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Microstructure and magnetic properties of bulk amorphous and nanocrystalline Fe61Co10Zr2.5Hf2.5Nb2W2B20 alloy
ARTICLE IN PRESS
Journal of Magnetism and Magnetic Materials 320 (2008) e787–e791 www.elsevier.com/locate/jmmm
Microstructure and magnetic properties of bulk amorphous and nanocrystalline Fe61Co10Zr2.5Hf2.5Nb2W2B20 alloy M. Nabia"eka, J. Zbroszczyka, J. Olszewskia, M. Hasiakb,, W. Ciurzyn´skaa, K. Sobczyka, J. S´wierczeka, J. Kaletab, A. Łukiewskaa b
a Institute of Physics, Cz˛estochowa University of Technology, Al. Armii Krajowej 19, 42-200 Cz˛estochowa, Poland Institute of Materials Science and Applied Mechanics, Wroc!aw University of Technology, Smoluchowskiego 25, 50-370 Wroc!aw, Poland
Available online 12 April 2008
Abstract The microstructure and magnetic properties, i.e. the initial magnetic susceptibility, its disaccommodation, core losses and approach to ferromagnetic saturation of the bulk amorphous and partially crystallized Fe61Co10Zr2.5Hf2.5Nb2W2B20 alloy are studied. From X-ray, Mo¨ssbauer spectroscopy and electron microscopy studies we have stated that all samples in the as-quenched state are fully amorphous. However, after annealing the samples at 850 K for 30 min the crystalline a-FeCo grains embedded in the amorphous matrix are found. Moreover, from Mo¨ssbauer spectra analysis we have stated that the crystalline phase in those samples exhibits the long-range order. The alloy in the as-quenched state shows good thermal stability of the initial magnetic susceptibility. Furthermore, the intensity of the magnetic susceptibility disaccommodation in the rod is lower than in the ribbon. It is due to low quenching rate during the rod preparation which involves the reduction of free volumes. From the analysis of the isochronal disaccommodation curves, assuming the Gaussian distribution of relaxation times, we have found that activation energies of the elementary processes responsible for this phenomenon range from 1.2 to 1.4 eV. After the annealing of the samples the initial susceptibility slightly enhances and disaccommodation drastically decreases. From high-field magnetization studies we have learned that the size of structural defects depends on the quenching rate (the shape of the samples) and changes after annealing. r 2008 Elsevier B.V. All rights reserved. PACS: 75.50.y; 75.50.Kj; 75.50.Tt; 75.50.Lr Keywords: Bulk amorphous alloys; Mo¨ssbauer spectroscopy; Disaccommodation; Ferromagnetic saturation
1. Introduction Recently, it has been found that multicomponent amorphous alloys exhibit excellent glass-forming ability and may be produced in the form of thick ribbons or cores [1]. Due to lower quenching rate in comparison with classical amorphous alloys atomic arrangement during the preparation of bulk amorphous materials leads to their good thermal stability of structure and magnetic properties in the as-received state. Moreover, one can suppose that microstructure and magnetic properties of those alloys are highly influenced by the shape of samples. The temperature and time stability of magnetic properties for Corresponding author. Tel.: +48 71 320 34 96; fax: +48 71 321 12 35.
