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NMR of 27 Al in paramagnetic single-crystal DyAl2
Volume 95A, number 5
PHYSICS LETTERS
2 May 1983
NMR OF 27A1 IN PARAMAGNETIC SINGLE-CRYSTAL DyAI2 Y.B. BARASH, J. BARAK Soreq Nuclear Research Centre, Yavne, Israel
and A. GRAYEVSKY The Racah Institute of Physics, The Hebrew University, Jerusalem, Israel
Received 14 December 1982
NMR of 27A1was observed in a spherical single crystal of DyAI2 in the paramagnetic state. The well split quadrupole spectrum yielded a quadrupole frequency VQ = 561 kHz and hyperfine constant HhPf -~ -3.17 kOe/n B. The anisotropy of the spectrum is well explained by the dipolar interaction.
A detailed study of the nuclear magnetic resonance (NMR) spectra of 27Al in powder samples of DyAl 2 and GdA12 in the paramagnetic phase has recently been reported [ I ]. In analyzing NMR powder patterns of magnetic materials one encounters two main problems. First it is difficult to interpret the spectrum directly, without comparing it with spectra simulated by a computer code which usually considers the hamiltonian of the nuclear spin to second-order perturbation only [ 1]. The second problem is the extensive line broadening in a powder, resulting from the distribution of the demagnetization fields due to the different shapes of the particles [ 1]. A way around these problems is to use a single crystal in the form of a well polished sphere [2]. Such a sample is much more difficult to prepare than a powder sample. Furthermore, the NMR signal from a bulk metal is expected to be small due to the "skin effect". Coping with these difficulties, however, is rewarded as the spectrum yields the parameters of the nuclear spin hamiltonian directly and precisely. In addition, in a sphere the demagnetization field exactly compensates for the Lorentz field, thus the contribution of the dipolar interaction is reduced to the local dipolar field. The sample used in our experiments was a DyA12 single crystal grown by the Bridgman technique and mechanically shaped into a polished sphere with a 252
diameter of 6.2 ram. Unfortunately a small piece cleaved out of the sample and this gave rise to line broadening. The cleavage plane, however, was a (111) plane and it helped in orienting the sample. DyAl 2 has a cubic Laves phase structure. All the aluminum sites are crystallographically equivalent and their point symmetry is axial (3m), with the symmetry axis parallel to one of the four equivalent [111] directions: group a (axis parallel to [ 111]), group b 1 ( J i l l ] ) , b2 ([1111) and b 3 ([1111) [3]. The equivalence of these four groups is removed by applying a magnetic field H 0 in a general, low-symmetry direction, In certain H 0 directions, however, some or all of the groups are equivalent and the NMR spectrum is easier to interpret. When H 0 is parallel to [ 111 ] all three b i groups (i = 1, 2, 3) are equivalent with an angle 0 between their symmetry axes and H 0 such that kcos 0[ = 1/3. Group a then has 0 = 0, an angle for which the hamiltonian is diagonal and the quadrupole splitting is maximal. We used a conventional spin-echo spectrometer operating at a fixed frequency v 0 = 17 MHz. The 27Al spin-echo signal as a function o f H 0 was detected using a boxcar integrator and accumulated by a Fabritek 1070 signal averager. Fig. i shows the spectrum obtained after 30 h of signal accumulation for H 0 parallel to [ 111 ]. The sample was at a temperature T = 330 K
0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
Volume 95A, number 5
PHYSICS LETTERS
where for 27AI we take the standard ~0 = 1109.4 Hz/Oe. The system is thus characterized by two ~ values: f o r H 0 in the z direction, ~11 = ~0( 1 + K p + 2Kax) and perpendicular to z, ~± = ~0(1 + Kp - Kax ). When H 0 is parallel to [111], hamiltonian (1) is diagonal for 27A1 nuclei in the a sites and the transitions m ~ m - 1 are observed at fields
T= 330K
rio II [t11] b o 65Cu
o
2 May 1983
o
Ho(O = 0) = [v0 + vQ(m - 1 )]/~ II I
I
15
I
15.5
t6
I 16.5
The calculation is not that simple for the b sites which have Icos 01 = 1/3. Following the similar case of an anisotropic g factor in EPR [4], the effective gyromagnetic ratio as a function of 0 is given by
I 17
H0 (kOe)
Fig. 1. NMRspin-echo spectrum of 27A1 in a paramagnetic
~(0) = ( ~ cos20 + ~2 sin20)l/2 .
