Central potentials

Figure A3.11.3. Coordinates for scattering of a particle from a central potential.

The basic assumption here is the existence over the inelastic scattering region of a connnon classical trajectory R(t) for the relative motion under an appropriately averaged central potential y[R(t)]. The interaction V r, / (t)] between A and B may then be considered as time-dependent. The system wavefiinction therefore satisfies... 

Fig. 14. Energy levels calculated for an infinitely deep spherical potential well of radius with an infinitely high central potential barrier with a radius the zigzag line...

As we show in Fig. 2 this relation holds as well for the two Al-based alloys studied here. This finding has consequences on the nature of the inter-atomic interactions. From a fee point of view, the hep structure has a stacking fault every second layer. The fact that relation (1) holds means that these stacking faults weakly interact, and therefore the range of the inter-atomic interactions should not go beyond the second neighbor shell whereas conventional central potentials require at least three atomic shells to differentiate the fee and hep stacking sequences. 

The summation for the coefficient of the P2 term likewise includes phase insensitive contributions where I m = Im, but also now one has terms for which I = I 2, which introduce a partial dependence on relative phase shifts— specifically on the cosine of the relative phase shift. Again, this conclusion has long been recognized for example, by an explicit factor cos(ri j — ri i) in one term of the Cooper-Zare formula for the photoelectron (3 parameter in a central potential model [43]. 

The way in which tunnelling affects the energy levels of a system is illustrated well by the behaviour of a particle in a one-dimensional box with a central potential barrier. 

The charges map directly 2,, i, with the change on the particle at the center of mass mapping to a central potential. 

Transition metal ions have an incompletely filled d-shell, i.e. their electron configuration is d". The optically active electrons are thus bound to central potential as well as experiencing crystal field potential, and are not shielded by outer electrons. Most transition metal ions are multi-valent. Mainly the number of 3d electrons and the crystal field determine their optical properties. Thus the groups below have similar optical behavior ... 

The Schrodinger equation for a single particle of mass (I moving in a central potential (one that depends only on the radial coordinate r) can be written as... 

Qualitatively, internal orbitals are contracted towards the nucleus, and the Radon core shrinks and displays a higher electron density. External electrons are much more efficiently screened from the nucleus, and the repulsion described by the (Hartree-Fock) central potential U(rj) in Eq. (7) of the preceding subsection is greatly enhanced. In fact, wave functions and eigenstates of external electrons calculated with the relativistic and the non-relativistic wave equations differ greatly. 

Fig. 12a-c. Schematic representation of the effective potential Vejf and of different possibilities of localized and itinerant states for electrons of high 1 quantum number, a) The solid line d represents the periodic potential set-up by the cores R and R +i, which is a superimposition of central potential a dashed line). The dashed line b represents the centrifugal potential of kinetic origin 1(1 + l)/2 R in an atom, and c dashed line) the effective potential V f for an atom (compare Fig. 6) and full line) for a solid, b) Relative to two shapes of the effective potential Ve, two examples of localized state are given 1. resonant state 2. fully localized state. Notice that 1. is very near to Ep. h and t represent hopping and tunneling processes, c) A narrow band is formed (resonance band), pinning Ep 3. narrow band... 

If the localized electron tunnels out through the barrier (state 1 in Fig. 12 b) a certain amount of f-f overlapping is present. States like 1 in Fig. 12 b are called sometimes resonant states or "virtually bound" states. In contrast with case 2 in Fig. 12b, which we may call of full localization , the wave function of a resonant state does not die out rapidly, but keeps a finite amplitude in the crystal, even far away from the core. For this reason, overlapping may take place with adjacent atoms and a band may be built as in ii. (If the band formed is a very narrow band, sometimes the names of localized state or of resonance band are employed, too. Attention is drawn, however, that in this case one refers to a many-electron, many-atoms wave function of itinerant character in the sense of band theory whereas in the case of resonant states one refers to a one-electron state, bound to the central potential of the core (see Chap. F)). 

