Lattice Parameter

[Cripin Plascencia]

Achemical substance in the solid state may form crystals in which the atoms, molecules, or ions are arranged in space according to one of a small finite number of possible crystal systems (lattice types), each with fairly well defined set of lattice parameters that are characteristic of the substance. These parameters typically depend on the temperature, pressure (or, more generally, the local state of mechanical stress within the crystal),[2] electric and magnetic fields, and its isotopic composition.[3] The lattice is usually distorted near impurities, crystal defects, and the crystal's surface. Parameter values quoted in manuals should specify those environment variables, and are usually averages affected by measurement errors.

The lattice parameters of a crystalline substance can be determined using techniques such as X-ray diffraction or with an atomic force microscope. They can be used as a natural length standard of nanometer range.[4][5] In the epitaxial growth of a crystal layer over a substrate of different composition, the lattice parameters must be matched in order to reduce strain and crystal defects.

The volume of the unit cell can be calculated from the lattice constant lengths and angles. If the unit cell sides are represented as vectors, then the volume is the scalar triple product of the vectors. The volume is represented by the letter V. For the general unit cell

Matching of lattice structures between two different semiconductor materials allows a region of band gap change to be formed in a material without introducing a change in crystal structure. This allows construction of advanced light-emitting diodes and diode lasers.

For example, gallium arsenide, aluminium gallium arsenide, and aluminium arsenide have almost equal lattice constants, making it possible to grow almost arbitrarily thick layers of one on the other one.

An alternative method is to grade the lattice constant from one value to another by a controlled altering of the alloy ratio during film growth. The beginning of the grading layer will have a ratio to match the underlying lattice and the alloy at the end of the layer growth will match the desired final lattice for the following layer to be deposited.

I am collecting the lattice parameters for multiple materials. I started with BiFeO3 and wanted to look up the lattice parameter of its room-temprature phase (space group R3c, rhombohedral). I found this record with a material_ID of mp-23501, and it says a = 5.59 A, c = 5.65 A. However, this is quite different from the values reported from published work, for a example, a = 3.96 A (Structural Instability of Epitaxial (001) BiFeO3 Thin Films under Tensile Strain Scientific Reports). I am confused about this result, how could there be such a big difference? Which one is the correct one? Going forward, should I trust materials project as a reliable source to collect massive lattice parameters? Any input is appreciated!

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Small palladium particles with mean diameters ranging from 1.4 to 5 nm have been prepared within a plasma polymer matrix from a vinyltrimethylsilane monomer. Electron diffraction has shown a decrease of the Pd lattice parameter with decreasing size of palladium cluster. These experimental data are used to deduce the value of the surface stress coefficient f=6.00.9 N/m. A contraction of the lattice constant of small palladium particles with decreasing of cluster size, to our best knowledge, has not been reported in the literature up to now. The results suggest that the dilatation of the Pd lattice constant, often reported in the literature, can be explained in terms of incorporation of impurities like carbon, hydrogen, and oxygen, or pseudomorphism in the case of crystalline supports.

An experimental law of the x-ray diffraction of polycrystalline martensite is found, which is satisfactorily explained from x-ray-diffraction theory. The experimental phenomenon is well consistent with the hypothesis that at low carbon content martensite is of cubic structure. The cubic-tetragonal transition of martensite concerned with the distribution of carbon atoms is observed, and an experimental critical point of the transition is also obtained. Dependences of the lattice parameter of low-carbon martensite upon carbon content are established. With the dependences, it becomes possible to determine the carbon content of the carbon-supersaturated α-phase of steel.

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Experimentally, AC is used to discriminate the effects of the local and global structure on the physical properties; if some property is observed only in QCs, then its origin should be ascribed to the quasiperiodicity; otherwise, the property can be considered as a characteristic of the common building block, i.e., the icosahedral Tsai-type cluster. Theoretically, on the other hand, AC is useful to compare between calculation and experiment19 because any calculation such as a band structure calculation is almost impossible for QCs.

