Investigation of magnetic phase transitions using the Laue technique - - PDF document

Institut Laue-Langevin International Student Summer Programme at ILL/ESRF 2017

Project Report

Investigation of magnetic phase transitions using the Laue technique

4 – 29 September 2017 Author Guido Homann∗ Supervisor

• Dr. Bachir Ouladdiaf

This report presents the analysis of Laue diffraction patterns measured by the multiple CCD neutron diffractometer CYCLOPS at ILL. The magnetic structure of the metal oxide CaBaFe4O7 is studied between 158 and 301 K. Two magnetic phase transitions can be observed at the ordering tempera- tures T1 ∼ 278 K and T2 ∼ 205 K. They are related to the magnetic prop- agation vectors k1 = (0, 0, 0) and k2 = 1

3, 0, 0

• , respectively. The phase

transitions are revealed by evaluating the temperature dependence of the intensities of selected reflections.

∗Universität Hamburg, guido.homann@physik.uni-hamburg.de

1 Introduction

The first quantitative neutron diffraction experiments were carried out by Shull et al. in the 1940s [Shu95]. Still today, neutron scattering is used to study the atomic and magnetic structure of materials. Due to its magnetic moment [Alv40], the neutron does not only interact with the atomic nuclei but also with the un- paired electrons inside a magnetic sample. Since this electromagnetic interaction is comparable in strength to the strong interaction with the nuclei, neutron diffrac- tion is a powerful tool for the investigation of magnetic structures in condensed matter. This report aims to demonstrate the analysis of Laue diffraction patterns in order to examine the magnetic behavior of a CaBaFe4O7 single crystal. The experimen- tal data were gained by Qureshi [Qur17] using the CCD neutron diffractometer CYCLOPS [Oul11] at the Institut Laue-Langevin in Grenoble. As shown in [Rav08] the compound CaBaFe4O7 belongs to a class of metal oxides with a kagome sublattice. The magnetic properties of these materials range from 3D long-range order of both antiferro- and ferrimagnetism to spin-liquid behav- ior [Sch07]. This variety of phenomena motivates to study the magnetic phase transitions of CaBaFe4O7 in detail.

2 Theoretical background

After providing a summary of the fundamental aspects of neutron scattering, this section introduces the concept of magnetic propagation vectors. A more elaborate formulation of the underlying theory can be found in [Mar71, Squ78, Res14].

2.1 Coherent elastic neutron scattering

Elastic neutron scattering is the dominant process in diffraction experiments. This means that the scattering event does not cause any energy transfer. Hence, the wave vectors of the incident neutron ki and the scattered neutron kf have the same

• modulus. There can be, however, a momentum transfer

q = ki − kf . (1) The following considerations are limited to the case of coherent scattering which is most relevant for the determination of nuclear and magnetic structures. Besides, the neutron beam is assumed to be unpolarized. 1

2.2 Nuclear neutron scattering

Because of the very short-range interaction between a neutron and a nucleus, this type of scattering is isotropic. Therefore, it can be characterized by only one parameter, the scattering length b, and the potential takes the form V1N(r) = 2π2 m b δ(r − R) (2) for a neutron at r and a nucleus at position R. Moreover, denotes the reduced Planck constant, and m the neutron mass. The potential stated above leads to a simple relation for the differential cross section: dσ dΩ

• 1N

= b . (3) Now, one can proceed to the case of a single crystal where the scattering poten- tial transforms into a sum over all the contained nuclei. Their positions can be expressed through the position of their unit cell Rα and their position inside the unit cell rj: Rα,j = Rα + rj . (4) Taking the temperature dependent Debye-Waller factor exp(−Wj) into account, the potential reads VN(r) = 2π2 m

• α,j

exp(−Wj) bj δ(r − Rα,j) . (5) The resulting differential cross section can be written as a sum over all reciprocal lattice vectors: dσ dΩ

• N

= (2π)3 V0 N

• h

|FN(q)|2 δ(q − h) (6) with the volume of a unit cell V0 and the number of unit cells inside the crystal

• N. Also the nuclear structure factor was introduced in this step:

FN(q) =

• j

exp(−Wj) bj exp(i q · rj) . (7) The structure factor is affected by all the atoms in one unit cell and can become zero for certain configurations. The differential cross section in Eq. 6 implies a condition for the occurrence of

• reflections. It vanishes unless the scattering vector q equals a reciprocal lattice
• vector. In fact, the condition

q = h (8) is equivalent to Bragg’s law and merely a consequence of the symmetry of the crystal lattice. 2

2.3 Magnetic neutron scattering

Beyond nuclear scattering, the neutron magnetic moment µn gives rise to magnetic scattering from the unpaired electrons of an atom. In terms of its gyromagnetic ratio γ = −1.91 [Ber12] and the nuclear magneton µN, the neutron magnetic moment has a magnitude of µn = γ µN . (9) The scattering potential is given by a sum of the dipole-dipole interactions of the neutron magnetic moment µn with all the unpaired electrons of one atom: Vm(r) = −µn · Bel = −µn

• i
• ∇ ×

µi × (r − Ri) |r − Ri|3

• − 2µB pi × (r − Ri)

|r − Ri|3

• .