single crystal of DyAI2. At first glance, the spectrum appears to be symmetric around the largest line with an unusual quadrupole structure. This " s y m m e t r y " is incidental. As will be shown, the spectrum is a superposition of 5 lines of site a and 5 lines, three times greater in intensity, of sites b with splitting that is about a third of that of the a site lines. The line denoted as "63 Cu" comes from the copper wire of the rf coil and is used for field calibration. In analyzing the spectrum we follow ref. [1 ]. The paramagnetic shift of 27AI in DyA12 has two parts. The isotropic shift Kp originates from the transferred hyperfine interaction with the dysprosium ions. The anisotropic shift Kax is due to the dipolar field induced by these ions or to the anisotropic part of the hyperfine interaction. For the axial symmetry, taking z along the symmetry direction of the AI group under observation, the nuclear spin hamiltonian is given by
~= -hl.~.1t
0 + ~hVQ[Iz 2 - ~ I ( I + I)] ,
(1)
where I = 5/2 for 27A1, VQis the quadrupole-frequency constant and ~ is the effective gyromagnetic ratio tensor (in units of Hz/Oe) which includes the influence of the paramagnetic shifts. Since, in the general case, for A1 in the Laves phase structure the dipolar field is not collinear with the external field, the dipolar interaction is represented by a tensor [1,2]. If we take a similar tensor for the anisotropic shift, ~ is given by
~t=~ 0
ii p x0 k
l+Kp-Kax
0
]
0
1 + Kp + 2Kax
0
(3)
,
(4)
The use of this expression is equivalent to the calculation of the vector sum of the external field, the hyperfine field and the dipolar field, as was done for 27A1 in ferromagnetic GdA12 [2]. In the ferromagnetic state the dipolar field is much smaller than the other contributions and it would be a good approximation to consider only the projection of the dipolar field on the direction of the external field (or of the hyperfine field in the zero-field experiments in ferromagnets). The parallel case in the paramagnetic phase is given by Kax <~ 1 + Kp. This is the situation in our experiments at temperatures far from the Curie point [1 ]. In this case ~(0) may be approximated by the usual expression ~(0) = ~0[1 + K p +Kax(3COS20 - 1)] .
(5)
Hamiltonian (1) might be solved for each 0 by diagonalization of the 6 × 6 matrix. For VQ ~ v0, however, it is enough to solve ( I ) to a second-order perturbation [5] and the NMR lines which correspond to the rn ~ m - 1 transition are expected at fields
Ho(O ) = Iv0 + ½VQ(3COS20 - 1)(m - ~) - 8V(2m)~m_l]/~(O),
(6)
where By(2) is the second-order correction of the quadrupole interaction, and for I = 5/2
8v(2-~/2~ 1/2 = (v~/2Vo)(1 - c°s20)( 1 - 9 c ° s 2 0 ) , (2) = 8v(2) ,~ t~3/2~ 1/2 -1/2 -3/2
= (v~/16Vo)(1 -- cos20)(5 -- 21 cos20),
(7)
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Volume 95A, number 5
PHYSICS LETTERS
5(2) v5/2.~3/2 = ~v(_2~/2,_,_5/2
= --(v~/4Vo)(l - cos20)(l - 33 cos20). (7 cont'd) For the a sites (0 = 0) the approximate solution (6) coincides with the exact solution (3). For these sites as well as for the b sites (with Icos01 = ~), 6v(__2)1/2~1/2 = 0. Thus the external fields H~ and H b, corresponding to the central lines (1/2 4,, - 1 / 2 ) of the a and b groups, are both functions of only Kp and Kax and are defined by v0 = Y0H(](1 +Xp + 2Xax), and v0 = $0Hb(1 + Xp - ~Kax ) . From these expressions and fig. 1, one obtains Kax = -0.0058 +- 0.0002 and Kp = -0.0322 -+ 0.0002. The value of VQ may be found directly, using eq. (3), from the splitting between any two consecutive lines of the a group spectrum, or from the b group spectrum. From (7), the/iv(2) are equal for each pair of satellite lines and the distance between the 3/2 ,e, 1/2 and the - 1 / 2 ~, - 3 [2 lines of group b is AH 0 = 2VQHb/3Vo and twice as much is the distance between the 5/2 o 3/2 and the - 3 ] 2 ~ - 5 / 2 lines of this group. All these calculations give VQ = (561 +- 5) kHz. This value is in very good agreement with (557 -+ 5) kHz, found for the powder sample [1]. The hyperfine field constant in the paramagnetic state Hl~f for the single crystal was found by measuring Kp as a function of T. The sample was oriented with H 0 parallel to [ 100]. For this direction all 27A1 sites are equivalent and (since 3 cos20 - 1 = 0) there is only one line at H 0 = v0/~¢0(1 +Kp) with a small additional shift and broadening, both due to the second-order perturbation (7). Using the measured susceptibility of DyAl 2 as a function of T [6], we obtained (from the Kp versus X plot), HPf = -3.17 kOe/n B + 0.5%, where n B is the polarization of each Dy 3+ ion in Bohr magnetons. This is in good agreement with the value H f f = -(3.2 -+ 0.1) kOe/n B cal-
