It is difficult to give a General recommendation. The common behavior of the magnetization curve and the sequence of magnetization of individual particles to saturation can be very different depending on the ratio of the parameters of the particles of different types and their mutual concentration. In addition to these characteristics, the coercive force of the particles and the kind of their magnetic anisotropy, the texture of the core and even the initial state of the subsystems in the demagnetized state (the initial mutual arrangement of the magnetic moments in them) are also important. For example, a combination of a relatively weakly magnetic material (with low magnetization and coercitivity [Ms1,Hc1]) and a material with high intrinsic saturation magnetization and higher coercivity [Ms2,Hc2] and a composite of type [Ms1,Hc2]+[Ms2,Hc1]* should give different results and should lead to different results (especially given the possible difference in concentrations). The resulting magnetization curve may have fractures (steps).

* Here Ms1 < Ms2, Hc1 < Hc2

Prof. Morchenko.Thanks a lot for your answer. How about the Fe based amorphous materials particle such as FeSiBCrC with intrinsic Bs=1.5 T Hc= 2 A/m and MnZn ferrite with Bs=0.5 T Hc=6 A/m? if the amorphous particle is insulated by MnZn ferrite particle to increase the Bs of the composite powder, do you think the resulting M-H curves may have fractures? or the MnZn will become saturated before amorphous particle?

It should also take into account that the magnetic susceptibility of these materials is important as well as the size, shape of the particles, structure of the composite (including its texture). Depending on the type of MnZn ferrite, it can be expected muinitial, as in FeSiBCrC (~5*103), or 2 times less. Value of mumax ~5*104 for FeSiBCrC is an order of magnitude higher than that of ferrite. Taking into account the higher HC value of ferrite we can probably expect the appearance of a stepped magnetization curve (with early magnetization of the FeSiBCrC particles) at a low content of the amorphous phase. In another case, the magnetized amorphous phase will accelerate the magnetization process ferrite particles too, and we will obtain a stepless curve.

Then, what do you think of the nanocrystalline magnetic materials which composed of nanocrystalline in amorphous matrix? there are two phases namely amorphous and nanocrystalline? do you think there will be step curves?

It's very difficult to predict the behavior of such systems, especially not knowing the specific parameters of the initial components of the composite. I tried, tried, tried to do it only in General.

Think about just two crystals of exactly the same kind and size, one oriented with its easy magnetization axis along the exciting field while the other with easy axis perpendicular to the field. It is easy to guess that the first one will be saturated earlier (in lower field). The resulting hysteresis curve of this system will differ from individual curves, taken separately. In a powder, the orientations of all small crystals will be distributed randomly and the sample will appear to VSM as a homogeneous one, but different than the original single crystal. This is even more true for your system composed of nanocrystalline material in amorphous matrix (presumably also magnetic). Here - additionally - individual crystals do interact between themselves but also with the amorphous matrix they are in. Long-range, dipolar interactions, operate as usually, but when the matrix is conductive then the long-range exchange interactions (of RKKY type) will develop at every interface. In result we have something like homogeneous sample and the idea of two different saturation fields becomes poorly defined. This is to say that the usual domain structure is observed in such materials, with any single domain covering a great number of individual nanocrystalls. So what is the conclusion?

1. We are dealing with still another magnetic material, neither the first nor the second one.

2. One may suspect that its initial permeability will be closer to the higher one, while saturation field should be closer to the lower of the two. But no warranty, sorry.

Dear Zhimin Zhou,

Dr Gutowski has given you a perfectly comprehensive answer

thanks for your answers. Prof. Gutowski and Morchenko. but i still have a question. for the nanocrystalline in amorphous matrix, such as Finemet, it is said that the materials have small hc due to the fact that the Hc of nanocrystalline is proportion to its size and it is single domain. but you said the domain cover many nanocrystalline. so how to explain the difference?

Dear Zhimin Zhou,

In a pretty wide temperature range the FINEMET and similar materials behave like superparamagnets. This means, among other things, that individual nanocrystals act like non-interacting superspins, each of order 105-106 Bohr's magnetons. Ideally, the coercivity should be exactly null. It is not - due to magnetic anisotropy of individual nanocrystals, which indirectly defines the value of Hc for each grain individually. Indeed, below the diameter d of roughly 20 nm the anisotropy behaves like d6, but larger nanocrystals are no longer single-domain and this 'law' ceases to be valid. What's more, the nanocrystals are not isolated. Some influence of the amorphous matrix was investigated in Magnetoresistance in Nanocrystalline Fe-Based Metallic Glass by Marek Kuźmiński, Anna Ślawska-Waniewska, Henryk K. Lachowicz, Giselher Herzer, IEEE TRANSACTIONS ON MAGNETICS, VOL. 30, NO. 2, MARCH 1994, p. 533. As far as I remember, the domains of ~10 cm long were observed in thin tapes of similar materials.

thanks for your answers. Prof. Gutowski. i still have a question. in the soft magnetic powder, each particle is isolated from each other by insulation coatings, so there are distributed air gap in the core. for each particle, there will be free poles and the demagnetization factors Nd. if there are two particles with different size diameter, then when they are mixed each other. both big particle and small particle will have different surrounding environment. the free poles will be different distributed. i think big particle and small particle will have different Nd. so the internal field Hin=Hext-Nd*M for both particle will be different. as a result, one size particle will be saturated before the other. How about your opinions? or according to your answers above, if the two particle system are mixed uniformly, the system could be looked as a new materials composed of new structure element in which the big particle and the surrounding particle are repeated in the systems.

