Calculations – Amplitude Factors

Calculations – Amplitude Factors

Hydrogen to Calcium

The amplitude factor is a measurement of constructive or destructive wave interference.  When one or more particles are located in two groups at a single distance (r), the rule for amplitude factor is simple.  The waves are added or subtracted based on the positive or negative charge of the particle where a single proton-electron combination is one.  In this configuration, the amplitude factor is Z-e+1 where Z is the number of protons and e is the number of electrons.

Beyond the 1s orbital, it becomes more complicated as electrons have varying distances, yet they affect each other. However, the amplitude factor resembles the shape and structure of the orbitals.  There is a pattern for s subshells and p subshells, and further, the p subshell is split into two parts based on the spin of the proton 2p^[1-3] (spin up) versus 2p^[4-6] (spin down).  The following are the equations for determining the amplitude factors.

Amplitude Factor Equations

There is a special case for amplitude factors for the 1s orbital that does not require distance to be known.  Instead, distance can be set to the Bohr radius when using the following equation (only valid for the 1s orbital). In this equation, e_x is the total number of electrons in each shell. See the Atomic Orbitals paper for details.

Amplitude Factor Equation – 1s orbital

Neutral Atoms

Using the above amplitude factor equations, the following table was produced. The last row in the table (1s*) is a special Amplitude Factor Equation – 1s Orbital that is used when the distance is not calculated (and the Bohr radius is used instead).

Ionized Atoms

Using the amplitude factor equations, ionized atoms can be determined by modifying the number of protons (Z) and electrons (e).  These tables for all of the ionized atoms are not inserted here, but they can be found in the spreadsheet with all of the calculations from this web site.

Example Calculations

Boron 1s*

* Using the Amplitude Factor Equation – 1s Orbital that only requires one variable for the Transverse Energy Equation, which is fixed as the Bohr radius as the distance.

Equation: Amplitude Factor Equation – 1s Orbital (from above)
Variables:

• Z = 5 (protons)
• e₁ = 2 (electrons in the first shell)
• e₂ = 3 (electrons in the second shell)

Result: 14.69
Comments: The value was inserted into the table above in the 1s* row for boron (B). The example calculation of ionization energy for boron 1s, using this special equation setting distance to Bohr radius, is here.

Boron 2p

* Using the Amplitude Factor Equation – 1s Orbital that only requires one variable for the Transverse Energy Equation, which is fixed as the Bohr radius as the distance.

Equation: Amplitude Factor 2p [1-3] (from table above)
Variables:

• Z = 5 (protons)
• e = 5 (total electrons)
• r_x = 1.41 (from Boron 2p distance in distance table)

Result: 0.8546
Comments: The value was inserted into the table above in the 2p row for boron (B). The example calculation of ionization energy for boron 2p is here.

Source Data: All graphs and tables shown here for orbitals distances can be found in the downloadable spreadsheet.  The solutions for orbitals were first generated with Mathcad, available on the same page, and then placed into the spreadsheet. Further information on the derivation of the equations and how to replicate them is in the Atomic Orbitals paper.