254
2 May 1983
culated from NMR measurements in the ferromagnetic state and with the value found for the paramagnetic powder, HhPf = - ( 3 . 2 4 +- 0.06) kOe/n B [1]. Finally, the value of Kax in the present study is much more precise than that found from the experiments with the powder sample. It is thus of interest to compare the value obtained here with what is expected due to a pure dipolar interaction. The dipolar field constant for the ferromagnetic state was calculated to beHd0 = 5.171 kOe by taking n B = 9.89 [1,3]. The corresponding parameter in the paramagnetic phase may be denoted by Ka0x. The internal field in the ferromagnetic phase is Hin t ==-nBHff = (31.5 + 0.5) kOe. It corresponds to Kp in the paramagnetic phase. We expect a simple relation K 0 / H 0 = KoP/Hin t to hold. For Kp 2. -0.0322, this relation gives Kax = -0.0053. The experimental value Kax = -0.0058 shows a small enhancement of the dipolar field [3] by a factor of e = 1.1. Equivalently the difference between Kax and KOx might be considered as a small anisotropy of the hyperfine interaction with a value of about 10% of the dipolar field for DyAl 2. This is to be compared with 16% found for GdA12 [3]. This research was supported by a grant from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel.
References [1] Y.B. Barash, J. Barak and N. Kaplan, Phys. Rev. B25 (1982) 6616. [ 2] D. Fekete, A. Grayevskey, N. Kaplan and E. Walker, Solid State Commun. 17 (1975) 573. [3] N. Kaplan, E. Dormann, K.H.J. Buschow and D. Lebenbaum, Phys. Rev. B7 (1973) 40. [4] A. Abragam and B. Bleany, Electron paramagnetic resonance of transition ions (Clarendon, Oxford, 1970) p. 135. [5] A. Abragam,The principles of nuclear magnetism(Oxford Univ. Press, London, 1961), [6] B. Barbara and M.F. Rossignol, private communication (1979).
PHYSICS LETTERS
2 May 1983
NMR OF 27A1 IN PARAMAGNETIC SINGLE-CRYSTAL DyAI2 Y.B. BARASH, J. BARAK Soreq Nuclear Research Centre, Yavne, Israel
and A. GRAYEVSKY The Racah Institute of Physics, The Hebrew University, Jerusalem, Israel
Received 14 December 1982
NMR of 27A1was observed in a spherical single crystal of DyAI2 in the paramagnetic state. The well split quadrupole spectrum yielded a quadrupole frequency VQ = 561 kHz and hyperfine constant HhPf -~ -3.17 kOe/n B. The anisotropy of the spectrum is well explained by the dipolar interaction.
A detailed study of the nuclear magnetic resonance (NMR) spectra of 27Al in powder samples of DyAl 2 and GdA12 in the paramagnetic phase has recently been reported [ I ]. In analyzing NMR powder patterns of magnetic materials one encounters two main problems. First it is difficult to interpret the spectrum directly, without comparing it with spectra simulated by a computer code which usually considers the hamiltonian of the nuclear spin to second-order perturbation only [ 1]. The second problem is the extensive line broadening in a powder, resulting from the distribution of the demagnetization fields due to the different shapes of the particles [ 1]. A way around these problems is to use a single crystal in the form of a well polished sphere [2]. Such a sample is much more difficult to prepare than a powder sample. Furthermore, the NMR signal from a bulk metal is expected to be small due to the "skin effect". Coping with these difficulties, however, is rewarded as the spectrum yields the parameters of the nuclear spin hamiltonian directly and precisely. In addition, in a sphere the demagnetization field exactly compensates for the Lorentz field, thus the contribution of the dipolar interaction is reduced to the local dipolar field. The sample used in our experiments was a DyA12 single crystal grown by the Bridgman technique and mechanically shaped into a polished sphere with a 252
