Nuclear total angular momentum

Besides the magnetic dipole moment, nuclei with spin higher than 1/2 also possess an electric quadrupole moment. In a semiclassical picture, the nuclear electric quadrupole moment informs about the deviation of nuclear charge distribution from spherical symmetry. Nuclei with spin 0 or 1/2 are therefore said to be spherical, with zero electric quadrupole moment. On the other hand, if the nuclear spin is higher than 1/2, the nuclei are not spherical, assuming cylindri-cally symmetrical shapes around the symmetry axis defined by the nuclear total angular momentum [17]. Within the subspace [/,m), the nuclear electric quadrupole moment operator is a traceless tensor operator of second rank, with Cartesian components written is terms of the nuclear spin [2] ... 

Initially, we neglect tenns depending on the electron spin and the nuclear spin / in the molecular Hamiltonian //. In this approximation, we can take the total angular momentum to be N(see (equation Al.4.1)) which results from the rotational motion of the nuclei and the orbital motion of the electrons. The components of. m the (X, Y, Z) axis system are given by ... 

The quantum numbers tliat are appropriate to describe tire vibrational levels of a quasilinear complex such as Ar-HCl are tluis tire monomer vibrational quantum number v, an intennolecular stretching quantum number n and two quantum numbers j and K to describe tire hindered rotational motion. For more rigid complexes, it becomes appropriate to replace j and K witli nonnal-mode vibrational quantum numbers, tliough tliere is an awkw ard intennediate regime in which neitlier description is satisfactory see [3] for a discussion of tire transition between tire two cases. In addition, tliere is always a quantum number J for tire total angular momentum (excluding nuclear spin). The total parity (symmetry under space-fixed inversion of all coordinates) is also a conserved quantity tliat is spectroscopically important. 

Nuclear spin 1 = Total angular momentum quantum number 7 = 0,1,2,., ... 

The phrase total angular momentum is commonly used to refer to a number of different quantities. Here it implies orbital plus electron spin but it is also used to imply orbital plus electron spin plus nuclear spin when the symbol F is used. 

We consider a nuclear wave function describing collisions of type A + BC(n) AC(n ) + B, where n = vj, k are the vibrational v and rotational j quantum numbers of the reagents (with k the projection of j on the reagent velocity vector of the reagents), and n = v, f, k are similarly defined for the products. The wave function is expanded in the terms of the total angular momentum eigenfunctions t X) [63], and takes the form [57-61]... 

Equation (4.15) would be extremely onerous to evaluate by explicit treatment of the nucleons as a many-particle system. However, in Mossbauer spectroscopy, we are dealing with eigenstates of the nucleus that are characterized by the total angular momentum with quantum number 7. Fortunately, the electric quadrupole interaction can be readily expressed in terms of this momentum 7, which is called the nuclear spin other properties of the nucleus need not to be considered. This is possible because the transformational properties of the quadrupole moment, which is an irreducible 2nd rank tensor, make it possible to use Clebsch-Gordon coefficients and the Wigner-Eckart theorem to replace the awkward operators 3x,xy—(5,yr (in spatial coordinates) by angular momentum operators of the total... 

Angular momentum plays an important role in both classical and quantum mechanics. In isolated classical systems the total angular momentum is a constant of motion. In quantum systems the angular momentum is important in studies of atomic, molecular, and nuclear structure and spectra and in studies of spin in elementary particles and in magnetism. 

Let us now illustrate the discretization process using the vibration of a triatomic molecule (ABC) as an example. The nuclear Hamiltonian with zero total angular momentum (J = 0) can be conveniently written in the Jacobi coordinates (h = 1 thereafter) ... 

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 the electro-nuclear model, it is the charge the only homogeneous element between electron and nuclear states. The electronic part corresponds to fermion states, each one represented by a 2-spinor and a space part. Thus, it has always been natural to use the Coulomb Hamiltonian Hc(q,Q) as an entity to work with. The operator includes the electronic kinetic energy (Ke) and all electrostatic interaction operators (Vee + VeN + Vnn)- In fact this is a key operator for describing molecular physics events [1-3]. Let us consider the electronic space problem first exact solutions exist for this problem the wavefunctions are defined as /(q) do not mix up these functions with the previous electro-nuclear wavefunctions. At this level. He and S (total electronic spin operator) commute the spin operator appears in the kinematic operator V and H commute with the total angular momentum J=L+S in the I-ffame L is the total orbital angular momentum, the system is referred to a unique origin. 

For nearly all molecules with A = 0, we have case (b) coupling there are also a few molecules with A= 0 that fall in this case. In case (b), the angular momentum A combines with the molecular rotational angular momentum O to give a total angular momentum apart from electronic and nuclear spin, which is called N. The magnitude of N is... 

The older literature uses K instead of N.) For the most common case-(b) case, we have A = 0 here N has the possible values 0,1,2,... and represents just rotational angular momentum. The angular momentum N then adds to the electronic spin angular momentum S to give a total angular momentum apart from nuclear spin, which, as usual, is called J. The quantum number J has the possible values [Equation (1.265)]... 

The angular momentum 0 then combines with the angular momentum O due to the rotation of the molecule to give a total angular momentum (apart from nuclear spin) J. (In the older literature, the rotational angular momentum of the nuclei is called N instead of O.) The magnitude of J is... 

For 2 molecules, we used J as the rotational angular-momentum quantum number since 2 molecules have no electronic spin or orbital angular momentum, J is also the total angular-momentum quantum number, exclusive of nuclear spin, for such molecules. Recall that for atoms J is also used as the total angular-momentum quantum number apart from nuclear spin.) The rotational energy in case (a) is given approximately by... 

The interaction between the nuclear electric quadrupole moment and the electrons of the molecule couples the nuclear spin I to the rotational angular momentum J, giving a resultant total angular momentum F, of... 

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