Dimensionless reduced energy

The results for total backscattering yields and backscattered energy of hydrogen ions on different materials are shown in Fig. 1063 6S> 70 7 . The values are plotted as a function of a dimensionless reduced energy e given by7 ... 

The harmonic oscillator has V = and the harmonic-oscillator Schrodinger equation contains the three constants k, m, and h. We seek to find a dimensionless reduced energy , and a dimensionless reduced x coordinate x, that are defined by... 

Problems 4.30-4.38 apply the Numerov method to several other one-dimensional problems. In solving these problems, you need to (a) find combinations of the constants in the problem that will give a dimensionless reduced energy and length [Eq. (4.68)] ... 

Here, p = AEq o/ eff is the dimensionless energy gap between the upper state and the closest lower-energy state in units of the effective vibrational energy, Veff (cm ). C is the electronic factor, and S is the Huang-Rhys dimensionless excited-state distortion parameter in units of vibrational quanta v ff. As shown in Eq. (2), /c ,p is strongly dependent onp. Additionally, for a given reduced energy gap p, the introduction of even small excited-state distortions, S, can rapidly enhance the radiationless multiphonon relaxation rate such that this dominates the total 0 K relaxation. This model is easily extended to elevated temperatures, where substantial increases in may be observed [7,8]. 

The values for the parameters in a given equation of state must be determined before it can be applied. The best course is to fit these parameters with measured experimental data. When measured data are not available, we can use the principle of corresponding states. On a molecular scale, the principle of corresponding states asserts that the dimensionless potential energy is the same for all species. On a macroscopic scale, it translates to the statement that all fluids at the same reduced temperature and reduced pressure have the same compressibility factor. We applied the principle of corresponding states to relate the parameters of equations of state to the critical temperature and pressure by noting that there is an inflection point on the critical isotherm... 

The moment of inertia 1 determines the rotational constant 0 = h /IT, which is the parameter that controls the strength of quantum effects. The other parameter of the model, which is the quadrupolar coupling constant K, can be conveniently taken as the energy and temperature scale. We can thus reduce all quantities related to energies by K, and define, e.g., the dimensionless temperature = k T/K, energy = E/K, and rotational... 

Figure 11-8. Single crystal of CoO after exposure (30 h) to an oxygen potential gradient at T= 1200°C. a) SEM picture looking onto the initially flat reducing (100) surface, b) Computer simulation iv = 0.05 E = 0.5 w dimensionless force (= increase of jump probability in forward direction) e surface energy, normalized to RT. Cross section of the crystal represented in Figure 11-7 [M. Martin (1991)].

As pointed out in Chapter 8, for the case where D > L the dimensionless energy balance reduces to... 

Note that temperature may be expressed in terms of kinetic energy of molecules. Heat, a form of energy, may be expressed in terms of M, L and T, [M L2 T-2]. However, in most heat transfer problems, heat is conserved and is not transformed into other forms of energy. Here we consider heat and temperature as new fundamental dimensions, H and . This has the advantage of increasing the number of fundamental dimensions, thus reducing the number of dimensionless groups required to describe the problem. 

Figure 5.12 Reduced Madelung Energy A/d as a function of dimensionless interionic distance, for CsCl, rocksalt and zincblende crystals.

These seven quantities must always be known in order to describe the pressure generation and energy behavior of screw machines. Dimensional analysis reduces the seven influencing variables to three dimensionless groups ... 

We have thus far obtained the distribution functions for the incoming and reflected particles of reduced mass in different regions in terms of the unknown constants A and B and the unknown dimensionless surface temperature 8. The surface temperature was then related to the internal energy and the number density of the fictitious particle at the reflecting surface. Now, in order to determine the constants A and B, wc must specify the boundary conditions for the mass and the energy flux at the sphere of influence. 

Figure 2.4. The energy-gap dependence of the nuclear Franck-Condon factor, which incorporates the role of the high-frequency intramolecular modes. Sc = A/2 is the dimensionless electron-vibration coupling, given in terms which reduce replacement (A) between the minimum of the nuclear potential surfaces of the initial and final electronic states. (Bixon and Jortner, 1999) Reproduced with permission.

In dealing with half-cell reactions, we note that n is usually expressed with respect to the compound of interest for example, 2 mol of electrons are used to reduce 1 mol of NADP+. In this case n is dimensionless and equal to 2, so AG has units of FAE or energy mol-1, just as for the AG of chemical reactions. 

A set of VLE data lias here been reduced to a simple mathematical equation for tire dimensionless excess Gibbs energy ... 

Systematic studies of particle-solid interactions began with Bohr (1948) and culminated in the general description of atomic collisions based on the Lindhard-Scharff-Schiott (LSS) theory (Lindhard et al. 1963, 1968). The universal treatment is described by a set of dimensionless or reduced variables that are proportional to the energy, distance and time functions. 

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