Directed angular-momentum vectors

Figure 1.5 Direction of the angular momentum vector p for an electron in an orbit...

Hyperfine structure arises through the interaction of the electron spin with a nuclear spin. Consider first the interaction of the electron spin with a single magnetic nucleus of spin , In an applied magnetic field the nuclear spin angular momentum vector, of magnitude (/ / -f l)]l/2, precesses around the direction of the field in an exactly analogous way to that of the electron spin. The orientations that the nuclear spin can take up are those for which the spin in the z-direction, M, has components of ... 

Spin I> 0 nuclei possess a magnetic dipole or dipole moment, n, which arises from a spinning, charged particle. Nuclei that have a nonzero spin will also have a magnetic moment, and the direction of that magnetic moment is collinear with the angular momentum vector associated with the nucleus. This can be expressed as... 

After the separation of the kinetic energy operator due to the center-of-mass motion from the Hamiltonian, the Hamiltonian describes the internal motions of electrons and nuclei in the system. These in the BO approximation can be separated into the vibrational and rotational motions of the nuclear frame of the molecule and the electronic motion that only parametrically depends on the instantenous positions of the nuclei. When the BO approximation is removed, the electronic and nuclear motions become coupled and the only good quantum numbers, which can be used to quantize the stationary states of the system, are the principle quantum number, the quantum number quantizing the square of the total (nuclear and electronic) squared angular momentum, and the quantum number quantizing the projection of the total angular momentum vector on a selected direction (usually the z axis). The separation of different rotational states is an important feamre that can considerably simplify the calculations. 

For most nuclides the nuclear angular momentum vector L and the magnetic moment vector (L point in the same direction, i.e. they are parallel. However, in a few cases, for example, N, they are antiparallel. 

Fig. IV-3.—Diagram representing orientations of the spin vectors of the two electrons and the orbital angular momentum vectors of the two electrons in the extreme Paschen-Back effect for an atom with two 2 electrons. The two spins orient themselves separately in the vertical magnetic field, as do also the two orbital angular momentum vectors. Each electron spin can assume orientations such that the component of angular momentum along the field direction is represented by the quantum number m, + or — and each orbital angular momentum may orient itself in such a way that the component of the orbital angular momentum along the field direction is represented by the quantum number m +1, 0, or —1.

A negatron emitted during beta decay has its spin aligned away from the direction of its emission (its angular momentum vector is antiparallel to its momentum vector) and hence has a negative helix, but an emitted positron has positive helix. It is because of the absence of beta particles with both positive and negative helix in both types of beta-emission processes that parity is not conserved in beta decay. 

Fig. 2. Direction of angular momentum vector is perpendicular to plane formed by radial vector r and momentum vector p.

Fig. 3. Magnitudes and directions of the angular momentum vectors for an / = I. s = 5 electron in an atomic energy state...

Fig. 4. Possible orientations of die total angular momentum vector i relative to the direction of an externally applied magnetic field B and the magnitudes of the associated magnetic quantum state vectors my...

Quantum Number (Magnetic), A quantum number that describes the component of the angular momentum vector of an atomic electron or group of electrons in the direction of an externally applied magnetic field. The values of these components are restricted, i.e., quantized, The symbol for the magnetic quantum number is m. 

The third technique for establishing a reference axis for angular correlations can be applied to nuclear reactions when the direction of a particle involved in the reaction is detected. This direction provides a reference axis that can be related to the angular momentum axis, but each nuclear reaction has its own pecu-larities and constr aints on the angular momentum vector. For example, the direction of an a particle from a decay process that feeds an excited state can be detected as indicated in Figure 9.7, but, as is discussed in Chapter 7, the energetics of a decay... 

Fig. 3. 15 The angular momentum vector in the partfde-on-a-sphere level where 1=2. Each cone shows ihe range of directions possible tor a given m value. The z component Is well defined, but the xand y. components are indefinite...

Since gm is related to the orbital angular momentum vector L the second term is often written in terms of this operator. The first term is the Langevin term it is the expectation value of a one-electron operator 2 (xt2 + jV), and can therefore be obtained directly from the ground-state wavefunction. It has formed the basis for a quite successful additivity scheme, Pascal s rules, which work well except for the conjugated hydrocarbons where non-additivity is ascribed to ring currents. Nevertheless, as it depends on the square of the electron co-ordinates, xL will be sensitive to the basis set used in variational calculations. 

The vector of the electromagnetic field defines a well specified direction in the laboratory frame relative to which all other vectors relevant in photodissociation can be measured. This includes the transition dipole moment, fi, the recoil velocity of the fragments, v, and the angular momentum vector of the products, j. Vector correlations in photodissociation contain a wealth of information about the symmetry of the excited electronic state as well as the dynamics of the fragmentation. Section 11.4 gives a short introduction. Finally, we elucidate in Section 11.5 the correlation between the rotational excitation of the products if the parent molecule breaks up into two diatomic fragments. 

A second correlation exists between the transition dipole moment fi, which is preferentially aligned parallel to Eo, and the angular momentum vector of the photofragment, j. This correlation concerns the direction of the final angular momentum vector j with respect to the space-fixed axis Eo- In quantum mechanics the projection of j on the axis defined by Eo is quantized with quantum numbers mj = —j, + —... 

Note that even when mi has its maximum value (mi = /), L2Z < L. Thus we never know the exact direction of the angular momentum vector. An atom does not really have any preferred direction in space, so all of the different mi levels have exactly the same energy such levels are called degenerate. In fact, for a hydrogen atom, states with different values of /, mi, and ms (but the same value of n) are degenerate as well. 

When molecules arrive at the state with rotation quantum number J" from the state J in the process of spontaneous radiation at rate (see Fig. 3.14), a photon possessing unit spin is emitted in an arbitrary direction. Let us assume that the angular momentum carried away by the emitted photon is small, as compared with both 3 [ and Jr. This means that the angular momentum vector of each separate molecule does not change its value and does not turn in space as a result of the spontaneous transition. Consequently, the angular momenta distribution pji(0,ip) is... 

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