Neutron Diffraction: A tool for the Magnetic Properties

Abstract

We start this chapter with an introduction of basic information of nuclear scattering, elastic neutron scattering by polycrystals, magnetic scattering, and some applications of neutrons. In the main part, it has the basic information for determining magnetic structures of any compound through analysis from neutron powder diffraction data. As an example, we will analyze the data collected for the double perovskite Sr ₂ Y RuO ₆ at variable step in scattering angle 2θ in the D1B instrument/Institut Laue-Langevin (ILL). We collected one pattern in the paramagnetic state (40 K), and we refined the crystal structure using a PCR file. Another pattern, in the magnetic state (1.6 K), was collected in order to observe the magnetic peaks. To analyze the data, we used the FullProf Suite program and other subprograms as WinPLOTR-2006 to select the peaks, K_Search to determine the propagation vector K, and BasIreps to obtain the magnetic structure symmetry of the compound in coefficients of the basis functions and in Cartesian or spherical components. We used the FullProf Studio for visualizing crystal and magnetic structures. Furthermore, we show how to use the Le Bail method to check the propagation vector calculated. For determining the magnetic structure of a material, required from the reader is the minimal experience with crystallography and structure refinement using X-ray powder diffraction and/or neutron diffraction through Rietveld analysis.

1.1 Introduction

Experimental techniques based on neutron scattering have a special relevance in the study of new materials due to some basic aspects of the neutrons as particles in free state, that is, outside the atomic nucleus. They are relatively stable, with a mean lifetime of about 14 min; as indicated in their own name, do not have a net electric charge, that is, they are neutral; have no measurable electrical dipole moment; but have a relevant magnetic dipole moment associated with an intrinsic angular momentum or spin.

Because they do not have an electric charge, or an electric dipole, they do not electrically interact directly with atoms, ions, or molecules that make up matter. In general, they are poorly absorbed in direct interaction with matter when compared to other charged particles or even with X- and gamma-type photons. The predominant interaction is with the atomic nuclei. The exact form of the interaction between neutrons and nuclei is not known, but it is possible to theoretically treat the problem with Fermi’s pseudo-potential. We will detail this later. This interaction can occur in an elastic regime where the kinetic energy of the neutron is conserved in the collision with the nucleus. We usually call the elastic scattering regime as neutron diffraction, and the phenomenon of the collective neutron scattering of a set of atoms can be treated in wave bases, involving notions, such as constructive and destructive interference.

Neutron diffraction experiments are widely used for the determination, and refinement, of crystalline structures. They are applied in materials in the form of monocrystals or polycrystals (powder) through experimental methods, and analysis procedures, that are completely similar to those used in X-ray diffraction. This correspondence can be appreciated with the use of structural refinement programs of polycrystalline materials by the Rietveld method, such as FullProf and GSAS where, in general, it is possible to process neutron diffraction and X-ray diffraction data on the same platform and with the same algorithms.

An important feature in neutron diffraction is that there is not much contrast in the efficiency of scattering, if so we can express it, among the various chemical elements in the periodic table. In X-ray diffraction, the scattering efficiency, as depicted by the atomic scattering factor, increases with the number of electrons in the atom. This fact allows it to be much easier to locate light atoms in association with heavy atoms through neutron diffraction experiments than X-ray diffraction. The advantages of neutron diffraction in relation to X-ray diffraction in the location of hydrogen atoms present, for example, in organic materials or in hydrated materials, are well-known.

In the inelastic scattering regime, neutrons can transfer momentum to the atoms in the lattice. In a nuclear reactor, we can produce neutrons with energy very close to the vibrational energy of the atoms (thermal neutrons) and yet with wavelength of the order of the interatomic distances. In this way, an expressive interaction with normal crystal vibration modes (phonons) is possible, and the result of the scattering can be very informative. Because they work with very close energies between neutrons and phonons, neutron diffraction is a very skillful technique for the study of lattice dynamics.

Another form of the interaction of neutrons with matter is the magnetic interaction. Whenever atoms with permanent magnetic dipole moments are present, an interaction between the atomic magnetic dipole and the magnetic dipole of the neutron is established. Particularly important is the result of this interaction when we have materials with some sort of magnetic ordering. The result of a neutron diffraction experiment, mediated by magnetic interaction, has the potential to reveal the spatial arrangement of the magnetic dipoles of the magnetic sublattice present in the material. On the other hand, in the inelastic scattering regime intermediated by the magnetic interaction, we can observe energy transfers between neutrons and spin waves present in the material in a manner analogous to that occurring in the inelastic nuclear scattering with the lattice phonons.

In this chapter, we intend to give some practical information on how this characterization can be achieved through a case study. We begin with a simple description of the scattering of neutrons with nuclei where we try to give an overview of the most important concepts involved, and then we will discuss the magnetic scattering. Naturally, it would be far from the scope of this chapter to attempt to synthesize all the details of the theoretical modeling of the neutron scattering process, and for those who would appreciate a greater depth, we remember that there are good texts that develop all the treatment from the first principles [1, 2].

1.2 Nuclear Scattering

Shortly after the discovery of the neutron by Chadwick in 1932, the first studies on the possibility of the diffraction of a beam of neutrons by crystals, in the same way that occurs with an electron or X-ray beams, appeared. The analogy is very direct and very plausible. Neutrons are very massive particles with mass comparable to the hydrogen atom, and because they do not have a net electric charge, they have a remarkable ability to penetrate the solid matter. According to the quantum mechanics, each particle of momentum p =  mv (m – mass, v – velocity) a wave of wavelength λ given by the relation of de Broglie,

$$\displaystyle \begin{aligned} \lambda = \frac{h}{p} \end{aligned} $$

(1.1)

where h is the Planck constant. If we consider a beam of neutrons generated by a radioactive source incident on a paraffin-moderating target, we will have a dispersion of the kinetic energy of the neutrons by elastic collisions with the hydrogen atoms, abundant in paraffin. This dispersion closely follows the distribution of Maxwell–Boltzmann velocities for an ideal gas. If the absolute temperature of the moderating target is T, the average kinetic energy of the neutrons (k) is given by

$$\displaystyle \begin{aligned} K = \frac{p^{2}}{2m} = \frac{3}{2}kT \end{aligned} $$

(1.2)

where k is the Boltzmann constant. For a paraffin-moderating target at room temperature (∼300 K), the average kinetic energy of the neutrons is about 0.04 eV with a De Broglie wavelength of approximately 1.5 . This value closely approximates the average wavelength of the Kα doublet of an X-ray tube with copper anode, widely used in the X-ray diffractometers of polycrystals for structural characterization of the most diverse materials.

In 1930s the first evidence of neutron diffraction [3,4,5] emerged through experimental assemblies based on low intensity radioactive sources. The applications of neutron diffraction in crystallography as an accessory technique in the structural characterization of materials had their great impetus in the post-II War period with the advent of nuclear reactors and accelerators of particles.