These two cases are both cases of localization, in the sense that both are relative to an atomic central potential. However, one practically fills independently of the chemical potential pp, the other may be emptied easily by perturbing fields, or by thermal motion. 

Most modern band-calculations for non-magnetic solids are performed in the LDA approximation, which is extended also to the relativistic formulation (see Chap. F). Care is taken in the choice of the set of o )i s. A particular problem exists in connecting the atomic wave functions ipi s, calculated in a central potential, in the inter-core region (see Fig. 12) of the solid. It is beyond the scope of this Chapter to go deeper into these details, which will be discussed further in Chap. F. 

Ef ) so defined is different from the atomic eigenvalues of the Schrodinger (or Dirac) equation in a central potential, which we discussed in Part II, since the latter are defined for integral occupation of the orbitals.)... 

It is known (Chap. A) that Koopmans theorem is not vahd for the wavefunctions and eigenvalues of strongly bound states in an atom or in the cores of a solid, i.e. for those states which are a solution of the Schrodinger (or Dirac) equation in a central potential. In them the ejection (or the emission) of one-electron in the electron system means a strong change in Coulomb and exchange interactions, with the consequent modification of the energy scheme as well as of the electronic wavefunction, in contradiction to Koopmans theorem. 

It can be seen that the doublet separation increases and the height of the potential hill decreases rapidly with decreasing of the bridge length or v0H, With very short hydrogen bonds, the central potential hill between the two minima is almost perfectly flattened out. 

NUCLEAR POTENTIAL. The potential energy V of a nuclear particle as a function of its position in the field of a nucleus or of another nuclear particle. A central potential is one that is spherically symmetric, so that V is a function only of the distance r of the particle from the center of force. A noncentral potential, on the other hand, is one that is not spherically symmetrical, or one that depends upon the relative directions of the angular momenta associated with the particle and the center of force, as well as upon the distance r. A negative potential corresponds to an attractive force, while a positive potential corresponds to a repulsive force. 

A more detailed model can be constructed for the nucleons in terms of a central potential that holds all the nucleons together plus a residual potential or residual interaction that lumps together all of the other nucleon-nucleon interactions. Other such important one-on-one interactions align the spins of unlike nucleons (p-n) and cause the pairing of like nucleons (p-p, n-n). The nucleons are then allowed to move independently in these potentials, that is, the Schrodinger equation is solved for the... 

The central potential can be a simple harmonic oscillator potential/(r) kr2 or more complicated such as a Yukawa function f(r) (e a,/r) 1 or the Woods-Saxon function that has a flat bottom and goes smoothly to zero at the nuclear surface. The Woods-Saxon potential has the form... 

As we have seen, the nucleons reside in well-defined orbitals in the nucleus that can be understood in a relatively simple quantum mechanical model, the shell model. In this model, the properties of the nucleus are dominated by the wave functions of the one or two unpaired nucleons. Notice that the bulk of the nucleons, which may even number in the hundreds, only contribute to the overall central potential. These core nucleons cannot be ignored in reality and they give rise to large-scale, macroscopic behavior of the nucleus that is very different from the behavior of single particles. There are two important collective motions of the nucleus that we have already mentioned that we should address collective or overall rotation of deformed nuclei and vibrations of the nuclear shape about a spherical ground-state shape. 

It is a reasonably good approximation to consider an alkali atom as a single electron moving in the modified Coulomb field of the ionic core, and this approximation has been made in almost all theoretical investigations of positron scattering by the alkali atoms. The interaction of the electron with the core is expressed as a local central potential of the general form... 

The wavefunctions of discrete orbitals in a central potential factorize according to... 

In spherical polar coordinates, for one mass point moving in a central potential V = V(r),... 

The 5-matrix is unitary and symmetric, while the T-matrix is symmetric. This particular definition of the T-matrix reduces for scattering by a central potential to the phase-shift factor in the scattering amplitude,... 

Aashamar, K., Luke, T.M. and Talman, J.D. (1978). Optimized central potentials for atomic ground-state wavefunctions, At. Data Nucl. Data Tables 22,... 

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