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Review of singular point detection techniques
ARTICLE IN PRESS
Physica B 346–347 (2004) 524–527
Review of singular point detection techniques F. Bolzoni, R. Cabassi* Istituto IMEM del CNR, Parco Area delle Scienze 37 A, Fontanini, 43010 Parma, Italy
Abstract The singular point detection (SPD) is the most powerful technique for the study of singularities in magnetization curves of polycrystalline samples. According to the SPD theory, the reversible magnetization curve of a polycrystalline ferromagnetic sample has an hidden singularity that shows up in the derivative, the derivative order depending on the particular crystal symmetry. The SPD technique finds its natural experimental use in the domain of pulsed high magnetic fields, where it can be conveniently used to perform measurements on high anisotropy materials. We present here a review of the various practical applications of SPD which have been exploited in magnetism laboratories: anisotropy field measurements, texture determination of polycrystalline magnets, measurement of the critical field in first-order magnetization processes (FOMPs) and double FOMPs. r 2004 Published by Elsevier B.V. PACS: 70; 75; 75.30.Gw; 07.55.-w; 75.50.Ww Keywords: Magnetic anisotropy; FOMP; Pulsed magnetic fields; Singular point detection; Magnetic measurements
1. Introduction The magnetization curve of a ferromagnetic single crystal shows, in general, a singularity when it reaches saturation at the anisotropy field Ha . The transition is, in principle, of the second order and occurs only if the magnetic field is perfectly aligned with the hard direction. For field orientations other than hard direction, the reversible magnetization curve is a regular function for H ¼ Ha : When we apply a magnetic field to a polycrystalline material we obtain, in general, a smooth curve MðHÞ which, at first sight, seems to give no indication of the singularity at H ¼ Ha . However the averaging over all the crystal *Corresponding author. E-mail address: [email protected] (R. Cabassi). 0921-4526/$ - see front matter r 2004 Published by Elsevier B.V. doi:10.1016/j.physb.2004.01.140
orientations does not cancel the singularity coming from the contribution of the crystallites oriented in such a way that their hard directions are nearly parallel to H: The singularity can be detected by observation of the successive derivatives dn M=dH n ; this is the principle at the basis of the singular point detection (SPD) techniques Ref. [1] that allows one to reveal the singularity at H ¼ Ha in polycrystalline samples and, of course, to measure the anisotropy field itself. The order of derivation depends on the particular symmetry of the specimen, common cases are uniaxial an cubic crystals, where the singularity becomes apparent in d2 M=dH 2 and d3 M=dH 3 ; respectively. The most important case is that of uniaxial materials having hard direction in the basal plane; the singularity appears in d2 M=dH 2 ; and has the shape of a cusp on the left-hand side of the peak at H ¼ Ha :
ARTICLE IN PRESS F. Bolzoni, R. Cabassi / Physica B 346–347 (2004) 524–527
2. H a and FOMPs detection The SPD technique has been widely applied during 30 years to measure the magnetocrystalline anisotropy field Ha ; mainly in rare earth and transition metals intermetallic compounds, particularly at IMEM-CNR Institute in Parma and at Technical University of Vienna [2]. In order to reach the region of physical interest, the use of an high magnetic pulse is convenient: the magnetic field gives rise to two branches corresponding to the rise and decrease of field. Each branch shows a peak at H ¼ Ha ; with amplitude proportional to the square of the corresponding field speed, see Fig. 1. Another well-developed application of SPD is the detection of first-order magnetization processes (FOMPs), that are discontinuous jumps occurring in the magnetization curve of some ferromagnetic crystals [3]. When the magnetization jump reaches the Ms value, we have a FOMP of type 1, otherwise we have a FOMP of type 2. The origin of FOMPs can be understood if one writes the crystal energy in terms of phenomenological anisotropy constants:
525
respectively. It turns out that, for some combinations of the anisotropy constants, two different points of minimum energy occur at different values of y; and the two points have the same energy if the applied magnetic field has a certain critical value H ¼ Hcr depending on the Ki values. This results in an irreversible rotation of the magnetization vector M; i.e. a FOMP. The SPD technique allows a clear and accurate determination of the critical field Hcr even for polycrystalline specimens [4]. This provides in principle a powerful method for detailed investigation of the anisotropy properties of ferromagnetic and ferrimagnetic materials. Fig. 2 shows the SPD results for the case of polycrystalline Nd2 Fe14 B at various temperatures. The peak shape evolves from the case of anisotropy field Ha to critical field Hcr of FOMP. In the case of a FOMP of type 2, Hcr and Ha positions can be detected simultaneously from a single SPD measurements [5].