With the new sources, it is possible to obtain beams with abundant thermal neutrons. These neutron beams have a continuous spectrum of kinetic energy, which is to say, in wavy terms, a “white” radiation. From a white beam of neutrons, we can obtain a monochromatic beam using monocrystals exactly as we proceed with X-rays. Monocrystals of Be, Si, Ge, Cu, Pb, or even laminar pyrolytic graphite are used for this purpose. Figure 1.1 shows, schematically, an experimental setup using a monochromatic beam for the study of a polycrystalline sample, in a configuration equivalent to Debye–Scherrer geometry used by X-ray diffractometers for polycrystals. The differences are more related to the dimensions: the neutron diffractometers are significantly larger than their equivalents for X-rays. In part, this is due to the dimensions of the monochromatic neutron beam and the neutron scattering cross section itself which are smaller than that of X-photons, requiring larger amounts of sample. The neutral diffractometer installations are also larger due to the need to provide safety for users and technical personnel.

Fig. 1.1

Schematic representation of the typical experimental configuration of a neutron diffractometer for polycrystalline samples

The most efficient experimental setup makes use of position-sensitive detectors as the arrangement of multiple detectors aligned in an arc concentric with the sample, as indicated in Fig. 1.1. As detection units for thermal neutrons (low energy), proportional gas detectors containing³He or BF₃ are used. Both ³He and B have high cross section of shock for thermal neutrons, and by absorbing a neutron, they decay in ionized particles that are detected by the proportional detector [1].

Other processes that do not involve a nuclear reactor can also produce neutrons. High-energy charged particles obtained from particle accelerators are used for the production of neutrons by collision with solid targets. Usually, because they are more efficient, proton beams are used instead of electrons, which when they collide against a solid target produce high-energy neutrons. These neutrons need to be moderate to energies in the range of a few tens of meV, so they can be applied in a useful way in the structural analysis of materials. As in the case of the reactors, the moderators are materials consisting of chemical elements with expressive section of shock to collision with the neutrons. It is common to use water maintained at controlled temperatures with a common cooling system. The neutron diffractometers based on accelerators bring some advantages due, among other things, to the characteristics of the primary neutron beam. Unlike the nuclear reactor where the neutron beam is continuous, in the particle accelerators, the beam is pulsed, discontinuous. Since the pulses have very short duration, it is possible to measure the flight time from the moderator to the detector with a very high precision, which allows to measure the speed of the neutrons (typically of a few kilometers per second) and, consequently, its wavelength, through de Broglie’s equation, very accurately. In this case monochromators are not used which allows the use of all produced neutrons.

Because it is a neutral particle, not carrying liquid electric charge, the interaction of the neutron with the atoms is mainly with the atomic nucleus. If we consider thermal neutrons with de Broglie wavelengths of the order 1 Å (10⁻¹⁰ m) compared to the dimensions of the atomic nucleus of the order of 10⁻¹⁴ m, we can conclude that the atomic nuclei are basically punctiform scattering centers. If the energy of the incident beam of neutrons is different from the energy of the excited states of the target nuclei, resonance does not occur, and the scattering is essentially elastic. Thus, in this context, we can describe the wave function for the incident neutron beam as a plane wave of the type $e^{(-ik_{0}.z)}$, and we can describe the scattering wave by an atomic nucleus as − b∕r.e ^(−ik.r),¹ where k ₀ is the incident wave vector, k ₀ = k = 2π∕λ, and λ is the associated wavelength (Fig. 1.2). The parameter b is a constant for each nuclide and is called scattering length.

Fig. 1.2

The atomic nucleus behaves like a point scattering center for neutrons. The incident neutron can be described as a plane wave with wave vector k ₀. The scattered neutron is well described as a spherical wave

Since there is no theory about the interaction between the nucleus and the neutron, it is not possible to determine b from the first principles, and its measurement is strictly obtained experimentally. To describe the scattering, a phenomenological approach is used where a potential interaction function V (r) between neutron and atomic nucleus is introduced, which only simulates the main characteristics of this interaction, such as short range, centrality, and high intensity. This potential is known as Fermi’s pseudo-potential,

$$\displaystyle \begin{aligned} V (\mathbf{r}) = \frac{2\pi \hbar^{2}}{m} b\delta(\mathbf{r}) \end{aligned} $$

(1.3)

where m is the mass of the neutron and is the Dirac delta function for three dimensions,

$$\displaystyle \begin{aligned} \int \delta(\mathbf{r}) dV = 1 \end{aligned} $$

(1.4)

The formalism that describes the scattering is too extensive to be summarized here, and we refer the interested reader to other sources [1, 6]. It is possible to show that if we produce the neutron collision with a fixed nucleus at the origin of a coordinate system, the fraction of the total number of neutrons scattered at a solid angle d Ω, in the direction given by the angles θ and φ in spherical coordinates, is given by

$$\displaystyle \begin{aligned} \frac{d\sigma}{d\Omega} = \left(\frac{m}{2\pi \hbar^{2}}\right)^{2} \left|\int V(\mathbf{r}).e^{i\mathbf{k}.\mathbf{r}}dV \right|{}^{2} {} \end{aligned} $$

(1.5)

where $\frac {d\sigma }{d\Omega }$ is the so-called differential cross section. If we substitute pseudo-potential in this expression, we obtain

$$\displaystyle \begin{aligned} \frac{d\sigma}{d\Omega} = b^{2} \end{aligned} $$

(1.6)

i.e., a constant for any direction of scattering. This picture is very peculiar to neutron scattering, and a rapid comparison with what occurs in X-ray diffraction might be useful.

In X-ray scattering, we have an electromagnetic plane wave that interacts with the N electrons of an atom. The atomic electrosphere has dimensions much greater than the size of the nucleus being of the same order of the wavelength of the X-rays. The general approach to treat this scattering is to take an elemental partition of electronic cloud. In the quantum-mechanical description, each infinitesimal volume dV , at position R of the electronic cloud, contains a charge dq mediated by the probability density of finding an electron in this position and given by

$$\displaystyle \begin{aligned} dq = \rho dV = -e.\sum_{j=1}^{N} |\Psi_{j}(\mathbf{R})|{}^{2}.dV \end{aligned} $$

(1.7)

where ρ is the density of electrons at point R, e is charge of the electron, and Ψ_j(R) is the wave function of the electron-j. Each charge element dq oscillates under the action of the incident wave and forms an elemental electric dipole that consequently emits radiation at the same frequency of the incident wave (Thomson scattering). The wave scattered by the atom is a superposition of each elemental wave. Figure 1.3 shows schematically how we should add the contributions, where a phase difference between the elementary waves has to be taken into account. If we consider the wave amplitude A(θ) (electric field or magnetic field) at a point far from the atom,

$$\displaystyle \begin{aligned} A(\theta) = \left( \int_{\mathrm{{atom}}} dq.e^{i[\mathbf{k} - {\mathbf{k}}_{0}].\mathbf{R}} \right) .A_{0}(\vartheta) = f(\theta).A_{0}(\theta) \end{aligned} $$

(1.8)

$$\displaystyle \begin{aligned} f(\theta) = \int_{\mathrm{{atom}}} \rho e^{i[\mathbf{k} - {\mathbf{k}}_{0}].\mathbf{R}}.dV \end{aligned} $$

(1.9)

Fig. 1.3

To compute the X-ray scattering by the electron cloud of the atom, we must consider the contribution of each charge element dq = ρdV  of an infinitesimal volume partition dV . Each charge element forms a small dipole that generates an elemental spherical wave. The interference of these waves generates an anisotropic scattering that varies according to the 2θ angle

A ₀(θ) represents the amplitude of the wave scattered by an oscillating electron supposedly placed at the origin of the reference system. Thus, in a closer view of the Huygens principle for wave propagation, the function f(θ) measures the interference of the elementary waves emitted by the different parts of the electron cloud of the atom and is called the atomic scattering factor. Figure 1.4 shows the behavior of f(θ) for the cobalt atom (N = 27). We see a noticeable attenuation of the scattering intensity of the X-rays with the scattering angle θ, due to the interference in the electronic cloud. As we have seen, the same effect does not occur with the neutron–nucleus interaction. In the same plot, we represent the behavior of scattering length for the nuclide ⁵⁹Co.

