Nuclear Zeeman Hamiltonian

H is the total Hamiltonian (in the angular frequency units) and L is the total Liouvillian, divided into three parts describing the nuclear spin system (Lj), the lattice (Ll) and the coupling between the two subsystems (L/l). The symbol x is the density operator for the whole system, expressible as the direct product of the density operators for spin (p) and lattice (a), x = p <8> ci. The Liouvillian (Lj) for the spin system is the commutator with the nuclear Zeeman Hamiltonian (we thus treat the nuclear spin system as an ensemble of non-interacting spins in a magnetic field). Ll will be defined later and Ljl... 

In quantum mechanical terms the energy is given by the Hamiltonian operator, which in this case is called the nuclear Zeeman Hamiltonian... 

We introduced the field-free nuclear Hamiltonian in section 3.10. Again by analogy with the electronic Hamiltonian, we include the effects of external magnetic fields by replacing P, by P, — Z,eA l in equation (III.248) and the effects of an external electric field by addition of the term Y,a Zae(pa, this treatment is only really justified if the nuclei behave as Dirac particles. The nuclear Zeeman Hamiltonian is thus ... 

A nucleus in a state with spin quantum number 7 > 0 will interact with a magnetic field by means of its magnetic dipole moment p. This magnetic dipole interaction or nuclear Zeeman effect may be described by the Hamiltonian... 

If the electric quadrupole splitting of the 7 = 3/2 nuclear state of Fe is larger than the magnetic perturbation, as shown in Fig. 4.13, the nij = l/2) and 3/2) states can be treated as independent doublets and their Zeeman splitting can be described independently by effective nuclear g factors and two effective spins 7 = 1/2, one for each doublet [67]. The approach corresponds exactly to the spin-Hamiltonian concept for electronic spins (see Sect. 4.7.1). The nuclear spin Hamiltonian for each of the two Kramers doublets of the Fe nucleus is ... 

The leading term in T nuc is usually the magnetic hyperfine coupling IAS which connects the electron spin S and the nuclear spin 1. It is parameterized by the hyperfine coupling tensor A. The /-dependent nuclear Zeeman interaction and the electric quadrupole interaction are included as 2nd and 3rd terms. Their detailed description for Fe is provided in Sects. 4.3 and 4.4. The total spin Hamiltonian for electronic and nuclear spin variables is then ... 

In Equation (6) ge is the electronic g tensor, yn is the nuclear g factor (dimensionless), fln is the nuclear magneton in erg/G (or J/T), In is the nuclear spin angular momentum operator, An is the electron-nuclear hyperfine tensor in Hz, and Qn (non-zero for fn > 1) is the quadrupole interaction tensor in Hz. The first two terms in the Hamiltonian are the electron and nuclear Zeeman interactions, respectively the third term is the electron-nuclear hyperfine interaction and the last term is the nuclear quadrupole interaction. For the usual systems with an odd number of unpaired electrons, the transition moment is finite only for a magnetic dipole moment operator oriented perpendicular to the static magnetic field direction. In an ESR resonator in which the sample is placed, the microwave magnetic field must be therefore perpendicular to the external static magnetic field. The selection rules for the electron spin transitions are given in Equation (7)... 

The nuclear Zeeman term describes the interaction of the nuclear spins with the external magnetic field. Just as the hyperfine splitting, this term is not incorporated in the original purely electronic Breit-Pauli Hamiltonian as presented in Eqs. (59) and (60) but becomes relevant for ESR spectroscopy. 

For a hydrogen atom in an external field of 10,000 G, draw a figure that shows the effect on the original 1 s energy level of including first the electron Zeeman term, then the nuclear Zeeman term, and finally the hyperfine coupling term in the Hamiltonian. 

The nuclear spin hamiltonian (H) for the Zeeman (Hz) and the quadrupole (Hq) interactions may be written... 

In the limit where the nuclear Zeeman term in the nuclear spin hamiltonian is much larger than the quadrupole interaction, it is only the secular part of Hq that contributes to the time-independent hamiltonian, H0. 

We dealt with the effects of applied static fields on the electronic Hamiltonian in section 3.7. In this section we first give the relevant terms for the nuclear Zeeman and Stark Hamiltonians and then perform the same coordinate transformations that proved to be convenient for the field-free molecular Hamiltonian. 

We calculate the effects of the Hamiltonian (8.105) on these zeroth-order states using perturbation theory. This is exactly the same procedure as that which we used to construct the effective Hamiltonian in chapter 7. Our objective here is to formulate the terms in the effective Hamiltonian which describe the nuclear spin-rotation interaction and the susceptibility and chemical shift terms in the Zeeman Hamiltonian. We deal with them in much more detail at this point so that we can interpret the measurements on closed shell molecules by molecular beam magnetic resonance. The first-order corrections of the perturbation Hamiltonian are readily calculated to be... 

The matrix elements of the Zeeman Hamiltonian given in equation (9.70) are evaluated most appropriately in an /-decoupled basis set because the nuclear spin is decoupled in almost all situations in quite modest magnetic fields. The matrix elements of the simpler Zeeman Hamiltonian have been given earlier in equations (9.56) to (9.60). For the sake of completeness, we give the matrix elements of the full Zeeman Hamiltonian here ... 

