Presentation Information
[P2-45]Expansion of the STEP method to magnetically isotropic magnets
*Kurima Kobayashi1, Tomoko Kuno1, Hiroshi Fujiwara1 (1. Ritsumeikan University (Japan))
Keywords:
The STEP method,SDG,MDG,SMDG,ThMn12 type magnet
The STEP method was developed for categorizing the grains and/or group of grains in magnets into the single-domain grains (SDG), multi-domain grains (MDG) and saturable multi-domain grains (SMDG) (1).The method was initially applied to highly orientated magnets, such as Nd-Fe-B sintered magnets and orientated Sm-Fe-N magnets (2), and successfully determined the volume fractions (vol.%) of each type of grains in the magnets. However, when the method applied to magnetically isotropic magnets, such as the ThMn12 type magnets, the method for determining of vol.% of SDG, MDG, and SMDG in the samples has not been established. In this study, we show the calculation method of vol.% of above three types of grains in magnetically isotropic magnet such as (Sm0.8Zr0.2)1.1(Fe0.9Co0.1)11.3Ti0.7 of comparatively large coercivity of 413 kA/m prepared at 850 ℃ for 30 minutes annealing.We started from the basic equation (1), (first term) magneto-anisotropic energy and (second term) magnetic energy under applied field (H), here Ku is anisotropy constant, Js is saturation polarization,θ0 is initial canting angle of c-axis from the direction of applied field, and θ is the rotation angle from the original canting angle.
E = Kusin2θ - JsHcos(θ0 - θ) (1)
As a typical isotropic magnet, we suppose the magnet in which the c-axes of grains isotropic point to whole surface of distribution sphere. Therefore, if the sphere is divided in certain width rings after the decision of direction of applied field, the existence probability of grains existing in a certain θ0 ring is easily calculated by the ordinary method using polar coordinates, then the total surface area of distribution sphere is unity. In this study, the sphere was divided into 18 rings, the width of angle delta θ0 is 10 degrees, the mathematical presentation of surface area (S) of distribution sphere is the following equation.
Sam (j = 1 - 18)Sj / 4π = 1 (2)
By differentiation of Eq. (1), the relation between H and θ in the grains having certain θ0 value can be calculated using the equation (3). For an example, in the real calculation, typical θ0 = 65° was used forθ0 = 60° ~ 70° grains.
H = (Kusin2θ) / Jssin(θ0 - θ) (3)
The isotropic distributed SDG shows the magnetic reversal, a jump of magnetic moment when the second differentiation of the equation (1) is zero, ∂2 E / ∂ θ 2 = 0. Therefore, the magnetization in first quadrant (1Q) and demagnetization (2,3Q) curves show prominent change of slope in the initial stage, which can be obviously detected in the measured magnetization curve in Fig.1 (a). The MDG, however, show the monotonous reversible curves in both 1Q and 2,3Q. The magnetization changes are originated by domain wall motion (DWM) and rotation of magnetic moments in grains. The SMDG is separated into two groups, first group is the same behavior with SDG after the magnetization in 1Q, and second group shows the reproduction of domain wall structure, ie DWM in the demagnetization process which disappears after the magnetization to inverse direction.Using the isotropic STEP method in this study, three categorized grains explained above can be clearly separated, and the vol.% of grains belonging to each category can be determined. For the case of (Sm0.8Zr0.2)1.1(Fe0.9Co0.1)11.3Ti0.7 magnet prepared at 850 ℃ for 30 minutes annealing, MDG is 17.0 vol.% and SDG+SMDG is 83.0 vol.% in the magnetization process, and MDG is 17.6 vol.% and SDG+SMDG is 82.4 vol.% in the demagnetization process. Therefore, the agreement of results in both processes is good enough for the purpose of this study.In the presentation, the details of the isotropic STEP method will be explained including the effects of rotation of magnetic moments in all grains.
References
Kobayashi, “Role of domain structure change in the coercivity mechanism of sintered permanent magnets”, J. Jpn. Soc. Powder Powder Metallurgy, 68 (2021) pp. 523-535.
Kobayashi and D, Givord, J. Magn. Soc. Japan, vol.21 (1997), No.10 pp.1175-1180.
E = Kusin2θ - JsHcos(θ0 - θ) (1)
As a typical isotropic magnet, we suppose the magnet in which the c-axes of grains isotropic point to whole surface of distribution sphere. Therefore, if the sphere is divided in certain width rings after the decision of direction of applied field, the existence probability of grains existing in a certain θ0 ring is easily calculated by the ordinary method using polar coordinates, then the total surface area of distribution sphere is unity. In this study, the sphere was divided into 18 rings, the width of angle delta θ0 is 10 degrees, the mathematical presentation of surface area (S) of distribution sphere is the following equation.
Sam (j = 1 - 18)Sj / 4π = 1 (2)
By differentiation of Eq. (1), the relation between H and θ in the grains having certain θ0 value can be calculated using the equation (3). For an example, in the real calculation, typical θ0 = 65° was used forθ0 = 60° ~ 70° grains.
H = (Kusin2θ) / Jssin(θ0 - θ) (3)
The isotropic distributed SDG shows the magnetic reversal, a jump of magnetic moment when the second differentiation of the equation (1) is zero, ∂2 E / ∂ θ 2 = 0. Therefore, the magnetization in first quadrant (1Q) and demagnetization (2,3Q) curves show prominent change of slope in the initial stage, which can be obviously detected in the measured magnetization curve in Fig.1 (a). The MDG, however, show the monotonous reversible curves in both 1Q and 2,3Q. The magnetization changes are originated by domain wall motion (DWM) and rotation of magnetic moments in grains. The SMDG is separated into two groups, first group is the same behavior with SDG after the magnetization in 1Q, and second group shows the reproduction of domain wall structure, ie DWM in the demagnetization process which disappears after the magnetization to inverse direction.Using the isotropic STEP method in this study, three categorized grains explained above can be clearly separated, and the vol.% of grains belonging to each category can be determined. For the case of (Sm0.8Zr0.2)1.1(Fe0.9Co0.1)11.3Ti0.7 magnet prepared at 850 ℃ for 30 minutes annealing, MDG is 17.0 vol.% and SDG+SMDG is 83.0 vol.% in the magnetization process, and MDG is 17.6 vol.% and SDG+SMDG is 82.4 vol.% in the demagnetization process. Therefore, the agreement of results in both processes is good enough for the purpose of this study.In the presentation, the details of the isotropic STEP method will be explained including the effects of rotation of magnetic moments in all grains.
References
Kobayashi, “Role of domain structure change in the coercivity mechanism of sintered permanent magnets”, J. Jpn. Soc. Powder Powder Metallurgy, 68 (2021) pp. 523-535.
Kobayashi and D, Givord, J. Magn. Soc. Japan, vol.21 (1997), No.10 pp.1175-1180.