Fig. 1.4

Schematic chart showing the behavior of the atomic scattering factor for X-rays and the scattering length for neutron scattering of element ⁵⁹Co (Z = 27). We can see that for X-rays the intensity of scattered radiation decreases sharply with the scattering angle (θ), while the scattering length remains constant

For certain nuclides b is a complex number that varies strongly with the energy of the incident neutron. For these cases, in the interaction between the neutron and the nuclide occurs the formation of a new nucleus, involving the original nucleus and the incident neutron, with energy close to an excited state of the new system. This interaction is strongly dependent on neutron energy. The imaginary part of parameter b is linked to the absorption of the neutron. Under these conditions, b can assume significant values. Examples of nuclides where these resonant effects occur are ¹⁰³Rh, ¹¹³Cd, ¹⁵⁷Gd, and ¹⁷⁶Lu [1]. For most cases the parameter b does not depend on the energy.

Since the neutron has spin 1/2, in the condition that the nuclide has a spin I ≠  0, the integrated neutron +  nuclide system will have two possible states for the total spin, I +  1/2 and I − 1/2, and for each of these states we will have a value of b. Table 1.1 presents some values. If the spin of the nuclide is zero, only one value of b occurs.

Table 1.1 Values of b for the interaction between the neutron and some nuclides [7, 8]

In Fig. 1.5 we see the absolute value of b as a function of the atomic mass of the chemical elements. As we see, the behavior of b is unpredictable. For each isotope of a given element, other values of b are found. In Table 1.2 we see the values of b for the different isotopes of the iron element.

Fig. 1.5

Variation of the scattering length (b) with the atomic number for the most stable isotopes. The measurements are based on experimental results. Variation with atomic number is not smooth and predictable

Table 1.2 The values of b for the different isotopes of the iron element [7, 8]

From Fig. 1.5 we can conclude that there is no big variation between the values of b and the atomic number, with the majority of values being in the range of 5 to 10 fm. This result contrasts sharply with what occurs in X-ray diffraction where the diffracted intensities are regulated by the number of electrons in the atoms. The consequence of this phenomenon is clear when we try to determine or refine, by XRD, a structure where very different atomic number elements occur, for example, in the superconducting ceramic YBACU where oxygen contributes very little compared to the cations, Y, Ba, and Cu. Only with the diffraction of neutrons, a correct determination of the structure of this material was possible. This same property shows why neutron diffraction is the most appropriate technique when it is necessary to locate hydrogen atoms in the crystal lattice with higher precision.

1.3 Elastic Neutron Scattering by Polycrystals

The applications of neutron scattering in the structural characterization of materials are similar to the applications of X-ray scattering. In a sense, we can say that both techniques present many conceptual similarities and, in what they have specifically, they complement each other. In elastic scattering the neutrons collide with the nuclei without losing their kinetic energy. In X-ray scattering we also observe the elastic scattering of photons, and in this condition, in both cases, we say that a diffraction phenomenon occurs. However, in the case of thermal neutrons, the typical energy is of the order of some meV, and in this case the neutron energies are of the same order of magnitude of the thermal vibrations of the atoms in the crystalline lattice. It is thus possible for efficient energy exchange interactions between neutrons and network phonons. In this interaction, the neutron can receive or donate energy. This interaction path gives rise to an inelastic scattering, which does not preserve the energy (wavelength) of the neutron and does not give rise to constructive interference formation (Bragg lines). In turn, neutron inelastic scattering has been used, among other things, to characterize the phonon spectrum of a given material of enormous importance to materials science and solid-state physics [9].

In the case of crystalline materials, one of the most widely used methods of neutron diffraction is the polycrystalline method where the sample is reduced to a very fine granulation powder and subjected to a well-known neutron beam, that is, wave determined. This is the experimental setup discussed in the introduction and illustrated in Fig. 1.1. The formation of elastically scattered beams in precise directions can be predicted through the Bragg equation, just as in X-ray diffraction,

$$\displaystyle \begin{aligned} 2d.\mathrm{sin}(\theta) = n\lambda {} \end{aligned} $$

(1.10)

where 2θ is the scattering angle, n is an integer, d is the distance between planes of a given family of crystallographic planes, and λ is the wavelength of the neutron beam. With the randomly distributed powder sample on the target, we ensure that an appropriate amount of small grains of the material will be conveniently oriented at the angle θ that satisfies the strict condition established by the Bragg equation. The scattered beams, also called Bragg lines or reflections, have intensities that relate to crystalline symmetries and the distribution of atoms in the crystal lattice. Each atom of the structure gives rise to a spherical wave, and the observed result is a superposition of these waves:

$$\displaystyle \begin{aligned} \sum_{i=1}^{N} \left( \frac{b_{i}}{r} \right) e^{ik.r}.e^{i{\mathbf{R}}_{i}.(\mathbf{k} - {\mathbf{k}}_{0})} {} \end{aligned} $$

(1.11)

In this expression the exponential term $e^{i{\mathbf {R}}_{i}.(\mathbf {k} - {\mathbf {k}}_{0})}$ takes into account the phase difference between the spherical waves emitted by the N atoms because they are in different positions of the crystal. R _i is the position vector of the atom-i in relation to an origin taken in the crystalline lattice. The intensity of the Bragg line is proportional to the square of the wave amplitude and depends in some way on the square of the scattering lengths of the atoms involved.

For the crystalline structures, where the atoms are distributed periodically, the sum in (1.11) can be rearranged. We can show that in the Bragg condition, constructive interference maxima can be written as

$$\displaystyle \begin{aligned} \frac{d\sigma}{d\Omega} \propto | F_{N}(hkl)|{}^{2} \end{aligned} $$

(1.12)

$$\displaystyle \begin{aligned} F_{N}(hkl) = \sum_{i=1} b_{i}e^{i{\mathbf{R}}_{i}.(\mathbf{k} - {\mathbf{k}}_{0})} {} \end{aligned} $$

(1.13)

where F _N is called the nuclear structure factor for the family of planes hkl. The sum in (1.13) extends over all atoms in a unit cell of the crystal lattice. The nuclear structure factor is completely analogous to the structure factor used in the X-ray diffraction. As we know, due to the translational symmetries present in the crystal lattice, the sum (1.13) can cancel. The symmetry conditions for the extinctions of certain reflections are the same in the neutron and X-ray diffraction.

