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In contrast to the traditional derivation based on microscopic reversibility, that given here does not invoke equilibrium. It thus shows that no net circular reaction is possible even under non-equilibrium conditions. However, there is one very important qualification A net circular reaction does occur if it entails the conversion of co-reactants to co-products of lower Gibbs free energy. For example, the cycle 2.13 (next page) converts reactants A and B to product P while undergoing a net circular reaction K— L— M — N — K. This is a typical catalytic cycle. Such a cycle is not a loop, a term to be reserved for circular pathways in which any... 

To understand this, consider a free electron with energy, e, and wave vector, k. As shown on the next page in 5.3.1., e varies with k. This curve is a potential energy diagram of the free electron. Note that we have converted sin 9 (the linear sinusoidal function) into jt (the radial function). If we apply a mono-atomic lattice, arranged in a linear manner, and having a lattice constant of a, to our electron, we can show that Bragg reflection occurs at ... 

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