Spins in an external magnetic field

Free fermions in a magnetic field Pauli paramagnetism 

We take the magnetic moment of the fermions to be mo, hence their energy in an external magnetic field H will be given by 

The canonical partition function will be given as the sum over all configurations consistent with the above expressions, which is equivalent to summing over all values of Nf from zero to N, with 

We can take advantage of the definition of the partition function of spinless fermions. 

We will approximate the above summation over by the largest term in it, with the usual assumption that in the thermodynamic limit this term dominates to find the value of N-  

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Unlike these conventional techniques, NSE measures the neutron velocities of the incident and scattered neutrons using the Larmor precession of the neutron spin in an external magnetic field, whereby, the neutron spin vector acts like the hand of an internal clock, which is linked to each neutron and stores the result of the velocity measurement at the neutron itself. The velocity measurement is thus performed individually for each neutron. For this reason, the... 

Previously, the dipole-dipole interaction between the two qubits through magnetic dipoles (nuclear spins in an external magnetic field) was seen through NMR logic. Qubits can also be seen to interact through their electric dipoles (single-electron quantum dots). Mathematically the two cases are identical [109]. While the subjects of this subsection are not molecules as such, the relatively small size of quantum dots justifies their brief discussion here. 

Electron paramagnetic resonance spectroscopy relies on microwave radiation induced resonance of electron spins in an external magnetic field, B (Fig. 2). The effective magnetic field experienced by the electron is, however, influenced by... 

As the ENDOR spectrum is in effect an NMR spectrum obtained by devious means, when considering the response of a nucleus with spin / one must keep track of all interactions that it and its coupled electron spin undergo. The energy levels of an unpaired electronic spin S in an externally applied magnetic field, and the associated EPR spectra, are described in detail elsewhere in this volume. As described therein, the interaction of an electronic spin in an external magnetic field is characterized by the parameter g. In a molecule, the magnitude of the electron spin interaction depends on the orientation of the molecule with respect to the external magnetic field, and thus g is a tensor quantity. 

In the next part of this work we will show how the thermodynamic formalism can be applied for the system of spins in an external magnetic field and will discuss the existence of two spin temperatures for one spin system. It is a real good example when one has to remember all conditions being used to... 

Let a consider the regular lattice of spins in an external magnetic field. The corresponding Hamiltonian of spin system has the following form... 

So, by use very simple, model system - systems of interacting spins in an external magnetic field - we have demonstrated that the definition of temperature depends crucially on the assumptions made for this procedure. And one has to be very careful doing so. 

Figure 5.8 Precession of the magnetic moment in each of the two possible spin states of an 7 = 1/2 nucleus in an external magnetic field B0. After Macomber [160]. Reprinted from R.S. Macomber, A Complete Introduction to Modern NMR Spectroscopy, John Wiley Sons, Inc., New York, NY, Copyright (1998, John Wiley Sons, Inc.). This material is used by permission of John Wiley Sons, Inc.

As already observed for some isotropic polynuclear clusters [30 - 32], slow relaxation of the magnetization in an external magnetic field can occur because of the inefficient transfer of energy to the environment, for example, the helium bath, and consequent reabsorption of the emitted phonon by the spin system. The phenomenon, also known as phonon bottleneck (PB), was first introduced by Van Vleck [33]. It is characteristic of low temperatures, where relaxation is dominated by the direct process between closely spaced levels, and results from the low density of phonons with such a long wavelength to match the small energy separation... 

Fig. 1. Energy levels of nucleus with 7 = in an external magnetic field. Arrows denote transitions caused by rf field wavy line denotes net transitions due to spin-lattice relaxation.

In an external magnetic field, Bq, the magnetization, Mq, of an ensemble of spins I is given by Curie s equation (10-13) ... 

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