Empty core

Similar to the case without consideration of the GP effect, the nuclear probability densities of Ai and A2 symmetries have threefold symmetry, while each component of E symmetry has twofold symmetry with respect to the line defined by (3 = 0. However, the nuclear probability density for the lowest E state has a higher symmetry, being cylindrical with an empty core. This is easyly understand since there is no potential barrier for pseudorotation in the upper sheet. Thus, the nuclear wave function can move freely all the way around the conical intersection. Note that the nuclear probability density vanishes at the conical intersection in the single-surface calculations as first noted by Mead [76] and generally proved by Varandas and Xu [77]. The nuclear probability density of the lowest state of Aj (A2) locates at regions where the lower sheet of the potential energy surface has A2 (Ai) symmetry in 5s. Note also that the Ai levels are raised up, and the A2 levels lowered down, while the order of the E levels has been altered by consideration of the GP effect. Such behavior is similar to that encountered for the trough states [11]. 

As is seen from the behaviour of the more sophisticated Heine-Abarenkov pseudopotential in Fig. 5.12, the first node q0 in aluminium lies just to the left of (2 / ) / and g = (2n/a)2, the magnitude of the reciprocal lattice vectors that determine the band gaps at L and X respectively. This explains both the positive value and the smallness of the Fourier component of the potential, which we deduced from the observed band gap in eqn (5.45). Taking the equilibrium lattice constant of aluminium to be a = 7.7 au and reading off from Fig. 5.12 that q0 at 0.8(4 / ), we find from eqn (5.57) that the Ashcroft empty core radius for aluminium is Re = 1.2 au. Thus, the ion core occupies only 6% of the bulk atomic volume. Nevertheless, we will find that its strong repulsive influence has a marked effect not only on the equilibrium bond length but also on the crystal structure adopted. 

We can understand the behaviour of the binding energy curves of monovalent sodium and other polyvalent metals by considering the metallic bond as arising from the immersion of an ionic lattice of empty core pseudopotentials into a free-electron gas as illustrated schematically in Fig. 5.15. We have seen that the pseudopotentials will only perturb the free-electron gas weakly so that, as a first approximation, we may assume that the free-electron gas remains uniformly distributed throughout the metal. Thus, the total binding energy per atom may be written as... 

Fig. 5.15 An ionic lattice of Ashcroft empty core pseudopotentials immersed in a free-electron gas.

Table 6.3 Contributions to the binding energy (in Ry per atom) of sodium, magnesium, and aluminium within the second order real-space representation, eqn (6.73), using Ashcroft empty-core pseudopotentials. L/gf is defined by eqn (6.75). The numbers in brackets correspond to the simple expression, eqn (6.77), for = 0) and to the experimental values of the binding energy and negative cohesive energy respectively.

The wave vector, k , and the screening length, 1/ , depend only on the density of the free-electron gas through the poles of the approximated inverse dielectric response function, whereas the amplitude, A , and the phase shift, a , depend also on the nature of the ion-core pseudopotential through eqs (6.96) and (6.97). For the particular case of the Ashcroft empty-core pseudopotential, where tfj fa) = cos qRc, the modulus and phase are given explicitly by... 

The resultant pair potentials for sodium, magnesium, and aluminium are illustrated in Fig. 6.9 using Ashcroft empty-core pseudopotentials. We see that all three metals are characterized by a repulsive hard-core contribution, Q>i(R) (short-dashed curve), an attractive nearest-neighbour contribution, 2( ) (long-dashed curve), and an oscillatory long-range contribution, 3(R) (dotted curve). The appropriate values of the inter-atomic potential parameters A , oc , k , and k are listed in Table 6.4. We observe that the total pair potentials reflect the characteristic behaviour of the more accurate ab initio pair potentials in Fig. 6.7 that were evaluated using non-local pseudopotentials. We should note, however, that the values taken for the Ashcroft empty-core radii for Na, Mg, and Al, namely Rc = 1.66, 1.39, and... 

Fig. 6.17 The structural-energy differences of a model Cu-AI alloy as a function of the band filling N, using an average Ashcroft empty-core pseudopotential with / c = 1.18 au. The dashed curves correspond to the three-term analytic pair-potential approximation. The full curves correspond to the exact result that is obtained by correcting the difference between the Lindhard function and the rational polynomial approximation in Fig. 6.3 by a rapidly convergent summation over reciprocal space. (After Ward (1985).)...

