Nucleus wave function

In principle, the products of a nuclear reaction can be any species permitted by conservation laws. In practice, direct reaction final channels will be strong if they possess substantial overlap with those of the initial state. Similarly, if the reaction proceeds aU the way to a CN, strong channels will be those that capture large portions of the available phase space. Compound nucleus wave functions are intractable objects. This, coupled with the myriad of equally complex final states, allows a statistical analysis to be employed. Even so, the full 6N dimensional phase space is far too large to cope with and so insight must be used to calculate the phase space area of relevant parts (e.g., the part of phase space well described by two large clumps of matter, rather than one, will be proportional to the fission yield.)... 

For projectile-nucleus scattering we will only be interested in matrix elements of T between initial and final projectile-nucleus wave functions, representing physical states of the system. The symmetry properties of (S,- u,) [Fe 71] result in intermediate states in the second term of eq. (2.5) which span only the physical, antisymmetric states of the nucleus and the physical states of the projectile. Likewise, from eq. (2.4), only the projections of If and (f) onto physical states of the system need be retained. Thus we consider and (f) from here on to be expanded in terms of all antisymmetric states of the target and all physical states of the projectile. Note that these states do not form complete sets [Ke 59]. Antisymmetrization between the projectile and target nucleon labels (in the case of nucleon projectiles) is, for the moment, neglected. TTie total wave function is therefore expanded according to... 

The wave function T i oo ( = 11 / = 0, w = 0) corresponds to a spherical electronic distribution around the nucleus and is an example of an s orbital. Solutions of other wave functions may be described in terms of p and d orbitals, atomic radii Half the closest distance of approach of atoms in the structure of the elements. This is easily defined for regular structures, e.g. close-packed metals, but is less easy to define in elements with irregular structures, e.g. As. The values may differ between allo-tropes (e.g. C-C 1 -54 A in diamond and 1 -42 A in planes of graphite). Atomic radii are very different from ionic and covalent radii. 

Also, the notations of the wave functions are to be changed. We shall denote the Gaussian function centered at nucleus A as 111,5), and the function centered at nucleus B as tig). 

As was shown in the preceding discussion (see also Sections Vin and IX), the rovibronic wave functions for a homonuclear diatomic molecule under the permutation of identical nuclei are symmetric for even J rotational quantum numbers in and E electronic states antisymmeUic for odd J values in and E elecbonic states symmetric for odd J values in E and E electronic states and antisymmeteic for even J values in Ej and E+ electeonic states. Note that the vibrational ground state is symmetric under pemrutation of the two nuclei. The most restrictive result arises therefore when the nuclear spin quantum number of the individual nuclei is 0. In this case, the nuclear spin function is always symmetric with respect to interchange of the identical nuclei, and hence only totally symmeUic rovibronic states are allowed since the total wave function must be symmetric for bosonic systems. For example, the nucleus has zero nuclear spin, and hence the rotational levels with odd values of J do not exist for the ground electronic state f EJ") of Cr. 

Spherically symmetric (radial) wave functions depend only on the radial distance r between the nucleus and the election. They are the Is, 2s, 3s. .. orbitals... 

Diffuse functions are those functions with small Gaussian exponents, thus describing the wave function far from the nucleus. It is common to add additional diffuse functions to a basis. The most frequent reason for doing this is to describe orbitals with a large spatial extent, such as the HOMO of an anion or Rydberg orbitals. Adding diffuse functions can also result in a greater tendency to develop basis set superposition error (BSSE), as described later in this chapter. 

Likewise, a basis set can be improved by uncontracting some of the outer basis function primitives (individual GTO orbitals). This will always lower the total energy slightly. It will improve the accuracy of chemical predictions if the primitives being uncontracted are those describing the wave function in the middle of a chemical bond. The distance from the nucleus at which a basis function has the most significant effect on the wave function is the distance at which there is a peak in the radial distribution function for that GTO primitive. The formula for a normalized radial GTO primitive in atomic units is... 

FIGURE 1 3 Boundary surfaces of the 2p orbitals The wave function changes sign at the nucleus The two halves of each orbital are indicated by different colors The yz plane is a nodal surface for the Ip orbital The probability of finding a electron in the yz plane is zero Anal ogously the xz plane is a nodal surface for the 2py orbital and the xy plane is a nodal surface for the 2pz orbital You may examine different presentations of a 2p orbital on Learning By Modeling... 

Section 1 1 A review of some fundamental knowledge about atoms and electrons leads to a discussion of wave functions, orbitals, and the electron con figurations of atoms Neutral atoms have as many electrons as the num ber of protons m the nucleus These electrons occupy orbitals m order of increasing energy with no more than two electrons m any one orbital The most frequently encountered atomic orbitals m this text are s orbitals (spherically symmetrical) and p orbitals ( dumbbell shaped)... 