The difference we find with neutrons when compared to X-rays is that for the same chemical element, we have different values of b due to their natural isotopic composition and also with respect to the state of the spin of the nucleus which, as we have seen, non-zero nuclear spin implies two values, b ₊ and b ₋. Taking into account the influence of a given chemical element on the intensity of the Bragg reflection, we have to use a kind of average between the values of b of the different isotopes present and different spin states. This average is represented as b _i in the Eq. (1.13). If we consider that nuclei of the different natural isotopic varieties of a given chemical element are distributed randomly in the crystalline lattice, we can show that this gives rise to an incoherent component of the elastic scattering. This radiation spreads in all directions and contributes to the background. The effect of background increase on hydrogen samples due to their characteristic two spin states is well-known. In Table 1.1, we see that for the two states of total spin, the values of b are very different. Depending on the hydrogen content in a sample, this significant difference between b ₊ and b ₋ may lead to a significant increase of background in the powder diffractograms. In these cases, one way to soften the influence of this incoherent spin component is to replace the hydrogen by deuterium in the synthesis of these compounds, because as indicated in Table 1.1 the component b ₋ is small compared to b ₊.

1.4 Magnetic Scattering

Another form of interaction between neutrons and atoms of enormous practical importance is the magnetic interaction. As we know, neutrons have a magnetic dipole μ _n associated with spin I given by

$$\displaystyle \begin{aligned} \mu _{n} = -\gamma_{n}\mu_{N}I {} \end{aligned} $$

(1.14)

where γ _n = 1.913 is a constant and μ _N is the nuclear magneton, given by

$$\displaystyle \begin{aligned} \mu _{N} = \frac{e\hbar}{2m_{p}} {} \end{aligned} $$

(1.15)

where m _p is the mass of the proton. The magnetic dipole of the neutron interacts with the magnetic field B produced by a permanent atomic magnetic dipole. This dipole, in turn, is associated with the orbital motion and the spin of the electrons. The magnetic dipole moment due to the orbital motion of the electron is given by

$$\displaystyle \begin{aligned} \mu _{\mathrm{{orbital}}} = \frac{-e}{2m_{e}}L \end{aligned} $$

(1.16)

where L is the angular momentum and m _e is the mass of the electron. The magnetic dipole moment due to spin is given by

$$\displaystyle \begin{aligned} \mu _{\mathrm{{spin}}} = -2\mu_{B}S \end{aligned} $$

(1.17)

$$\displaystyle \begin{aligned} \mu _{B} = \frac{e\hbar}{2m_{e}} \end{aligned} $$

(1.18)

where μ _B is the magneton of Bohr and S is the spin of the electron. It is convenient to calculate separately the contributions to the total magnetic field due to electronic spin and orbital motion and to study the interaction of the magnetic dipole of the neutron with each of the components separately. Particularly when the atomic dipoles are oriented and ordered in the crystalline lattice, forming a kind of magnetic subnetwork, constructive interference beams (Bragg reflections) can be formed based on this interaction. This magnetic ordering is observed in ferromagnetic, antiferromagnetic, and ferrimagnetic materials to name but three well-known examples. The presence of new reflections will indicate a change of symmetry of the scattering system that starts to incorporate the distribution of magnetic dipoles and not just the spatial distribution of the nuclei. In Fig. 1.9 we can see the neutron diffractogram of a substance in the paramagnetic state and the diffractogram of the same sample after transit to the magnetic state. We can see the emergence of new Bragg lines and a change in the intensities of lines that were already present in the paramagnetic state. The characterization of the magnetic structure of materials has in the neutron diffraction technique its greatest experimental tool.

The potential V , which describes the magnetic interaction of the neutron with the magnetic field B of the atom, is given by

$$\displaystyle \begin{aligned} V = -\mu_{n}.B \end{aligned} $$

(1.19)

which differs greatly from the potential governing interaction with the nucleus. While the interaction with the nucleus is restricted to a very limited region, the magnetic interaction is a long-range interaction. In turn, it results from a superposition of contributions to the atomic magnetic field that comes from the entire electronic cloud. For these reasons, in the nuclear magnetic interaction, we cannot treat the atom as a particle, as a punctiform scattering object. The magnetic field originates in the magnetic dipoles associated with the spin and the orbital motion of the unpaired electrons, that is, we can write that

$$\displaystyle \begin{aligned} B(r) = B_{\mathrm{{spin}}}(r,S) + B_{\mathrm{{orbital}}}(r,L) \end{aligned} $$

(1.20)

Using Eqs. (1.14) and (1.15) and using the same procedure that allowed the derivation of Eq. (1.5), we obtain the expression of the differential cross section for the magnetic scattering. We are omitting the details (see [1]), but we can add that B is calculated from the atomic model of quantum mechanics and the formalism of classical electrodynamics.

A similar behavior to X-ray scattering is observed in the magnetic interaction. A term analogous to the atomic scattering factor f(θ), now of magnetic origin, most commonly termed magnetic form factor is used to describe the contribution of each atom:

$$\displaystyle \begin{aligned} f(\theta) = \frac{\langle q| \int_{\mathrm{{atom}}} M.e^{-i(k-k_{0})R}d^{3}r | q \rangle} {\langle q | \int_{\mathrm{{atom}}} Md^{3}r | q \rangle} {} \end{aligned} $$

(1.21)

The letter q symbolizes the fundamental state of the atom, the brackets represent the spatial averaged value, and M represents the magnetization operator. The magnetization can be separated with origin in the spin and orbital motion of electrons:

$$\displaystyle \begin{aligned} M = M_{\mathrm{{spin}}} + M_{\mathrm{{orbital}}} \end{aligned} $$

(1.22)

Magnetization is a measure of the amount of magnetic dipole moment per unit volume and can be separated by contributions due to spin and orbital motion of the electrons. The calculation of magnetization takes into account all the electrons, but it turns out that the relevant contributions come from the unpaired electrons. For transition metals, for example, the electrons of the 3d-orbital are those relevant [10]. Figure 1.6 shows the behavior of the magnetic form factor for the Nd³⁺ obtained by values provided by International Tables of Crystallography, Vol. C. The graph shows separately the contributions due to the spin magnetic dipole and the magnetic dipole associated with the orbital movement of the unpaired electrons.

Fig. 1.6

Magnetic form factor of the Nd ³⁺ with partial contributions due to spin and orbital motion of the unpaired electrons

The nuclear interaction is completely isotropic and the dipole–dipole magnetic interaction is strongly anisotropic. However, both interactions are, basically, of the same magnitude. As a consequence of the anisotropy of the magnetic interaction, it is verified that only the parallel component of the magnetic dipole moment of the atom at a given crystallographic plane of indices hkl contributes to the magnetic scattering (“reflection”). In other words, for those cases where magnetization is perpendicular to a given plane hkl, a magnetic contribution does not result to the scattering by this plane. This is an intrinsic extinction condition, different from the extinction conditions due to the symmetries of the crystalline lattice and is a very important factor to determine the orientation of the magnetic dipoles in relation to the crystallographic axes. Figure 1.7 schematically shows this arrangement.