E-mail address: [email protected] (M. Hasiak). 0304-8853/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2008.04.046
bulk amorphous alloys may be better after their partial crystallization. The aim of this paper is to study the microstructure and magnetic properties, i.e. the initial magnetic susceptibility, its disaccommodation, core losses and high-field magnetization of the bulk amorphous and nanocrystalline Fe61Co10Zr2.5Hf2.5Nb2W2B20 alloy. 2. Experimental procedure Amorphous ribbons 90 mm thick and 3 mm wide were produced by a melt spinning method. However, the amorphous rods and tubes 2 cm long have been obtained by a suction casting method in a protective argon atmosphere. The rods were 1 mm in diameter, and outer and inner diameters of tubes were equal to 4 and 3 mm,
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respectively. The microstructure of the as-quenched and nanocrystalline samples was investigated by X-ray diffractometry, Mo¨ssbauer spectroscopy and transmission electron microscopy (TEM). The initial susceptibility and its disaccommodation were measured using a completely automated set up in the temperature region from 130 K up to 550 K. Using a hysteresis loop tracer the core losses of the samples were studied. The high-field magnetization was measured by a vibrating sample magnetometer. All investigations were carried out for the samples in the asquenched state and after annealing at 850 K for 30 min in vacuum. 3. Results and discussion Fig. 1 shows representative transmission Mo¨ssbauer spectra measured at room temperature for the powdered rod of the Fe61Co10Zr2.5Hf2.5Nb2W2B20 alloy. The spectrum of the as-quenched sample (Fig. 1a) exhibits the broad six-line pattern characteristic of ferromagnetic amorphous alloys. In the hyperfine field distribution P(B)
(Fig. 1c) obtained from this spectrum one can distinguish at least three components related to the regions with the different iron concentration. After the annealing of the sample at 850 K for 0.5 h (Fig. 1b) the obtained Mo¨ssbauer spectrum shows more complex structure and it is seen a sextet with narrow lines attributed to a-FeCo phase superimposed on the broad line feature corresponding to the amorphous intergranular phase. The high-field component in the hyperfine field distribution (Fig. 1d) corresponds to the crystalline a-FeCo phase. From Mo¨ssbauer spectra analysis we have evaluated that the volume fraction of the crystalline Fe64Co36 phase is equal to about 0.1. Moreover, it has been found that this phase exhibits long-range order with Bragg–Williams parameter S ¼ 0.72. The similar results were obtained for the samples in the form of ribbons and tubes. TEM studies confirm the amorphicity of the asquenched sample (Fig. 2a) and the presence of the crystalline grains in the annealed alloy. It is seen (Fig. 2b) that in the partially crystallized sample, except small grains (2), agglomerations of crystals
Fig. 1. Transmission Mo¨ssbauer spectra (a, b) and corresponding hyperfine field distributions (c, d) for the powdered rod of the Fe61Co10Zr2.5Hf2.5Nb2W2B20 alloy in the as-quenched state (a, c) and after annealing at 850 K for 0.5 h (b, d).
ARTICLE IN PRESS M. Nabia!ek et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e787–e791
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Fig. 2. Microstructure and corresponding electron diffraction patterns of the Fe61Co10Zr2.5Hf2.5Nb2W2B20 alloy in the form of rod; as-quenched state (a), after annealing at 850 K for 0.5 h (b).
Fig. 3. Initial magnetic susceptibility as a function of temperature for the Fe61Co10Zr2.5Hf2.5Nb2W2B20 alloy in the form of rod: the as-quenched state (a), and after annealing at 850 K for 0.5 h (b).
Fig. 4. Isochronal disaccommodation curves D(1/w) ¼ f(T) for the amorphous Fe61Co10Zr2.5Hf2.5Nb2W2B20 alloy (second run) in the form of rod (a) and ribbon (b).
(1) embedded in the amorphous matrix are observed. Moreover, in some areas the large grains are found. The amorphous and partially crystallized samples are magnetically soft ferromagnets as compared to classical crystalline ones. The initial magnetic susceptibility versus temperature for the Fe61Co10Zr2.5Hf2.5Nb2W2B20 alloy is presented in Fig. 3. The initial magnetic susceptibility of the amorphous sample slightly increases with temperature and then rapidly drops after reaching the maximum at about 475 K. The occurrence of maximum observed in w ¼ f(T) curve is associated with the reduction of anisotropy and the decrease of magnetization with temperature. After partial crystallization of the sample at temperature higher than 320 K the initial susceptibility monotonically decreases with temperature. This behaviour is connected with the inhomogeneity of an amorphous matrix in that sample which is confirmed by Mo¨ssbauer spectroscopy studies.
