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In an external magnetic field, a magnetic material may respond and become magnetized. The material's mass magnetization can be seen as the density of induced magnetic moments per unit mass. The proportionality factor between the external magnetic field H and the response magnetization M is called the magnetic susceptibility χ.

𝑴𝑴=𝜒𝜒𝑯𝑯 (2.7)

Different magnetic systems result in different behaviours for χ due to the interactions between other magnetic ions within the material. Magnetically ordered systems typically have non-linear M-H responses, and χ is a tensor. Temperature also affects χ, and critical temperatures where thermal energy can overcome exchange interactions and the material reverts to paramagnetism (TC for ferromagnets and TN for antiferromagnets) can be seen. Some simple examples are depicted in Figure 2.3 below.

Figure 2.3. Part (a) shows typical M-H responses for materials with different magnetic interactions (‘dia-’ = ‘diamagnetic’ etc.) and (b) illustrates the temperature dependence of χ for

different magnetic interactions. This illustration is based on information from [ 311],[ 312] and [ 24].

In this thesis, the M-T and M-Hdata are collected to get information about the type of ordering present and any transition points within the sample. Then 1/χ is fitted with the Curie-Weiss model within the paramagnetic temperature ranges to obtain some information about the size of moments and the nature of the exchange interaction, and the M-H data are fitted with the Langevin function to see if any clustered behaviour is occurring.

2.4.1.1 M-T Modelling: Curie-Weiss

The susceptibility χ is temperature dependent, as shown in Figure 2.3 above. At weak fields, paramagnetism can be modelled using the Curie-Weiss law313_{. According to this law, the} magnetic susceptibility determined from the mass magnetization plot can be expressed as:

𝜒𝜒=𝑥𝑥𝑁𝑁_3𝑘𝑘𝐴𝐴𝜇𝜇0𝜇𝜇𝑠𝑠𝑒𝑒𝑒𝑒2

𝐵𝐵𝑀𝑀𝑠𝑠𝑇𝑇 = 𝐶𝐶

𝑇𝑇 − 𝜃𝜃𝑤𝑤 (2.8)

where x is the molar fraction of the magnetic species in the material, NA is Avogadro’s number,

μ0 is the permeability of free space, kB is the Boltzmann constant, Mm is the molar mass of the material, C is the Curie constant and θw is the Weiss constant. This means, by plotting 1/χ vs. temperature and applying a linear fit to the paramagnetic part of the curve, the slope is the Curie constant C and the intercept is the Weiss constant θw. Weiss constants of – θw K and +

θw K are typical for materials with some antiferromagnetic and ferromagnetic interactions respectively.

Rearranging and applying the fitted slope C, an estimate of the effective magnetic moment (µeff), in bohr magnetons (μB), can be obtained via:

𝜇𝜇𝑠𝑠𝑒𝑒𝑒𝑒 =�_{𝑥𝑥 ×}3𝑘𝑘𝐵𝐵_𝑁𝑁×𝐶𝐶×𝑀𝑀𝑠𝑠

𝐴𝐴×𝜇𝜇0×𝜇𝜇𝐵𝐵2 (2.9)

This determination of effective moments from fitting allows comparison to expected values for different magnetic ions, as μeff is also equal to:

𝜇𝜇𝑠𝑠𝑒𝑒𝑒𝑒=𝜇𝜇𝐵𝐵𝑔𝑔�𝑱𝑱(𝑱𝑱+ 1) (2.10)

where g is the Landé g-factor and J is the total angular momentum.

A special case arises for the ions with partially filled d-levels such as in Fe3+_{. They are prone} to a quenching of orbital angular momentum L314 _{and the µeff instead becomes:}

𝜇𝜇𝑠𝑠𝑒𝑒𝑒𝑒=𝜇𝜇𝐵𝐵𝑔𝑔�𝑺𝑺(𝑺𝑺+ 1) (2.11)

where S is the spin angular momentum. This is often referred to as the ‘spin-only’ moment.

2.4.1.2. M-H fitting: Langevin

At low temperature and high field, paramagnetic behaviour can be modelled using the classical depiction of magnetization312_:

𝑴𝑴=𝑀𝑀0∙ 𝐿𝐿(𝛽𝛽) (2.12)

Where M0 = NgμBJ, N is the number of atoms, and L(β) is the Langevin function:

𝐿𝐿(𝛽𝛽) =𝑡𝑡𝑡𝑡𝑡𝑡ℎ−1₋ 1

𝛽𝛽 (2.13)

Fitting for β will gives the cluster moment ‘m’ (not equal to effective moment above).

𝛽𝛽 = 𝜇𝜇0∙ 𝑚𝑚 ∙ 𝑯𝑯

𝑘𝑘𝐵𝐵∙ 𝑇𝑇 (2.14)

Therefore the Langevin function does not describe the behaviour of individual independent atoms/ions but the behaviour of some kind of macroscopic spin in a magnetic field315_{. Such} macroscopic spins occur in systems with ferromagnetic or ferrimagnetic clusters in a nonmagnetic matrix or in magnetic nanoparticles which can be aligned in an external magnetic field.

2.4.1.3 Experimental Methods

The SQUID M-H and M-T magnetization measurements in Chapter 3 were conducted with a Quantum Design Magnetic Property Measurement System (MPMS) at the ISIS Neutron and Muon Source, UK. Low temperature field-cooled (FC) and zero field-cooled (ZFC) M-T data were collected from 2 – 324 K. The M-H data were collected at 10, 80, 150, 200 and 300 K in a field with intensity up to 70 kOe. The high temperature M-T from this Chapter was conducted with a Quantum Design MPMS with oven option (Durham University, UK), and FC data was collected from 300 – 535 K in a field of 175 Oe.

In Chapter 4, the BFNTO DC M-H data were measured using a Quantum Design Physical Properties Measurement System (PPMS) at UNSW Canberra, Australia to a field of 90 kOe and temperatures 2, 10, 100, 200 and 300 K.

The remaining M-H measurements in Chapter 4, 5 and 7 were conducted using the Vibrating Sample Magnetometer (VSM) mode of a Quantum Design PPMS at the Australian Centre for Neutron Scattering (Lucas Heights Australia) at 3 K. The FC and ZFC M-T data were collected at various fields over a temperature range of 3 – 300 K, and 5 – 373 K for VTO50 in Chapter 7 specifically.

While these are the most basic methods of measuring magnetic responses in a material, information can often be difficult to extract if there are any magnetic impurities present. For this reason, other complementary techniques can be employed.