Two-state

In an electron spin resonance spectrometer, transitions between the two states are brought about by the application of the quantum of energy hv which is equal to g H. The resonance condition is defined when hv = g H and this is achieved experimentally by varying H keeping the frequency (v) constant. Esr spectroscopy is used extensively in chemistry in the identification and elucidation of structures of radicals. 

The integral A/, while expressible in terms of surface free energy differences, is defined independently of such individual quantities. A contact angle situation may thus be viewed as a consequence of the ability of two states to coexist bulk liquid and thin film. 

The preceding derivation, being based on a definite mechanical picture, is easy to follow intuitively kinetic derivations of an equilibrium relationship suffer from a common disadvantage, namely, that they usually assume more than is necessary. It is quite possible to obtain the Langmuir equation (as well as other adsorption isotherm equations) from examination of the statistical thermodynamics of the two states involved. 

Thus the kinetic and statistical mechanical derivations may be brought into identity by means of a specific series of assumptions, including the assumption that the internal partition functions are the same for the two states (see Ref. 12). As discussed in Section XVI-4A, this last is almost certainly not the case because as a minimum effect some loss of rotational degrees of freedom should occur on adsorption. 

One should also notice that resonant excitation completely cycles the population between the lower and upper state with a period of 2ji/0. Non-resonant excitation also cycles population between the states but never completely depopulates the lower state. Finally, one should notice that non-resonant excitation cycles population between the two states at a faster rate than resonant excitation. 

Let us consider first the two-state model of non-mteracting spin-i particles in a magnetic field. For a system... 

This behaviour is characteristic of any two-state system, and the maximum in the heat capacity is called a Schottky anomaly. 

Figure A2.2.1. Heat capacity of a two-state system as a function of the dimensionless temperature, lc T/([iH). From the partition fimction, one also finds the Helmholtz free energy as...

It is also instructive to start from the expression for entropy S = log(g(A( m)) for a specific energy partition between the two-state system and the reservoir. Using the result for g N, m) in section A2.2.2. and noting that E = one gets (using the Stirling approximation A (2kN)2N e ). 

Figure Bl.1.1. (a) Potential curves for two states with little or no difference in the equilibrium position of tire upper and lower states. A ttansition of O2, witli displacement only 0.02 A, is shown as an example. Data taken from [11]. Most of the mtensity is in the 0-0 vibrational band with a small intensity in the 1-0 band, (b) Potential curves for two states with a large difference in the equilibrium position of the two states. A ttansition in I2, with a displacement of 0.36 A, is shown as an example. Many vibrational peaks are observed.

If the experunental technique has sufficient resolution, and if the molecule is fairly light, the vibronic bands discussed above will be found to have a fine structure due to transitions among rotational levels in the two states. Even when the individual rotational lines caimot be resolved, the overall shape of the vibronic band will be related to the rotational structure and its analysis may help in identifying the vibronic symmetry. The analysis of the band appearance depends on calculation of the rotational energy levels and on the selection rules and relative intensity of different rotational transitions. These both come from the fonn of the rotational wavefunctions and are treated by angnlar momentum theory. It is not possible to do more than mention a simple example here. 

They are caused by interactions between states, usually between two different electronic states. One hard and fast selection rule for perturbations is that, because angidar momentum must be conserved, the two interacting states must have the same /. The interaction between two states may be treated by second-order perturbation theory which says that the displacement of a state is given by... 

These wavefiinctions are orthogonal. Assume that the system is initially in state 1, and that the interaction begins at = 0. Since there are only two states, at any later time the wavefiinction for the system is... 

The Landau-Zener transition probability is derived from an approximation to the frill two-state impact-parameter treatment of the collision. The single passage probability for a transition between the diabatic surfaces H, (/ ) and R AR) which cross at is the Landau-Zener transition probability... 

Because NH has an electron affinity of 0.4 eV, the total energies of the above two states can be equal only if the kinetic energy KE carried away by the ejected electron obeys... 

The introduction of EOMs for energy differences and for operators that connect two states appears first in the nuclear physics literature see for example ... 

As early as 1969, Wlieeler and Widom [73] fomuilated a simple lattice model to describe ternary mixtures. The bonds between lattice sites are conceived as particles. A bond between two positive spins corresponds to water, a bond between two negative spins corresponds to oil and a bond coimecting opposite spins is identified with an amphiphile. The contact between hydrophilic and hydrophobic units is made infinitely repulsive hence each lattice site is occupied by eitlier hydrophilic or hydrophobic units. These two states of a site are described by a spin variable s., which can take the values +1 and -1. Obviously, oil/water interfaces are always completely covered by amphiphilic molecules. The Hamiltonian of this Widom model takes the form... 

Moerner W E, Piakhotnik T, Irngartinger T, Croci M, Palm V and Wild U P 1994 Optical probing of single molecules of terryiene in a Shpol skii matrix a two-state single-molecule switch J. Chem. Phys. 98 7382-9... 

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