 200 K is dominated by the paramagnetic relaxation. In other words, we could find out the activation energy of the paramagnetic impurity ions by measuring the magnetization recovery curve of the 27Al nucleus by NMR. 4. Summary From the 27Al NMR relaxation study in the emerald single crystal, the spin–lattice relaxation mechanisms of 27Al were investigated. While the quadrupolar interaction through the Raman process is dominant above 200 K, the paramagnetic relaxation is dominant below 200 K. This was confirmed by the analysis of the activation energy of the paramagnetic ions in the transient region as well as those of the 27Al nucleus. We also investigated the nuclear magnetic resonance of the 27Al nucleus in the temperature range 120–420 K. A set of 27Al NMR spectra was recorded. From the angular dependence of the spectra, the quadrupole coupling constant and the asymmetry parameter of the 27Al nucleus were determined. The local site symmetry of 27Al nucleus is also determined to be three-fold. This result is consistent with the crystal symmetry of emerald. There is no appreciable temperature dependence of the e2 qQ=h and h in the whole temperature range studied. These results show that the crystal is very stable in the investigated temperature range.
Acknowledgements This work was supported by the Korea Science and Engineering Foundation through the RCDAMP at Pusan National University (1997–2000). One of the authors (I.G. Kim) is grateful to the support from KOSEF through Internship of Research Fellow (from May 1999). Also, the authors thank Dr J.K. Jung for his helpful discussion.
References [1] J. Buchert, A. Katz, R.R. Alfano, IEEE J. Quantum Electron. QE-19 (1983) 1477. [2] S.T. Lai, AIP Conf. Proc., USA, Vol. 160, 1986, p. 128. [3] L.C. Brown, D. Williams, Phys. Rev. 95 (1954) 1110. [4] L.C. Brown, D. Williams, J. Chem. Phys. 24 (1956) 751. [5] A. Edgar, D.R. Hutton, J. Phys. C 11 (1978) 5051. [6] R.M. Martirosyan, M.O. Manvelyan, G.A. Mnatsakanyan, V.S. Sevast’yanov, Sov. Phys.—Solid State 22 (1980) 563. [7] A.A. Akhumyah, R.M. Martirosyan, N.G. Pogosyan, Sov. Tech. Phys. Lett. 7 (1981) 371. [8] R.M. Martirosyan, M.O. Manvelyan, G.A. Mnatsakanyan, Sov. Phys.—Solid State 25 (1983) 904. [9] H. Ohkura, H. Hashimoto, Y. Mori, Y. Chiba, S. Isotarri, Jpn J. Appl. Phys. Part 1 26 (1987) 1422. [10] M.O. Manvelyan, R.M. Martirosyan, Sov, J. Contemp. Phys. 23 (1988) 279. [11] E.S. Choi, M.K. Kim, J.M. Lee, Y.P. Ahn, C.K. Seoh, C.J. Ahn, J. Kor. Assoc. Cryst. Growth 6 (1996) 44. [12] W.L. Bragg, J. West, Proc. R. Soc. Ser. A 111 (1926) 691. [13] R.W.G. Wyckoff, The Structure of Crystal, The Chemical Catalog Company, New York, 1931. [14] Powder Diffraction File, International Centre for Diffraction Data, Swarthmore, PA, 1981. Card No. 9-430. [15] M.J. Mombourquette, J.A. Weil, D.G. McGavin, Operation Instructions for Computer Program EPR-NMR, version 6.0, University of Saskachewan, Canada, 1995. [16] E. Fukushima, S.B.W. Roeder, Experimental Pulse NMR, A Nuts and Bolts Approach, Addison-Wesley, London, 1981 (chap. III). [17] W.E. Blumberg, Phys. Rev. 119 (1960) 79. [18] J. Van Kranendonk, Physica 20 (1954) 781. [19] A. Abragam, The Principles of Nuclear Magnetism, Oxford University Press, Oxford, 1961 (chaps I and IX). [20] R.L. Mieher, Phys. Rev. 125 (1962) 1537. [21] T.H. Yeom, K.S. Hong, I. Yu, H.W. Shin, S.H. Choh, J. Appl. Phys. 82 (1997) 2472. [22] T.H. Yeom, K.T. Han, S.H. Choh, K.S. Hong, J. Korean Phys. Soc. 28 (1995) 113. [23] T.T. Phua, B.J. Beaudry, D.T. Peterson, D.R. Torgeson, R.G. Barnes, M. Belhoul, G.A. Styles, E.F.W. Seymour, Phys. Rev. B 28 (1983) 6227.