Centrifuge potential

The rotational motion of a linear polyatomic molecule can be treated as an extension of the diatomic molecule case. One obtains the Yj m (0,(1)) as rotational wavefunctions and, within the approximation in which the centrifugal potential is approximated at the equilibrium geometry of the molecule (Re), the energy levels are ... 

Thus the potential matrix for even parity is identical to that for odd parity for K y 0. The centrifugal potential with the (J — ji2)2 term in Eq. (1), which is not diagonal in K in the BF representation, has matrix elements... 

Solutions for which J O require the vibrational wavefunction and energy to respond to the presence of the centrifugal potential given by -h2 J(J+l)/(2 oR2) these solutions obey the full coupled V/R equations given above. 

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... 

One can evaluate the effect of this centrifugal potential upon a-decay half-lives by simply adding this energy to the Coulomb barrier height. If we define... 

Figure 7.8 Modification of the potential energy in a decay due to the centrifugal potential. [From W. E. Meyerhof, Elements of Nuclear Physics, Copyright 1967 hy McGraw-Hill Book Company, Inc. Reprinted hy permission of McGraw-Hill Book Company, Inc.]...

The probability of fusion is a sensitive function of the product of the atomic numbers of the colliding ions. The abrupt decline of the fusion cross section as the Coulomb force between the ions increases is due to the emergence of the deep inelastic reaction mechanism. This decline and other features of the fusion cross section can be explained in terms of the potential between the colliding ions. This potential consists of three contributions, the Coulomb potential, the nuclear potential, and the centrifugal potential. The variation of this potential as a function of the angular momentum l and radial separation is shown as Figure 10.26. 

As shown in Figure 10.25, there is an /-dependent barrier to fusion that is the sum of the nuclear, Coulomb, and centrifugal potentials. This barrier is also a sensitive function of the relative deformation and orientation of the colliding ions. In Figure 10.27, we show the excitation function for fusion of 160 with various isotopes of Sm that span a wide range of deformations. 

Figure 10.26 Sum of the nuclear, Coulomb, and centrifugal potential for lsO + 120Sn as a function of radial distance for various values of the orbital angular momentum l.

This is called the radial wave equation. Apart from the term involving l, it is the same as the one-dimensional time-independent Schrodinger equation, a fact that will be useful in its solution. The last term is referred to as the centrifugal potential, that is, a potential whose first derivative with respect to r gives the centrifugal force. 

Fig. 2.4 Coulomb and centrifugal potentials (—) which are summed to give the effective radial potential (—) in which an electron of angular momentum i moves.

These results, Eqs. (6.35) and (6.36), are only valid for m = 0 states. In higher m states there is a l/(x2 + y2) centrifugal potential keeping the electron away from the z axis, and the centrifugal barrier raises the threshold field of m 0 states.16 Specifically, for m 0 states the fractional increase in the field required for ionization, compared to an m = 0 state of the same energy is16... 

Here, Vc is the sum of all the two-body interactions and A( 2C) is the angular momentum operator suitable for the five-dimensional angular space. The latter is referred to as the grand angular momentum operator. It introduces the centrifugal potential for the motion in p. It takes an explicit form... 

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