The typical isochronal disaccommodation curves for amorphous samples are presented in Fig. 4. As can be seen in this figure the intensity of maximum disaccommodation is lower for the sample in the form of rod. It is due to lower quenching rate during preparation of amorphous rods and the annealing out of some free volumes [2,3]. From the decomposition of the isochronal disaccommodation curves into three elementary processes [4] (each of them being described by a Gaussian distribution of relaxation times) we have found that activation energies of these processes range from 1.2 up to 1.4 eV and pre-exponential factor in the Arrhenius law is of order of 1015 s. This results indicate that the disaccommodation effect in the investigated alloy is connected with ordering of atom pairs in the vicinity of free volumes. It is worth adding that after partial crystallization of samples in D(1/w) ¼ f(T) curves only temperature independent background with very low intensity in the temperature range from 130 K up to 350 K is observed. The increase of
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Fig. 5. Total core losses of the Fe61Co10Zr2.5Hf2.5Nb2W2B20 alloy in the form of 70 mm thick ribbon: as-quenched state (a) and after annealing at 850 K for 30 min (b).
the disaccommodation intensity takes place near the Curie temperature of the amorphous matrix. The total core losses versus maximum induction for the as-quenched and partially crystallized Fe61Co10Zr2.5Hf2.5Nb2W2B20 alloy are shown in Fig. 5. The core losses of this alloy are the same order as in soft magnetic crystalline materials. However, after partial crystallization the slight increase of the losses is observed. It is due to the presence of grains which are pinning centres of domain walls during the sample magnetizing. The magnetization (M) as a function of magnetizing field (m0H) near ferromagnetic saturation may be expressed by the equation [5]: ai MðHÞ ¼ M s 1 (1) þ bðm0 HÞ1=2 ðm0 HÞi where Ms is the saturation magnetization, m0-vacuum magnetic permeability, i ¼ 12 for point-like defects, i ¼ 1 or 2 for quasi-dislocation dipoles and the last term describes so-called Holstein–Primakoff paraproces [6]. In the magnetic field (m0H) from 0.05 to 0.45 T the normalized magnetization (M/Ms) for the as-quenched sample in the form of 70 mm thick ribbon obeys the 1/(m0H)1/2 law. The same behaviour is observed for the asquenched tube in the magnetic field 0.14 T om0Ho1.00 T. As for the rod-shaped sample in the as-quenched state, in the magnetic field from 0.14 to 0.22 T the 1/m0H law is fulfilled. In higher magnetic field (up to 0.49 T) the 1/(m0H)2 law is held (Fig. 6). Taking into account the obtained results one may conclude that the size of defects in amorphous alloys depends on the quenching rate (shape of the samples). In the case of ribbons and tubes point-like defects influence the magnetization process in high magnetic fields. However, the increase of magnetization with the magnetic field in the amorphous rod near the ferromagnetic saturation is connected with rotation of the magnetic moments in the
Fig. 6. Magnetization curves of the as-quenched Fe61Co10Zr2.5Hf2.5Nb2W2B20 alloy in the form of ribbon (a) and rod (b, c).
vicinity of quasi-dislocation dipoles. In the magnetic field m0H 40.45 T the Holstein–Primakoff paraproces in both samples is observed.
ARTICLE IN PRESS M. Nabia!ek et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e787–e791
From the magnetization studies in high magnetic fields we have found that the width of quasi-dislocation dipoles in the as-quenched amorphous rod is equal to 3.33 nm. The structural defects in the investigated samples are instable and after the annealing of the rod at 850 K the width of the quasi-dislocation dipoles decreases to 2.70 nm. Moreover, this heat treatment leads to the transition from 1/(m0H)1/2 to 1/m0H law in the case of the tube. 4. Conclusions
The intensity of the disaccommodation is lower in amorphous rods than in ribbons. In the case of ribbons the point-like defects influence the magnetization in high magnetic fields, whereas in rods the quasi-dislocation dipoles play main role. During annealing of the amorphous samples the decrease of the width of the quasi-dislocation dipoles occur.
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Acknowledgements This work was partially supported by the Grants PBZMEiN-01/2006/09 (M. Hasiak) and PBZ-KBN-115/T08/04 (J. Kaleta).
References [1] B. Shen, A. Inoue, Mater. Trans. JIM 43 (2002) 1235. [2] H. Kronmu¨ller, N. Moser, R. Rettenmeier, IEEE Trans. Magn. 20 (1984) 1388. [3] H. Kronmu¨ller, Phys. Stat. Sol. (b) 127 (1985) 531. [4] W.H. Ciurzyn´ska, G. Haneczok, J. Zbroszczyk, J. Magn. Magn. Mater. 189 (1998) 384. [5] M. Vazquez, W. Fernengel, H. Kronmu¨ller, Phys. Stat. Sol. (a) 115 (1989) 547. [6] T. Holstein, H. Primakoff, Phys. Rev. 59 (1941) 388.
