Abstract
Small-angle scattering of X-rays and neutrons is a routine method for the determination of nanoparticle sizes. The so-called Guinier law represents the low-q approximation for the small-angle scattering curve from an assembly of particles. The Guinier law has originally been derived for nonmagnetic particle-matrix-type systems and it is successfully employed for the estimation of particle sizes in various scientific domains (e.g. soft-matter physics, biology, colloidal chemistry, materials science). An important prerequisite for it to apply is the presence of a discontinuous interface separating particles and matrix. Here, the Guinier law is introduced for the case of magnetic small-angle neutron scattering and its applicability is experimentally demonstrated for the example of nanocrystalline cobalt. It is well known that the magnetic microstructure of nanocrystalline ferromagnets is highly nonuniform on the nanometre length scale and characterized by a spectrum of continuously varying long-wavelength magnetization fluctuations, i.e. these systems do not manifest sharp interfaces in their magnetization profile. The magnetic Guinier radius depends on the applied magnetic field, on the magnetic interactions (exchange, magnetostatics) and on the magnetic anisotropy-field radius, which characterizes the size over which the magnetic anisotropy field is coherently aligned into the same direction. In contrast to the nonmagnetic conventional Guinier law, the magnetic version can be applied to fully dense random-anisotropy-type ferromagnets.
Highlights
The determination of particle sizes is one of the most important daily tasks in many branches of the natural sciences
Its application to magnetic materials, which is the subject of the present article, should be considered with special care; for instance, the Guinier law is certainly applicable to systems consisting of saturated and homogeneous magnetic particles in a nonmagnetic and homogeneous matrix or, likewise, to pores in a saturated matrix
We refer to the article by Burke (1981) who investigated the influence of magnetic shape anisotropy on the Guinier law of fine ferromagnetic single-domain particles
Summary
The determination of particle sizes is one of the most important daily tasks in many branches of the natural sciences. Its application to magnetic materials, which is the subject of the present article, should be considered with special care; for instance, the Guinier law is certainly applicable to systems consisting of saturated and homogeneous magnetic particles in a nonmagnetic and homogeneous matrix or, likewise, to pores in a saturated matrix In this context, we refer to the article by Burke (1981) who investigated the influence of magnetic shape anisotropy on the Guinier law of fine ferromagnetic single-domain particles. Equation (1), with a constant and field-independent RG, does not describe the low-q region of the magnetic small-angle neutron scattering (SANS) cross-section. It may be clear from the previous considerations that an effective magnetic Guinier radius is expected to depend on the applied magnetic field as well as on the magnetic interactions (e.g. exchange, anisotropy, magnetostatics). In the Supporting information for this article, the two- and onedimensional total SANS cross-sections and a graphical representation of the relative error of the magnetic Guinier approximation are featured
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