NORMAS TECNICAS PARA EL DISEÑO DE LOCALES ESCOLARES DE

In this section, a brief review of several theoretical descriptions of near-threshold ionization wiU be given, covering classical, semi-classical and quantum-mechanical approaches to the problem.

2.2.1 The W annier Threshold Law

In 1948 Wigner derived a general threshold law describing the dependence on projectile energy of the cross-section for any reaction in which two particles are present in the final state. Wigner's theory was based on the assumption that the form of the cross-section

depends only on the asymptotic interactions between the product particles and that detailed knowledge of the reaction mechanism in the early stages of the process, when the particles are within a small "inner" zone, is not required. Wannier (1953) extended Wigner's analysis to the double electron escape process, similarly arguing that the near-threshold behaviour of the cross-section was dependent only on the asymptotic configuration of the system in the final state. He was therefore able to employ a purely classical treatment of the problem, avoiding the difficulties associated with the quantum mechanical analysis of three bodies interacting via long-range Coulomb forces.

Wannier assumed the remnant ion to be stationary and restricted his analysis to a final state of zero total orbital angular momentum (Z,=0) and zero spin (S=0) in the expectation that this state represents the dominant contribution near threshold (extension of the theory to states of higher Z-values will be discussed later in this chapter). This simplification made it possible to describe the system entirely with two vectors and (the positions of the outgoing electrons relative to the ion) and the angle, between these two vectors, as illustrated in figure 2.1.

e •

+Zg

Figure 2.1 Schematic representation of the coordinate system Wannier used in his analysis of two electrons escaping from an ion of charge +Ze.

Wannier's classical approach to the problem can be justified by considering the coUinear system in which the two outgoing electrons move in opposite directions (0i2=ti) away

from the ion of charge +Ze. Such a system may evolve into a final state where both electrons are free (ionization) or where only one escapes and the other remains bound to the ion (excitation). For a final state of total energy, E \ Wannier showed that, in the vicinity of h e threshold, ionization occurs only for a system starting in, or close to, the symmetric condition defined by r\ = -r^ =r and ri = - r^. The Coulomb energy of such a system is given in atomic units by

(2.1)

The total kiaetic energy of the two electrons is

K = E ' - V (2.2)

and each electron has the de Broglie wavelength

A, = (2.3)

In order to illustrate the behaviour of the evolving system, it is convenient to define a quantity caLed the "critical" radius, as

rr. = 2

(z -l)

E' (2.4)

Manipulaticn of equations (2.1) to (2.4) then yields

V 1

k =---— ---r r (2.7)

1

The physical interpretation of (2.5) and (2.6) is that separates the "Coulomb" zone, in which K and F have similar magnitudes, from the "outer" zone in which K dominates over

V and the two electrons are effectively free. Equation (2.7) shows that for sufficiently small E', X « r for For this reason the electron can be treated as a classical particle, its wavelength being much smaller than the distance over which the forces acting on it vary appreciably. The above condition can be expressed more accurately as

A » ï ( z - \

The Coulomb zone is defined to extend from the minimum value of r for which (2.8) is satisfied, up to r^. The three divisions in "r-space" are represented schematically in figure 2.2.

Electrons in the Coulomb and outer zones therefore behave classically and, if spin-dependent effects can be ignored, the behaviour of the system in these zones is determined for small E ' by integrating the classical equations of motion. This, however, relies on knowledge of the initial conditions at the boundary between the Coulomb zone and the inner zone. Wannier circumvented this requirement by making the quasi-ergodic

INNER or REACTION ZONE Quantum Mechanics COULOMB ZONE

Potential energy dominates

Classical Mechanics OUTER or FREE ZONE Kinetic energy dominates Classical Mechanics r

Figure 2.2 The three regions of r-space through which the system passes as it evolves.

assumption, derived from the earlier work of Wigner, which states that no strong singularities exist in the distribution of the representative points of phase space leading to double escape and that the density of such points is approximately uniform.

For convenience Wannier described the system in terms of the hyperspherical coordinates defined by

R = [r\ +^2) \

012 = cos Mt, .r (2.9)

a = tan

These coordinates define the geometry of the triangle formed by the vectors and shown in figure 2.1. The restriction of this analysis to systems in the isotropic S-state obviates the need for coordinates describing the orientation of this triangle in space and therefore greatly simplifies the problem.

The Coulomb energy of the system in terms of these hyperspherical coordinates is

where

(2. 10)

The shape of C(a, plotted as a function of a and cosG^^ in figure 2.3 (after Fano and Lin 1975), helps to understand the expected behaviour of the system.

C is symmetrical about the point 8^2=% and a=7i/4, which is known as the Wannier point (for clarity C is not shown for between n and 2n) and represents the two electrons escaping in opposite directions and at equal distances fiom the ion. The Wannier point has saddle structure, being stable along 8^^ about 8i2=7t and unstable in the direction of a about

a=n/4. The region centred on 8^2^0 and a=n/4 represents the physically inaccessible

situation in which the two electrons escape the nucleus on identical trajectories, and the valleys at a=G (^2=8) or n il (r^=G) correspond to one of the electrons remaining bound in the potential well of the ion (i.e. excitation). It can be seen fiom this that a system originating at the Wannier point will remain there as R increases, ultimately leading to an ionization event. This can also occur for a system deviating fiom the Wannier point providing the system stays in the vicinity of the saddle point long enough for both electrons to reach the outer zone, where they are essentially fiee. If the deviation fiom

the Wanner point is too large, however, the system will become trapped in one o f the two

valleys ard ionization will not occur.

0^ -1

Figure 2.3 The Coulomb energy function C as a function of a and cos0,2, showing cleirly the saddle structure of the Wannier point (0,2=7i, a=7r/4).

By sobing the classical Hamiltonian for this system, Wannier deduced the equations o f

motion fo' trajectories leading to double escape and found that