Journal of Magnetism and Magnetic Materials 320 (2008) e787–e791 www.elsevier.com/locate/jmmm
Microstructure and magnetic properties of bulk amorphous and nanocrystalline Fe61Co10Zr2.5Hf2.5Nb2W2B20 alloy M. Nabia"eka, J. Zbroszczyka, J. Olszewskia, M. Hasiakb,, W. Ciurzyn´skaa, K. Sobczyka, J. S´wierczeka, J. Kaletab, A. Łukiewskaa b
a Institute of Physics, Cz˛estochowa University of Technology, Al. Armii Krajowej 19, 42-200 Cz˛estochowa, Poland Institute of Materials Science and Applied Mechanics, Wroc!aw University of Technology, Smoluchowskiego 25, 50-370 Wroc!aw, Poland
Available online 12 April 2008
Abstract The microstructure and magnetic properties, i.e. the initial magnetic susceptibility, its disaccommodation, core losses and approach to ferromagnetic saturation of the bulk amorphous and partially crystallized Fe61Co10Zr2.5Hf2.5Nb2W2B20 alloy are studied. From X-ray, Mo¨ssbauer spectroscopy and electron microscopy studies we have stated that all samples in the as-quenched state are fully amorphous. However, after annealing the samples at 850 K for 30 min the crystalline a-FeCo grains embedded in the amorphous matrix are found. Moreover, from Mo¨ssbauer spectra analysis we have stated that the crystalline phase in those samples exhibits the long-range order. The alloy in the as-quenched state shows good thermal stability of the initial magnetic susceptibility. Furthermore, the intensity of the magnetic susceptibility disaccommodation in the rod is lower than in the ribbon. It is due to low quenching rate during the rod preparation which involves the reduction of free volumes. From the analysis of the isochronal disaccommodation curves, assuming the Gaussian distribution of relaxation times, we have found that activation energies of the elementary processes responsible for this phenomenon range from 1.2 to 1.4 eV. After the annealing of the samples the initial susceptibility slightly enhances and disaccommodation drastically decreases. From high-field magnetization studies we have learned that the size of structural defects depends on the quenching rate (the shape of the samples) and changes after annealing. r 2008 Elsevier B.V. All rights reserved. PACS: 75.50.y; 75.50.Kj; 75.50.Tt; 75.50.Lr Keywords: Bulk amorphous alloys; Mo¨ssbauer spectroscopy; Disaccommodation; Ferromagnetic saturation
1. Introduction Recently, it has been found that multicomponent amorphous alloys exhibit excellent glass-forming ability and may be produced in the form of thick ribbons or cores [1]. Due to lower quenching rate in comparison with classical amorphous alloys atomic arrangement during the preparation of bulk amorphous materials leads to their good thermal stability of structure and magnetic properties in the as-received state. Moreover, one can suppose that microstructure and magnetic properties of those alloys are highly influenced by the shape of samples. The temperature and time stability of magnetic properties for Corresponding author. Tel.: +48 71 320 34 96; fax: +48 71 321 12 35.
