On-shell T matrix

In the derivation (10.34) we have used (10.27), the transformation (10.19—10.21) from electron coordinates and momenta to relative and centre-of-mass coordinates and momenta, the definition corresponding to the time reversal of (4.112) for the half-on-shell T matrix, and the representation reciprocal to (3.30) of <5(K — K). 

The direct amplitudes involving are analogous to the distorted-wave Born approximation and are calculated by (10.31). The T-matrix element in the second amplitude of (10.51), which has the observed resonances, is calculated by solving the problem of electron scattering on He" ". The solution consists of half-on-shell T-matrix elements at the quadrature points for the scattering integral equations (6.87). The same points are used for the k integration of (10.51). 

Expression (70) has no direct computational interest. Its usefulness is to provide a general framework to investigate cross sections. The physical quantity of interest is not the transition operator (60), but its matrix elements between the continua, the scattering states, whose projector is Q, that are peaked within a narrow range of energy. These matrix elements define the on-shell T matrix... 

In microscopic nuclear structure calculations, the off-shell behavior of the NN potential is important (see Section 4 for a detailed discussion). The fit of NN potentials to two-nucleon data fixes them on-shell. The off-shell behavior cannot, by principle, be extracted from two-body data. Theory could determine the off-shell nature of the potential. However, not any theory can do that. Dispersion theory relates observables (equivalent to on-shell T-matrices) to observables e.g., nN to NN. Thus, dispersion theory cannot, by principle, provide any off-shell information. The Paris potential is based upon dispersion theory thus, the off-shell behavior of this potential is not determined by the underlying theory. On the other hand, every potential does have an off-shell behavior. When undetermined by theory, then the off-shell behavior is a silent by-product of the parametrization chosen to fit the on-shell T-matrix, with which the potential is identified, by definition. In summary, due to its basis in dispersion theory, the off-shell behavior of the Paris potential is not derived on theoretical grounds. This is a serious drawback when it comes to the question of how to interpret nuclear structure results obtained by applying the Paris potential. 

However, the off-shell potential does not really play any role in free-space NN scattering. The reason for this is simply the procedure by which NN potentials are constructed. The parameters of NN potentials are adjusted such that the resulting on-shell T-matrix fits the empirical NN data. For our later discussion, it is important to understand this point. Let us consider a case in which the off-shell contributions are particularly large, namely the on-shell T-matrix in the Si state ... 

The on-shell T-matrix is related to the observables that are measured in experiment. Thus, potentials which fit the same NN scattering data produce the same on-shell T-matrices. However, this does not imply that the potentials are the same. As seen in Eqs. (1) and (2), the T-matrix is the sum of two terms, the Born term and an integral term. When this sum is the same, the individual terms may still be quite different. 

Let us consider an example. The T-matrix in the Si state is attractive below 300 MeV lab. energy. If a potential has a strong (weak) tensor force, then the integral term in Eq. (5) is large (small), and the negative Born term will be small (large) to yield the correct on-shell T-matrix element. 

The subscript os refers to the on-shell value. The on-shell t-matrix element in terms of the proton-nucleus COM system kinematics for P = 0 is given by [Me 83a, Wa 75]... 

For applications at intermediate energies it seems more appropriate to use Lorentz transformations to obtain the relative momenta in the NN COM frame in terms of the momenta (k, p k p ), rather than rely on the nonrelativistic results in eq. (3.13). The effective NN COM energy, e, in eq. (3.14) also should be computed relativistically. Wigner rotations should also be included. However, at present it is not clear how to do such a calculation, since the correct Lorentz invariant form of the fully off-shell t-matrix is unspecified. 

In order to work with the on-shell (physical) scattering T matrix, we must consider the kinetic equation in the space of the asymptotic states which is the direct sum of the channel subspaces In this space we have the following completeness relation... 

The radial integral equations (4.121) are solved for each partial wave L and the half-off-shell solutions substituted in the equivalent of (4.118) for the T matrix. The on-shell solutions are in fact the Tl of (4.115), from which the scattering amplitude and cross sections can be calculated. 

According to the residue theorem applied to the k" integral the scattering is determined by the poles of the partial T-matrix element in the complex k" plane. The existence and positions of the poles are of course determined by the details of the potential V, but we will assume that there is a pole corresponding to complex energy Cr — iTr. The magnitude of the partial T-matrix element varies rapidly with values of E near the pole and we can consider er as the resonance energy. For the cross section we need only consider the on-shell partial T-matrix element... 

Ford (1964) has obtained the half-on-shell Coulomb T-matrix element as the limit of the T-matrix element for the screened potential e /r as A —> 0. It is... 

The first amplitude in the integrand of the second term of (10.50) is a half-on-shell element of the time-reversed distorted-wave T matrix T+ for the electron—ion collision (6.87). The approximation calculated for... 

Qo is the projector onto the subspace of the scattering states. Prom (37) the on-shell matrix elements of T in the basis of the scattering states can be written as... 

For the on-shell tp calculations, eq. (3.25), one needs to deal only with the transformation of an on-shell proton-nucleon t-matrix. In this case it has been customary in the literature (e.g., [Ah 81,... 

The Wigner rotation mixes the spin dependent amplitudes in the NN COM system to form the resulting spin dependent terms in the pA COM frame [Me 83a]. The on-shell proton-nucleon t-matrix... 

The on-shell tp form of the optical potential may now be specified using these properly transformed t-matrix amplitudes. Using eq. (3.45), the optical potential in eq. (3.25) becomes... 

In eqs. (3.101)-(3.103) (vf) is the local, static Coulomb term and should include the relativistic correction factor, rj (see section 3.4). In eq. (3.101) ( ,cn,m) interpreted as the empirical, on-shell Coulomb distorted nuclear t-matrix obtained from phase shift analyses. The (, cn.m) contain... 

At any rate, the relativistic NN t-matrix, Vj, introduced in eq. (4.8) for the optical potential, can also be expanded in terms of the five Lorentz invariants, where for on-shell matrix elements... 

We have provided a pedagogical derivation of the traditional, nonrelativistic form of multiple scattering theory based on the optical potential formalism. We have also discussed in detail each of the important advances made over the past ten years in the numerical application of the NR formalism. These include the full-folding calculation of the first-order optical potential, off-shell NN t-matrix contributions, relativistic kinematics and Lorentz boost of the NN t-matrix, electromagnetic effects, medium corrections arising from Pauli blocking and binding potentials in intermediate states, nucleon... 

The familiar set of the three t2g orbitals in an octahedral complex constitutes a three-dimensional shell. Classical ligand field theory has drawn attention to the fact that the matrix representation of the angular momentum operator t in a p-orbital basis is equal to the matrix of — if in the basis of the three d-orbitals with t2g symmetry [2,3]. This correspondence implies that, under a d-only assumption, l2 g electrons can be treated as pseudo-p electrons, yielding an interesting isomorphism between (t2g)" states and atomic (p)" multiplets. We will discuss this relationship later on in more detail. 

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