Kramers states

At thermal equilibrium, Boltzmann statistics yields the familiar relation between the populations of the Kramers states... 

According to Kramers model, for flat barrier tops associated with predominantly small barriers, the transition from the low- to the high-damping regime is expected to occur in low-density fluids. This expectation is home out by an extensively studied model reaction, the photoisomerization of tran.s-stilbene and similar compounds [70, 71] involving a small energy barrier in the first excited singlet state whose decay after photoexcitation is directly related to the rate coefficient of tran.s-c/.s-photoisomerization and can be conveniently measured by ultrafast laser spectroscopic teclmiques. 

The transition from k to on the low-pressure side ean be eonstnieted using iiiidtidimensional unimoleeular rate theory [1, 44], if one knows the barrier height for the reaetion and the vibrational frequeneies of the reaetant and transition state. The transition from to k y ean be deseribed in temis of Kramers theory [45]... 

Berezhkovskii A M and Zitserman V Yu 1992 Generalization of the Kramers-Langer theory decay of the metastable state in the case of strongly anisotropic friction J. Phys. A Math. Gen. 25 2077-92... 

If all the PES coordinates are split off in this way, the original multidimensional problem reduces to that of one-dimensional tunneling in the effective barrier (1.10) of a particle which is coupled to the heat bath. This problem is known as the dissipative tunneling problem, which has been intensively studied for the past 15 years, primarily in connection with tunneling phenomena in solid state physics [Caldeira and Leggett 1983]. Interaction with the heat bath leads to the friction force that acts on the particle moving in the one-dimensional potential (1.10), and, as a consequence, a> is replaced by the Kramers frequency [Kramers 1940] defined by... 

Kramer MH, Sorhage FE, Goldstein ST et al (1998) First reported outbreak in the United States of cryptosporidiosis associated with a recreational lake. Clin Infect Dis 26 27-33... 

In other words, we have expressed the interaction between the adsorbate and the metal in terms of A(e) and /1(e), which essentially represent the overlap between the states of the metal and the adsorbate multiplied by a hopping matrix element A(e) is the Kronig-Kramer transform of A(e). Let us consider a few simple cases in which the results can be easily interpreted. 

From the Kronig-Kramer relation it immediately follows that A(e) = 0, as the function to be integrated is odd, and hence the resulting projected density of states becomes... 

Hamm UW, Kramer D, Zhai RS, Kolb DM. 1998. On the valence state of bismuth adsorbed on a Pt(lll) electrode—An electrochemistry, LEED and XPS study. Electrochim Acta 43 2969-2978. 

If the electric quadrupole splitting of the 7 = 3/2 nuclear state of Fe is larger than the magnetic perturbation, as shown in Fig. 4.13, the nij = l/2) and 3/2) states can be treated as independent doublets and their Zeeman splitting can be described independently by effective nuclear g factors and two effective spins 7 = 1/2, one for each doublet [67]. The approach corresponds exactly to the spin-Hamiltonian concept for electronic spins (see Sect. 4.7.1). The nuclear spin Hamiltonian for each of the two Kramers doublets of the Fe nucleus is ... 

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