Fig. 1.7

In the magnetic scattering, the intensity of the diffracted beam depends on the component of the magnetization vector perpendicular to the scattering vector Δk

The intensity of the Bragg reflections due to magnetic interaction can be calculated in a similar way to the way we calculate for nuclear scattering. In this case, the differential cross section is directly proportional to the square of the magnetic structure factor (F _M). The most important difference is that the magnetic structure factor is a vector quantity, as opposed to a nuclear structure factor that is scalar. This property is also related to the anisotropic nature of magnetic interaction. The magnetic structure factor, in the Bragg condition, for a simple magnetic structure in which the magnetic ordering is entirely defined in a unit cell of the crystal can be written as

$$\displaystyle \begin{aligned} F_{M}(hkl) = p\sum_{i}^{\mathrm{cell}}f_{i}(\theta _{hkl}).\mu _{i}e^{-iR_{i}(k-k_{0})} {} \end{aligned} $$

(1.23)

where p is a constant and μi is the magnetic dipole moment of the atom-i.

Other forms of magnetic structures, such as helical magnetic structures, cannot be described by a simple translation of the magnetic arrangement contained in a unit cell. These cases can be analyzed using propagation vectors (k). The idea is based on the fact that if there is a periodic ordering in the spatial distribution of the atomic magnetic dipoles, these can be described as a Fourier series expansion as

$$\displaystyle \begin{aligned} \mu_{ji} = \sum_{k}C_{k,i}e^{-2\pi ik.R_{j}} \end{aligned} $$

(1.24)

where μ _ji is the magnetic dipole moment of the l-atom in the j-cell. C _k,i are complex vector coefficients, k are vectors defined in the reciprocal space, called propagation vectors, and R _j are position vectors of the origin of the unit j-cell. A similar procedure is used in the description of modulated structures, where instead of μ _ji, we will have, for example, a vector that measures the displacement of the l-atom in the j-cell in relation to the position of these atoms in an original cell, taken as reference . As in modulated structures, these magnetic structures need few propagation vectors, typically no more than three. The simplest situation is that in which there is only one propagation vector k = 0. In this situation C _0,i = μ _ji for all cells. This is the case described by Eq. (1.23).

In the case of magnetic structures, the scattering of neutrons is given by the sum of the nuclear and magnetic contributions. In the situation where a beam of neutrons with non-polarized spins, that is, not aligned in a particular direction, which impinges on a crystal with magnetic ordering, it can be shown that the intensity of the interference maxima is given as

$$\displaystyle \begin{aligned} \frac{d \sigma}{d \Omega} \propto | F_{N}|{}^{2} + |(F_{M})\perp |{}^{2} {} \end{aligned} $$

(1.25)

In the magnetic part of (1.25), we take into account the perpendicular component of the magnetic structure factor to the scattering vector, defined as Δk = k −k ₀ . In the Bragg condition, the scattering vector is parallel to the normal direction to the plane hkl considered.

In vector notation, the Bragg condition given by Eq. (1.10) can be written as

$$\displaystyle \begin{aligned} \Delta {\boldsymbol{k}} = {\boldsymbol{h}} {} \end{aligned} $$

(1.26)

where h is a vector of the reciprocal lattice representing a family of hkl planes. For the case of magnetic structures, the formalism with the propagation vectors allows us to write a more general Bragg condition,

$$\displaystyle \begin{aligned} \Delta {\boldsymbol{k}} = {\boldsymbol{h}} + {\boldsymbol{k}} {} \end{aligned} $$

(1.27)

In the case where k = 0 (1.26), the reflections correspond to the points of the reciprocal lattice, and the magnetic contribution overlap with the nuclear contribution. In the case where k ≠ 0 (1.27), new lines are observed close to the points of the reciprocal lattice. These lines are generally called satellite lines.

1.5 Applications

Neutron diffraction is the application of neutron scattering to the determination of the atomic and/or magnetic structure of a material. As we have already seen, neutrons are uncharged and carry a spin and therefore interact with magnetic moments, including those arising from the electron cloud around an atom. Neutron diffraction can therefore reveal the microscopic magnetic structure of a material [2].

Magnetic scattering does require an atomic form factor as it is caused by the much larger electron cloud around the tiny nucleus. The intensity of the magnetic contribution to the diffraction peaks will therefore decrease toward higher angles. The use of neutrons as research tools gives scientists unprecedented insight into the structure and properties of materials important in biology, chemistry, physics, and engineering. Neutron scattering shows directly where atoms are and what they are doing. It allows researchers to see in real time how the structure of a material shifts with changes in temperature, pressure, and magnetic or electronic fields. It also traces the atomic motions and electron states that give materials properties such as magnetism or the ability to conduct electricity – essential information in the pursuit of energy savings.

Neutrons have wavelengths ranging from 0.1 to 1000 Å, and they have the power to reveal what cannot be seen using other types of radiation. The specific properties of neutrons to behave as particles or as microscopic magnetic dipoles enable them to give us some information which is often impossible to obtain using other techniques. For instance, the neutrons scattering off gases, liquids, and solid matter give us information about the structure and magnetism of these materials.

Neutrons are nondestructive radiation and can penetrate deep into matter. For this reason, they are ideal for applications in biological materials and samples under extreme conditions of pressure, temperature, and magnetic field. Furthermore, neutrons are sensitive to hydrogen atoms. It is a powerful tool for analyzing hydrogen storage materials, organic molecular materials, and biomolecular samples or polymers. Some applications of neutrons in different fields of research are [11]:

Condensed-matter physics, materials science, and chemistry:

• Examination of the structure of new materials, for example, new ceramic high-temperature superconductors or magnetic materials (important for electronic and electrical applications).

• Clarification of still unknown phenomena in processes such as the recharging of electric batteries.

• Storing of hydrogen in metals, an important feature for the development of renewable energy sources.

• Analysis of important parameters (e.g., elasticity) in polymers (e.g., plastic material).

• Colloid research gives new information on such diverse subjects as the extraction of oil, cosmetics, pharmaceuticals, food industry, and medicine.

Biosciences:

• Biological materials, naturally rich in hydrogen and other light elements, are ideal samples for analysis with neutrons.

• Cell membranes and proteins

• Virus investigations

• Photosynthesis in plants

Nuclear and elementary particle physics:

• Experiments on the physical properties of the neutron and the neutrino.

• Production of extremely slow neutrons down to 5 m/s (the velocity of the neutrons which leave the reactor is generally about 2200 m/s).

• Experiments on atomic fission and the structure of nuclei.

Engineering sciences:

• Since neutron diffraction is nondestructive, it is ideal for the analysis of different technical phenomena in materials.

• Visualization of residual stress in materials.

• Hardening and corrosion phenomena in concrete.

• Inhomogeneity and trace elements in materials.

1.6 Instrumental

1.6.1 Institut Laue-Langevin (ILL)

The Institut Laue-Langevin (France) provides scientists with a very high flux of neutrons feeding some 40 state-of-the-art instruments, which are constantly being developed and upgraded. As a service institute, the ILL makes its facilities and expertise available to visiting scientists. Every year, some 1400 researchers from over 40 countries visit the ILL. More than 800 experiments selected by a scientific review committee are performed annually. Research focuses primarily on fundamental science in a variety of fields: condensed matter physics, chemistry, biology, nuclear physics and materials science, etc. ILL can specially tailor its neutron beams to probe the fundamental processes that help to explain how our universe came into being, why it looks the way it does today, and how it can sustain life. ILL is funded and managed by France, Germany, and the United Kingdom, in partnership with 10 other countries.

