## Calculations – Photon Wavelengths

**Hydrogen Wavelengths (Ionization)**

Using the photon wavelength equation, the wavelengths of photons absorbed during hydrogen ionization were calculated for the ground state and each of the excited states (orbitals 1, 2, 3, 4, 5, 6). Calculations are compared to measured results for photons wavelengths of hydrogen.

**Hydrogen Wavelengths (Shell Transition)**

Using the photon wavelength equation, the wavelengths of photons emitted during hydrogen for an electron transitioning from an excited state (orbitals 3, 4, 5, 6, 7, 8, 9) to the second (2) orbital. Calculations are compared to measured results for photons wavelengths of hydrogen. For example, as the electron transitions from the 3s orbital to 2s orbital (3->2) it emits a photon calculated to be 6.56E-07 meters.

## Calculations – Photon Energy

**Ionization Energy of the 1s Electron – Neutral Element (from Spectroscopy Experiments)**

In this section, the photon energy of the 1s electron is described. A pattern emerges for the first orbital of elements from hydrogen to calcium that allow a simplified method reducing two variables to one. In these calculations, the Amplitude Factor Equation – 1s Orbital Ionization is used to solve one variable without knowing orbital distance. Instead, the Bohr radius is used as the orbital distance when this equation is used. *This equation only applies to the 1s orbital.* *

**Ionization Energy of Hydrogen to Calcium – All Orbitals**

The photon energies for orbitals beyond the 1s orbital requires the electron distance to be known. These calculations are placed in the Atoms section of this web site after an explanation and calculation of electron distances. The complete list of calculated photon ionization energies from hydrogen to calcium, for all orbitals, is here.

## Example Calculation – Photon Wavelength

**Hydrogen – Orbital Transition from 3s -> 2s**

In the Photon Wavelength page, an example was provided for hydrogen ground state ionization. In this example, an electron changes orbitals but does not leave the atom in the case of ionization. When the electron transitions to a lower shell, it emits a photon. The calculation in this case yields a positive result, noting that a photon is emitted. When a photon is absorbed and the electron transitions to a higher shell or is ejected from the atom (ionization), the calculation yields a negative result.

The hydrogen photon wavelength calculations graph and table (above) contains calculations of electron transitions from various orbitals to the second orbital. An example calculation is provided below, as an example of transitioning from the third orbital (3) to the second orbital (2). This is represented by: 3->2. The amplitude factor for hydrogen is one. The distances of the orbitals are a function of the Bohr radius and the principal quantum number. The distances are 3^{2}(a_{0}) for the third orbital and 2^{2}(a_{0}) for the second orbital. Both the wave format and classical formats arrive at the same result, so only one version is shown here (wave format).

**Equation: **Photon Wavelength Equation – Wave Format

**Variables:**

- δ
_{H}= 1 - r
_{0}= 3^{2}(a_{0}) = 4.7626E-10 meters (m) - r = 2
^{2}(a_{0})**=**2.1167E-10 meters (m)

**Result:**6.561E-7 meters (m)

**Comments:** 0.026% from measured results.

## Example Calculation – Photon Energy

**Boron – 1s (***using Amplitude Factor Equation – 1s orbital*)

*using Amplitude Factor Equation – 1s orbital*)

In photon energy calculations (above), calculations are charted for ionized and neutral elements. Energy levels of photons are calculated using the Transverse Energy Equation. The three variables in the equation are the initial distance (r_{0}), final distance (r) and the amplitude factor (δ).

The Transverse Energy Equation requires distance to be known. An alternative method for calculating the 1s electron ionization can be performed using the Amplitude Factor Equation – 1s Orbital equation without knowing the distance. Instead, distance is set to the Bohr radius with this special equation for 1s electrons. The results for neutral atoms are found in the last row in the Amplitude Factor table, but the equation will also be shown again to solve for the amplitude factor for the 1s^{2} electron for boron. For boron, Z=5 for the number protons, e_{1}=2 and e_{2}=3 for the first and second orbital electrons.

Using this method, the initial orbital distance is set to the Bohr radius (a_{0}=5.2918E-11 m) in the Transverse Energy Equation. The final distance (r) is set to infinity because the electron is ionized (leaves the atom). The remaining are wave constants. A negative sign in the result indicates that a photon is absorbed.

**Equation: **Transverse Energy Equation – Wave Format

**Variables:**

- δ
_{B1s}= 14.69 (*from above)* - r
_{0}= a_{0}= 5.2918E-11 m (*Bohr radius used as distance only for Amp Factor Eq. – 1s orbital)* - r = infinity (
*leaves the atom*)

**Result: **-19.3 megajoules per mole (MJ/mol)

**Comments:** It is converted from joules to megajoules per mole using **Avogadro’s number (N _{A}) and dividing by 10^{6}**. This matches the measured result for boron 1s spectroscopy which is -19.3 Mj/mol.

**Source Data: **All graphs shown here, and the calculations for all of the photon wavelengths and ionization energies can be found in the downloadable spreadsheet. Further information on the derivation of the equations and how to replicate them are in the Particle Energy and Interaction and Atomic Orbitals papers.