diameter of 6.2 ram. Unfortunately a small piece cleaved out of the sample and this gave rise to line broadening. The cleavage plane, however, was a (111) plane and it helped in orienting the sample. DyAl 2 has a cubic Laves phase structure. All the aluminum sites are crystallographically equivalent and their point symmetry is axial (3m), with the symmetry axis parallel to one of the four equivalent [111] directions: group a (axis parallel to [ 111]), group b 1 ( J i l l ] ) , b2 ([1111) and b 3 ([1111) [3]. The equivalence of these four groups is removed by applying a magnetic field H 0 in a general, low-symmetry direction, In certain H 0 directions, however, some or all of the groups are equivalent and the NMR spectrum is easier to interpret. When H 0 is parallel to [ 111 ] all three b i groups (i = 1, 2, 3) are equivalent with an angle 0 between their symmetry axes and H 0 such that kcos 0[ = 1/3. Group a then has 0 = 0, an angle for which the hamiltonian is diagonal and the quadrupole splitting is maximal. We used a conventional spin-echo spectrometer operating at a fixed frequency v 0 = 17 MHz. The 27Al spin-echo signal as a function o f H 0 was detected using a boxcar integrator and accumulated by a Fabritek 1070 signal averager. Fig. i shows the spectrum obtained after 30 h of signal accumulation for H 0 parallel to [ 111 ]. The sample was at a temperature T = 330 K
0 031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
Volume 95A, number 5
PHYSICS LETTERS
where for 27AI we take the standard ~0 = 1109.4 Hz/Oe. The system is thus characterized by two ~ values: f o r H 0 in the z direction, ~11 = ~0( 1 + K p + 2Kax) and perpendicular to z, ~± = ~0(1 + Kp - Kax ). When H 0 is parallel to [111], hamiltonian (1) is diagonal for 27A1 nuclei in the a sites and the transitions m ~ m - 1 are observed at fields
T= 330K
rio II [t11] b o 65Cu
o
2 May 1983
o
Ho(O = 0) = [v0 + vQ(m - 1 )]/~ II I
I
15
I
15.5
t6
I 16.5
The calculation is not that simple for the b sites which have Icos 01 = 1/3. Following the similar case of an anisotropic g factor in EPR [4], the effective gyromagnetic ratio as a function of 0 is given by
I 17
H0 (kOe)
Fig. 1. NMRspin-echo spectrum of 27A1 in a paramagnetic
~(0) = ( ~ cos20 + ~2 sin20)l/2 .
single crystal of DyAI2. At first glance, the spectrum appears to be symmetric around the largest line with an unusual quadrupole structure. This " s y m m e t r y " is incidental. As will be shown, the spectrum is a superposition of 5 lines of site a and 5 lines, three times greater in intensity, of sites b with splitting that is about a third of that of the a site lines. The line denoted as "63 Cu" comes from the copper wire of the rf coil and is used for field calibration. In analyzing the spectrum we follow ref. [1 ]. The paramagnetic shift of 27AI in DyA12 has two parts. The isotropic shift Kp originates from the transferred hyperfine interaction with the dysprosium ions. The anisotropic shift Kax is due to the dipolar field induced by these ions or to the anisotropic part of the hyperfine interaction. For the axial symmetry, taking z along the symmetry direction of the AI group under observation, the nuclear spin hamiltonian is given by
~= -hl.~.1t
0 + ~hVQ[Iz 2 - ~ I ( I + I)] ,
(1)
where I = 5/2 for 27A1, VQis the quadrupole-frequency constant and ~ is the effective gyromagnetic ratio tensor (in units of Hz/Oe) which includes the influence of the paramagnetic shifts. Since, in the general case, for A1 in the Laves phase structure the dipolar field is not collinear with the external field, the dipolar interaction is represented by a tensor [1,2]. If we take a similar tensor for the anisotropic shift, ~ is given by
~t=~ 0
ii p x0 k
l+Kp-Kax
0
]
0
1 + Kp + 2Kax
0
(3)
,
(4)
The use of this expression is equivalent to the calculation of the vector sum of the external field, the hyperfine field and the dipolar field, as was done for 27A1 in ferromagnetic GdA12 [2]. In the ferromagnetic state the dipolar field is much smaller than the other contributions and it would be a good approximation to consider only the projection of the dipolar field on the direction of the external field (or of the hyperfine field in the zero-field experiments in ferromagnets). The parallel case in the paramagnetic phase is given by Kax <~ 1 + Kp. This is the situation in our experiments at temperatures far from the Curie point [1 ]. In this case ~(0) may be approximated by the usual expression ~(0) = ~0[1 + K p +Kax(3COS20 - 1)] .
