Detennination

In the example shown a covermeter is used to detennine the distance to a known single bar, and this information is in turn used to establish a reference for the radar measurements. A wave speed of 110 m/ps is calculated for this concrete. 

Starting with the quantum-mechanical postulate regarding a one-to-one correspondence between system properties and Hemiitian operators, and the mathematical result that only operators which conmuite have a connnon set of eigenfiinctions, a rather remarkable property of nature can be demonstrated. Suppose that one desires to detennine the values of the two quantities A and B, and that tire corresponding quantum-mechanical operators do not commute. In addition, the properties are to be measured simultaneously so that both reflect the same quantum-mechanical state of the system. If the wavefiinction is neither an eigenfiinction of dnor W, then there is necessarily some uncertainty associated with the measurement. To see this, simply expand the wavefiinction i in temis of the eigenfiinctions of the relevant operators... 

Regardless of how many single-particle wavefiinctions i are available, this number is overwhelmed by the number of n-particle wavefiinctions ( ) (Slater detenninants) that can be constructed from them. For example, if a two-electron system is treated within the Flartree-Fock approximation using 100 basis fiinctions, both of the electrons can be assigned to any of the % obtained m the calculation, resulting in 10,000 two-electron wavefimctions. For water, which has ten electrons, the number of electronic wavefiinctions with equal numbers of a and p spin electrons that can be constructed from 100 single-particle wavefimctions is roughly... 

There can be subtle but important non-adiabatic effects [14, ll], due to the non-exactness of the separability of the nuclei and electrons. These are treated elsewhere in this Encyclopedia.) The potential fiinction V(R) is detennined by repeatedly solving the quantum mechanical electronic problem at different values of R. Physically, the variation of V(R) is due to the fact that the electronic cloud adjusts to different values of the intemuclear separation in a subtle interplay of mutual particle attractions and repulsions electron-electron repulsions, nuclear-nuclear repulsions and electron-nuclear attractions. 

One can detennine the total number of electrons in the system by integrating the density of states up to the highest occupied energy level. The energy of the highest occupied state is called the Eermi level or Eermi energy, E ... 

This expression is not orbitally dependent. As such, a solution of the Hartree-Fock equation (equation (Al.3.18) is much easier to implement. Although Slater exchange was not rigorously justified for non-unifonn electron gases, it was quite successfiil in replicating the essential features of atomic and molecular systems as detennined by Hartree-Fock calculations. 

This equation is usually solved self-consistently . An approximate charge is assumed to estimate the exchange-correlation potential and to detennine the Flartree potential from equation Al.3.16. These approximate potentials are inserted in the Kolm-Sham equation and the total charge density is obtained from equation A 1.3.14. The output charge density is used to construct new exchange-correlation and Flartree potentials. The process is repeated nntil the input and output charge densities or potentials are identical to within some prescribed tolerance. 

The electronic energy, as detennined from must be added to tire ion-ion interactions to obtain the structural energies. This is a straightforward calculation for confined systems. For extended systems such as crystals, the calculations can be done using Madelimg summation techniques [2]. 

Because (k) = (k + G), a knowledge of (k) within a given volume called the Brillouin zone is sufficient to detennine (k) for all k. In one dimension, G = Imld where d is the lattice spacing between atoms. In this case, E k) is known once k is detennined for -%ld < k < %ld. (For example, m the Kronig-Peimey model (fignre Al.3.6). d = a + b and/rwas defined only to within a vector 2nl a + b).) In tlnee dimensions, this subspace can result in complex polyhedrons for the Brillouin zone. 

In fignre A1.3.9 the Brillouin zone for a FCC and a BCC crystal are illustrated. It is a connnon practice to label high-synnnetry point and directions by letters or symbols. For example, the k = 0 point is called the F point. For cubic crystals, there exist 48 symmetry operations and this synnnetry is maintained in the energy bands e.g., E k, k, k is mvariant under sign pennutations of (x,y, z). As such, one need only have knowledge of (k) in Tof the zone to detennine the energy band tlnoughout the zone. The part of the zone which caimot be reduced by synnnetry is called the irreducible Brillouin zone. 

Since and depend only on die valence charge densities, they can be detennined once the valence pseudo- wavefiinctions are known. Because the pseudo-wavefiinctions are nodeless, the resulting pseudopotential is well defined despite the last temi in equation Al.3.78. Once the pseudopotential has been constructed from the atom, it can be transferred to the condensed matter system of interest. For example, the ionic pseudopotential defined by equation Al.3.78 from an atomistic calculation can be transferred to condensed matter phases without any significant loss of accuracy. 

As with any basis method the a(k, G) coefficients are detennined by solving a secular equation. 

Other methods for detennining the energy band structure include cellular methods. Green fiinction approaches and augmented plane waves [2, 3]. The choice of which method to use is often dictated by die particular system of interest. Details in applying these methods to condensed matter phases can be found elsewhere (see section B3.2). 

Several factors detennine how efficient impurity atoms will be in altering the electronic properties of a semiconductor. For example, the size of the band gap, the shape of the energy bands near the gap and the ability of the valence electrons to screen the impurity atom are all important. The process of adding controlled impurity atoms to semiconductors is called doping. The ability to produce well defined doping levels in semiconductors is one reason for the revolutionary developments in the construction of solid-state electronic devices. 

The expansion is done around the principal axes so only tliree tenns occur in the simnnation. The nature of the critical pomt is detennined by the signs of the a. If > 0 for all n, then the critical point corresponds to a local minimum. If < 0 for all n, then the critical point corresponds to a local maximum. Otherwise, the critical points correspond to saddle points. 

Similarly, with the nuclear-fixed axis convention, we detennine that... 

To explain the vanishing integral nile we first have to explain how we detennine the synnnetry of a product. G fold degenerate state of energy and synnnetry T, with eigenfunctions - / -fold degen... 

The coordinates are redundant since they can be detennined from the condition drat the X, Y, Z)... 

In order to detennine the relationship between R translation where A X is infinitesimally small. In this case we can approximate the right hand side of... 

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