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The Spatial Structure of Electrons in Atoms

The theory of the energy quantization of electrons in hydrogen-like atoms is well worked out. What is less certain is the spatial arrangement of those electrons. It is clear that the electrons are arranged in shells and that each of these shells is associated with an integer n called the principal quantum number. There are three other quantum numbers beside n which uniquely identify each state: 1. The orbital quantum number l 2. The magnetic quantum number m 3. The spin quantum number s. The orbital quantum number l can take on any of the 2m+1 integer values between −m and +m. The magnetic quantum number m can range from 0 to a possible maximum of (n-1). The spin quantum number can take on only the two values ±½. If the maximum value of m is reached the total number of states for a principal quantum number n would be twice the sum of the first (n-1) odd numbers. This would be 2n².

In practise in atoms m does not reach that maximum. For example energy of the state for n=3 and m=2 is greater than the energy for n=4 and m=0 so after the state for n=3 and m=1 is filled the next state to be filled is n=4 and m=0. A similar thing occurs for n=4. The energy of n=4 and m=3 is greater than n=5 and m=0. Thus the occupancies of the first through sixth electron shells are 2, 8, 8, 18, 18, 32.

The orbit radius r of an electron in a hydrogen-like atom, according to the Bohr model, is

r = n²h²/(KZ)

where n is the principal quantum number, h is Planck's constant divided by 2π, K is constant for the electrostatic force and Z is the net positive charge experienced by the electron. The net positive charge Z is the charge in the nucleus of the atom less the shielding due to the electrons in inner shells and the shielding by the other electrons in the same shell. The shielding factor for electrons in inner shells is roughly one and roughly one half for the other electrons in the same shell. Thus for the tenth electron in an atom the value of Z is 10-2-½(8-1)=4.5. The values of n and Z for the filled shells are given below.

ShellPrincipal
Quantum
Number n
Charge
in Nucleus
Shielding Due
to Inner Shell
Electrons
Shielding Due
to Same Shell
Electrons
Net Charge
Z
n²/Z
First1200.51.52/3
Second21023.54.58/9
Third318103.54.52
Fourth436188.59.532/19
Fifth554368.59.550/19
Sixth6865415.516.572/33

The last column n²/Z represents a quantity proportional to shell radius. The diagram below shows the relationship between shell occupancy and shell radius.

If the arrangement of electrons were annular (ring-like) then the occupancy would to proportional to the first power of the shell radius. If it were spherical or cylindrical the occupancy would be proportional to the second power of the shell radius. The exponent of radius in the relationship can be discerned by plotting the logarithm of occupancy versus the logarithm of radius. Such a plot is shown below.

The regression of the logarithm of occupany on the logarithm of radius give the following result.

ln(Shell Occupancy) = 1.75 + 1.48·ln(Shell Radius)

The regression coefficient of 1.48 is not statistically significantly different from 3/2. So the empirical results indicate a spatial structure for electrons in atoms that is somehow between ring-like and two dimensional like a spherical or cylindrical shell. Alternatively, and more likely, the results may be due to two dimensional arrangement like a spherical or cylindircal shell with decreasing areal densities at increasing order numbers for the shells.


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