3. Multidomain theory
being W and g the angles formed by the magnetization vector M with the crystallographic easy axis and with the external magnetic field H;
Real-world experiments are usually carried out on ferromagnetic materials having relatively large crystallites, so that the single domain assumption of SPD theory can easily be violated. For this reason, the SPD theory was extended to multidomain crystallites [6], describing the individual crystallites according to the Neel phase theory [7]. Multidomain SPD theory is able to closely
Fig. 1. SPD measure of the anisotropy field on a Nd2 Fe14 B commercial permanent magnet.
Fig. 2. Presence of FOMPs in polycrystalline Nd2 Fe14 B below 200 K observed by means of SPD method.
E ¼ K1 sin2 W þ K2 sin4 W þ K3 sin6 W HMs cosg ð1Þ
ARTICLE IN PRESS 526
F. Bolzoni, R. Cabassi / Physica B 346–347 (2004) 524–527
Fig. 3. Calculated multidomain SPD peaks for different grain shape distribution functions (see text).
Fig. 4. Angular dependence of SPD peak amplitudes in an extruded MnAl magnet at different angles of field with respect to the easy direction.
describe the true experimental peaks. The exact position and shape of the SPD peak result to depend on the distribution function of the crystallite shapes: see Fig. 3 in which gaussian distribution functions pðeÞ of the grain ellipticity e are choosen with /eS ¼ 1:5; 0.67, 0.5, 0.4 for the cases a; b; c; d; respectively, with the same standard deviation of 0.2. The more pðeÞ approaches to a Dirac delta, the sharper the peaks become.
different orientations of the angle a formed by the magnet alignment axis with respect to H and evaluating their amplitudes (Fig. 4), one can plot them vs. a: Theory [9] explains how, from a best-fit of the obtained points in term of powers of sinðaÞ : Gðsin2 aÞ ¼ Si ai sin2i a; one can obtain a texture function F ðcos2 WÞ ¼ Si bi sin2i W by means of a mathematical transformation linking the coefficients of the two polynomial expansions: ai ¼ bi ð2iÞ!=½22i ði!Þ2 :
4. Polycrystal texture determination
5. FOMPs in non-collinear systems
Hard magnetic materials of common use are polycrystalline structures made of many grains each one with its own easy magnetization axis. The grain axis are not collinear, but are distributed in space according to an angular distribution function which is given the name of ‘‘texture’’. The preparation methods of good quality uniaxial magnets look for texture functions sharp peaked around the magnet easy axis. SPD technique gives a method for the determination of the texture function [8], which is important in order to assess the quality of a magnet. The amplitude of the SPD peak is proportional, in its non regular part, to the volume fraction of the crystallites having the easy axis oriented perpendicular to the applied magnetizing field H: Measuring the SPD peaks at
The above analysis refers to the sublattice magnetizations as rigidly collinear coupled (see Eq. (1)). Some materials are better described by means of a two sublattice model, with the possibility of a canting angle between the two sublattice magnetization vectors. In this case the total energy of the system becomes Eðya ; yb ; HÞ ¼ Jab Ma . Mb þ
3 X
ðKna sin2n Wa
n¼1 2n
þ Knb sin Wb Þ H ðMa þ Mb Þ; where a and b refer to the two sublattices, Wa and Wb are the angles that the two sublattices Ma and Mb form with the c-axis and Jab is the exchange constant [10]. Minimizing the energy expression,
ARTICLE IN PRESS F. Bolzoni, R. Cabassi / Physica B 346–347 (2004) 524–527
527
Fig. 5. SPD signal of the compound Pr2 ðCo0:1 Fe0:9 Þ17 presenting a double FOMP. Inset: corresponding single crystal magnetization curve along the hard direction.
Fig. 6. Measured SPD peak shape at 220 K for Pr2 ðFe0:6 Co0:4 Þ14 B: Inset: SPD signal for a Y-Co compound containing both 2:7 and 5:19 phases.
we can find different situations: simple approach to saturation, normal FOMP as described in Section 2, and double FOMP. The latter is characterized by a double jump in the magnetization curve of the single crystal in hard direction, with two different critical fields (see inset in Fig. 5). The first jump does not reach the saturation state, while the second one always reach saturation. In Fig. 5 is reported the experimental result of the polycrystalline Pr2 ðCo0:1 Fe0:9 Þ17 intermetallic compound at 149 K:
peaks can appear in multiphase specimens, thus allowing the detection of the anisotropy field of each phase, as in the case of a double hard magnetic phase Y-Co compound (see inset of Fig. 6). For the technical data of the experimental apparatus used for SPD measurements, we encourage the reader to see the IMEM web pages [12].
