The raw data is in the form of the flux of scattered neutrons as a function of momentum and energy transfer [3]. In Figure 1.1 we have defined the incident ray by its de Broglie wave vectork⃗i and energyE⃗i.
Where λi and λfar are the initial and final states of the sample, respectively, and σ is the scattering cross section of the neutron beam. The double differential cross section could also be related to the Van Hove function, a time-dependent pair correlation function.
Crystal Lattice
We define a unit cell by integer linear combinations of three independent vectors a1,a2 and a3, the primitive lattice vectors in the crystal. We define the unit cell as the smallest cell that can generate the entire crystal.
![Figure 2.1: All 5 two-dimensional Bravais lattices [1]](https://thumb-ap.123doks.com/thumbv2/123dok/10412615.0/17.918.210.711.214.519/figure-2-1-all-two-dimensional-bravais-lattices.webp)
Phonons
This is usually done by diagonalizing the dynamic matrix from the original, coupled equations of motion. In fact, phonons in periodic lattices are more precisely defined by the uniformly distributed energy levels of these normal modes.
Harmonic approximation
The potential energy, φ, is the sum of the contributions from each pair of atoms in the crystal. We can substitute the displacements of the propagating waves into the equations of motion, or alternatively, we can note that this is the same as taking the spatial Fourier transform of the right-hand side.
Thermodynamics
Extending this description of lattice modes to quantized vibrational excitations allows us to capture the evanescent nature of the heat capacity (Debye went a step further and was able to capture the T3 power law where the heat capacity decays by assuming that the frequencies in a material can vary as a function of the wave vector). Furthermore, the harmonic approximation correctly predicts that the phonon free energy will change as a function of temperature.
Failures of the Harmonic Approximation
So even though the DOS phonon harmonic approximation products will remain fixed with temperature, which we know is not the case, it still allows the higher energy phonon modes to populate more often at higher temperatures . In fact, the phonon entropy in the harmonic approximation actually turns out to be quite accurate even when we consider higher-order terms in the potential between atoms, especially when we substitute the DOS for the exact experimental DOS at any given temperature .
Quasi-harmonic Approximation
To underline, the QHA differs from the harmonic approach in that, although we still assume a harmonic potential at each temperature, this potential varies as a function of temperature, generating the shifting phonon DOS. Moreover, the QHA also avoids the nonsensical vanishing equilibrium pressure and bulk modulus that the harmonic approximation predicts. The QHA is therefore very similar to the harmonic approach, only the spring constants between the atoms vary as a function of temperature, leading to shifting frequencies.
Alternatively, we can separate the entropy of dilation from the entropy at zero thermal expansion. The entropy of expansion can be found by integrating the difference between the heat capacity at constant pressure and volume. We expect that these two methods of calculating the entropy will give the same answer, but this is not always true.
Anharmonicity
Furthermore, although the QHA improves the accuracy of thermodynamic models to account for experimental thermal expansion, it is not a completely rigorous approach consistent with the underlying physics of the material. We can use a second quantization expression for the atomic displacements to determine the Hailtonian of any order in this second quantization form. Experimental characterization of phonons at high temperatures is extremely important to elucidate the consequences of the anharmonicity in materials.
At high temperatures, more anharmonic parts of the potentials are investigated, so it makes sense that we. It is not entirely clear which materials can be well characterized by QHA alone and which materials need a lot of help from anharmonic effects. Understanding the validity or invalidity of the QHA for the element Germanium is the focus of much of this thesis.
Motivation and Goals
We noted previously in Chapter 2 that most of the entropy stems from the vibrational entropy of phonons. Even if the volume remains fixed, the frequencies shift due to the increased excursion distance of the atoms around their equilibrium positions, leading to phonon-phonon interactions. High temperature studies of phonon behavior are therefore critical to our understanding of the role of anharmonicity in materials.
High temperature studies are also important because most of the entropy of materials is derived from phonons. It is therefore well known, at least at low to moderate temperatures [7], that most of the entropy of materials is derived from Svib. It will allow us to 1: allow us to better distinguish and elucidate the quasiharmonic and anharmonic contributions to the damping of phonon frequencies (because these differences are more apparent at high temperature) and 2: allow us to test whether the statement that most of the total entropy in Ge can be calculated by harmonic theory that the vibrational entropy remains valid at higher temperatures.
Experimental Setup
Time-of-Flight Methods
Sampling Region
Time-of-flight techniques allow us to simultaneously sample large regions of q-space (reciprocal space). Collecting data over such a large range of mutual space is one of the main advantages of time-of-flight techniques over three-axis spectrometers. Using time-of-flight, we can perform phonon counting to reveal fundamental thermodynamic quantities.