The ILL’s powder diffraction instruments are D2B, D20, D1B (CRG), D4, SALSA and D7. All of our measurements were performed in the D1B instrument. It uses a large crystal monochromator to select a particular neutron wavelength, just as the different wavelengths of light can be separated using a prism or fine grating. The material to be studied is placed in this monochromatic neutron beam, and the scattered neutrons are collected on a large 2D detector. The sample can be a liquid, a bunch of fibers, a crystal, or a polycrystal. A polycrystal is the usual form of solid matter, such as a lump of metal or ceramic, and is made up of millions of tiny crystals. (site)

Neutron diffraction experiments at ILL are thus really quite simple and available to a wide variety of users – materials scientists, chemists, physicists, and biologists. The simplest is called “powder diffraction,” when a polycrystalline lump of material, often ground to a fine powder, is placed in the beam. Neutrons are scattered at specific angles, corresponding to the spacing between atomic planes, and by measuring these angles and intensities, the atomic structure of the material can be deduced. If instead of a crystalline powder an amorphous or liquid sample is used, there are only broad peaks at specific angles corresponding to average interatomic distances.

To obtain more data, short neutron wavelengths are used, and sometimes one type of atom is replaced by its isotope – chemically identical, but with a different nucleus and different neutron scattering power. This difference then gives information specific to that atom.

D1B (Fig. 1.8) is a high intensity powder diffractometer with a PSD covering the angular range 0.8^∘ to 128^∘. A cryostat 1.6–320 K or even a dilution cryostat can be mounted on the D1B diffractometer enabling to investigate magnetic structures on powder samples. It has always been in very high demand for real-time experiments and for very small samples because of its high-efficiency position-sensitive detector (PSD). The new PSD makes it an even more valuable tool. Run as a CRG-A instrument by a CNRS/CSIC team, it is available 50% of the time for scheduled ILL experiments (Fig. 1.8).

Fig. 1.8

This layout shows the new D1B (ILL) setup with the new position-sensitive detector (PSD) with 1280 cells covering a total of 128^∘ and the radial oscillating collimator (ROC) (Figure taken from [https://www.ill.eu/instruments-support/instruments-groups/instruments/d1b/description/instrument-layout/])

1.7 Magnetic Pattern

The magnetic structure for the most of magnetic compounds can be determined using neutron powder diffraction (NPD) data and complementary magnetic measures. In this chapter, we will determine the magnetic structure of the double perovskite Sr ₂ Y RuO ₆ using FullProf Suite [12], a crystallographic program developed for Rietveld analysis (structure profile refinement) of neutron (constant wavelength, time of flight, nuclear and magnetic scattering) or X-ray powder diffraction data collected at constant or variable step in scattering angle 2θ. The NPD measurements were performed in Institut Laue-Langevin (ILL), and the basic steps for determining the magnetic structure were based on a tutorial made by Juan Carvajal [13]. This procedure will be easier if the reader has experience with crystallography and structure profile refinement using X-ray powder diffraction and/or neutron diffraction through Rietveld analysis. For determining the magnetic structure of Sr ₂ Y RuO ₆, we will take the following steps:

• The first step is to perform a NPD measure of the sample in paramagnetic state (in general, at room temperature if you do not know the magnetic transition temperature). The data must be refined in order to obtain the sample structural parameters and check if any impurities are present.

• Perform a NPD measure of the sample in magnetic state; in other words, we have to know the magnetic transition temperature (T _C or T _N). Below this temperature, additional magnetic peaks will appear, specially for lower 2θ values (if additional peaks are not observed, go to the next step). Refine with the same PCR file used in the paramagnetic state. Using WinPLOTR-2006, open both diffraction pattern data, and save the additional peaks in a K_Search Format.

• At this point, we have to determine the propagation vectors (K) of the magnetic structure (more details about K vector can be found in the references [13, 14]). If additional peaks are not observed in magnetic state, the magnetic structure has a K = (0,0,0). Otherwise, we will use the program K_Search to calculate the propagation vector. The propagation vectors must be tested with Le Bail fit mode [12, 15].

• After the determination of the propagation vector, the program BasIreps is used to calculate the basis vector of the irreducible representations (irreps). It is a basic information about the magnetic structure symmetry that we have to insert in the PCR file (for more details, see [13]).

• Finally, we can refine the magnetic structure and get all the possible magnetic informations about the sample given by the Rietveld method.

1.7.1 Magnetic Structure of the Double Perovskite A ₂ BB′O ₆

NPD measurements of the sample Sr ₂ Y RuO ₆ were performed for various temperatures at the D1B instrument in ILL with . For determining the magnetic structure of this compound, we will use just two powder diffraction patterns, one in the paramagnetic state (40 K) and another in the magnetic state (1.6 K). All the basic informations about the magnetic and structural state of this compound were obtained in the references [16,17,18,19]. Initially, we will use the monoclinic P2₁∕n space group and lattice parameters with a = 5.7698(2), b = 5.7814(1), and c = 8.1646(4), along with β = 90.21(1) [16]. The data obtained at 1.6 K and 40 K corresponds to the format XY SIGMA (Ins =10), Fig. 1.9.

Fig. 1.9

Powder diffraction patterns at 1.6 K (blue line) and 40 K (red line)

In order to obtain the sample structural parameters and check if any impurities are present, we refined the crystal structure of Sr ₂ Y RuO ₆ at 40 K in a folder called Temp40 K. The complete PCR file (Sr2YRuO6_40 K.pcr) is shown in Fig. 1.10, and the calculated and observed patterns are shown in Fig. 1.11. Also, we put Pcr =  2 (a new input file is generated conserving the old one) to generate Sr2YRuO6_40 K.new.

Fig. 1.10

PCR file at 40 K for the structural pattern of the Sr ₂ Y RuO ₆

Fig. 1.11

Calculated (red line) and observed (black line) patterns at 40 K

After creating a folder called test, we can observe the magnetic peaks using the PCR file (Sr2YRuO6_40 K.pcr) at 40 K (after the refinement) into a test PCR file (test.pcr) and refine the data at 1.6 K. At this moment, we can just leave zero, background coefficients, and scale to be refined. All the other parameters must be fixed. Also, we can put Nba = −3 (read 6 additional polynomial background coefficients). After the refinement, we can see the magnetic peaks, not calculated by the refinement, at low angles (Fig. 1.12). Now we are able to calculate the propagation vector K that represents the magnetic peaks observed.

Fig. 1.12

Magnetic peaks at low angles not calculated by the refinement

1.7.2 Magnetic Peaks Identification and Propagation Vector K

The next step is to open the last PRF file (test.prf) using the WinPLOTR-2006 program. This file is generated using Prf = 1 or 3, into PCR file. Then, we can select the most salient peaks and save them in K_Search Format. For this, first go in the menu Calculation – Peak detection – Enable. Then, Calculation – Peak detection – Insert peak. After insert the magnetic peaks (two in this case), and go in the menu Calculation – Peak detection – Save peaks – K_Search format (Fig. 1.13).