6. The fake SPD peak In Fig. 6 an SPD experimental result is given for the Pr2 ðFe0:6 Co0:4 Þ14 B: The measurements become more and more difficult because the SPD peak amplitude decreases rapidly and disappears below 100 K: Another remarkable effect at low temperature is the onset of a strong distortion of the magnetization curve that produces a ‘‘fake’’ SPD peak, i.e. a strong wide peak in the SPD signal well below the sharp anisotropy field peak, originated by the contribution of higher order anisotropy constants [11]. The difficulty of interpreting this strongly anomalous behavior could lead to possible errors, since decreasing temperature the true SPD peak tends to vanish while the ‘‘fake’’ SPD peak gets sharper. Conversely, true double SPD
References [1] G. Asti, S. Rinaldi, J. Appl. Phys. 45 (1974) 3600. [2] K.Y. Guslienko, E.H.C.P. Sinnecker, R. Grossinger, Phys. Rev. B 55 (1997) 380. [3] G. Asti, F. Bolzoni, J. Magn. Magn. Mater. 20 (1980) 29. [4] G. Asti, F. Bolzoni, J. Appl. Phys. 58 (1985) 1924. [5] L. Pareti, A. Paoluzi, F. Albertini, M.R. Ibarra, L. Morellon, P.A. Algarabel, J. Appl. Phys. 76 (1994) 7473. [6] G. Asti, F. Bolzoni, R. Cabassi, J. Appl. Phys. 73 (1993) 323. [7] L. Neel, R. Pauthenet, G. Rimet, V.S. Giron, J. Appl. Phys. 31 (1960) 27S. [8] G. Asti, F. Bolzoni, R. Cabassi, M. Ghidini, J. Magn. Magn. Mater. 128 (1993) 58. [9] G. Asti, F. Bolzoni, L. Pareti, J. Magn. Magn. Mater. 83 (1990) 270. [10] F. Albertini, F. Bolzoni, A. Paoluzi, E. Zannoni, Physica B 294–295 (2001) 172. [11] G. Asti, F. Bolzoni, J.M. Le Breton, M. Ghidini, A. Morel, M. Solzi, F. Kools, P. Tenaud, ICM 2003 Proceedings, J. Magn. Magn. Mater., submitted. [12] www.imem.cnr.it/bolzoni/highfields.html.
Physica B 346–347 (2004) 524–527
Review of singular point detection techniques F. Bolzoni, R. Cabassi* Istituto IMEM del CNR, Parco Area delle Scienze 37 A, Fontanini, 43010 Parma, Italy
Abstract The singular point detection (SPD) is the most powerful technique for the study of singularities in magnetization curves of polycrystalline samples. According to the SPD theory, the reversible magnetization curve of a polycrystalline ferromagnetic sample has an hidden singularity that shows up in the derivative, the derivative order depending on the particular crystal symmetry. The SPD technique finds its natural experimental use in the domain of pulsed high magnetic fields, where it can be conveniently used to perform measurements on high anisotropy materials. We present here a review of the various practical applications of SPD which have been exploited in magnetism laboratories: anisotropy field measurements, texture determination of polycrystalline magnets, measurement of the critical field in first-order magnetization processes (FOMPs) and double FOMPs. r 2004 Published by Elsevier B.V. PACS: 70; 75; 75.30.Gw; 07.55.-w; 75.50.Ww Keywords: Magnetic anisotropy; FOMP; Pulsed magnetic fields; Singular point detection; Magnetic measurements
1. Introduction The magnetization curve of a ferromagnetic single crystal shows, in general, a singularity when it reaches saturation at the anisotropy field Ha . The transition is, in principle, of the second order and occurs only if the magnetic field is perfectly aligned with the hard direction. For field orientations other than hard direction, the reversible magnetization curve is a regular function for H ¼ Ha : When we apply a magnetic field to a polycrystalline material we obtain, in general, a smooth curve MðHÞ which, at first sight, seems to give no indication of the singularity at H ¼ Ha . However the averaging over all the crystal *Corresponding author. E-mail address: [email protected] (R. Cabassi). 0921-4526/$ - see front matter r 2004 Published by Elsevier B.V. doi:10.1016/j.physb.2004.01.140