Explain the concept of a pseudopotential. Aluminium is fee with a lattice constant of a = 7.7 au. It is well described by an Ashcroft empty core pseudopotential of core radius 1.1 au. Show that the lattice must be expanded by 14% for the 2n/a(200) Fourier component of the pseudopotential to vanish. 

The intensive fluctuation region is concentrated between the outlets of the drawing tubes and around the flow axis, and takes the form of a couple of truncated cones with an empty core and with coinciding bottoms. The region is symmetrical with respect to both the impingement plane and the axis of the streams and is essentially independent of nn. The space surrounded by the external surface of the cones is defined as the impingement zone. 

When we strip off as layers of convention all differences among ways of describing it what is left The onion is peeled down to its empty core, (ibid., 118)... 

Superimposed ionie potentials in aluminum, and the empty core approximation, which includes the cITccts of the repulsive kinetic energy term. 

The combined effect of the potential energy and the extra kinetic energy has been to add a potential + for r < i to each of the point-ion potentials we used for the evaluation of the electrostatic energy. Thus the empty-core contribution to the total energy per ion is simply... 

The free-clectron approximation described in Chapter 15 is so successful that it is natural to expect that any effects of the pseudopotential can be treated as small perturbations, and this turns out to be true for the simple metals. This is only possible, however, if it is the pscudopotential, not the true potential, which is treated as the perturbation. If we were to start with a frcc-electron gas and slowly introduce the true potential, states of negative energy would occur, becoming finally the tightly bound core states these arc drastic modifications of the electron gas. If, however, we start with the valence-electron gas and introduce the pscudopotential, the core states arc already there, and full, and the effects of the pseudopotential arc small, as would be suggested by the small magnitude of the empty-core pseudopotential shown in Fig. 15-3. 

Let us systematically discuss the effect of the pseudopotential in terms of the empty-core form, Eq. (15-13). We first write the total pseudopotential in the metal as a superposition of the individual pseiidopotcntials w (r — r,) centered at the ion positions r,... 

A plot of the form factor for the empty-core pseiidopoteiitial, from Eq. (16-7), for aluminum. For comparison, the points give the model potential (Animalu and Heine, 1965). The choice of is such that the two curves cross the axis at the same point. 

Core radii for the empty-core model, in A. These are listed also in the Solid State Table. 

It should be noticed that curves like those shown in Fig. 16-1 arcdclined only for q 0. The q = 0 value is best obtained by combining all of the terms in the energy, as we did in the discussion in Chapter 15. Notice, however, that in the discussion of cohesion in Chapter 15 we used the same that we use here for the discussion of w, for q 0. This remarkable feature appears to be special to the empty-core pseudopotential. 

The lowest-lying s state for the empty-core pseudopotential is illustrated in F ig. 16-13,a. Its energy f can be computed by numerical integration and obtained as a function of Z and c. We shall make an approximate solution. 

Solutions of the empty-core problem can be obtained for this particular set of energies by fitting the above solutions, beyond the last node, to the solution Ap sinh p/2) which holds for r < i. The Ll are polynomials containing terms up to order ii - I, but the last two (p" and ) " ) dominate the fit at i. If we in fact keep only these two terms, we can simply let I) become nonintcgral (corresponding to intermediate energies) and obtain a... 

There have been a very large number of applications of this theory to the vibration spectra (see Heine and Weairc, 1970, for a review) since the earliest studies (Harrison, 1964, and Sham, 1965). Perhaps the most relevant here is the calculation by Ashcroft (1968) for the alkali metals that calculation makes use of the empty-core model of the pseudopolential, described here. The results of his calculation, along with experimental points, are shown in Fig. 17-4 for potassium similar results were obtained for sodium. 

The vibration spectrum for potassium calculated by Aslicrofl (1968), wlio used the empty-core pseudopolenlial from Eq. (15-13) and = 1.13 A (rather than the value of 1.20 A from Table 16-1). Experimental points are from Cowley, Woods, and Dolling (1966). 

The speed of sound (in units of 10 cm/scc) calculated. by using the Bohm-Staver formula, by including the effect of the empty core (liq. 17-2.3), and from experiment. 

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