These shortcomings of the static lattice model point to the importance of lattice vibrations in understanding the properties of materials. In the next subsection, we present a brief overview of the role of lattice vibrations in state-of-the-art materials research.
Role of Lattice Vibrations
Important quantities commonly calculated in the quasiharmonic approximation include equilibrium lattice parameters, elastic constants, specific heat, and coefficients of thermal expansion as a function of temperature [6, 7]. In temperature-induced phase transitions, the vibrational entropy Sαvib, which is a measure of the average stiffness of a material, plays a key role in stabilizing one particular phase over another.
Experimental Techniques
Neutron scattering cross-sections
On the other hand, in the incoherent case, any macroscopic interference is prevented by randomness in the phase and the scattering intensity is effectively the sum of the scattering intensity from each nucleus. The parentheses hi denote the thermal average, and the quantity in parentheses the space-time correlation function.
Theoretical Developments
Basic Approximations
- The Harmonic Approximation
- The Adiabatic Approximation
Let Ms be the mass of atom s, Rls be the position of the s atom in the lth lattice cell and us(Rl)=Rls - R0ls be its displacement away from its equilibrium positions. The adiabatic approximation that allows the separation of the electronic and ionic degrees of freedom provides a very useful simplification to tackle such difficult problems.
Models of Lattice Vibration
- Einstein Model
- Debye Model
- Born-von Karman Model
- Ab-initio Methods
This results in atomic vibrations that remain completely unaffected by the motion of the neighbors. This basic shortcoming of the Einstein model results from the oversimplified assumption that the oscillators do not interact.
Lattice dynamics of Alloys
- The mass effect
- The size mismatch effect
- The charge transfer effect
- The short-range order effect
In the disordered state of the system, the atoms undergo significant relaxations away from their ideal lattice sites. The Fermi energy (EF) of the bound elements is also relevant in connection with charge transfer.
Outline of the thesis
In this thesis, we use this method to calculate the configuration-averaging of the necessary quantities. The fundamental equation that allows a great simplification of the problem is Dyson's equation which states that the configuration averages the Green's function ≪ G.
Virtual crystal approximation (VCA)
Average t-matrix approximation (ATA)
Coherent potential approximation (CPA)
Consider an alloy of finite concentration, C, of type B atoms embedded in a host material of type A. Mi j = M0(1−ǫi)δi j, (2.11) where,ǫi is the fractional mass error that becomes zero at the atom sites of the host and has a value of ǫα at the sites occupied by α-type atoms.
Generalizations of the CPA
Generalizations of CPA 27 environmental disorder, as in the case of alloys with short-range order [79, 80] CPA was found to be inadequate. First, it is computationally very cumbersome and second, it yields a super-lattice efficient medium that violates the translational invariance of a single site of a given lattice.
Itinerant coherent potential approximation (ICPA)
The Augmented Space Formalism
The masses m, force constant Φ, and Green's function G are defined in the dynamic Hilbert space Ψ in which the Hamiltonian of the system operates. In the configurational representation withinΘ, the state of site i is specified by the single-site state|Aiiif.
Multiple-scattering picture
The entire set of atoms undergoes fluctuations in the force constants as the location of the fluctuation site changes. The quantity V indicates all disturbances in the average medium, and F contains the intensity of the fluctuation in the average medium. Summary 41 In Fig 2.2, the scheme of calculations in the ICPA formalism is represented by a.
Summary
The quest is thus to find a reliable source of interatomic force constants in arbitrary alloys. In this chapter we present a new approach to calculate the interatomic force constants in arbitrary alloys. In the next two sections we discuss the functional density theory (DFT) and the functional density perturbation theory (DFPT) [112], which we use for ab initio calculations of interatomic force constants.
Density functional theory (DFT)
First-principles-based modeling of inter-atomic force constants A major problem in the direct application of this remarkably simple result, however, is that the form of the functional, F[n], is unknown. In particular, when the strength of the electron-electron interaction vanishes, F [n] defines the ground state kinetic energy of a system of non-interacting electrons as a functional of its ground state charge density distribution T0[n]. Variation of the energy functional with respect to n (r) with the restriction that the number of electrons is kept fixed leads formally to the same equation as applies to a system of non-interacting electrons subject to an effective potential, also called the self-consistent field (SCF) potential, whose form is, .
Density functional perturbation theory (DFPT)
The first-principles modeling of interatomic force constants, the Hellmann-Feynman theorem, is used to calculate the elements of the force constant matrices. According to the Hellmann-Feynman theorem, the first and second derivatives of the ground state energy are In the next section, we present an approach to modeling interatomic force constants in random alloys using results obtained from first-principles DFPT calculations.
Modeling the inter-atomic force constants
- Computational details
- Results
The weak perturbation of the force constant was due to the fact that the Fe-Fe and Pd-Pd force constants were almost of the same magnitude, and the Fe-Pd force constants were the concentration averages of the above two. As a first approximation, the Fe-Pd force constants are taken as simple concentration averages of the Fe-Fe and Pd-Pd force constants, as was done in calculations with model potentials [119, 120]. Anomalies in the phonon spectrum completely disappear with a 20% decrease in the Fe-Pd force constants relative to the average concentration values.
Limitations of the modeling strategy
Finally, we calculate the elastic constants in these alloys from the slopes of the phonon dispersion curves to verify the accuracy and quality of the phonon frequencies obtained by the modeling strategy adopted here. Results obtained from model potential calculations [119] and experiments [123] are also presented for comparison. We observe an overall better agreement of our results with the experimental values than those from the potential model approach.
Transferable force-constant (TFC) model
Transferable Force-Constant (TFC) Model 63 bond-length dependent transferable force constants across different environments, this model. Here the coordinate system is transformed such that the z-axis is aligned along the direction connecting atoms i and j. This symmetrization ensures that the force constants never have a lower symmetry than the environment to which it is transferred.
