Energy dimensionless unit

Here r is the distance between the centers of two atoms in dimensionless units r = R/a, where R is the actual distance and a defines the effective range of the potential. Uq sets the energy scale of the pair-interaction. A number of crystal growth processes have been investigated by this type of potential, for example [28-31]. An alternative way of calculating solid-liquid interface structures on an atomic level is via classical density-functional methods [32,33]. 

The universality of certain dimensionless quantities is often taken for granted. For example, the exponent 2 in the last term of Eq. (2-6) has no dimensions and hence has the same magnitude regardless of the scale or units used for measurement. Likewise, the kinetic energy per unit mass of a body moving with a velocity v is given by... 

Looking at the Bernoulli equation, we see that the friction loss (ef) can be made dimensionless by dividing it by the kinetic energy per unit mass of fluid. The result is the dimensionless loss coefficient, K ... 

It is convenient to present the energy spectrum of subexcitation electrons in dimensionless units as a function hwm rj(E) of the variable Elhwater vapor each dot represents the average value for a small energy interval. One can note that the spectra... 

Both Qj and Ax are in dimensionless units and therefore the force constants kj are equal to the excited state vibrational energies in wavenumbers. The latter are obtained from the spectrum (Fig. 10 and Table 1). The At denote the excited state distortions and are the parameters to be determined. The values of both E00 (22613 cm-1) and F(19 cm-1) are obtained from the spectrum in Fig. 10. The initial wavepacket 0 is calculated by using the literature values for the ground state vibrational energies along the 3 modes. 

Let us express in terms of dimensionless units [Eq. (41)] the quantities referring to the law of motion of a dipole. The period of the function i) (cp) is equal to the time interval between consecutive reflections of a dipole. During this interval it turns on the angle 2a, the square of its angular velocity being equal to a normalized energy h. Hence,... 

Figure 5.4 Homonuclear diatomic energy curve in dimensionless units...

By fixing the characteristic radii of all elements the dissociation energies and interatomic distances of diatomic covalent interactions are converted into dimensionless units and predicted to generate a set of points within the narrow band that defines covalence in Figure 5.6(a). 

Figure 5.8 Potential-energy curve of first-order C-C interaction, in dimensionless units, calculated by the Heitler-London method.

The formula is converted into dimensionless units by defining a dimensionless distance based on a characteristic spacing for each compound. It is noted that the closest interionic distance can be specified as the sum of two ionic radii, d = r + r+. Using the well established anionic radii for halide and chalconide ions [75], a conversion factor R = /r 1 r is calculated in each case, and used to define the dimensionless distance d = d/R, such that the Madelung energy,... 

From the above relationship, we can sense that molecular distrihution has something to do with entropy generation during mixing. R, the Gas Constant, has the dimensions of entropy (energy per unit absolute temperature) and and Ng are dimensionless mole fractions defined by... 

Dimensionless Dimensionless u Internal energy per unit mass j- g Btu/lbm... 

Here Sg and are the tetragonal and orthorhombic components of the low symmetry field, sometimes called strain terms. The signs of Sg, are chosen to conform with the signs for the two coordinates Qg and respectively. All parameters in Eqs. (1) and (2) can have units of energy if dimensionless units are used for Qg and [2,8]. 

Here Jq is the incident light irradiance (measured in units of energy per unit time per unit area), c is the light velocity, is the dimensionless efficiency fac-... 

Some solvent effects on entropy and enthalpy contributions to the free energy of activation of Sif2 reactions are expressed in dimensionless units in Table 23. As already noted, much more could be said about these data if appropriate enthalpies of transfer for ions and molecules were known. However it is significant that in Sn2 reactions of anions at a saturated carbon atom, the large increase in rate, on transfer from water or methanol to DMF, is due to a decreased enthalpy of activation, which is only partially compensated for by a small increase in entropy of activation. For 8 2 reactions of anions at aromatic carbon, the change in both the enthalpy and entropy of activation, upon solvent transfer, favours reaction in DMF. As at saturated carbon, the solvent effect on AH is considerably larger than that on J-S, with the exception of one reaction of SON". 

As Eq. (4.42) cannot be solved explicitly for pp in terms of pb and T, we must have recourse to a numerical method, which is detailed below. In the remainder of this section, we express all quantities in the customary dimensionless (i.e., reduced ) units, where length is given in units of face-centered cubic lattice for substrate atoms. Then in dimensionless units we have Ob = 87t/3 and Pb = 1... 

Table 5.2 Comparison of interpolation [truncated version of Eq. (5.68)] and direct evaluation [Eq. (5.71)] of fluid substrate potential energy for various properties of a fluid confined by substrates consisting of single, chemically homogeneous planes. Entries, given in dimensionless units defined in Table 1, refer to simulations based on either direct evaluation (D) or interpolation (I).

The ground state and first excited state wave functions for this problem are shown in fig. 3.1. Each wave function has been plotted such that the zeroes of the wave function occur at the value on the vertical energy axis corresponding to that particular state. Thus, in the dimensionless units favored in the figure, iAi(x) has an energy Ei =, while i/2(x) is associated with the energy E2 = 4. 

Fig. 4.17. Band energy associated with rectangular band density of states. Energy is plotted in dimensionless units, scaled by the width of the band, W. Similarly, the band filling is plotted as a fraction of the full band.

Note that the dimensionless units defined in Table I are used, so that the curvature along the X direction is renormalized to 1. Here Uq is the two-body interaction potential defined in Eq. (2.13). The two terms linear in X are the dipolar interaction energy (with Uj and U2 two unit vectors, respectively, along the z-axis of the fixed frame for the solute and the solvent body, cf. Fig. 1). Finally a quadratic term in X has been added in order to confine the fluctuations of the stochastic field. 

Fig. 20. Comparision of experimental measured values for the Lamb shift in hydrogenlike ions and the theoretical predicition. The energy shift is presented in the dimensionless unit F Za) similar to Fig. 10.

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