(10) µB represents the Bohr magneton whereas µi is the spin magnetic moment of one unpaired electron. The momentum and the position of such an electron are de- noted by pi and Ri, respectively. The differential cross section caused by this interaction is governed by two prop- erties of the scattering atom: the so-called magnetic form factor f(q) and that component of its magnetic moment µ⊥ which is perpendicular to the scattering

• vector. Defining the constant

p = 2m 2 γ µB µn (11)

• ne finds

dσ dΩ

• 1M

= p2 f 2(q) µ2

⊥ .

(12)

2.4 Magnetic propagation vectors

Before discussing the final expression for the magnetic neutron scattering from a crystal, it is useful to introduce the concept of magnetic propagation vectors. The magnetic moments of an atom at the position Rα,j can be expanded in a Fourier series over all reciprocal lattice vectors: µα,j =

• k

mj,k exp(−i k · Rα) . (13) Those reciprocal lattice vectors with a non-vanishing Fourier component mj,k are referred to as the magnetic propagation vectors of a material. In this sense, the magnetic propagation vectors characterize the periodicity of a magnetic structure. The determination of the complete magnetic structure requires knowledge of the corresponding Fourier components as well. 3

Furthermore, the magnetic propagation vectors allow for a compact formulation of the differential cross section for magnetic neutron scattering from a single crystal: dσ dΩ

• M

= (2π)3 V0 N

• h,k

|FM,⊥(q, k)|2 δ(q − h − k) (14) Comparing the relation above with the differential cross section for the nuclear scattering in Eq. 6, there is one apparent difference, namely the additional sum

• ver the magnetic propagation vectors involving a modified Bragg condition. It

states that magnetic reflections occur for q = h + k . (15) This modified Bragg condition implies the emergence of additional peaks for mag- netic propagation vectors k = 0. In the case of k = 0, the magnetic periodicity equals the nuclear one so that the intensity of the lattice reflections will increase. Actually, both contributions simply superimpose each other: dσ dΩ

• =

dσ dΩ

• N

+ dσ dΩ

• M

. (16) Of course, the magnetic structure factor FM,⊥ differs from the nuclear one. It contains the magnetic form factor of each atom in the unit cell and vanishes for distinguished scattering vectors. FM,⊥(q, k) = p

• j

fj(q) mj,k exp(i q · rj) (17) The notation with a ⊥ symbol indicates that only the magnetic moments with a projection perpendicular to q affect the scattering process.

3 Experimental setup

3.1 Laue technique

The Laue technique [Fri12] is a wide-spread method for structure resolution with both X-rays and neutrons. As already mentioned, only reflections with q = h can be observed for the diffraction from a single crystal. They are determined by Bragg’s law [Bra13] which reads nλ = 2dhkl sin θ (18) 4

in its original form. It relates the wavelength of the incoming radiation λ to the scattering angle θ for a reflection at parallel crystal planes with interplanar dis- tance dhkl (specified in Miller notation). n is a positive integer that gives the order

• f diffraction.

The Laue technique has two key features. First, the incident X-ray/neutron beam is polychromatic. Second, the detection of the diffraction pattern covers a large solid angle. Thus, Bragg’s law is fulfilled for many pairs of wavelengths and angles at the same time. Due to the simultaneous detection of numerous reflections in

• ne diffraction image, the Laue technique permits a highly efficient data acquisi-
• tion. According to [McI15], further benefits of area detectors include an optimal

delineation of peak and background plus the volumetric surveying of reciprocal space. Figure 1: The Ewald spheres and the resolution sphere in a Laue experiment. The acces- sible region of reciprocal space is the region contained be- tween the Ewald spheres of radii 1/λmax and 1/λmin and the resolution sphere of radius 1/dmin. Aside from the limits of the wavelengths the resolution dmin determines which Bragg reflections are experimentally accessible. The Ewald spheres for λmin and λmax as well as the resolution sphere are visualized in Fig. 1. The observable reflections correspond to the reciprocal lattice vectors in the region between the Ewald spheres and the resolution sphere.

