travel at a significant fraction of the speed of light. Second, relativistic effects give rise to indirect consequences that are especially evident for d and f atomic orbitals. Qualitative treatment One of the most important and familiar results of relativity is that the relativistic mass of the electron increases as where are the electron rest mass, velocity of the electron, and speed of light respectively. The figure at the right illustrates this relativistic effect as a function of velocity. This has an immediate implication on the Bohr radius (), which is given by where is the reduced Planck's constant, and α is the fine-structure constant (a relativistic correction for the Bohr model). Arnold Sommerfeld calculated that, for a 1s orbital electron of a hydrogen atom with an orbiting radius of 0.0529 nm, α ≈ 1/137. That is to say, the fine-structure constant shows the electron traveling at nearly 1/137 the speed of light. One can extend this to a larger element with an atomic number Z by using the expression for a 1s electron, where v is its radial velocity, i.e., its instantaneous speed tangent to the radius of the atom. For gold with Z = 79, v ≈ 0.58c, so the 1s electron will be moving at 58% of the speed of light. Plugging this in for v/c in the equation for the relativistic mass, one finds that mrel = 1.22me, and in turn putting this in for the Bohr radius above one finds that the radius shrinks by 22%. If one substitutes the "relativistic mass" into the equation for the Bohr radius it can be written It follows that At right, the above ratio of the relativistic and nonrelativistic Bohr radii has been plotted as a function of the electron velocity. Notice how the relativistic model shows the radius decreases with increasing velocity. When the Bohr treatment is extended to hydrogenic atoms, the Bohr radius becomes where is the principal quantum number, and Z is an integer for the atomic number. In the Bohr model, the angular momentum is given as . Substituting into the equation above and solving for gives From this point, atomic units can be used to simplify the expression into Substituting this into the expression for the Bohr ratio mentioned above gives At this point one can see that a low value of and a high value of results in . This fits with