Experimental study on the thermodynamic interactions of phonons and magnetism in

Introduction

Metallic bonding and structure

Atoms consist of a small nucleus with most of the atomic mass and an essentially massless electron cloud around them. The electron configuration of the valence shell of Ni has two unpaired electrons in the outer shell.

Figure 1.1: Electronic wavefunction of fcc transition metals. (a) Radial distribution of the 3d zz and 4s wavefunctions of an isolated Ni atom [1]

Elementary excitations

In the meV range, it is absorbed in the form of oscillations of atoms. By increasing the energy transfer to about 0.1 eV, the electron spins can be excited in the form of spin waves, called magnons.

Phonons

Each oscillation mode has an angular frequency ωj and propagates with the wave vector k. The displacement of the nucleus in a lattice with N atoms is given by. The force F is proportional to the displacement u of the oscillator (F =k u, where k is the spring constant).

Figure 1.4: Phonon spectrum of Fe 65 Ni 35 (Invar). (a) Phonon density of states (DOS) at am- am-bient conditions

Magnetism

If more electrons have 'spin-up', the overall Coulomb repulsion is lower, but the kinetic energy is greater. The temperature dependence of the magnetization, M(T), is shown in Fig.1.10 for Ni and the two Invar alloys.

Figure 1.7: Models for magnetism in solid-state systems and their temperature depen- depen-dence, where M denotes the net magnetization [4]

Thermodynamics

Analogous to the magnetic entropy, it can be seen as a measure of disorder in atomic configurations or the entropy of mixing. As the temperature increases (or the pressure decreases), the interatomic forces soften, lowering the vibrational energies and causing an increase in the phonon entropy.

Figure 1.12: Qualitative connection between entropy and thermal expansion on a 2- 2-dimensional lattice

Phase stability of iron

The intensities of the S(Q, E) scattering can be integrated over all momenta to give the energy spectrum of the atomic vibrations, the phonon density of states (DOS). For the x direction (and correspondingly in y and z) it contributes to the intensities as. A.6) The resulting effect of the number of diffraction planes, N, on the shape factor and consequently on the shape of the diffraction peaks is illustrated in Fig.A.3.

Figure 1.13: Phase stability of Fe. (a) Structural phase diagram of iron showing its different crystal structures as a function of temperature and pressure

Experimental methods

High pressures in diamond-anvil cells

As the pressure increases, the diamonds compress the packing and reduce the sample chamber volume. (f) Final state of a packing with a shrunken chamber after high pressures (~20GPa) are achieved. The gas it trapped in the sample chamber medium by engaging the DAC screws which seal the packing with the diamonds.

Figure 2.1: Operation of a diamond-anvil cell (DAC). (a) Schematic of a symmetric-type DAC [2]

Inelastic neutron scattering (INS)

The inelastic spectrum, i.e., the amount of energy transferred between neutrons and phonons, is measured with time. The data cuts (white parts of the plot) correspond to the kinematic limits of the ARCS instrument.

Figure 2.4: Phonon spectrum of polycrystalline Fe-36%Ni (Invar) measured by inelas- inelas-tic neutron scattering at ambient conditions

Mössbauer spectroscopy

States with different spinsmI with respect to the magnetic field shift by different amounts, leading to a splitting of the energy levels. (b) Mössbauer spectrum of ferromagnetic 57Fe65Ni35 at ambient conditions measured with a conventional Doppler drive (velocity of the drive is the lower axis) . Since the distribution of the spectral lines is directly proportional to the magnetic field experienced by the nucleus, called the hyperfine magnetic field (HMF), Mössbauer spectroscopy is an effective method to measure this field [22].

Figure 2.5: Mössbauer spectrum of Fe-Ni Invar. (a) Shifts of the nuclear energy levels of

Synchrotron radiation - Advanced Photon Source

Such a wavy inserter was used for nuclear resonant scattering experiments at beamlines 3 ID-D and 16 ID-D of the APS. A camera with a microscope and a Raman spectrometer can be moved in and out of the beam.

Figure 2.9: Experimental setup for nuclear resonant experiments at beamlines 3 ID-D and 16 ID-D of the APS

Synchrotron Mössbauer spectroscopy

57Fe due to the magnetic field (from Figure 2.5). (b) The sum of two sine waves with slightly different frequencies results in a wave with intensity modulations or "beats". (c) Measured NFS spectra showing the intensity modulation due to the interference of up to six energy lines. The period of the modulations is used to infer the strength of the hyperfine magnetic field.