3.2 CYCLOPS

The multiple CCD neutron detector CYCLOPS was developed by Ouladdiaf et al. [Oul11] and manufactured by Photonic Science Ltd. (http://photonic-science. co.uk/). The instrument combines the Laue technique with the fast readout of CCD detectors. It is located at the end of the H24 thermal guide at the Insti- tut Laue-Langevin where the incoming neutrons have wavelengths in the range 0.8 Å λ 5.0 Å. 5

Figure 2: The setup of CYCLOPS installed at the end of the H24 thermal guide at ILL [Oul11]. CYCLOPS has an octagonal geometry maintained by a monolithic annulus with eight 45◦ segments. Behind each inner octagonal face a large-area neutron scintil- lator converts scattered neutrons into photons. These are detected by two image- intensified CCD cameras (200 mm apart vertically) via two f0.95 close-focus lenses. Thin tubes let the neutron beam pass through the vessel and one pair of opposing scintillator faces without impairing the imaging process. Fig. 2 illustrates the experimental setup of CYCLOPS. Having a diameter of 400 mm and a height of 412 mm, this integrating detector subtends 270◦ horizontally by 92◦ vertically. An intensity image of the entire ac- tive scintillator area is saved in a 16 bit TIFF file divided into 7680 × 2400 pixels being 172 µm on edge each. The total readout time amounts to ∼1 s. Table 1: Techni- cal specifications of CYCLOPS [Oul11]. 6

A heavy-duty rotation table enables the sample to be rotated about the vertical axis of the detector. In addition, the free diameter of 390 mm around the sample allows for the use of various devices, e.g. cryostats or pressure cells. The technical specifications of the instrument are summarized in Table 1.

3.3 Sample

The material studied in the frame of this project is the metal oxide CaBaFe4O7. A single crystal of this compound belonging to the orthorombic space group Pbn21 (33) [Rav08] was recently examined at CYCLOPS [Qur17]. The lattice constants are known to be a = 6.36 Å, b = 10.98 Å, and c = 10.36 Å while the angles between them equal 90◦ (per definition of a orthorombic lattice). Taking the unit cell into account, one finds trigonal and kagome layers of FeO4 tetrahedra which supposedly cause the observed magnetic ordering [Rav08].

4 Data analysis

In the following, the analysis of a CaBaFe4O7 single crystal is described. Diffraction images are available in the temperature range between 158 and 301 K, 128 files in

• total. The analysis is performed using the software program Esmeralda Laue Suite

(http://lauesuite.com/).

4.1 Geometry conversion and peak search

First of all, the diffraction images are converted from the octagonal symmetry

• f CYCLOPS into a cylindrical one. This task can be done by the embedded

application Cyclops2cyl. A cylindrical geometry yields a more natural pattern and is the basis for further analysis. Another fundamental step is the determination of the diffraction peaks. Therefore, Esmeralda Laue Suite offers a threshold algorithm for automatic peak finding in the submenu Peak Tools. Its parameter values should be chosen so that only actual reflections are found. It is, however, positive to have as many peaks as possible for the later indexation. In order to avoid multiple identification of a strong reflection, the minimum distance between peaks is set to 50. Concerning the other parameters, a satisfying result is obtained for a cut-off of 2, a block size

• f 200, and a 2D peak area of 100.
• Fig. 3 shows the diffraction image at T = 301 K after the geometry conversion and

a peak search. One can clearly recognize the eight segments of CYCLOPS and the hole in the center where the neutron beam exits the instrument. The shaded semicircles at the lateral margins of the outer units are related to the entry of the 7

Figure 3: Diffraction image of CaBaFe4O7 at 301 K after the geometry conversion and a peak search. The hole in the center corresponds to the exit of the neutron beam from CYCLOPS whereas the identified peaks are marked by black circles.

• neutrons. Moreover, the curved edges at the top and bottom indicate that the

image was converted into a cylindrical geometry.

4.2 Indexation of nuclear reflections

After finding the nuclear reflections one can proceed to their indexation. This requires the determination of the crystal orientation in addition to the knowledge

• f the lattice parameters. Again, Esmeralda Laue Suite provides a helpful tool,

namely Automatic Orientation. Once the algorithm is started, the program sys- tematically rotates a calculated pattern (based on the lattice parameters) until an

• ptimal match of observed and calculated peaks is achieved. Such an indexation

is depicted in Fig. 4. Figure 4: Indexed diffraction pattern of CaBaFe4O7 at 301 K after a refinement of the orientation. The white squares mark the calculated reflections. 8

At this point, the orientation angles are given by φx = 91.0◦, φy = 319.1◦, and φz = 92.9◦. According to the Bragg condition q = h, every calculated reflection corresponds to one or several reciprocal lattice vectors. Only few peaks cannot be indexed in this manner. As discussed in the previous section, the accessible part of the reciprocal lattice depends on the wavelengths λmin, λmax and the resolution dmin. These parameters need to be adjusted for the computed diffraction pattern. They should be chosen so that the number of calculated reflections exceeds the number of visible reflections by a minimum. On the other hand, it is important that indexed peaks remain

• indexed. Manual variation of the parameters yields the best agreement for λmin =

0.51 Å, λmax = 5.0 Å, and dmin = 0.69 Å. Now, one can make use of the function Orientation Refinement to improve the accuracy of the computed pattern. This completes the indexation of the nuclear reflections (cf. Fig. 4). The final orientation angles φx = 90.8◦, φy = 319.1◦, and φz = 92.6◦ allow for the indexation of 375 reflections whereas 28 peaks cannot be matched any reciprocal lattice vector.