(5)
Hamiltonian (1) might be solved for each 0 by diagonalization of the 6 × 6 matrix. For VQ ~ v0, however, it is enough to solve ( I ) to a second-order perturbation [5] and the NMR lines which correspond to the rn ~ m - 1 transition are expected at fields
Ho(O ) = Iv0 + ½VQ(3COS20 - 1)(m - ~) - 8V(2m)~m_l]/~(O),
(6)
where By(2) is the second-order correction of the quadrupole interaction, and for I = 5/2
8v(2-~/2~ 1/2 = (v~/2Vo)(1 - c°s20)( 1 - 9 c ° s 2 0 ) , (2) = 8v(2) ,~ t~3/2~ 1/2 -1/2 -3/2
= (v~/16Vo)(1 -- cos20)(5 -- 21 cos20),
(7)
(2) 253
Volume 95A, number 5
PHYSICS LETTERS
5(2) v5/2.~3/2 = ~v(_2~/2,_,_5/2
= --(v~/4Vo)(l - cos20)(l - 33 cos20). (7 cont'd) For the a sites (0 = 0) the approximate solution (6) coincides with the exact solution (3). For these sites as well as for the b sites (with Icos01 = ~), 6v(__2)1/2~1/2 = 0. Thus the external fields H~ and H b, corresponding to the central lines (1/2 4,, - 1 / 2 ) of the a and b groups, are both functions of only Kp and Kax and are defined by v0 = Y0H(](1 +Xp + 2Xax), and v0 = $0Hb(1 + Xp - ~Kax ) . From these expressions and fig. 1, one obtains Kax = -0.0058 +- 0.0002 and Kp = -0.0322 -+ 0.0002. The value of VQ may be found directly, using eq. (3), from the splitting between any two consecutive lines of the a group spectrum, or from the b group spectrum. From (7), the/iv(2) are equal for each pair of satellite lines and the distance between the 3/2 ,e, 1/2 and the - 1 / 2 ~, - 3 [2 lines of group b is AH 0 = 2VQHb/3Vo and twice as much is the distance between the 5/2 o 3/2 and the - 3 ] 2 ~ - 5 / 2 lines of this group. All these calculations give VQ = (561 +- 5) kHz. This value is in very good agreement with (557 -+ 5) kHz, found for the powder sample [1]. The hyperfine field constant in the paramagnetic state Hl~f for the single crystal was found by measuring Kp as a function of T. The sample was oriented with H 0 parallel to [ 100]. For this direction all 27A1 sites are equivalent and (since 3 cos20 - 1 = 0) there is only one line at H 0 = v0/~¢0(1 +Kp) with a small additional shift and broadening, both due to the second-order perturbation (7). Using the measured susceptibility of DyAl 2 as a function of T [6], we obtained (from the Kp versus X plot), HPf = -3.17 kOe/n B + 0.5%, where n B is the polarization of each Dy 3+ ion in Bohr magnetons. This is in good agreement with the value H f f = -(3.2 -+ 0.1) kOe/n B cal-
254
2 May 1983
culated from NMR measurements in the ferromagnetic state and with the value found for the paramagnetic powder, HhPf = - ( 3 . 2 4 +- 0.06) kOe/n B [1]. Finally, the value of Kax in the present study is much more precise than that found from the experiments with the powder sample. It is thus of interest to compare the value obtained here with what is expected due to a pure dipolar interaction. The dipolar field constant for the ferromagnetic state was calculated to beHd0 = 5.171 kOe by taking n B = 9.89 [1,3]. The corresponding parameter in the paramagnetic phase may be denoted by Ka0x. The internal field in the ferromagnetic phase is Hin t ==-nBHff = (31.5 + 0.5) kOe. It corresponds to Kp in the paramagnetic phase. We expect a simple relation K 0 / H 0 = KoP/Hin t to hold. For Kp 2. -0.0322, this relation gives Kax = -0.0053. The experimental value Kax = -0.0058 shows a small enhancement of the dipolar field [3] by a factor of e = 1.1. Equivalently the difference between Kax and KOx might be considered as a small anisotropy of the hyperfine interaction with a value of about 10% of the dipolar field for DyAl 2. This is to be compared with 16% found for GdA12 [3]. This research was supported by a grant from the United States-Israel Binational Science Foundation (BSF), Jerusalem, Israel.
References [1] Y.B. Barash, J. Barak and N. Kaplan, Phys. Rev. B25 (1982) 6616. [ 2] D. Fekete, A. Grayevskey, N. Kaplan and E. Walker, Solid State Commun. 17 (1975) 573. [3] N. Kaplan, E. Dormann, K.H.J. Buschow and D. Lebenbaum, Phys. Rev. B7 (1973) 40. [4] A. Abragam and B. Bleany, Electron paramagnetic resonance of transition ions (Clarendon, Oxford, 1970) p. 135. [5] A. Abragam,The principles of nuclear magnetism(Oxford Univ. Press, London, 1961), [6] B. Barbara and M.F. Rossignol, private communication (1979).