Bohrs Model of the Nuclear Atom Thu, 04 Oct 2018 | Silicon Technology The structure of the atom is completely nanophysical, requiring quantum mechanics for its description. The radially diverging Coulomb electric field of a point charge Q, E = kQ/r2, leads to Gauss's Law of electrostatics: outward electric flux, where Q is the total electric charge inside the closed surface S (and the Coulomb constant kC = 1/(4peo)). However, the model assumed electrons in the shells didn't interact with each other and couldn't explain why electrons seemed to stack in an irregular manner. The details of the observed diffraction patterns could be fitted if the wavelength of the electrons was exactly given by h/p. The factor in brackets above (4.32) relates to the speed of the light wave. Following Maxwell, in free space, the relevant equations (assuming no currents and no charge density) are (i) VxE = -3B/9t, and (ii) VxB = eoio9E/9t. It also explained why the noble gases were inert and why atoms on the left side of the periodic table attract electrons, while those on the right side lose them. Schrodinger Equation, Quantum States and Energies, Barrier Tunneling. So a possible linear combination solution is, W1+3(x,t) = A[sin(px/L) exp(-iE1t/") + sin(3px/L) exp(-iE3t/")]. Expectation Values. Figure 4.1 Sketch of transverse electric modeTE10 in a rectangular metallic waveguide. According to the Bohr model, an atom consists of a nucleus of protons and neutrons surrounded by orbiting electrons at various distances from the nucleus called shells. The simplest problem is a trapped particle in one dimension. Bohr's postulate of 1913 was ofthe quantization ofthe angular momentum L ofthe electron of mass m circling the nucleus, in an orbit of radius r and speed v: Here n is the arbitrary integer quantum number n = 1,2 Note that the units of Planck's constant, J-s, are also the units of angular momentum. Overview of the Bohr Model Niels Bohr proposed the Bohr Model of … W = A[cos(k1x-x1t) + cos(k2x-x2t)] = 2Acos[(1/2)Akx-(1/2)Axt]cos(kavx-xavt). Thus, the Bohr model for heavier atoms described electron shells. In addition, the first time-derivative 9 W(x,t)/9t =-ix W(x,t) is needed, in order to produce the "x in the statement of conservation of energy, which in (4.33) appears in the bracket factor. (4.38), This has the same form as the simple harmonic oscillator equation (2.1) discussed earlier, so the solutions (2.2) and (2.3), similarly, can be written as, The infinite potential walls at x = 0 and x = L require w(0) = W(L) = 0, which means that B = 0. In one branch of modern silicon device development, "wave transistor" devices are fabricated in nearly perfect semiconductors, where electrons behave as waves predicted by Schrodinger's equation (in the absence of scattering) and have electron probability plots that qualitatively resemble those shown for electromagnetic waves in Figure 4.2. Silicon, a nonmetallic chemical element in the carbon family that makes up 27.7 percent of Earth’s crust; it is the second most abundant element in the crust, being surpassed only by oxygen. where h is Planck's constant, and p = mv is the momentum. The Bohr model is equally applicable to donor impurity atoms in semiconductor physics, where the conduction band corresponds to the vacuum, and the loosely bound electron orbiting a singly charged core has a hydrogen-like spectrum according to the usual Bohr–Sommerfeld formula, shifted to the far-infrared because of the small effective mass and high dielectric constant. This is true because the kinetic energy is always -0.5 times the (negative) potential energy in a circular orbit, as can be deduced from the mentioned force balance. Figure 4.1 shows the electric E and magnetic H fields in a rectangular waveguide, for propagation of the lowest frequency TE10mode, in the positive z-direction. (4.42). A point of fixed phase (such as a peak in the real part of the wave) moves as x = (x/k)t, so (x/k) is called the phase velocity, W(x,t) = L-1/2e(ikx-ixt) = L-1/2exp(ikx-ixt) (4.16). The Bohr Model is a planetary model in which the negatively charged electrons orbit a small, positively charged nucleus similar to the planets orbiting the sun (except that the orbits are not planar). The optical fiber presents a situation for the electric field E in which the analogy with the wavefunction of quantum mechanics is clear. Energy Bands Consider an isolated silicon atom; its energy levels are quantized (see the Bohr model for Hydrogen). Maxwell's equations also accurately describe electromagnetic modes in confined geometries such as waveguides for microwaves. (4.20b), J. C. Maxwell discovered (by restoring a missing current density term, eo9E/9t, into the Ampere's Law expression) that propagating electromagnetic waves are implied by these experimental laws. For light p = E/c, so the relation k = h/p can be read as k = hc/E = c/m. Particle-wave Nature of Light and Matter, DeBroglie Formulas k = h/p, E = hm, One of the most direct indications of the wave nature of light is the sinusoidal interference pattern of coherent light falling on a screen behind two linear slits of small spacing, d. The rule for appearance of maxima at angular position h in the interference pattern, nk= dsinh, (4.5). The radius of the possible orbits increases as n2, where n is the principal quantum number. Complex Numbers x+iy. 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