The formula you cite, Hin=Hext-Nd*M, describes the whole sample, it is not applicable to a single nanocrystal inside the sample. This is because, as seen by any given nanocrystal, the field Hext consists of external field but also contains components generated by all other neighboring nanocrystals. Even worse: the demagnetization tensor describes (again) the whole sample. And going to the properties of Nd tensor: the values of its components depend on sample shape rather than its size. The previous sentence is true when the sample shape is bounded by second degree surface. In this case the sample is uniformly magnetized in uniform external field. In every other case the components of Nd change their values from point to point. And one more comment: if your nanocrystals are small enough to be in single-domain state then they are already saturated, even in absence of external field. All the external field can introduce is the reorientation of (local) magnetization, just like in paramagnets.

I have just found another article related to this thread:

Properties of polycrystalline nanoparticles with uniaxial and cubic types of magnetic anisotropy of individual grains, V.A.Bautin, G.Seferyan, M.S.Nesmeyanov, N.A.Usov, Journal of Magnetism and Magnetic Materials Volume 460, 15 August 2018, Pages 278-284, available at: Article Properties of polycrystalline nanoparticles with uniaxial an...

Maybe this is what you are looking for ...

It is not possible to give an unambiguous or more definite answer to your question even after clarification of the information and additional explanations about your system.

I'm interested in a similar problem with ferromagnetic microwires in a glass shell. Such wires are usually in an amorphous state, but under certain conditions of their preparation or after heat treatment under special conditions they acquire a nanocrystalline or amorphous crystal structure with the presence of several magnetically ordered phases(see, for example, our articles:

Structural and magnetic transformations in amorphous ferromagnetic microwires during thermomagnetic treatment under conditions of directional crystallization in Journal of Alloys and Compounds, 2016,

https://www.researchgate.net/publication/311778563_Structural_and_magnetic_transformations_in_amorphous_ferromagnetic_microwires_during_thermomagnetic_treatment_under_conditions_of_directional_crystallization?_sg=Ux4X34lB-ibp5UdDP0Tl99TnJEw7T78_v69HwAJMcE_mKGvfWI1GxCyIQ8u6jfFcMHLS-BNGgG_wJ651dSNcI4DV5tvdMPAQRLX7tmr3.qFYe7aPDhiEwUqMUSj5bzyJQ-k49zE2VS9TiUE6nlfuy8RRQg1iLV0-W_wT8_KMtq0FfFPD3pQoNwKWkDw1YdQ);

X-ray and Mössbauer study of amorphous and crystalline ferromagnetic microwires (in Russian), https://www.researchgate.net/publication/311983308_X-ray_and_Mossbauer_study_of_amorphous_and_crystalline_ferromagnetic_microwires_in_Russian?_sg=Ux4X34lB-ibp5UdDP0Tl99TnJEw7T78_v69HwAJMcE_mKGvfWI1GxCyIQ8u6jfFcMHLS-BNGgG_wJ651dSNcI4DV5tvdMPAQRLX7tmr3.qFYe7aPDhiEwUqMUSj5bzyJQ-k49zE2VS9TiUE6nlfuy8RRQg1iLV0-W_wT8_KMtq0FfFPD3pQoNwKWkDw1YdQ;

Effect of Mechanical Stresses and Annealing on the Magnetic Structure and the Magnetic Impedance of Amorphous CoFeSiBCr Microwires (in Physics of the Solid State, 2018,

https://www.researchgate.net/publication/323478573_Effect_of_Mechanical_Stresses_and_Annealing_on_the_Magnetic_Structure_and_the_Magnetic_Impedance_of_Amorphous_CoFeSiBCr_Microwires?_sg=Ux4X34lB-ibp5UdDP0Tl99TnJEw7T78_v69HwAJMcE_mKGvfWI1GxCyIQ8u6jfFcMHLS-BNGgG_wJ651dSNcI4DV5tvdMPAQRLX7tmr3.qFYe7aPDhiEwUqMUSj5bzyJQ-k49zE2VS9TiUE6nlfuy8RRQg1iLV0-W_wT8_KMtq0FfFPD3pQoNwKWkDw1YdQ);

articles by I. Baraban, et al. The role of structural properties on magnetic characteristics of glass-coated microwires in J. Magn. and Magn. Mater. 459 (2018),

https://doi.org/10.1016/j.jmmm.2017.12.048

and by G.N. Elmanov, et al. Effect of heat treatment on phase transformations and

magnetization of amorphous Co69Fe4Cr4Si12B11 microwires, Journal of Alloys and Compounds

(2018), doi: 10.1016/j.jallcom.2018.01.114.

