Particle in a sphere

Here nf and nf are the effective masses of the electron and the positive hole created when an electron is excited from the valence band to the conduction band. In bulk CdSe, m Jm and nfjm have been determined to be about 0.12 aud 0.5, respectively. The reduced mass p, = nfgwl l nfg + ml) = 0.091 m is thus much smaller than the mass of an electron. It is seen that the form of equations (3) and (4) is similar to that for a free particle in a sphere but with a uegative sign for the energies of the hole. 

In a time t, An at particles escape from the exterior into a spherical hole of radius . The hole will he filled by these particles if this number is equal to the number of particles in a sphere of radius , thatis, (4/3)7t c. Then the time for... 

Hence, the average number of particles in a sphere of radius RM (excluding the particle at the center) is... 

For R 0, the polarizability will tend to that of a particle in a sphere. For deducing the dipole polarizability in this case, we use the expression in Equation (3.9). The equation for the first order perturbation in the wave function for the ground state can be written as... 

Larger clusters are more open (more tenuous or rarefied). This is shown as follows. Equation (13.12) gives the number of particles in a cluster. We can also calculate the number of sites Ns in the cluster, i.e., the number of particles in a sphere of radius R if closely packed ... 

To treat the quantum size effect, an electron or hole inside a spherical particle is modeled as a particle in a sphere. " " The electron is considered to be confined to a spherical potential well of radius a with potential energy V(r) = 0 for r < u and V(r) = oo for r > u. Solving the Schrodinger equation yields the wave functions" ... 

For Eg O (nonmetal), the optical properties can generally be successfully modeled using the particle in a sphere model and its extensions." " Consideration of an electron and hole with negligible spatial correlation in a potential well, with neglect of polarization effects, leads to the following expression for the shift of the optical bandgap (l hlAe excited state) with the radius of the sphere ... 

The theory of cyciization in condensation polymerization was first investigated by Kuhn in the 1930s with the introduction of the concept of effective concentration (Ceff), which is the local concentration of two chain ends of the same molecule for a Gaussian chain. This measurement provides a relationship between the end-to-end length of a polymer chain and that same chain s propensity to cyclize. Therefore, C s provides a method of quantification for the propensity of intramolecular interactions and cyciization, and Kuhn predicted that the cyciization probability decreases as where N is the number of bonds in the chain. Several other treatments have addressed the calculation of C s as a function of chain length using either random-flight statistics or a particle-in-a-sphere approximation. ... 

Considering now the scattering law, we have made use of the diffusion model of a particle in a sphere of radius Ri. 

A richer zoo of magic numbers can be obtained by filling the sphere. Then the qualitative types (a)-(d) in Figure 15.1 can be imagined. Case (a) is that of the thin attractive shell or particle on a sphere. Case (b) is a wine bottle potential, with a repulsive excess at the centre. Case (c) is the classic problem of a particle in a sphere. Case (d) is a parabolic one, possibly flattened at the bottom. 

An example on a radial potential is the jellium one, see for example the review by de Heer [6]. The radial potential of the jellium model rather is of the wine-bottle type. The level order of the particle in a sphere is qualitatively similar to the jellium one. See Figure 2 of de Heer [6]. It gives the level order... 

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