Kronig-Penney potential

Antiferromagnetic one-dimensional Kronig-Penney potentials, 747 Antiferromagnetic single particle potential, 747 energy band for, 747 Antilinear operator, 687 adjoint of, 688 antihermitian, 688... 

Paramagnetic one-dimensional Kronig-Penney potentials, 748 Paramagnetic single particle potential, 747... 

The assumption of ZDO introduces periodicity into an otherwise constant potential free electron scheme, in the same way that the Kronig-Penney potential modifies the simple Sommerfeld model. 

In the original paper of Tamm (1932), the concept of surface state is demonstrated with a Kronig-Penney potential (Kittel, 1986) with a boundary, as shown in Fig. 4.5. By solving the Schrodinger equation, exphcit expressions for the surface states and their energy levels can be obtained. In... 

Fig. 4.5. Surface states. By solving the Schrodinger equation for a cut-off Kronig-Penney potential, it is found that in the energy gaps of the corresponding Kronig-Penney solid, there are surface states that decay exponentially into the vacuum and into the solid (Tamm, 1932). The explicit wavefiinction of a Tamm state with P = 15 and a = 3Aat = 5eV below the vacuum level is shown. The shaded areas represent allowed energy bands in the bulk.

Figure 6.3 The one-dimensional Kronig-Penney potential. The rectangular wells model of the interionic distances, whereas the barriers of a Vq height model the one-dimensional ion lattice.

One of the first models to describe electronic states in a periodic potential was the Kronig-Penney model [1]. This model is commonly used to illustrate the fundamental features of Bloch s theorem and solutions of the Schrodinger... 

Band Theory of Metals, Three approaches predict the electronic band structure of metals. The first approach (Kronig-Penney), the periodic potential method, starts with free electrons and then considers nearly bound electrons. The second (Ziman) takes into account Bragg reflection as a strong disturbance in the propagation of electrons. The third approach (Feynman) starts with completely bound electrons to atoms and then considers a linear combination of atomic orbitals (LCAOs). 

Figure 3.7. (a) Potential energy of an electron in a one-dimensional crystal (b) Kronig-Penney model of the potential energy of an electron in a one-dimensional crystal (square-well periodic potential model). 

Fermi-level DOS 115 Jellium model 92—97 failures 97 schematic 94 surface energy 96 surface potential 93 work function 96 Johnson noise 252 Kohn-Sham equations 113 Kronig-Penney model 99 Laplace transforms 261, 262, 377 and feedback circuits 262 definition 261 short table 377 Lateral resolution... 

Figure 5.4 shows the one-dimensional potential V(x) of the Kronig-Penney model, which comprises square wells that are separated by barriers of height,... 

Consider the simplest possible case, a monatomic crystalline solid. The potential at each lattice site is represented by a single square well in the Kronig-Penney model (Kronig and Penney, 1931) by Ralph Kronig (1904-1995) and William G. Penney (1909-1991). For a perfect monatomic crystaHine array (Fig. 7.3a), all the potential... 

The properties of electrons in a periodic potential are demonstrated with the use of a simple square well potential, the Kronig-Penney model. The potential is zero for 0 < x < a and Vq for —b < x < 0, i.e. it has a period (a + b). The Schrodinger equations for the two regions follow directly and lead on substitution of a Bloch function to ... 

The potential utilised in the Kronig-Penney model is not a very realistic representation of the potential function of an atomic lattice. A model that can incorporate more realistic potentials is the tight-binding model. The lattice... 

For the square well, taking into account that a variable separation is possible for a potential of the form V(r) = Vo(0(x)+0(y)), where 0 is the step-like function which is equal to 0 and 1 inside and outside the well respectively, one has two independent ID set of equations. Moreover, assuming that (r) = f(x) f y), the Bloch s boundary conditions can be split into two ones for f x) and as well. Finally, one has two independent ID Kronig-Penney problems for and Ky. Thus, for x-direction ... 

The influence of the periodic potential of the crystal lattice on the electronic structure is introduced through the one-dimensional Kronig-Penney model that illustrates the essential features of band theory of solids. 

Let us consider this simple but significant enough physical model of the electron movement in a field of periodic potential. Figure 6.3 illustrates the distribution of the one-dimensional potential V x) in the Kronig-Penney model. This potential comprises square wells that are separated by barriers of height Vo and thickness b. The potential is periodic with the period a so that... 

The model of a periodic crystalline potential is considered as a generalization of the effectively null potential model of Figure 3.13, by the alternation of the wells with potential barriers of a finite altitude V, as illustrated in Figure 3.21 (Kronig-Penney model), see (Further Readings on Quantum Solid, 1936-1967),... 

An important simple model that demonstrates some of the properties of electron states in periodic solids is the so called Kronig-Penney model. In this model, a particle of mass m experiences a one-dimensional periodic potential with period a ... 

This result was used by Kronig and Penney to define a one-dimensional square well potential for solids. 

Kronig and Penney made the assumption that potential energy is zero near the nucleus and is maximum V0 at the point half way between two neighbouring nuclei. Further, they assumed the product of the potential height and the width (V0w) as constant. They obtained the relation... 

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