Optically pure (Section 7 4) Descnbing a chiral substance in which only a single enantiomer is present Orbital (Section 1 1) Strictly speaking a wave function i i It is convenient however to think of an orbital in terms of the probability i i of finding an electron at some point relative to the nucleus as the volume inside the boundary surface of an atom or the region in space where the probability of finding an electron is high... 

In Figure 1.8 the real wave functions for the f, 2p and 3d orbitals are plotted in the form of polar diagrams, the construction of which may be illustrated by the simple case of the 2p orbital. The wave function in Equation (f.43) is independent of 4> and is simply proportional to cos 6. The polar diagram consists of points on a surface obtained by marking off, on lines drawn outwards from the nucleus in all directions, distances proportional to I cos 6 at a constant value of R2i(r). The resulting surface consists of two touching spheres. 

If / = 1 for each nucleus, as in H2 and N2, the total wave function must be symmetric to nuclear exchange. There are nine nuclear spin wave functions of which six are symmetric and three antisymmetric to exchange. Figure 5. f 8 illustrates the fact that ortho- ll2 (or N2)... 

The basis of constructing the MOs is the linear combination of atomic orbitals (LCAO) method. This takes account of the fact that, in the region close to a nucleus, the MO wave function resembles an AO wave function for the atom of which the nucleus is a part. It is reasonable, then, to express an MO wave function 1/ as a linear combination of AO wave functions Xi on both nuclei ... 

Another consequence of the quantum theory of the atomic and nuclear systems is that no two protons, or two neutrons, can have exactly the same wave function. The practical appHcation of this rule is that only a specific number of particles can occupy any particular atomic or nuclear level. This prevents all of the electrons of the atom, or protons and neutrons in the nucleus, from deexciting to the single lowest state. 

In almost all cases X is unaffected by any changes in the physical and chemical conditions of the radionucHde. However, there are special conditions that can influence X. An example is the decay of Be that occurs by the capture of an atomic electron by the nucleus. Chemical compounds are formed by interactions between the outer electrons of the atoms in the compound, and different compounds have different electron wave functions for these outer electrons. Because Be has only four electrons, the wave functions of the electrons involved in the electron-capture process are influenced by the chemical bonding. The change in the Be decay constant for different compounds has been measured, and the maximum observed change is about 0.2%. 

Eor specific models of the nucleus, it is possible to compute theoretical wave functions for the states. Eor a model that assumes that the nucleus is spherical, the general properties of these wave functions have been used to compute theoretical estimates of the half-hves for y-rays of the various multipolarities. Some values from the Weisskopf estimate of these half-hves are shown in Table 7. These half-fives decrease rapidly with the y-ray energy, namely, as and, as Table 7 shows, increase rapidly with E. This theoretical half-life applies only to the y-ray decay, so if there are other modes of... 

Decay Schemes. Eor nuclear cases it is more useful to show energy levels that represent the state of the whole nucleus, rather than energy levels for individual atomic electrons (see Eig. 2). This different approach is necessary because in the atomic case the forces are known precisely, so that the computed wave functions are quite accurate for each particle. Eor the nucleus, the forces are much more complex and it is not reasonable to expect to be able to calculate the wave functions accurately for each particle. Thus, the nuclear decay schemes show the experimental levels rather than calculated ones. This is illustrated in Eigure 4, which gives the decay scheme for Co. Here the lowest level represents the ground state of the whole nucleus and each level above that represents a different excited state of the nucleus. 

From electronic structure theory it is known that the repulsion is due to overlap of the electronic wave functions, and furthermore that the electron density falls off approximately exponentially with the distance from the nucleus (the exact wave function for the hydrogen atom is an exponential function). There is therefore some justification for choosing the repulsive part as an exponential function. The general form of the Exponential - R Ey w function, also known as a ""Buckingham " or ""Hill" type potential is... 

The optimum value of c is determined by the variational principle. If c = 1, the UHF wave function is identical to RHF. This will normally be the case near the equilibrium distance. As the bond is stretched, the UHF wave function allows each of the electrons to localize on a nucleus c goes towards 0. The point where the RHF and UHF descriptions start to differ is often referred to as the RHF/UHF instability point. This is an example of symmetry breaking, as discussed in Section 3.8.3. The UHF wave function correctly dissociates into two hydrogen atoms, however, the symmetry breaking of the MOs has two other, closely connected, consequences introduction of electron correlation and spin contamination. To illustrate these concepts, we need to look at the 4 o UHF determinant, and the six RHF determinants in eqs. (4.15) and (4.16) in more detail. We will again ignore all normalization constants. 

A single determinant MO wave function for the H2 molecule within a minimum basis consisting of a single s-function on each nucleus is given as (see also... 

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