Effects of external electric and magnetic fields

The equations derived thus far take no account of the effects of applied electric or magnetic fields, or even of the fields created by the motion of the nuclei and electrons. We shall discuss these effects explicitly in the derivation of the electronic Hamiltonian, but for the moment we aim to correct our equations for the motion of the nuclei by appealing to classical mechanics. We here sketch the main points which are covered in detail by Landau and Lifshitz [15], among others. 

In non-relativistic classical mechanics a mechanical system can be characterised by a function called the Lagrangian, S(q, q) where q denotes the coordinates, and the motion of the system is such that the action S, defined by 

Now for a single particle in an external field, the Lagrangian is given by 

These equations are modified when we turn to relativistic mechanics. The 

An electromagnetic field is described in relativistic theory by a four-vector A, where the three space components Aij2,3 = Aare called the vector potential A and the fourth (time) component A4 is equal to i / where p is called the scalar potential. The Lagrangian for a particle in an electromagnetic field is now given by 

These equations are modified when we turn to relativistic mechanics. The Lagrangian for a free particle is now given 1  

An elecU-omagnetic field is described in relativistic theory by a four-vector A,, where the three space components Ai, 2,3 are called the vector potential A and 

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The first term, Hj, is the spin-orbit (one electron term) and spin-other-orbit (two electron term) couplings, which are the topic of the following subsection. The second term Hf contains the spin-spin coupling term and Fermi contact interaction. Both the Hj and f/ can lift degeneracy in multiplets. The parameter Hf is the Dirac correction term for electron spin and Ff is the classical relativistic correction to the interaction between electrons due to retardation of the electromagnetic field produced by an electron. The parameter H is the so-called mass-velocity effect, due to the variation of electron mass with velocity. Finally, H is the effect of external electric and magnetic fields. 

The structural effect found in a polymer under the effect of extomal electric and magnetic fields should also be included in the features of the polymeric state of liquid crystals. In addition to orientation unda the effect of Helds (discussed in Section 8.2), an additional level of order, the layered structure, appears in the initially nematic polymer. The increase in the correlation length ( y) by 1.5 order of magnitude under the effect of an external Held has not yet found a physical explanation within the framework of current theoretical concepts on induced phase transitions. However, this reflects the specific polymeric feature of the behavior of LC systems in fields. 

The birefringence in external electric and magnetic fields (the Kerr and Cotton-Mouton effects) can be explained by the anisotropy of the properties of the medium that is due to either the orientation of anisotropic molecules in the external field (the Langevin-Bom mechanism) or the deformation of the electric or magnetic susceptibilities by this field, i.e., to hyperpolarizabilities (Voight mechanism). The former mechanism is effective for molecules that are anisotropic in the absence of the field and... 

The ab initio computation of molecular properties - including those associated with time-dependent external electric and magnetic fields - has advanced significantly in the last several decades, yielding accurate models for linear, quadratic, and higher-order response functions. When electron correlation effects play a pivotal... 

For weak anchoring there is a competition between the torque in the bulk due to one plate, and the torque resulting from the other surface this is usually confined to a boundary region as indicated in the figure. External electric and magnetic fields will also affect the equilibrium director distribution, and this is the basis of many liquid crystal applications these effects will be considered later. 

We are interested only in thermal vibrations from the equilibrium point (minimum) in the direction of a raddle point of the energy hyperplane. There is no friction in the hydrodynamical sense. However a change of the kinetic and the potential energy of the thermal vibrations does take place, due to the fact that in all relaxation and flow processes the kinetic energy of all oszillating systems combined will increase, this the effect of external stress, external electric or magnetic fields. In all cases we observe an excess of vibration energy in relaxation processes. 

The title of the book, Optical Rheometry of Complex Fluids, refers to the strong connection of the experimental methods that are presented to the field of rheology. Rheology refers to the study of deformation and orientation as a result of fluid flow, and one principal aim of this discipline is the development of constitutive equations that relate the macroscopic stress and velocity gradient tensors. A successful constitutive equation, however, will recognize the particular microstructure of a complex fluid, and it is here that optical methods have proven to be very important. The emphasis in this book is on the use of in situ measurements where the dynamics and structure are measured in the presence of an external field. In this manner, the connection between the microstructural response and macroscopic observables, such as stress and fluid motion can be effectively established. Although many of the examples used in the book involve the application of flow, the use of these techniques is appropriate whenever an external field is applied. For that reason, examples are also included for the case of electric and magnetic fields. 

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