orientations does not cancel the singularity coming from the contribution of the crystallites oriented in such a way that their hard directions are nearly parallel to H: The singularity can be detected by observation of the successive derivatives dn M=dH n ; this is the principle at the basis of the singular point detection (SPD) techniques Ref. [1] that allows one to reveal the singularity at H ¼ Ha in polycrystalline samples and, of course, to measure the anisotropy field itself. The order of derivation depends on the particular symmetry of the specimen, common cases are uniaxial an cubic crystals, where the singularity becomes apparent in d2 M=dH 2 and d3 M=dH 3 ; respectively. The most important case is that of uniaxial materials having hard direction in the basal plane; the singularity appears in d2 M=dH 2 ; and has the shape of a cusp on the left-hand side of the peak at H ¼ Ha :
ARTICLE IN PRESS F. Bolzoni, R. Cabassi / Physica B 346–347 (2004) 524–527
2. H a and FOMPs detection The SPD technique has been widely applied during 30 years to measure the magnetocrystalline anisotropy field Ha ; mainly in rare earth and transition metals intermetallic compounds, particularly at IMEM-CNR Institute in Parma and at Technical University of Vienna [2]. In order to reach the region of physical interest, the use of an high magnetic pulse is convenient: the magnetic field gives rise to two branches corresponding to the rise and decrease of field. Each branch shows a peak at H ¼ Ha ; with amplitude proportional to the square of the corresponding field speed, see Fig. 1. Another well-developed application of SPD is the detection of first-order magnetization processes (FOMPs), that are discontinuous jumps occurring in the magnetization curve of some ferromagnetic crystals [3]. When the magnetization jump reaches the Ms value, we have a FOMP of type 1, otherwise we have a FOMP of type 2. The origin of FOMPs can be understood if one writes the crystal energy in terms of phenomenological anisotropy constants:
525
respectively. It turns out that, for some combinations of the anisotropy constants, two different points of minimum energy occur at different values of y; and the two points have the same energy if the applied magnetic field has a certain critical value H ¼ Hcr depending on the Ki values. This results in an irreversible rotation of the magnetization vector M; i.e. a FOMP. The SPD technique allows a clear and accurate determination of the critical field Hcr even for polycrystalline specimens [4]. This provides in principle a powerful method for detailed investigation of the anisotropy properties of ferromagnetic and ferrimagnetic materials. Fig. 2 shows the SPD results for the case of polycrystalline Nd2 Fe14 B at various temperatures. The peak shape evolves from the case of anisotropy field Ha to critical field Hcr of FOMP. In the case of a FOMP of type 2, Hcr and Ha positions can be detected simultaneously from a single SPD measurements [5].
3. Multidomain theory
being W and g the angles formed by the magnetization vector M with the crystallographic easy axis and with the external magnetic field H;
Real-world experiments are usually carried out on ferromagnetic materials having relatively large crystallites, so that the single domain assumption of SPD theory can easily be violated. For this reason, the SPD theory was extended to multidomain crystallites [6], describing the individual crystallites according to the Neel phase theory [7]. Multidomain SPD theory is able to closely
Fig. 1. SPD measure of the anisotropy field on a Nd2 Fe14 B commercial permanent magnet.
Fig. 2. Presence of FOMPs in polycrystalline Nd2 Fe14 B below 200 K observed by means of SPD method.
E ¼ K1 sin2 W þ K2 sin4 W þ K3 sin6 W HMs cosg ð1Þ
ARTICLE IN PRESS 526
F. Bolzoni, R. Cabassi / Physica B 346–347 (2004) 524–527
Fig. 3. Calculated multidomain SPD peaks for different grain shape distribution functions (see text).
Fig. 4. Angular dependence of SPD peak amplitudes in an extruded MnAl magnet at different angles of field with respect to the easy direction.
describe the true experimental peaks. The exact position and shape of the SPD peak result to depend on the distribution function of the crystallite shapes: see Fig. 3 in which gaussian distribution functions pðeÞ of the grain ellipticity e are choosen with /eS ¼ 1:5; 0.67, 0.5, 0.4 for the cases a; b; c; d; respectively, with the same standard deviation of 0.2. The more pðeÞ approaches to a Dirac delta, the sharper the peaks become.