4.3 Identification of magnetic propagation vectors

In analogy to the indexation of the nuclear reflections, the magnetic propagation vectors can be determined for a measurement a low temperature. The diffraction image at 158 K reveals some additional peaks, especially two strong reflections close to the center. Following the theoretical description , these reflections are due to a magnetic ordering with a propagation vector k = 0. There is also an option to add a pattern of magnetic reflections to the analysis in Esmeralda Laue Suite. By varying the underlying propagation vector manually, it is possible to index the additional peaks in agreement with the relation q = h+k. Figure 5: Indexation of mag- netic reflections at 158 K. The two peaks

• n

the left

• f

the center are in agreement with the calculated magnetic diffraction pattern represented by green boxes. 9

For CaBaFe4O7 the non-zero magnetic propagation vector can be identified with k = 1

3, 0, 0

• as shown in Fig. 5.

Furthermore, some nuclear reflections exhibit a considerably higher intensity. Hence, a propagation vector k = 0 might be involved in the magnetic structure at low tem-

• perature. A quantitative evaluation of the temperature dependent peak intensities

should clarify this aspect.

4.4 Temperature dependence

The temperature dependence of the magnetic order gives rise to a variation of the intensity of a magnetic reflection with temperature. In the case of k = 0, this ef- fect involves nuclear peaks with non-vanishing magnetic structure factors, too. For each kind of reflection one representative peak is selected to investigate its tem- perature dependence. The peak with nuclear indexation refers to q = (1, −5, −2) while the magnetic one is indexed by q = (1, 2, 2) − 1

3, 0, 0

• .

Esmeralda Laue Suite facilitates the determination of peak intensities via the tool Integrate Peaks (Calculated Positions). It integrates the intensity of a reflection within a cutting circle around its calculated position and accounts for the back- Figure 6: Temperature dependence of the intensities of two selected reflections. The data sets reveal magnetic phase transitions at T1 ∼ 278 K and T2 ∼ 205 K. 10

ground in a ring surrounding that circle. Adequate parameter values are 30 for the size of the cutting circle and 5 for the size of the border.

• Fig. 6 displays the temperature dependence of the two considered peaks. At high

temperatures, the magnetic reflection exhibits a plateau close to zero intensity. Slightly above 200 K, the intensity begins to increase severely. The qualitative behavior of the nuclear reflection is similar albeit the transition around 280 K is less sharp. Because of the contribution of nuclear scattering its intensity never

• vanishes. The magnetic diffraction contributes as soon as the temperature falls

below a certain value. In order to find the ordering temperatures for the two phase transitions, linear functions can be fitted to the intensity plateaus and slopes of both peaks. Identi- fying the intersections of these lines with the transition temperatures, one obtains T1 ∼ 278 K for k1 = (0, 0, 0) and T2 ∼ 205 K for k2 = 1

3, 0, 0

• . The fluctuations

in the data points, especially in the plateau regions, do not permit a reasonable statement about the uncertainty of these results.

5 Discussion and conclusion

To sum up, the metal oxide CaBaFe4O7 exhibits two distinct magnetic phase tran-

• sitions. The ordering temperatures T1 ∼ 278 K and T2 ∼ 205 K correspond to the

magnetic propagation vectors k1 = (0, 0, 0) and k2 = 1

3, 0, 0

• , respectively. Only

few peaks in the outer segments cannot be indexed. This indicates that the sample was not a perfect single crystal but contained crystallites. The determination of the periodicity does, however, not complete the examina- tion of the magnetic properties of this compound. Gaining information about the magnetic moments of single atoms requires the resolution of the related Fourier

• components. This should be subject to further investigation. One possibility for

that is provided by CYCLOPS. A rotation of the sample about the detector axis effects a variation of the peak intensities since the scattering process is only af- fected by magnetic moments perpendicular to q. The difficulty of this method [Oul11] consists in the wavelength dependent intensity distribution of the incident

• neutrons. Therefore, it is necessary to record 30 to 40 diffraction patterns over the

course of one full rotation. As demonstrated in this report, the analysis of Laue diffraction patterns is a chal- lenging task in contrast to the fast and simple procedure of the measurement. Nonetheless, the Laue technique is a powerful tool for studying magnetic struc- tures. 11

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