Nuclear masses

Finally, we shall look briefly at the form of the non-adiabatic operators. Taking the kinetic energy operator in Cartesian form, and using mass-scaled coordinates where Ma is the nuclear mass associated with the ath... 

Physically, the strength of the spring representing the bond is affected by a subtle balance of nuclear repulsions, electron repulsions and electron-nuclear attractions. None of these is affected by nuclear mass and, therefore, k is not affected by isotopic substitution. 

Nuclear. Mass can be determined directly by measuring changes in the absorption, reflection, or transmission of alpha- or beta-rays, which changes in proportion to the amount of material present. This method is primarily used to determine the mass of bulk material moving on a conveyor. The advantages include the following ... 

Since nuclear masses are much greater than the electron mass we can treat the nucleus as if it were fixed in space. Taking the mass of the electron charge cloud as m, then k = mu>Q where angular frequency of the oscillator. 

In the Bom-Oppenheimer picture the nuclei move on a potential energy surface (PES) which is a solution to the electronic Schrodinger equation. The PES is independent of the nuclear masses (i.e. it is the same for isotopic molecules), this is not the case when working in the adiabatic approximation since the diagonal correction (and mass polarization) depends on the nuclear masses. Solution of (3.16) for the nuclear wave function leads to energy levels for molecular vibrations (Section 13.1) and rotations, which in turn are the fundamentals for many forms of spectroscopy, such as IR, Raman, microwave etc. 

Corrections involving nuclei (with the nuclear spin I replacing the electron spin s) are analogous to the above one- and two-particle terms in eqs. (8.29-8.30), with the exception of those involving the nuclear mass, which disappears in the Bom-Oppenheimer approximation (which may be be considered as the Mnucieus oo limit). 

The only term surviving the Bom-Oppenheimer approximation is the direct spin-spin coupling, as all the others involve nuclear masses. Furthermore, there is no Fermi-contact term since nuclei cannot occupy the same position. Note that the direct spin-spin coupling is independent of the electronic wave function, it depends only on the molecular geometry. 

The total energy in ab initio theory is given relative to the separated particles, i.e. bare nuclei and electrons. The experimental value for an atom is the sum of all the ionization potentials for a molecule there are additional contributions from the molecular bonds and associated zero-point energies. The experimental value for the total energy of H2O is —76.480 a.u., and the estimated contribution from relativistic effects is —0.045 a.u. Including a mass correction of 0.0028 a.u. (a non-Bom-Oppenheimer effect which accounts for the difference between finite and infinite nuclear masses) allows the experimental non-relativistic energy to be estimated at —76.438 0.003 a.u. ... 

It is first transfonned to mass-dependent coordinates by a G matrix eontaining the inverse square root of atomic masses (note that atomic, not nuclear, masses are used, this is in line with the Bom-Oppenheimer approximation that the electrons follow the nucleus). 

The reactions that we discuss in this chapter will be represented by nuclear equations. An equation of this type uses nuclear symbols such as those written above in other respects it resembles an ordinary chemical equation. A nuclear equation must be balanced with respect to nuclear charge (atomic number) and nuclear mass (mass number). To see what that means, consider an equation that we will have a lot more to say about later in this chapter ... 

In an ordinary chemical reaction, Am is immeasurably smalL In a nuclear reaction, on the other hand, Am is appreciable, amounting to 0.002% or more of the mass of reactants. The change in mass can readily be calculated from a table of nuclear masses (Table 19.3). 

Nuclear equation, 513 Nuclear masses, 521t Nuclear radiation. See Radioactivity Nuclear reactions biological effects, 527 equations, 513 fission, 523-526 fusion, 526-527,528 mass-energy relations, 520-523,... 

Mass number = number of protons and neutrons in the nucleus (fixes nuclear mass)... 

Let us do a little bookkeeping with the exact masses of these nuclei. Actually we will simplify a bit and use the exact masses of the atoms. This will make no difference. The masses of the atoms differ from the nuclear masses by the masses of the number of electrons in each atom. We have shown that electrons are conserved in nuclear changes. Exact masses of atoms (that is, exact masses of each isotopic species and not the chemical atomic weights shown on the inside back cover) are readily available. For our hydrogen-helium reaction we have... 

As can be seen in Fig. 2-1 (abundance of elements), hydrogen and oxygen (along with carbon, magnesium, silicon, sulfur, and iron) are particularly abundant in the solar system, probably because the common isotopic forms of the latter six elements have nuclear masses that are multiples of the helium (He) nucleus. Oxygen is present in the Earth s crust in an abundance that exceeds the amount required to form oxides of silicon, sulfur, and iron in the crust the excess oxygen occurs mostly as the volatiles CO2 and H2O. The CO2 now resides primarily in carbonate rocks whereas the H2O is almost all in the oceans. 

As stated in Chapter the periodic table lists elements in order of increasing masses, with a few exceptions. In fact, nuclear charge is the organizing feature of the periodic table. As nuclear charge increases, so also does nuclear mass (with few exceptions), so... 

A molecule is composed of positively charged nuclei surrounded by electrons. The stability of a molecule is due to a balance among the mutual repulsions of nuclear pairs, attractions of nuclear-electron pairs, and repulsions of electron pairs as modified by the interactions of their spins. Both the nuclei and the electrons are in constant motion relative to the center of mass of the molecule. However, the nuclear masses are much greater than the electronic mass and, as a result, the nuclei move much more slowly than the electrons. Thus, the basic molecular structure is a stable framework of nuclei undergoing rotational and vibrational motions surrounded by a cloud of electrons described by the electronic probability density. 

In the original mathematical treatment of nuclear and electronic motion, M. Bom and J. R. Oppenheimer (1927) applied perturbation theory to equation (10.5) using the kinetic energy operator Tq for the nuclei as the perturbation. The proper choice for the expansion parameter is A = (me/M) /", where M is the mean nuclear mass... 

Chemists were not able to use their methods to determine the structure of the atom. The discovery of radioactivity by Henri Becquerel and the work of Marie and Pierre Curie showed, however, that heavy elements were not stable. The earlier postulate of their indivisibility could no longer be maintained. In 1906 Ernest Rutherford made the next horrorif-ic revelation his scattering experiments showed that the atom was almost empty. A tiny nuclear mass was circled by electrons at a large distance. For comparison, if the nucleus were the size of a cherry pit and were placed in the center of a football field, the electrons would be circulating in the back rows of the stadium. If the nucleus were the size of a football, the first electrons would be circling it at a distance of one kilometer. Between them would be absolute emptiness. 

Consider the results of Problem 3.16 and the fact that l2C and 14C have very similar chemical properties. Which is more important in determining chemical properties, nuclear charge or nuclear mass ... 

Until recently the only satisfactory way to separate these molecular interferences has been on the basis of nuclear mass defects, i.e., the mass of molecules having the same mass number differs from that of the atoms of the same mass number. Figure 2 shows the resolution that is needed to resolve the molecular impurities present in the previous example. Clearly, an unambiguous identification can be made, and all molecular fragments can only be eliminated for an instrument with resolution M/AM approximately 20,000. Once again, the need for high resolution will cause the transmission efficiency to be low. 

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