Tunneling Landau-Zener

The problem of nonadiabatic tunneling in the Landau-Zener approximation has been solved by Ovchinnikova [1965]. For further refinements of the theory beyond this approximation see Laing et al. [1977], Holstein [1978], Coveney et al. [1985], Nakamura [1987]. The nonadiabatic transition probability for a more general case of dissipative tunneling is derived in appendix B. We quote here only the result for the dissipationless case obtained in the Landau-Zener limit. When < F (Xe), the total transition probability is the product of the adiabatic tunneling rate, calculated in the previous sections, and the Landau-Zener-Stueckelberg-like factor... 

Figure A.l. Schematic adiabatic potentials and various parameters used in the ZN formulas, (a) Landau-Zener type, (b) Nonadiabatic tunneling type. Taken from Ref. [9].

On the basis of a Landau-Zener curve crossing formalism, Borgis and Hynes derived the nonadiabatic rate constant k, which is similar to that expressed by the DKL model but where the tunneling term Cnm found in Eq. (4) is significantly modified due to the influence of the low-frequency promoting mode Q, with a frequency 0)q, on the tunneling rate. The dependence of Cnm on Q is given by [13]... 

Figure 7. Detail of the energy level diagram near an avoided level crossing, m and m are the quantum numbers of the energy level. Pm m- is the Landau-Zener tunnel probability when sweeping the applied field from the left to the right over the anticrossing. The greater the gap A and the slower the sweeping rate, the higher is the tunnel rate, Eq. (2).

In order to apply quantitatively the Landau-Zener formula Eq. (2), we first saturated the crystal of Fes clusters in a field of Hz = -1.4 T, yielding an initial magnetization Min = -Ms. Then, we swept the applied field at a constant rate over one of the resonance transitions and measured the fraction of molecules which reversed their spin. This procedure yields the tunneling rate P 1010 and thus the tunnel splitting A 1010 , Eq. (2), with n = 0, 1, 2,. 

An applied field in the xy-plane can tune the tunnel splittings Amm. via the Sx and Sy spin operators of the Zeeman terms that do not commute with the spin Hamiltonian. This effect can be demonstrated by using the Landau-Zener method (Section 3.1). Fig. 8 presents a detailed study of the tunnel splitting A 10 at the tunnel transition between m - +10, as a function of transverse fields applied at different angles (p, defined as the azimuth angle between the anisotropy hard axis and the transverse field (Fig. 4). 

Landau-Zener method can be used to probe the transverse dipolar distribution by measuring the tunnel splittings A around a topological quench. 

The basic assumptions of the Landau-Zener theory need to be satisfied. These involve the applicability of classical mechanics (e.g. the neglect of tunneling) for the nuclear dynamics and the locality of the curve crossing event. 

The calculation of the electronic-transmission factor currently involves three different methods, viz. the Landau-Zener formula, Fermi s golden mle [35], and electron tunneling formalism such as the Wentzel-Kramer-Brillouin method [36]. We used the Landau-Zener formula [37,38] to calculate it ... 

Figure 2 Two basic cases of curve crossing (a) Landau-Zener (LZ) case, (b) nonadiabatic tunneling (NT) case.

It can then be calculated by using the Landau-Zener theory as discussed in Sec.6.1.II. In general, however, nuclear motion should be treated quantum-mechanically hence, the tunneling of solvent molecules must be taken into account, as done in the considerations in Sec.6.2 II. The reaction probability, i.e., the probability of an electron... 

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