The theoretical framework for a discussion of the hyperfine interactions in radicals is given by the so called spin Hamiltonian, which describes the interaction between the unpaired electrons and the magnetic nuclei (I>l/2) in the sample. When the radical is placed in a static magnetic field, the electrons and the magnetic nuclei will interact with the field. These interactions give rise to the electronic and nuclear Zeeman terms,... 

Except for some quadrupolar effects, all the interactions mentioned are small compared with the Zeeman interaction between the nuclear spin and the applied magnetic field, which was discussed in detail in Chapter 2. Under these circumstances, the interaction may be treated as a perturbation, and the first-order modifications to energy levels then arise only from terms in the Hamiltonian that commute with the Zeeman Hamiltonian. This portion of the interaction Hamiltonian is often called the secular part of the Hamiltonian, and the Hamiltonian is said to be truncated when nonsecular terms are dropped. This secular approximation often simplifies calculations and is an excellent approximation except for large quadrupolar interactions, where second-order terms become important. 

We now turn to the last of the tree main magnetic interactions, the nuclear Zeeman term. The general approach (7) is to divide the spin Hamiltonian into two distinct parts ... 

The third term of the nuclear Hamiltonian contains two contributions. The nuclear Zeeman term couples the magnetic moment of the nucleus to the external magnetic field Bo. Furthermore, there is a term that describes the interaction of the nuclear spin with the internal magnetic hyperfine field. For paramagnetic samples this is often done in terms of the hyperfine coupling tensor, which multiplied by the spin... 

The most straightforward way to understand the origin of ESEEM and the physical chemistry behind its detection and analysis is to step back ft om this Cu(II) center and focns on an 5 = 1 /2, 7 = 1/2 coupled spin system. The spin Hamiltonian for this system consists of electronic Zeeman, nuclear Zeeman, and electron-nnclear hyperfine interaction terms. For the case of an isotropic electron -matrix and an axial hyperfine interaction, this Hamiltonian can be conveniently written in the laboratory reference Ifame,... 

The spin Hamiltonian used to model the deuterium ligand hyperfine interaction consisted of nuclear Zeeman, electron-nuclear hyperfine and nuclear quadrupole terms. 

The energy of an unpaired electron will now not only depend on the interactions of the unpaired electron (Zeeman level) and the nucleus (Nuclear Zeeman levels) with the applied external magnetic field, but also on the interaction between the unpaired electron and the magnetic nuclei. To explain how one derives the energy terms for such a system, a simple two-spin system (S = 1/2, I = 1/2) will be considered. The simplified spin Hamiltonian for this two-spin system (S = 1/2,... 

In Eq. (3-2), / is the value of the exchange integral between two electron spins (Si and S2) and the 1/2 term is put for convenience although it is not used in many textbooks. Eq. (3-3) is just the sum of spin Hamiltonian of one radical given by Eq. (2-22), but the nuclear Zeeman terms are omitted in Eq. (3-3) because their magnitude is much smaller than those of the electron Zeeman term and the HFC one. In Eq. (3-3), ga and gb are the isotropic g-values of two component radicals (radicals a and b) in a radical pair, respectively, and A, and A are the isotropic HFC constants with nuclear spins (/, and 4) in radicals a and b, respectively. 

Magnetic Hyperfine Interaction (MHI). This interaction arises from the interaction of the nuclear dipole moment with a magnetic field due to the atom s own electrons. The nuclear Zeeman effect may be described by the Hamiltonian (15)... 

The classical Zeeman interaction between a magnetic moment and an applied magnetic field was defined above. In a quantum mechanical description the operators for the quantities need to be used, such as the nuclear spin I. With an externally applied magnetic field B the Zeeman Hamiltonian is given by... 

The Zeeman Hamiltonian can be used to calculate the energy difference between the nuclear spin states. For a nucleus the energy eigenvalues can be obtained by taking the Hamiltonian operator given in Eq. 2.11 so that... 

The Zeeman Hamiltonian Hz describes the interaction between the nuclear magnetic moment, and the external magnetic field Bq. It determines the Larmor frequency coq of deuterium, which is, for example, 76.8 MHz at 11.7 T (corresponding to a 500-MHz spectrometer). 

In this Hamiltonian (5) corresponds to the orbital angular momentum interacting with the external magnetic field, (6) represents the diamagnetic (second-order) response of the electrons to the magnetic field, (7) represents the interaction of the nuclear dipole with the electronic orbital motion, (8) is the electronic-nuclear Zeeman correction, the two terms in (9) represent direct nuclear dipole-dipole and electron coupled nuclear spin-spin interactions. The terms in (10) are responsible for spin-orbit and spin-other-orbit interactions and the terms in (11) are spin-orbit Zeeman gauge corrections. Finally, the terms in (12) correspond to Fermi contact and dipole-dipole interactions between the spin magnetic moments of nucleus N and an electron. Since... 

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