For a certain scattering angle,φand incident energy,Ei, the maximum momentum transfer Qmax. and the energy transfer can be related by, . The problem with the former is that the instrument resolution may not be high enough to accommodate it.
![Figure 3.4: A distance vs time plot of the chopper timing. From [2]](https://thumb-ap.123doks.com/thumbv2/123dok/10412615.0/43.918.221.686.495.817/figure-3-distance-vs-time-plot-chopper-timing.webp)
Calculating the S(Q, E)
The 3He detectors used for neutron scattering may not always work properly. The mask removes pixels for which the number of neutrons is outside the limits we set. The efficiency of the detectors is determined by the prescribed coordinates (d, p and t), where d is the detector, p is the pixel and t is the time of flight.
This large solid angle means that pixels at different detectors and even different positions within a single detector will cover different levels of solid angle. As the scattered neutrons are detected, they are thrown in to calculate S(E, 2θ), where 2θ is the scattering angle and E is the energy transfer. We calculated the dynamical structure factor from the raw data for all the measurements taken on Ge.
S(Q, E) to Phonon DOS
Note that technically this should be I(E, 2θ) instead of S(E, 2θ) because we have not yet corrected for multiple scattering. To recover an accurate phonon DOS, we must ensure that we calculate a one-phonon DOS, which is obtained after correcting for multi-phonon and multiple scattering. Multiple scattering is scattering from a single neutron that scatters more than once before leaving the sample.
We then calculate the SQE of multiphonon scattering before calculating the SQE of multiple scattering. The multiple scattering and multiphonon scattering SQE are subtracted from the experimental SQE to obtain an approximation of the single phonon SQE. We calculated one-phonon DOSs for each of the S(Q, E)s we found in the previous section.
Results
Error Propagation Aside
In performing Gaussian fits to each of the DOSs, we made sure to preserve the covariance matrix for each of the fits. For each temperature, the square root of the diagonal of the covariance matrix provides the standard deviation of each of the peak positions. We then perform the standard Taylor method of error propagation by extending our formula for the fractional percentage shift.
Results cont
The isobaric Gruneisen parameter examines both the effects of quasiharmonicity and anharmonicity by elucidating the effect of both temperature and temperature-dependent volume on the frequency shifts. We also compare these values with values for the Gruneisen parameter that we would expect in the quasiharmonic approximation. It is often stated that the mode Gruneisen parameter should lie between 1 and 2 in the quasiharmonic approximation [9].
It is immediately apparent that the Gruneisen parameter is much higher than that predicted by QHA for all modes except the low-energy transverse optical mode. The vibrational entropy calculated in the harmonic approximation was overlaid with the total entropy. Further analysis of the relationship between total entropy and vibrational entropy, especially at higher temperatures, is left to debate.
Discussion
The data is in Gibbs free energy form and is differentiated to obtain the total entropy. We see that the harmonic approximation shows that the vibrational entropy accounts for most of the total entropy at low temperatures, but shows that the vibrational entropy accounts for less and less of the total entropy at higher temperatures. The vibrational entropy constitutes almost all of the total entropy at all temperatures found in phonon DOSs.
The vibrational entropy was able to account for almost all of the total entropy at each temperature. This is particularly noteworthy because the total entropy was calculated by calorimetry-like measurements, whereas our calculations were done by counting phonons. However, we also note that there is a small difference between the vibrational entropy and total entropy at temperatures above 1000 K.
Future Work
Melting at Caltech
The autumn period was spent testing the control of Sn, Bi and Pb under melting in quartz ampoules. Glass blowing and vacuum sealing had to be learned to create the quartz ampoules. We have also included an image of some example samples of Sn, Bi and Pb being melted.
For example, some of the quartz ampoules turned out to be too thin for the vacuum tubes. Tubes that allowed intake of the quartz ampoules while also allowing access to the vacuum had to be procured to create a multi-tube system (Fig. 4.3).
Future Work and Bayesian Analysis
- Bayesian Analysis
- Parameter Estimation
- Model Selection
- Algorithm
- Results
- Simulated Data
Therefore, we developed a method to determine the number of vertices that the maximum Bayesian evidence is in the Q-cut S(Q, E). The data from ARCS will, as we have seen, be a blurred and noisy version of the real spectrum. P(A0,{Ai,yi}|M,{Dk}) (4.2) We call this PDF the posterior PDF because it represents our beliefs about the probability densities of the parameter values after we have analyzed the data.
We need to discretize this space and calculate the value of the probability at each point in this space. The integrand is just the product of the prior and the likelihood function of the parameters. We also find estimates for the amplitude and width of the peaks, as well as the uncertainties of our estimates.