Summary
A significant insight into the lattice dynamics and related properties in this class of alloys can only be obtained if a microscopic picture of the role of each of the factors affecting the phonon spectra is obtained. Our results also show that consideration of the local lattice relaxations is necessary in these systems. We explain the reasons behind the discrepancy in light of the interatomic force constants and thus provide a physically reasonable picture of the microscopic situation in this system.
Computational Details
Results and Discussion
- Pd 0.96 Fe 0.04 , Pd 0.9 Fe 0.1 and Pd 0.28 Fe 0.72
- Pd 0.5 Fe 0.5
In the previous chapter we tried to model the interatomic force constants for these. F4.6: Partial and total structure factors calculated by ICPA for various ζ values in the [ζ,0,0] direction in Pd0.50Fe0.50 using the force constants extracted at the disordered alloy bond length (5.12 a.u. ). It was therefore argued that the force constants at 860 K correctly represent those of the disordered alloy due to.
- Cu 0.715 Pd 0.285
- Cu 0.75 Au 0.25
F4.10: Partial and total structure factors calculated by the ICPA for various ζ values in the [ζ,0,0] direction in Cu0.715Pd0.285 with force constants calculated at the alloy bond length. F4.14: Partial and total structure factors calculated by the ICPA for variousζ values in the [ζ,0,0] direction in Cu0.75Au0.25 with force constants calculated at the alloy bond length. F Partial and total structure factors calculated by the ICPA for various ζ values in the [ζ,0,0] direction in Cu0.75Au0.25 with force constants calculated at the relaxed bond lengths.
- Ni 0.95 Pt 0.05
- Ni 0.70 Pt 0.30
- Ni 0.50 Pt 0.50
- Ni 0.25 Pt 0.75
The circles are the experimental data from Reference [156]. value of the Ni-Pt force constants than the Ni-Ni ones. Our calculations of the structure factors show that the Ni-Pt pairs do indeed contribute to the structure factors near resonance. For ζ >0.55 the Pt-Pt pairs have practically no significant contributions and the structure factors are dominated by the vibrations of the Ni-Ni and Ni-Pt pairs.
Summary
156] can lead to the false splitting of the distribution curves, which therefore gives rise to the disagreement between our calculations and the experiments. Therefore, a theoretical calculation that includes short-range order in our formalism and a careful neutron scattering experiment must be performed for this composition of the alloy. Incorporation of the local environment affects phonon dispersion spectra of homogeneously disordered Ni0.5Pt0.5 alloy.
Methodology
Structural model
Here (k,m) corresponds to the figure defined by the number of k atoms located at its vertices (k=2, 3, 4.. are pairs, triangles, tetrahedra, etc.) with m being the order of neighbor distances that separate them (m=1, 2..are first, second neighbors etc.). This approach creates a distribution of distinct local environments whose mean corresponds to the random medium which is contrary to the requirement of mean-field theories (e.g. CPA in which each atom is affected by an identical mean medium. The main advantage of the SQS over a conventional supercell to model positional disorder is that the former uses the knowledge of the pair correlation functions, a key property of the random alloys, to determine the positions of the atoms in the unit cell, instead of randomizing them addition as it was done in the conventional supercell technique, and is therefore guaranteed to provide a better description of the environments in a real random alloy.
Averaging procedure for force-constants
The force constant matrix for two atoms separated by the vector (12,12,0)a has the form The other nearest neighbor force constant matrices are of the same form and are related by point group operations. We convert the SQS force constant matrix Φ to the FCC force constant matrix Φ' by performing the UTΦU operation on each of the 4 cases and then summing them.
Incorporating the local environmental effects, the SQS averaging scheme is more difficult than the Ni-Ni effects, a result inconsistent with the empirical scheme. Under the SQS averaging scheme, the actual stiffness of the Pt-Pt bonds in the alloy is found to be about 25% compared to the pure Pt. The Ni-Ni (Pt-Pt) force constants in the empirical model as shown in Table 5.1 are softer (harder) compared to the CPAs which explains the reason for better agreement of the phonon frequencies calculated by the empirical-ICPA model with the experimental results.
Summary
Substitutional alloy
F1.2: Interstitial alloy where some of the interstices (holes) in the structure of the host element are occupied by smaller atoms of the alloying element.
Interstitial alloy
The results clearly show that the use of the LDA exchange correlation function is justified in the present case. To understand the sources of nonphysical separations in our cases, we turn to partial and mean structure factors. This is the reason for the existence of additional longitudinal branches in the dispersion curves.
To see if fitting the L10 bond distances gets rid of the spurious peak in the structure factors in Fig. This observation together with the force constant results in Figure 4.20 proves that the resonance frequency is affected by the vibrations of the Ni-Pt pair. In the thesis, we used the version of ICPA, suitable only for disordered binary alloys.
The construction of SQS for multi-component systems and the subsequent calculations of force constants could be too demanding.
Multiple scattering picture in the CPA and the ICPA
Schematic representation of the ICPA formalism
Phonon dispersion curves for FCC Fe computed at the experimental lattice
Schematic representation of the DFPT-TFC-ICPA method
Thus, this is an artifact of the softer Cu-Au and Au-Au force constants due to the expansion of bonds associated with the Au atoms. A double peak structure of the line shapes obtained from the experiments for the transverse branches could thus be an artifact of the limitations in the experimental setup. The reasons behind the better agreement of the current calculations with the experiments compared to the results of Ref.
We demonstrate here a clever procedure to recover the mean force constants with the symmetry of the real binary alloy. 127 correct qualitative trends and that the reliability of the experimental results for this system is questionable.