E-mail address: [email protected] (M. Hasiak). 0304-8853/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2008.04.046
bulk amorphous alloys may be better after their partial crystallization. The aim of this paper is to study the microstructure and magnetic properties, i.e. the initial magnetic susceptibility, its disaccommodation, core losses and high-field magnetization of the bulk amorphous and nanocrystalline Fe61Co10Zr2.5Hf2.5Nb2W2B20 alloy. 2. Experimental procedure Amorphous ribbons 90 mm thick and 3 mm wide were produced by a melt spinning method. However, the amorphous rods and tubes 2 cm long have been obtained by a suction casting method in a protective argon atmosphere. The rods were 1 mm in diameter, and outer and inner diameters of tubes were equal to 4 and 3 mm,
ARTICLE IN PRESS e788
M. Nabia!ek et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e787–e791
respectively. The microstructure of the as-quenched and nanocrystalline samples was investigated by X-ray diffractometry, Mo¨ssbauer spectroscopy and transmission electron microscopy (TEM). The initial susceptibility and its disaccommodation were measured using a completely automated set up in the temperature region from 130 K up to 550 K. Using a hysteresis loop tracer the core losses of the samples were studied. The high-field magnetization was measured by a vibrating sample magnetometer. All investigations were carried out for the samples in the asquenched state and after annealing at 850 K for 30 min in vacuum. 3. Results and discussion Fig. 1 shows representative transmission Mo¨ssbauer spectra measured at room temperature for the powdered rod of the Fe61Co10Zr2.5Hf2.5Nb2W2B20 alloy. The spectrum of the as-quenched sample (Fig. 1a) exhibits the broad six-line pattern characteristic of ferromagnetic amorphous alloys. In the hyperfine field distribution P(B)
(Fig. 1c) obtained from this spectrum one can distinguish at least three components related to the regions with the different iron concentration. After the annealing of the sample at 850 K for 0.5 h (Fig. 1b) the obtained Mo¨ssbauer spectrum shows more complex structure and it is seen a sextet with narrow lines attributed to a-FeCo phase superimposed on the broad line feature corresponding to the amorphous intergranular phase. The high-field component in the hyperfine field distribution (Fig. 1d) corresponds to the crystalline a-FeCo phase. From Mo¨ssbauer spectra analysis we have evaluated that the volume fraction of the crystalline Fe64Co36 phase is equal to about 0.1. Moreover, it has been found that this phase exhibits long-range order with Bragg–Williams parameter S ¼ 0.72. The similar results were obtained for the samples in the form of ribbons and tubes. TEM studies confirm the amorphicity of the asquenched sample (Fig. 2a) and the presence of the crystalline grains in the annealed alloy. It is seen (Fig. 2b) that in the partially crystallized sample, except small grains (2), agglomerations of crystals
Fig. 1. Transmission Mo¨ssbauer spectra (a, b) and corresponding hyperfine field distributions (c, d) for the powdered rod of the Fe61Co10Zr2.5Hf2.5Nb2W2B20 alloy in the as-quenched state (a, c) and after annealing at 850 K for 0.5 h (b, d).
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Fig. 2. Microstructure and corresponding electron diffraction patterns of the Fe61Co10Zr2.5Hf2.5Nb2W2B20 alloy in the form of rod; as-quenched state (a), after annealing at 850 K for 0.5 h (b).
Fig. 3. Initial magnetic susceptibility as a function of temperature for the Fe61Co10Zr2.5Hf2.5Nb2W2B20 alloy in the form of rod: the as-quenched state (a), and after annealing at 850 K for 0.5 h (b).
Fig. 4. Isochronal disaccommodation curves D(1/w) ¼ f(T) for the amorphous Fe61Co10Zr2.5Hf2.5Nb2W2B20 alloy (second run) in the form of rod (a) and ribbon (b).
(1) embedded in the amorphous matrix are observed. Moreover, in some areas the large grains are found. The amorphous and partially crystallized samples are magnetically soft ferromagnets as compared to classical crystalline ones. The initial magnetic susceptibility versus temperature for the Fe61Co10Zr2.5Hf2.5Nb2W2B20 alloy is presented in Fig. 3. The initial magnetic susceptibility of the amorphous sample slightly increases with temperature and then rapidly drops after reaching the maximum at about 475 K. The occurrence of maximum observed in w ¼ f(T) curve is associated with the reduction of anisotropy and the decrease of magnetization with temperature. After partial crystallization of the sample at temperature higher than 320 K the initial susceptibility monotonically decreases with temperature. This behaviour is connected with the inhomogeneity of an amorphous matrix in that sample which is confirmed by Mo¨ssbauer spectroscopy studies.
The typical isochronal disaccommodation curves for amorphous samples are presented in Fig. 4. As can be seen in this figure the intensity of maximum disaccommodation is lower for the sample in the form of rod. It is due to lower quenching rate during preparation of amorphous rods and the annealing out of some free volumes [2,3]. From the decomposition of the isochronal disaccommodation curves into three elementary processes [4] (each of them being described by a Gaussian distribution of relaxation times) we have found that activation energies of these processes range from 1.2 up to 1.4 eV and pre-exponential factor in the Arrhenius law is of order of 1015 s. This results indicate that the disaccommodation effect in the investigated alloy is connected with ordering of atom pairs in the vicinity of free volumes. It is worth adding that after partial crystallization of samples in D(1/w) ¼ f(T) curves only temperature independent background with very low intensity in the temperature range from 130 K up to 350 K is observed. The increase of
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Fig. 5. Total core losses of the Fe61Co10Zr2.5Hf2.5Nb2W2B20 alloy in the form of 70 mm thick ribbon: as-quenched state (a) and after annealing at 850 K for 30 min (b).
