Zener solid

It is left as a problem for the reader to show that the alternative variant of the Zener model (Figure 4.14(b)) also leads to eqn 4.47. A solid which conforms to the predictions of this model is term a Zener solid. The model shows all the significant characteristics of polymer relaxations it has to be generalized slightly (see Section 4.3.2) to be a precise fit. The solutions which we now state for creep, stress relaxation, and dynamic response do however have considerable illustrative significance. 

By solving the differential eqn 4.47 show that the stress-strain relation crCy) for constant-strain-rate deformation of a Zener solid has the following form... 

Check that this is satisfied by J and G as given by eqns 4.49 and 4.53 respectively for the Zener solid. 

For the alternative generalization of the Zener solid shown below, derive the differential equations relating strain, stress, and time. Solve them to show that the stress relaxation modulus function G(r) is given by... 

Another model consisting of elements in series and parallel is that attributed to Zener. It is known as the Standard Linear Solid and is illustrated in Fig. 2.41. The governing equation may be derived as follows. 

Figure 3. Initial vibrational state specified cummulative reaction probabilities for v = 2. Dashed line exact quantum mechanical numerical solution. Solid hue TSH results with use of the Zhu-Nakamura formulas. Dash-dot hue TSH results with use of the Landau-Zener formula. Taken from Ref. [50].

The study of the effect of electric fields on the properties of solids dates back to Zener s (1934) investigation of electrical breakdown in solid dielectrics. Further pioneering work was carried out by Houston (1940) and Slater... 

C. Zener. Theory of growth of spherical precipitates from solid solution. J. Appl. Phys., 20 950-953, 1949. 

Solid-state diode (including zener diodes)... 

Figure 37. Variation with collision energy of cross section, leading by autoionization to main electron peak at 8.25 eV. Solid line represents result of Landau-Zener calculation (see text).

The contribution of the thermoelastic effect to energy dissipation in solids under transient or cyclic deformation was first studied by Zener and shown to account for mechanical relaxation peaks in some metals. 

Fig. 4.2.2 Potential energy diagram for the Landau-Zener model. Adiabatic potentials (solid lines) and diabatic potentials (dashed lines), with /3i < 0 and (32 > 0. The arrow illustrates the dynamics on the lower adiabatic (ground-state) potential.

C. Zener, A theory of the dielectric breakdown of solid dielectrics, Proc. Roy. Soc. London A145 523-529 (1934). 

The colinear collision problem of atom A colliding with a molecule BC was first attempted quantum mechanically by Zener [14,15] and then by Jackson and Mott [28] for the purpose of investigating thermal accommodation coefficients for atoms impinging on solid surfaces. An exponential repulsion was utilized, along with the harmonic-oscillator approximation. The distorted-wave (DW) method was employed to obtain a 1 — 0 transition probability of the form... 

Figure 1. The Slater-Pauling curve displaying saturation ferromagnetic moment for the first-row transition metal alloys. This figure shows a comparison of experimental values (solid curves) and predicted values (dashed lines) of the saturation ferromagnetic moment per atom, in Bohr magnetons, for Fe-Co, Co-Ni, and Ni-Cu alloys. The short vertical lines indicate change in crystal structure. When the Zener contribution is taken into account, the slope of the dashed line from Fe72Co28 to Ni44Cus6 changes from -1, as...

West, A. R. Solid State Chemistry and its Applications, John Wiley Sons, Chichester, 1985. Zener, C. Trans. AIME 1955, 203, 619. 

Fig. 8. Relative velocity dependence of integral cross sections calculated for Na + O collisions for the indicated exit channels. The solid curve is the charge transfer cross section calculated using a multichannel Landau-Zener formalism (see text). The dashed curve is the two-state Landau-Zener cross section. Charge transfer calculations by van den Bos are indicated by triangles. Full circles and squares are the respective excitation channels as determined using the multichannel Landau-Zener model.

What is commonly called the three-element standard, or simply the standard solid (or Zener s solid), is a combination of either a Kelvin-Voigt element in series with a spring or, alternatively, a Maxwell element in parallel with a spring (see Fig. 10.6). The strain response of the first model to the stress input CT = cjoH(t) can be written as... 

To describe the response of the medium on a stress suddenly applied to a solid and held constant, it was introduced unrelaxed (subscript V) and relaxed (subscript R) stiffness and compliance coefficients. The unrelaxed quantities relate to immediate response while the relaxed ones are the coefficients after relaxation occurs. The process of relaxation is characterized with the relaxation time r. Proposed by Zener... 

This low misfit accompanied with very low solid solubility of Zr in A1 favors the stability of the precipitates. The reason of stability was showed by the study of peritectoid reaction (998+12 °C) ((3Zr)+Zr2Al <- Zr3Al [1], These small precipitates act to inhibit recrystallization and pin grain boundaries through a Zener drag process. 

Figure 7. The cross section for D" formation calculated taking the quantum interference into account (solid line). The dotted curve represents the experimental cross section. LZS stands for Landau-Zener-Stuckelberg. (Reproduced with permission from Ref. [55].)...

Figure 12 The Landau-Zener transition probability p in the LZ case. Dashed line exact, solid line Eqs. (170) and (183). (From Ref. 23.)...

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