Single crystals are by definition extremely pure and homogeneous samples. The paper measures the properties of ß-Ti3Au, a crystalline material, and used microtribology, XRD, and computational tools to probe its properties. By definition, materials measurements of single crystals do not depend on experimental factors like preparation method. Two single crystals of a compound will have identical properties, so long as they share the same crystal structure, regardless of how you got them.
Actual fabrication processes are more nonuniform. Also, as with inclusion in other materials such as ceramics or glasses, where the quasi-crystalline borders are important to the way the material resists heat flux without mechanical degradation, irregularities become the site of the materials flaws. A practical consideration for aerospace use.
Hardness is an intrinsic property of covalent crystalline materials, as laid out in reference 23 of the paper under discussion, arising from the net strength of the covalent bonds between atoms in a crystal lattice.
Indeed. It is precisely that where the physics focus comes from here.
The interplay between the covalent and band theory, where specific valence plays a role.
From a physical standpoint, a single crystal is an infinitely repeating spacial tiling of a defined unit cell.
Metals can be amorphous, crystalline, or various "hybrids" of both.
In the paper, there is a lot of talk about valence electron density.
The paper does not speak to the physics of the bond, because the point is to communicate the nature of the material as for application.
The physics of the bond gives the use and nature of the materials as expressed in "valence electron density".
In the physics of this bond, it is the application of band theory for Group IV metals (and an analogous case in Group III metals).
I think you are misunderstanding this term. It is just the total number of valence shell electrons per volume; when considered in combination with the band gap one can estimate the strength of the chemical bonds in a covalent crystal and, with some assumptions and approximation, an estimate of the hardness.
My posts are intentionally abbreviated. They do not include definitions. Am using the term "valence" as it applies to band theory.
In this case with band theory it is referring to the
specific quantum states of those valence shell electrons, not a total number of a volume.
I am not dealing with a chemical situation but a quantum mechanical one. In fact, it was the characteristics of the possibility of such bonds that allowed the exploration of this material as a material (chemical and materials engineering). The value of this find to condensed matter physicists (and materials engineers) means that they can apply / extend band theory into specific cases of overlapping and partially filled shells with unique angular momentum cases. These can only occur with specific materials. (Physics and chemisty reinforce, and often intertwine, as is the case here.
However, they often butt heads over who's view dominates, the internal or the external. As appears to be the case here.)
Titanium has higher stability and lower density(45%) than steel. Ti(z=22) with a partially filled 3d shell is in the d2s2 configuration and appears in five phases of α(hcp), ω(hexagonal), γ(distorted hcp), δ(distorted bcc) and β(bcc) [1, 2, 3] in which γ and δ are instable phases.
Phases α and ω are at room temperature and atmospheric pressure and phase β is found to be up to 900C and 8GPa.
Because of the nature of this work in this paper, calculations show that ω phase is more stable than α phase at 0K. Observed x-ray diffraction have shown that the stability range of α phase varies between room temperature to around 923K. More on why that matters is below.
This work has nothing to do with defect density, higher-dimensional symmetry as found in quasicrystals, or "transitory valence states" (do you mean transition states?). Also I would say that the new physics you see in "valence grouping" has another name: century old types of chemical bonds, analyzed through a century old technique (x-ray crystallography).
No it doesn't. And I believe I've spotted the miscommunication - "chemical bonds".
Are you familiar with peculiarities of how high temperature coatings and alloys like mondaloy? That you can operate in extreme environments with them?
Are you also familiar with AMO work, and that with related semiconductor "chillers", that allow cancellation of thermal energy to achieve the localized effect of lower thermal lattice energies for constrained function?
The "new physics" aspects in condensed matter research focuses on using the above mentioned phases to achieve similar effects in materials like Ti, Al, and Mg.
Now we enter the proprietary area I won't descend further.
The "hardness" evidenced by the above bonds in this material (and similar ones) can be used to absorb the lattice energies allowing such desirable materials.So band theory is selectively extended to allow translation of angular momentum quantum states of mostly d orbital valences, only possible on certain covalent bond structures with metals with bands that work with slightly irregular (almost like a semiconductor) arrangement.
The above mentioned phases of Ti occur as states to alternate through.That is the point. Sorry if its not you "centuries old" use of chemical bond types. It's a necessary hybrid.
And its not pure chemistry, just like the other materials I've alluded to.
Sorry, am just a simple mathematician helping those awful condensed matter physicists to once again wreck the pristine, centuries old chemistry of common metals. Apologize for crossing disciplines again in pursuit of solving yet another applied math problem.
My interest is in the group theory (mathematics) of the bond structure as a quantum mechanical version of a "ring oscillator". So-called "dynamic stability".
Can you elaborate on this?
Sure.
In group theory we have the concept of a "ring" - a cyclic state structure that repeats in successive order. In electronics, we can use this to make an oscillator that likewise sequences in a pattern that can be used to impart a phased arrangement of discrete energy packets (overtones in electronic music synthesizers, where they mimic certain musical instruments/bells).
You can do the same in condensed matter to evoke properties of the materials. The benefit is to stochastically exceed the static durability of a substance.