Fig. 1.13

Using the WinPLOTR-2006 program to save the magnetic peaks in K_Search format

After selecting the K_Search format menu, open a dialog for the input parameters. If you are working with the refined pattern, the parameters will be introduced automatically (Fig. 1.14a). We can start selecting Short Output and Search only special K-vectors options. The magnetic peaks will be saved in a file called K-search.sat. Then, we have to run the K_Search program in order to get the solutions for the possible propagation vectors (Fig. 1.14b). If you want to select more peaks, delete peaks, or just rerun the program, press the enter key (↩) to close the program.

Fig. 1.14

(a) Is the dialog for the input parameters for K_Search program, and in (b) we can see the solutions for the possible propagation vectors K calculated by K_Search program

In Fig. 1.14b, we can see that K = (0.5 0.5 0.0) is the best solution, followed by K = (0 0 0). If the magnetic cell is the same as the nuclear cell, then the propagation vector K = (0 0 0). In our case, K = (0.5 0.5 0.0) is commensurate with magnetic cell 2a × 2b × c (compared with the nuclear unit cell a, b and c) [20]. At this time, we must be careful with a possible small shift in positioning when we insert the magnetic peaks. When it takes place, a wrong solution can be found for a better R-factor. However, we can check the propagations vectors by doing Le Bail fit method [12, 15].

1.7.3 Le Bail Method

In this chapter, we will just use the Le Bail method to check if it is possible to use the propagation vectors calculated by K_Search program. If a specific propagation vector generates the correct satellites (compared with the observed data), this propagation vector can be used to describe the magnetic structure of the compound. In the Le Bail method, we have to introduce a magnetic phase, with a propagation vector, into the PCR file at 1.6 K (test.pcr). For this, we can create a folder called LeBail, copy the data at 1.6 K, and make a copy of the file test.pcr into the file trestLE.pcr.

At this point, we have to be careful when we introduce the magnetic phase due to the large number of parameters we have to consider. Open the testLE.pcr file with a text editor and put Nph = 2 (two phases), and duplicate the description of the phase 1 to include it as phase 2. Change the phase 2 nome (Magnetic Lebail), remove the atoms, and put Nat = 0 and Jbt = 2.

After that, we recommend to open this file with FullProf PCR Editor, and in the phase 2, change the symmetry to P-1. Once the magnetic structure is unknown, we can use the space group P-1 for generating the fundamental reflections. Next, add a propagation vector, and to create a line with !Jvi [13], go in the menu Output – Phase Output Information – Overlapped peaks List(INT) and mark phase 2. Open again the testeLE.pcr file with a text editor. You will observe that the Nvk = 1, More = 1 and a line with Jvi = 11. Put Aut = 0 (the maximum number of parameters to be refined is fixed manually) and do not forget to put all parameters fixed, put Irf = −1, and now you can start the Le Bail method running FullProf with the testLE.pcr file (Fig. 1.15).

Fig. 1.15

Part of the testLE.pcr file for the Le Bail method to run in the FullProf program

Observing Fig. 1.14b, we have a list of the five best solutions for the magnetic peaks. After testing K = (0.5 0.5 0.0) and K = (0 0 0), we obtained the observed and calculated patterns shown in Fig. 1.16. We can observe that the magnetic peaks are described for both propagation vectors. Therefore, both K = (0.5 0.5 0.0) and K = (0 0 0) can be used to describe the magnetic structure.

Fig. 1.16

The observed (red line) and calculated (black line) patterns obtained by Le Bail method

The program has produced a file called testLE_cltr.int due to Jvi = 11, and if we put Ipr = −1, the program will generate a file called testLE.spr. Both files can be used for doing a simulated annealing for determining the magnetic structure in more complicated cases. For more details, we recommend the tutorials [13, 15, 21].

Although the two propagation vectors can be used to describe the magnetic structure, we have to be careful for choosing the correct propagation vector. In this case, we can use a trial and error method and previous information about the macroscopic magnetic results of the sample. Anticipating the result, we were able to refine the data using both propagation vectors. However, we obtained the lowest R-factor for K = (0.5 0.5 0.0). Furthermore, for the propagation vector K = (0 0 0), we observed some divergence conditions after running the program.

The next step is to calculate the basis vector of the irreducible representations (irreps) using the program BasIreps [12, 13, 22] for the best propagation vector. The program BasIreps calculates the irreducible representations (irreps) of the so-called little groups from which the full irreducible representations of space groups can be calculated using the induction formula [22].

1.7.4 Irreducible Representation of Space Groups (BasIrreps)

In this chapter, we are going to use the BasIreps program to obtain the information about the magnetic structure symmetry of the compound. Few information is required to calculate the irreducible representations using BasIrep, as shown in Fig. 1.17. More details about representation analysis for magnetic structures can be found in the references [13, 22].

Fig. 1.17

BasIreps window: basic information is required to calculate the irreducible representations

In BasIreps, we will need the space group (P2₁∕n), the propagation vector K (in Fig. 1.17 we used K _1∕2), and Wyckoff sites for the magnetic atom. The Ru atoms are in Wyckoff position 2c with two sublattices for this space group in (1/2, 0, 1/2) and (0, 1/2, 0). Axial vector or polar vector will select the behavior of the symmetry operators upon acting on vectors [22]. When explicit sublattice is marked, the “number of atoms” is the total number of atoms in the unit cell. For a particular site, they must share the same atom code, and the different sublattices are distinguished by an added underscore “_” followed by the number of the sublattice [22].

After run, the program will generate an output file called Sr2YRuO6.fp with a list for the possible irreducible representations (Fig. 1.18), two in this case, IRrep(2) and IRrep(4). In this way, we can describe the magnetic structure using directly the basis vectors.

Fig. 1.18

Part of the output file Sr2YRuO6.fp with a list for the possible irreducible representations IRrep(2) and IRrep(4)

1.7.5 Magnetic Structure Description Using the Coefficients of the Basis Functions

Now we can use a trial and error method applying the symmetry information of the output file. For this, we will complete the magnetic phase in testLE.pcr file (Fig. 1.19) and save the file with a different name (Mag.pcr). After that, put More = 0 and delete the line with Jvi = 11. Copy the block of lines from Sr2YRuO6.pf into Mag.pcr file and add the magnetic atoms.

Fig. 1.19

The magnetic phase in testLE.pcr file with the irreducible representations IRrep(2)

In the PCR file, the parameter Typ is used for identifying the name of the magnetic form factor. For transition metals the symbol is given by MCSV, CS is the chemical symbol, and V is a number corresponding to the valence of the magnetic ion [13]. For V, we used the magnetic form factor of the Ru ⁺¹ ion due to the absence of the magnetic form factor of the Ru ⁺⁵ in the FullProf program. It is a good approximation once the difference between the magnetic form factors is very small [23].

When we put Jbt = 1, the phase is treated with the Rietveld method, and it is considered as pure magnetic (only magnetic atoms are required). Jbt = −1, we have extra parameters in spherical coordinates. Isy = −1, the program will read symmetry instructions, and Isy = −2, we use the basis functions of irreducible representations of propagation vector group instead of symmetry operators.