Figure 2.10: Mössbauer spectroscopy in time domain. (a) Splitting of nuclear energies of

Nuclear resonant inelastic x-ray scattering (NRIXS)

Left axis: concentration dependence of Curie temperature (TC) and Néel temperature (TN) in the γ phase. By measuring the volume of the mesh at two different temperatures, we can determine the volumetric expansion, i.e. the thermal expansion.

Figure 2.11: NRIXS spectrum of 57 Fe 65 Ni 35 at ambient conditions. The resonant absorption of the 57 Fe through the Mössbauer effect gives the strong elastic line at E=0

Thermodynamic stability and heat absorption of

Thermodynamics of nanoparticles

Our low-temperature heat capacity data show an increase in heat absorbed by electrons in nanostructured materials of about 40%, which is reflected in higher electron entropy. Most of the magnetic disorder, and thus most of the magnetic entropy, increases near the Curie transition.

Prior work on vibrations in nanomaterials

An increased number of modes at low energies has been observed in calculations and measurements on isolated or rigidly confined nanocrystals [13–42]. The enhanced phonon DOS of nanoparticles at low energies increases the number of phonons that can be excited, resulting in higher vibrational amplitudes and higher entropies.

Figure 3.3: Size effects on phonons. (a) Phonon DOS from a polycrystalline powder of an fcc Ni-Fe alloy

Materials

Some of the smallest nanocrystal sizes in metals, around 6-7 nm, can be obtained in Ni3Fe prepared by high-energy ball milling [55]. Hereinafter, nanomaterials prepared by high-energy ball milling are referred to as "as-ground" samples.

Figure 3.4: Phase diagram of Ni-Fe compounds. Adapted with permission from [52].

X-ray diffraction

They confirm that most of the grain growth actually takes place between 580 and 800 K, corresponding to the heat release. If part of the highlighted signal in fig. 3.8 includes the magnetic transition, will our calculation of.

Figure 3.7: X-ray diffraction patterns of the as-milled material and of the bulk control sample after annealing at 873 K for 1 h.

Heat capacity

The heat capacity of the milled nanocrystalline material also increased at temperatures above 200 K. Low temperature measurements on disordered Ni3Fe [65] are in excellent agreement with the heat capacity of our control sample.

Figure 3.9: Heat capacity of Ni 3 Fe measured by calorimetry for the as-milled materials (+) and large-grained control samples (•), normalized by the number of moles of atoms.

Contributions at low temperatures

It is a first-order transition, so with partial order the change in heat capacity will be reduced proportionally. We therefore ignore the effects of partial chemical ordering on the heat capacity of our samples.

Phonon contributions

This resulted in an underestimation of the DOS at low energies, especially for large-grained materials, due to the sharper features in their spectra. Solid curves correspond to new measurements performed at ARCS, while markers are measurements from previous work [35].

Table 3.2: Low temperature fits the heat capacity data of in Fig. 3.11 b to Eq. 3.8, including and excluding the contributions from magnons α

Contributions to the heat capacity at higher temperatures

Nanomaterials therefore have a larger contribution from magnetism to the heat capacity and entropy, even at 300 K and below. The vibrational contribution to the heat capacity was calculated from the phonon DOS measured at 300 K using Eq.3.10, based on a harmonic model.

Figure 3.14: Individual contributions to the heat capacity of (a) the control and (b) nanocrystalline samples

Free energy and thermodynamic stability

Reduced lifetime is compatible with broadening the characteristics of the DOS as shown in Fig.3.12a. The excess entropy of the nanomaterial with respect to the control sample can be seen in Fig.3.15b.

Figure 3.15: Total enthalpy (a) and entropy (b) of nanocrystalline material and control sample computed from calorimetry measurements of the heat capacity using Eq

Conclusions

The resulting pressure dependence of the phonon spectrum of Invar is represented in Fig.5.17. The average of these distributions, proportional to the magnetization of the iron atoms, is shown in Fig.C.2a.

Magnetic quasi-harmonic model for Fe and Fe 3 C

Experiments

The phonon density of states (DOS) of iron in the bcc phase has been previously measured at temperatures between 30-1180 K on beamline 16 ID-D of the Advanced Photon Source [1]. However, as indicated by the dotted lines in Figure 4.1b, the QHA is unable to fully capture phonon behavior, especially when the temperature approaches the Curie transition.

Figure 4.1: Phonon energies of bcc Fe. (a) Phonon densities of states (DOS) of 57 Fe from 30 K to 1180 K

Standard phonon quasiharmonic approximation

Since we rely on the formalism of the harmonic model (eq. 4.2), phonons in the QHA still do not interact. Such spin-lattice coupling could account for the drift of the QHA as the spin disorder increases as the Curie transition approaches.

