Elastic constants magnetic field dependence

For TmCd and TmZn a variety of techniques have been applied to determine the important coupling constants and gj. they are listed in table 3. In addition to the temperature dependence of the symmetry elastic constant Cj (T), the parastriction method, the third-order susceptibility and the magnetic field dependence of the structural phase transition temperature Tq B) have been used. The different experimental methods have been described in sect. 2.4.1. It is seen from table 3 that the coupling constants determined with these different methods are in good agreement with each other. 

Magnetic field dependence of elastic constants In a magnetic field one obtains Zeeman-split CEF levels with new eigenstates of the total Hamiltonian,... 

The magnetic field dependence of the elastic constants in paramagnetic hep structures was studied by Goddings (1977). Furthermore a theory was developed... 

Magnetostriction experiments were performed in the paramagnetic phase of different RSb compounds (Liithi et al. 1977). Such measurement enables one to determine size and sign of some of the magneto-elastic coupling constants. The effects result from the magnetic field dependence of the lattice equilibrium positions (see section 5.2.1). 

Fig. 25. Magnetic field dependence of the elastic constant C33 at various temperatures in Dy (field along the n-axis). Open circles, 135K solid circles, 150K open triangles, 160K open squares, 168K solid squares,...

More difficult to calculate are the properties which depend on the response of the solid to an outside influence (stress, electric field, magnetic field, radiation). Elastic constants are obtained by considering the response of the crystal to deformation. Interatomic potential methods often provide good values for these and indeed experimental elastic constants are often used in fitting the potential parameters. Force constants for lattice vibrations (phonons) can be calculated from the energy as a function of atomic coordinates. In the frozen phonon approach, the energy is obtained explicitly as a function of the atom coordinates. Alternatively the deriva-tive, 5 - can be calculated at the equilibrium geometry. 

Since the applied magnetic field H is less than Hc no deformation occurs. If H is greater than Hc, ()rn depends on the value of H. The elastic constant may be determined through measuring The ratio of K to Kis can be evaluated from the curve of 0rn versus H, or accordingly from the transmittance versus H or the electric capacitance versus H. 

Recently, in analogy to the magnetic Kondo effect, a quadrupolar Kondo effect was suggested (Cox 1987). Prominent candidates for this effect to exist are the UBejj and Ce systems with Tg ground states. Qearly temperature- and magnetie-field-dependent elastic constants can help to clarify the situation of this effect. 

Fig. 9. The temperature dependence of the single-crystal elastic constants Cn and C33 of terbium at zero magnetic field, and in an applied field along the easy axis. The inset shows the details of Cu at the Neel and Curie temperatures. (After Palmer et al. 1974.)...

During their investigation Palmer et al. (1974) examined the effect of a 2.5 Tesla magnetic field on the temperature dependence of the elastic constants of terbium, Monfort and Swenson (1965) and Stephens and Johnson (1969) obtained the pressure dependence of the compressibility. 

We note immediately that, when H is strictly perpendicular to n, the magnetic field has no effect. Only fluctuations 8n in n allow the field to act upon the orientation of the director. The elastic energy can be simplified by observing that, for each case represented in Fig. 9.6, the deformation is associated with a single Frank-Oseen constant Ki, and that it depends only on z. Hence we can write... 

In CLCs with positive dielectric anisotropy, an electric field-induced cholesteric-nematic phase transition was theoretically predicted [45], [46] and experimentally observed [47], [48]. If the electric field E is applied perpendicular to the helix axis hot a CLC, the helix unwinds like in a magnetic field (Chapter 2). At sufficiently high field strengths, the homeotropic nematic structure is stabilized (Figure 6.3). The critical field strength E = Ecn depends on the pitch P, the dielectric anisotropy As, and the twist elastic constant K22 ... 

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