Bohr orbit, radius

The dimensions of the polarizability a are those of volume. The polarizability of a metallic Bphere is equal to the volume of the sphere, and we may anticipate that the polarizabilities of atoms and ions will be roughly equal to the atomic or molecular volumes. The polarizability of the normal hydrogen atom is found by an accurate quantum-mechanical calculation to be 4.5 ao that is, very nearly the volume of a sphere with radius equal to the Bohr-orbit radius a0 (4.19 a ). 

Here, an is the Bohr orbit radius of the isolated center and nc is the critical carrier density at the M-NM transition. Another way of viewing the transition is that of an electronic instability which ensues when the trapping of an electron into a localized level also removes one electron from the Fermi gas of electrons. This must clearly lead to a further reduction in the screening properties (which are themselves directly related to the conduction electron density) and a catastrophic situation then ensures the localization of electrons from the previously metallic electron gas. 

Solution of the Schrodinger equation for R i r), known as the radial wave functions since they are functions only of r, follows a well-known mathematical procedure to produce the solutions known as the associated Laguerre functions, of which a few are given in Table 1.2. The radius of the Bohr orbit for n = 1 is given by... 

Thus, for the hydrogen atom (Z = 1) the most probable distance of the electron from the nucleus is equal to the radius of the first Bohr orbit. 

The effect of the spin-orbit interaction term on the total energy is easily shown to be small. The angular momenta L and S are each on the order of h and the distance r is of the order of the radius ao of the first Bohr orbit. If we also neglect the small difference between the electronic mass We and the reduced mass the spin-orbit energy is of the order of... 

According to De Broglie an electron in a Bohr orbit is associated with a standing wave. To avoid self destruction by interference an integral number of wavelengts are required to span the orbit of radius r, which implies n — 2nr, or nh/2n = pr, which is the Bohr condition. As a physical argument the wave conjecture is less plausible, but not indefensible. One possible interpretation considers the superposition of several waves rather than a single monochromatic wave to simulate the behaviour of a particle. 

The unit of separation is the Bohr li orbit radius of hydrogen atom, that of energy is the ionization potential of atomic hydrogen. 

Electronic and nuclear energy in H2. a. Values for non-interacling electrons. 6, Coulomb energy of nuclear repulsion, c, Approximate electronic energy curve for interacting electrons. Units ordinates, 1 = Rydberg constant, abscissas, 1 = radius of first Bohr orbit in hydrogen atom. 

The atomic unit of length is the radius of the first Bohr orbit in the hydrogen atom when the reduced mass of the electron is replaoed by the rest mass tne. Thus the atomic unit of length is... 

These two equations are easily solved. It is found that the radius of the circular Bohr orbit for quantum number n is equal to W/4xaZnt. This can be written as n ao/Zy in which a0 has the value 0.530 A. The speed of the electron in its orbit is found to be v = 2irZe2/nh. For the normal hydrogen atom, with Z = 1 and n = 1, this speed is 2.18 X 108 cm/sec, about 0.7 percent that of the speed of light. 

If the relativistic effects are sufficiently large and therefore cannot be accounted for as corrections, then as a rule one has to utilize relativistic wave functions and the relativistic Hamiltonian, usually in the form of the so-called relativistic Breit operator. In the case of an N-electron atom the latter may be written as follows (in atomic units, in which the absolute value of electron charge e, its mass m and Planck constant h are equal to one, whereas the unit of length is equal to the radius of the first Bohr orbit of the hydrogen atom) ... 

Show by substitution in the formula given in the text ( Interaction of Light with Matter ) that ao, the radius of the first Bohr orbit for hydrogen, is 5.29 x 10-11 m. 

When an electron of mass m and charge — e moves with tangential velocity v and angular velocity [Pg.182]

From the point of view of the electron rotating with tangential velocity v in a circular Bohr orbit of radius r around a nucleus (or the center of mass), the nucleus appears to rotate around the electron with a tangential velocity —v and to generate a current )int ... 

Length Radius of first Bohr orbit (1 bohr = a0) 107(/i/27iec)2me 1 5.2917725 x 10-11m... 

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