the disaccommodation intensity takes place near the Curie temperature of the amorphous matrix. The total core losses versus maximum induction for the as-quenched and partially crystallized Fe61Co10Zr2.5Hf2.5Nb2W2B20 alloy are shown in Fig. 5. The core losses of this alloy are the same order as in soft magnetic crystalline materials. However, after partial crystallization the slight increase of the losses is observed. It is due to the presence of grains which are pinning centres of domain walls during the sample magnetizing. The magnetization (M) as a function of magnetizing field (m0H) near ferromagnetic saturation may be expressed by the equation [5]: ai MðHÞ ¼ M s 1 (1) þ bðm0 HÞ1=2 ðm0 HÞi where Ms is the saturation magnetization, m0-vacuum magnetic permeability, i ¼ 12 for point-like defects, i ¼ 1 or 2 for quasi-dislocation dipoles and the last term describes so-called Holstein–Primakoff paraproces [6]. In the magnetic field (m0H) from 0.05 to 0.45 T the normalized magnetization (M/Ms) for the as-quenched sample in the form of 70 mm thick ribbon obeys the 1/(m0H)1/2 law. The same behaviour is observed for the asquenched tube in the magnetic field 0.14 T om0Ho1.00 T. As for the rod-shaped sample in the as-quenched state, in the magnetic field from 0.14 to 0.22 T the 1/m0H law is fulfilled. In higher magnetic field (up to 0.49 T) the 1/(m0H)2 law is held (Fig. 6). Taking into account the obtained results one may conclude that the size of defects in amorphous alloys depends on the quenching rate (shape of the samples). In the case of ribbons and tubes point-like defects influence the magnetization process in high magnetic fields. However, the increase of magnetization with the magnetic field in the amorphous rod near the ferromagnetic saturation is connected with rotation of the magnetic moments in the
Fig. 6. Magnetization curves of the as-quenched Fe61Co10Zr2.5Hf2.5Nb2W2B20 alloy in the form of ribbon (a) and rod (b, c).
vicinity of quasi-dislocation dipoles. In the magnetic field m0H 40.45 T the Holstein–Primakoff paraproces in both samples is observed.
ARTICLE IN PRESS M. Nabia!ek et al. / Journal of Magnetism and Magnetic Materials 320 (2008) e787–e791
From the magnetization studies in high magnetic fields we have found that the width of quasi-dislocation dipoles in the as-quenched amorphous rod is equal to 3.33 nm. The structural defects in the investigated samples are instable and after the annealing of the rod at 850 K the width of the quasi-dislocation dipoles decreases to 2.70 nm. Moreover, this heat treatment leads to the transition from 1/(m0H)1/2 to 1/m0H law in the case of the tube. 4. Conclusions
The intensity of the disaccommodation is lower in amorphous rods than in ribbons. In the case of ribbons the point-like defects influence the magnetization in high magnetic fields, whereas in rods the quasi-dislocation dipoles play main role. During annealing of the amorphous samples the decrease of the width of the quasi-dislocation dipoles occur.
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Acknowledgements This work was partially supported by the Grants PBZMEiN-01/2006/09 (M. Hasiak) and PBZ-KBN-115/T08/04 (J. Kaleta).
References [1] B. Shen, A. Inoue, Mater. Trans. JIM 43 (2002) 1235. [2] H. Kronmu¨ller, N. Moser, R. Rettenmeier, IEEE Trans. Magn. 20 (1984) 1388. [3] H. Kronmu¨ller, Phys. Stat. Sol. (b) 127 (1985) 531. [4] W.H. Ciurzyn´ska, G. Haneczok, J. Zbroszczyk, J. Magn. Magn. Mater. 189 (1998) 384. [5] M. Vazquez, W. Fernengel, H. Kronmu¨ller, Phys. Stat. Sol. (a) 115 (1989) 547. [6] T. Holstein, H. Primakoff, Phys. Rev. 59 (1941) 388.