Finally, we have to use reasonable values of the coefficients of the basis functions C1, C2, and C3 to run the program. If we run the program with all parameters fixed and using any symmetry, IRrep(2) or IRrep(4), we will see a poor agreement between the magnetic peaks in the observed and calculated pattern.

After that, we can try to refine the coefficients of the basis functions C1, C2, and C3. If a divergence condition appears at some cycle, the symmetry used is wrong, and we have to change it. In the case of Mag.pcr file, both symmetries worked fine; however the best solution is IRrep(2), with the lowest R-factor. From this point, we are able to refine all parameters and the background, obtaining the pattern shown in Fig. 1.20.

Fig. 1.20

Observed (red line) and calculated (black line) patterns with all parameters refined after completing the information for the Mag.pcr file

1.7.6 Magnetic Structure Description in Cartesian Components or Spherical Components

Instead of describing the magnetic structure using the coefficients of the basis functions, we can describe it using the Cartesian components or the spherical components. Using Jbt = 1, Isy = −1, and a small change in the block of symmetry, we can refine the Cartesian components of the magnetic moments Rx, Ry, and Rz (Fig. 1.21a). If Jbt = −1, we can use spherical components and refine the magnetic moments Rm, Rphi, and Rtheta (Fig. 1.21b).

Fig. 1.21

(a) Is the information for the pcr file in Cartesian components of the magnetic moments Rx, Ry, and Rz. (b) Is the information in spherical components with the parameters Rm, Rphi, and Rtheta

Also, the symmetry block and the magnetic parameters in the Mag.pcr file can be easily edited using EdPCR program. For this, select Phase – Next – Symmetry, and put User defined in Symmetry Operators to refine the Cartesian components. For spherical components, just change Calculation to “Magnetic Phase with magnetic moments in spherical mode (Rietveld Method).”

From Fig. 1.21a, we can see the magnetic component Rx is larger than the other components and Rz is close to zero. For this reason, in spherical coordinates, we can start by putting Rphi = 180^∘ and Rtheta = 90^∘ before running the program.

Previous informations about the magnetic state of any sample is very important to conduct the magnetic refinement properly. Sr ₂ Y RuO ₆ has two peaks in the magnetic contribution to the specific heat, at T _N1 = 28 K and T _N2 = 24 K [16, 19]. For a long time, this compound was described as having a collinear type-I AFM order with K = (0,0,0). Furthermore, for this compound, the emergence of a canted spin state from the temperature dependence of the coercive field H _C (Fig. 1.22) is observed, which exhibits a maximum of about 2 kOe close to T _N1 followed by an abrupt decrease down to a few Oe near T _N2 [16, 19]. The collinear type-I AFM order cannot account for the H _C(T) behavior.

Fig. 1.22

The temperature dependence of the coercive field H _C [24]

Anticipating the result, if we refine the magnetic structure with just one magnetic ion (Fig. 1.19), only collinear structures will appear. For this reason, once the program generates the sublattices by applying the symmetry operators of the space group [13], in spherical coordinate, we can use the Wyckoff position 2c with two sublattices and a small block of magnetic symmetry operator (Fig. 1.23). Thereby, the program is able to generate a magnetic structure with canted spin state.

Fig. 1.23

Magnetic phase in spherical coordinate with two sublattices and a small block of magnetic symmetry operator

After completing the Mag.pcr file in spherical coordinate and refining all the parameters, we obtained Table 1.3 with the factors χ ² = 1.46.

Table 1.3 Results for the refinement of magnetic ions Ru_1 and Ru_2 using spherical coordinates

1.7.7 FullProf Studio

The program FullProf Studio has been developed for visualizing crystal and magnetic structures [25]. After refining the magnetic structure, we can open the Mag.pcr file and put the keyword magphn, being n the number of the magnetic phase (n = 2), in the line with the name of the crystallographic phase (1). By doing this, we are able to plot the magnetic structure together with the crystal structure [13]. Then, in Fig. 1.10, instead of Sr2YRuO6 in the name of the phase 1, we put Sr2YRuO6 magph2. The FullProf program generates an output file called Mag.fst, and the result of the magnetic structure description observed using FullProf Studio is shown in Fig. 1.24, which is a spin structure of the K ₂ NiF ₄ − type [17, 26].

Fig. 1.24

Magnetic structure observed using FullProf Studio. Spin structure of the K ₂ NiF ₄ − type

Top view from the c-axis is showing the orientation of the magnetic moments of the Ru ions for the canted spin state in Fig. 1.25. The canting angle η = |Rphi ₁ − Rphi ₂ − 180^∘| = 67.6^∘ is defined as the deviation from the AFM alignment of the magnetic moments for neighboring Ru(1) and Ru(2) ions along the c-axis.

Fig. 1.25

Top view from the c-axis showing the orientation of the magnetic moments of the Ru ions for the canted spin state. η is the canting angle

For Figs. 1.24 and 1.25, we are showing just the magnetic moments of the Ru ions. For this, in FullProf Studio, go in the menu Crystal Structure – Atoms – Display, and uncheck the box Show atom for all atoms. In Fig. 1.26a, we checked the box for the Ru and Y atoms, and then it is possible to see the atomic positions for these atoms. Also, if we uncheck all the atoms and edit the Mag.fst file, adding the line CONN Y O 0.000 2.500 RADIUS 1.000, after the list of atoms, and the line POLY Y COLOR 0.000 1.000 0.000 1.000, after the list of magnetic moments, we will obtain Fig. 1.26b showing a polyhedral representation with Y in the center of the octahedron.

Fig. 1.26

In addition to the positions of the magnetic moments of the Ru ions, figure (a) shows the Sr and Y atoms, and the figure (b) shows a polyhedral representation with Y in the center of the octahedron

1.8 Conclusions

After using some programs for the analysis of neutron diffraction data, we observed a spin canting structure like K ₂ NiF ₄ − type for the magnetic structure of the double perovskite Sr ₂ Y RuO ₆. We found the propagation vector K = (0.5 0.5 0.0) as being the best solution for the two magnetic peaks observed at low angles. Although we confirmed, using the Le Bail method, that the K = (0 0 0) can describe the magnetic structure, we were able to refine the magnetic structure just in the case when we put one magnetic Ru sublattice. This description is not convenient for the spin canting structure previously observed.

The BasIreps program is an easy tool to obtain the information about the magnetic structure symmetry of the compound. This program gives us a list with the possible irreducible representations and the basis vectors to be used directly in the pcr file. In this case, we have to be careful with the values of the coefficients of the basis functions C1, C2, and C3 if a divergence condition appears at some cycle.

The easiest way to obtain the magnetic parameters of the compounds is to refine the magnetic structure using the Cartesian components or the spherical components. The first one is better for analyzing the magnetic moment on each axis, in which we can refine more accurately if we have some information about the magnetic behavior of a single crystal of the compound. On the other hand, the second is fundamental to describe a magnetic structure with spin canting and gives us a global magnetic moment.

FullProf Studio is a program for visualizing crystal and magnetic structures using the output files from refinement. In this program we can easily see the magnetic moments direction and the atoms’ position. We should keep in mind that this chapter is just a basic way to describe a magnetic structure. We tried to approach the magnetic structure description as easily as we could. In more complicated cases, we should use the simulated annealing (SAnn) procedure in FullProf.