different orientations of the angle a formed by the magnet alignment axis with respect to H and evaluating their amplitudes (Fig. 4), one can plot them vs. a: Theory [9] explains how, from a best-fit of the obtained points in term of powers of sinðaÞ : Gðsin2 aÞ ¼ Si ai sin2i a; one can obtain a texture function F ðcos2 WÞ ¼ Si bi sin2i W by means of a mathematical transformation linking the coefficients of the two polynomial expansions: ai ¼ bi ð2iÞ!=½22i ði!Þ2 :
4. Polycrystal texture determination
5. FOMPs in non-collinear systems
Hard magnetic materials of common use are polycrystalline structures made of many grains each one with its own easy magnetization axis. The grain axis are not collinear, but are distributed in space according to an angular distribution function which is given the name of ‘‘texture’’. The preparation methods of good quality uniaxial magnets look for texture functions sharp peaked around the magnet easy axis. SPD technique gives a method for the determination of the texture function [8], which is important in order to assess the quality of a magnet. The amplitude of the SPD peak is proportional, in its non regular part, to the volume fraction of the crystallites having the easy axis oriented perpendicular to the applied magnetizing field H: Measuring the SPD peaks at
The above analysis refers to the sublattice magnetizations as rigidly collinear coupled (see Eq. (1)). Some materials are better described by means of a two sublattice model, with the possibility of a canting angle between the two sublattice magnetization vectors. In this case the total energy of the system becomes Eðya ; yb ; HÞ ¼ Jab Ma . Mb þ
3 X
ðKna sin2n Wa
n¼1 2n
þ Knb sin Wb Þ H ðMa þ Mb Þ; where a and b refer to the two sublattices, Wa and Wb are the angles that the two sublattices Ma and Mb form with the c-axis and Jab is the exchange constant [10]. Minimizing the energy expression,
ARTICLE IN PRESS F. Bolzoni, R. Cabassi / Physica B 346–347 (2004) 524–527
527
Fig. 5. SPD signal of the compound Pr2 ðCo0:1 Fe0:9 Þ17 presenting a double FOMP. Inset: corresponding single crystal magnetization curve along the hard direction.
Fig. 6. Measured SPD peak shape at 220 K for Pr2 ðFe0:6 Co0:4 Þ14 B: Inset: SPD signal for a Y-Co compound containing both 2:7 and 5:19 phases.
we can find different situations: simple approach to saturation, normal FOMP as described in Section 2, and double FOMP. The latter is characterized by a double jump in the magnetization curve of the single crystal in hard direction, with two different critical fields (see inset in Fig. 5). The first jump does not reach the saturation state, while the second one always reach saturation. In Fig. 5 is reported the experimental result of the polycrystalline Pr2 ðCo0:1 Fe0:9 Þ17 intermetallic compound at 149 K:
peaks can appear in multiphase specimens, thus allowing the detection of the anisotropy field of each phase, as in the case of a double hard magnetic phase Y-Co compound (see inset of Fig. 6). For the technical data of the experimental apparatus used for SPD measurements, we encourage the reader to see the IMEM web pages [12].
6. The fake SPD peak In Fig. 6 an SPD experimental result is given for the Pr2 ðFe0:6 Co0:4 Þ14 B: The measurements become more and more difficult because the SPD peak amplitude decreases rapidly and disappears below 100 K: Another remarkable effect at low temperature is the onset of a strong distortion of the magnetization curve that produces a ‘‘fake’’ SPD peak, i.e. a strong wide peak in the SPD signal well below the sharp anisotropy field peak, originated by the contribution of higher order anisotropy constants [11]. The difficulty of interpreting this strongly anomalous behavior could lead to possible errors, since decreasing temperature the true SPD peak tends to vanish while the ‘‘fake’’ SPD peak gets sharper. Conversely, true double SPD
References [1] G. Asti, S. Rinaldi, J. Appl. Phys. 45 (1974) 3600. [2] K.Y. Guslienko, E.H.C.P. Sinnecker, R. Grossinger, Phys. Rev. B 55 (1997) 380. [3] G. Asti, F. Bolzoni, J. Magn. Magn. Mater. 20 (1980) 29. [4] G. Asti, F. Bolzoni, J. Appl. Phys. 58 (1985) 1924. [5] L. Pareti, A. Paoluzi, F. Albertini, M.R. Ibarra, L. Morellon, P.A. Algarabel, J. Appl. Phys. 76 (1994) 7473. [6] G. Asti, F. Bolzoni, R. Cabassi, J. Appl. Phys. 73 (1993) 323. [7] L. Neel, R. Pauthenet, G. Rimet, V.S. Giron, J. Appl. Phys. 31 (1960) 27S. [8] G. Asti, F. Bolzoni, R. Cabassi, M. Ghidini, J. Magn. Magn. Mater. 128 (1993) 58. [9] G. Asti, F. Bolzoni, L. Pareti, J. Magn. Magn. Mater. 83 (1990) 270. [10] F. Albertini, F. Bolzoni, A. Paoluzi, E. Zannoni, Physica B 294–295 (2001) 172. [11] G. Asti, F. Bolzoni, J.M. Le Breton, M. Ghidini, A. Morel, M. Solzi, F. Kools, P. Tenaud, ICM 2003 Proceedings, J. Magn. Magn. Mater., submitted. [12] www.imem.cnr.it/bolzoni/highfields.html.