In such wires, there are both smooth magnetization curves, and with features, including stepped hysteresis curves. The shape and orientation of the crystallographic axes of individual particles in the matrix can be different or have a certain set of preferred orientations, so that even the magnetization of particles (or their groups and aggregates) can occur in different fields. (see, for example, article by V. Rodionova et al. in Journal of Superconductivity and Novel Magnetism: Tailoring of Magnetic Properties of Amorphous Ferromagnetic Microwires (2015) DOI 10.1007/s10948-014-2777-8

and our future article: Structural and magnetic anisotropy of directionally crystallized ferromagnetic microwires (to be published in EPJ Web of Conferences).

Each individual particle in the matrix can be single-domain, but the magnetic moments of all particles are consistent, oriented in the same direction. Also, the matrix or interlayer between the grains (if they are magnetically ordered) can also be magnetized in the same direction.

As for the domains that extend to several crystallites, it is useful to get acquainted with such fundamental papers by Giselher Herzer as:

Grain size dependence of coercivity and permeability in nanocrystalline ferromagnets \\ IEEE Trans. Magn. 26(5), 1397-1402 (1990)

Grain structure and magnetism of nanocrystalline ferromagnets \\ IEEE Trans. Magn. 25(5), 3327-3329 (1989)

Modern soft magnets: Amorphous and nanocrystalline materials \\ Acta Mater. 61(3), 718-734 (2013)

and other works by this author.

The answers are quite good in general, but if you have superparamagnetism (or others as micromagnetism in wires) there many other details to take into account. For instance, it is necessary to observe the zero field cooling, the field cooling, the blocking temperature, the size of the domains for the respective magnetic material, ...I think that is not too useful answer in general without knowing the kind of material employed, sizes etc

In addition to the above:

Too many factors influence the final result. In particular, it should be taken into account that not only the shape and orientation of the crystallographic axes of individual nanoparticles, but also their sizes can be different. Therefore, the resulting behavior of the system will also depend on the ratio of superparamagnetic and non-superparamagnetic particles.

Thanks for your answers. Prof. Morchenko, Boldmir and Gutowski. now i will give two cases A and B. in case A, for example nanocrystalline with size 10 nanometer sphere and 70 volume percent embedded in 30 volume percent amorphous matrix. they are sinter together on the border. in case B, 75 micrometer case A system sphere 70 vol percent and 25 micrometer amorphous sphere 30 volume percent are distributed uniformly and they are elastic point contact on the border. case A could be looked as new materials different from its component. how about case B?

I do not dare to give a complete answer and, again, will get off common perceptions and remarks. Of course, in the first case we are dealing with nanoscale particles, the properties of which are very different from the properties of objects of micrometer scale. Already because of this, systems A and B will have different characteristics. In addition, in models of effective medium (even regardless of the particle size) usually it is distinguishes two cases: 1-heterogeneous mixture of statistically distributed particles of different sorts and 2-matrix systems with particles distributed in a continuous medium. And even under other equal conditions, the results do not always turn out similar. However, even in case B, we can probably (though not always) assume that we have received new material, if the so-called synergistic effect will be observed, i.e. the properties of our composite will exceed the properties of the initial components, and not equal to their simple weighted average sum (sorry for the immodesty, I again refer to our work with large particles of ferrite in a continuous medium. In our case the compositions of the components, of course, are quite different from yours, and we do not describe magnetic characteristics: Dielectric and Magnetic Properties of Two-Phase Composite System: Mn-Zn or Ni-Zn ferrites in Dielectric Matrices:

www.researchgate.net/publication/290055764_Dielectric_and_Magnetic_Properties_of_Two-Phase_Composite_System_Mn-Zn_or_Ni-Zn_ferrites_in_Dielectric_Matrices).

Your volume concentration of the crystalline component particles is close to the maximum which is possible for the closest packing (~74% vol.). Thus, in both cases, crystal particles practically contact each other, and there is a magnetodipole interaction between them. Consequently, the behavior of magnetic moments of even individual single crystal spheres themselves will take place in both cases, and this will affect the coordinated consistent behavior of their magnetic moments during magnetization or in electromagnetic field, etc. In addition, it is necessary to take into account the properties of both phases, through which at such close proximity it is also possible (though not always and not to a large extent) the transfer of exchange interactions with the corresponding consequences.

Another problem: how the voids between the round particles of the amorphous and crystalline phase are supposed to be filled. For example, there may be pores. The result will depend on that too.

However, we must also take into account the already said that in system A superparamagnetism is possible, and in system B, probably, there will not even be a single-domain state. But the character of the systems' remagnetization will be different for an ensemble of single-domain (even non-superparamagnetic) particles and multi-domain particles.

I'm sorry, but I'm not sure I can add anything more sensible to your question in the future.

Thanks for your answers.Prof. Morchenko. i will read the papers you cited above.