The spectrum consists of two components, a resonant component yielding the double-peaked structure, and a double-step-like fluorescence component contributing to the background spectrum, as observed in other IV compounds27,28. The resonant and fluorescence components were assumed to be described by a double-pseudo-Voight and double-Sigmoid function, respectively. See SI for another choice of functions. The resonant component is decomposed into two components coming from \(\textYb^2+\) and \(\textYb^3+\) configuration, each forming a peak at 8938 and 8945.4 eV (denoted by green and orange lines in Fig. 2b), respectively. Using the intensities of the resonant components of the \(\textYb^2+\) and \(\textYb^3+\) configurations, \(I_2\) and \(I_3\), we define an Yb mean-valence \(\nu\) as follows27,

We stress that for both forms, the spectral line shape and hence the Yb mean-valence varies drastically with elemental substitutions of Au/Ga, Au/Cu and Al/Ga. A similar variation in PFY spectral shape was observed in heavy-fermion compounds under pressure, which was ascribed to a change in c-f hybridization strength associated with the pressure-induced lattice-parameter compression28. (Here, c and f indicates conduction and 4f electrons, respectively.) By analogy with this, we attribute the spectral-shape change observed here primarily to the hybridization-strength change via the lattice parameter change (see discussion below for more details). Hereafter, the lattice-parameter change caused by the constituent-element substitution is referred to as a chemical pressure effect.

What should be stressed here is the similarity between the QCs and the ACs. This similarity suggests that the presence of the critical lattice parameters is not related to the absence/presence of the periodicity but rather ascribed to the local structure such as the size of the Tsai-type cluster. Structural analysis must be performed to measure the cluster size, but it is natural to assume that the cluster size is proportional to the lattice parameter.

Figure 5a shows the temperature dependence of the uniform magnetic susceptibility \(\chi =M/H\) of representative QCs between \(\sim 1.8 \,\textK\) and room temperature, where M and H denote magnetization and external magnetic field, respectively. We find that the magnetism strongly varies with alloy composition; sample A shows a strong temperature dependence over a wide temperature range (note a logarithmic scale in the horizontal axis), whereas sample C only shows a weak temperature dependence at low temperatures; note that the low-temperature increase in \(\chi (T)\) for both samples is related to the quantum criticality as described below.

We stress the close resemblance between Figs. 4a and 5b, indicating that the magnetism is tightly related to the Yb mean-valence anticipated from the fact that \(\textYb^3+\) is magnetic whereas \(\textYb^2+\) is nonmagnetic. This indicates that the valence and magnetic instabilities occur simultaneously.

The interrelationship observed here between valence, magnetism and lattice parameter is compatible with the pressure effect for Yb-based heavy fermions; when a non-magnetic Yb-based heavy fermion is pressurized and, as a result, the lattice shrinks, a magnetic order usually appears due to the emergence of the magnetic \(\textYb^3+\) ions with a smaller ionic radius.

Figure 5b,d shows the close resemblance between the QCs and the ACs; when \(a_\mathrm6D\) \((a_\mathrm3D)\) is very slightly increased above \(a_\mathrm6D^\mathrmc\) (\(a_\mathrm3D^\mathrmc\)), the \(\chi\) suddenly drops by one order of magnitude at \(T=2\) K. Similar to the valence instability, this resemblance suggests that the magnetic instability is not related to the absence/presence of the periodicity but rather ascribed to a change in the local structure such as the size of the Tsai-type cluster.

Inverse magnetic susceptibility of Yb-based QCs and ACs. \(T^\zeta \) dependence of inverse magnetic susceptibility (\(1/\chi\)) of the QCs (a) and ACs (b). The critical exponent \(\zeta\) is given in the figure. Dotted lines indicate extrapolation to absolute zero. When the extrapolation goes through the origin of the figure, the magnetic susceptibility diverges toward zero temperature.

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