Figure 4.3: Temperature dependence of magnetism in Fe 3 C (cementite). (a) Nuclear for- for-ward scattering (NFS) spectra of Fe 3 C at several temperatures

Magnetic quasiharmonic approximation

Here the phonon energy E remains as kBT in the mode and is unchanged by magnetism. These are the same roles that E and S play in a standard QHA, but in a magnetic QHA it is M instead of V that changes the phonon entropy and free energy.

Magnetic QHA for Fe and Fe 3 C

To further test the magnetic QHA, magnetic Grüneisen parameters were obtained by fitting Eq.4.16 to the phonon free energy differences for iron and cementite. Nevertheless, the magnetic QHA approximately predicts the shape of the phonon free energy curves beyond the standard QHA for phonons.

Figure 4.5: Deviation of the phonon free energy from the QHA (∆F , left axes) and evo- evo-lution of the magnetization (M/M 0 , right axes) with temperature for (a) Fe and (b) Fe 3 C.

Conclusions

We observe a precise cancellation of the phonon and magnetic contributions to the thermal expansion, leading to near-zero thermal expansion known as the Invar effect. Due to the resonant nature of the NRIXS technique, there is no background signal (other than a constant background from detector noise).

Thermodynamic origin of the Invar effect

Anomalous thermal expansion

Due to the skewed potential, the atomic separation tends to increase a bit, causing the thermal expansion. The anomalous properties of Invar are by no means limited to the anomalous thermal expansion [1].

Figure 5.2: Linear thermal expansion coefficient α of Fe-Ni alloys measured by Guillaume.

Alloying iron and nickel

This is in striking contrast to the expected thermal expansion of the 'lattice' of the thermally amplified phonon vibrations in an anharmonic potential. Above the Curie temperature, TC = 515K, the Invar behavior disappears, and the measured thermal expansion corresponds to that of the 'lattice'.

Figure 5.3: Anomalous thermal expansion of Fe-Ni alloys, compared to the expected ther- ther-mal expansion from anharmonic phonons, labeled ‘lattice’ [14]

Interpretations and theories of the Invar effect

Understanding the γ-phase of iron is key to understanding previous models of the Invar effect. The anti-invar transition, from the LM minimum to the HM saddle point, appears to be the precursor to the martensitic transformation.

Figure 5.5: States of fcc γ-Fe. (a) Hyperfine field of fcc Fe as a function of volume from Mössbauer spectroscopy, giving the strength of magnetic moments

Thermal expansion from the entropy

X-ray diffraction

The downstream diamond seat blocked part of the signal, resulting in a vertical strip of data. Note that the pressure dependence of the thermal expansion shows a very similar behavior to the temperature dependence shown in Figure 5.3.

Figure 5.9: Diffraction experimental setup at beamline 16 BM-D at the APS. (a) Beamline schematic [43]

Magnetism

It can be compared to the heat flow measured on a reference material, with a known heat capacity, to quantify the heat capacity of the sample. The contribution from phonons, CPph, can be calculated from measurements of the phonon DOS, g(ε), which The resulting phonon contribution is included in Fig.5.11a (and the NRIXS measurements are discussed further below).

Figure 5.11: Heat capacity of Invar and its contributions. (a) Total heat capacity measured by DSC, compared the phonon contribution and fitted electronic heat capacity

Phonons

The average energy of each Lorentzian (vertical line) is plotted against pressure in Fig. 5.18c. Longitudinal branch calculations show no energy changes up to about 4 GPa. (d) Calculated phonon DOS.

Figure 5.15: Magnetic entropy of Invar. Bottom axis: Pressure dependence of magnetic en- en-tropy calculated from the measured M (P ) with Eq

Electronic contribution

Thermal expansion & Invar effect

Conclusion

The resulting grain sizes and RMS strains are shown in Figure A.5 as a function of annealing temperature. Due to the geometric limitations of the cryogenic setup, only two detectors (instead of three) can be used.

Figure 6.1: Anomalous thermal expansion behavior of different transition metals. (a) Fe- Fe-35%Ni, Fe-32%Pd, Fe-28%Pt [10]

Conclusion and future directions

Lattice parameters

In practice, however, the accuracy of the lattice parameter calculated using Bragg's law may suffer due to experimental errors. According to the Nelson-Riley method, the error in the lattice parameter, ∆a0, has the following angular dependence:.

Grain growth

The error bars of the calculated phonon DOS (shown in Figure 5.17) were propagated from the standard deviation of consecutive NRIXS scans. Fits of the Mössbauer spectra using the CONUSS software package [6,7] are shown as solid lines in Figure C.1b.

Figure A.1: Grain sizes vs. lattice parameters. The fitted curve shows an inverse grain size dependency of the lattice parameters in our Ni 3 Fe samples.