Nucleus energy levels

Energy levels Energy levels are the volume of space where certain electrons of specific energy are restricted to move around the nucleus. Energy levels consist of one or more orbitals. Energy levels are categorized by the letter n using whole numbers (n = 1,2, 3,4. ..). 

The essence of NMR spectroscopy is to measure the separation of the magnetic energy levels of a nucleus. 

The dissociation energy is unaffected by isotopic substitution because the potential energy curve, and therefore the force constant, is not affected by the number of neutrons in the nucleus. However, the vibrational energy levels are changed by the mass dependence of 03 (proportional to where /r is the reduced mass) resulting in Dq being isotope-... 

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. 

Figure 4-3. Energy level splitting for a nucleus with / = 5 in applied field Hn. The energy separation is proportional to Ho-...

An atom consists of a positively charged nucleus surrounded by one or more negatively charged electrons. The electronic structure of an atom can be described by a quantum mechanical wave equation, in which electrons are considered to occupy orbitals around the nucleus. Different orbitals have different energy levels and different shapes. For example, s orbitals are spherical and p orbitals are dumbbell-shaped. The ground-state electron configuration of an... 

The decrease in atomic radius moving across the periodic table can be explained in a similar manner. Consider, for example, the third period, where electrons are being added to the third principal energy level. The added electrons should be relatively poor shields for each other because they are all at about the same distance from the nucleus. Only the ten core electrons in inner, filled levels (n = 1, n = 2) are expected to shield the outer electrons from the nucleus. This means that the charge felt by an outer electron, called the effective nuclear charge, should increase steadily with atomic number as we move across the period. As effective nuclear charge increases, the outermost electrons are pulled in more tightly, and atomic radius decreases. 

K-electron capture, in which an electron in the innermost energy level (n = 1) falls into the nucleus. 

Consider the lowest energy level of a hydrogen atom, n = 1. We have just learned that there are /7s levels with this energy, and since n = 1, there is but one level. It corresponds to an electron distribution that is spherically symmetrical around the nucleus, as shown in Figure 15-8. It is called the Is orbital. An electron moving in an s orbital is called an s electron. 

Every energy level with n above 1 has three p orbitals. As n increases, the np orbitals place the electron, on the average, farther and farther from the nucleus, but always with the axial directional properties shown in Figure 15-9. 

The properties of 7-rays are indistinguishable from those of x-rays of the same wavelength, but they do differ in origin. A 7-ray is emitted by a nucleus upon the occurrence of a quantum transition between two energy levels of the nucleus. For our purposes, only the 7-rays originating in radioactive nuclei need be considered. 

If a monoarylacetylene (ArC = CH) is taken as a model for a transition state of an arenediazonium ion with a nucleophile Nu, two types of transition state can be visualized the first, 7.13, leads to the (Z)-azo compound 7.14, whereas the second, 7.15, results in the (E )-isomer 7.16 (Scheme 7-3). If the transition state is reactantlike (i.e., early on the reaction coordinate), repulsive interaction between the nucleophile and the aryl nucleus is small because the distance Nu-Np is still large. Therefore, the repulsion between the lone pair on Np and the aryl nucleus becomes the decisive factor. It favors an (E )-configuration of the Np lone pair with respect to the aryl nucleus (obviously it is energetically dominant compared with the repulsion between the lone pairs on Na and Np) therefore, transition state 7.13 is at a lower energy level, and Nu attacks NB in the (Z)-configuration. 

An electron in an atom is like a particle in a box, in the sense that it is confined within the atom by the pull of the nucleus. We can therefore expect the electron s wavefunctions to obey certain boundary conditions, like the constraints we encountered when fitting a wave between the walls of a container. As we saw for a particle in a box, these constraints result in the quantization of energy and the existence of discrete energy levels. Even at this early stage, we can expect the electron to be confined to certain energies, just as spectroscopy requires. 

Many atomic nuclei behave like small bar magnets, with energies that depend on their orientation in a magnetic field. An NMR spectrometer detects transitions between these energy levels. The nucleus most widely used for NMR is the proton, and we shall concentrate on it. Two other very common nuclei, those of carbon-12 and oxygen-16, are nonmagnetic, so they are invisible in NMR. 

Figure 1.44 Transitions between various energy levels of an AX spin system. A, and Aj represent the single-quantum relaxations of nucleus A, while Xi and Xj represent the single-quantum relaxations of nucleus X. W2 and are double- and zero-quantum transitions, respectively.

Figure 4.2 Energy levels and populations for an IS system in which nuclei I and S are not directly coupled with each other. This forms the basis of the nuclear Overhauser enhancement effect. Nucleus S is subjected to irradiation, and nucleus I is observed, (a) Population at thermal equilibrium (Boltzmann population).

Bohr s hypothesis solved the impossible atom problem. The energy of an electron in orbit was fixed. It could go from one energy level to another, but it could not emit a continuous stream of radiation and spiral into the nucleus. The quantum model forbids that. 

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