The energy differences between levels in the Bohr model, and hence the wavelengths of emitted or absorbed photons, is given by the Rydberg formula: \frac{1}{\lambda} = Z^2 R_M \left( \frac{1}{{n_1}^2} - \frac{1}{{n_2}^2} \right) where For helium, Z=2, the Pickering-Fowler series is for n_1=4 and the reduced mass for {}_2^4\text{He}^{+} is \mu=\frac{1}{\frac{1}{m_e}+\frac{1}{2m_p+2m_n}} thus \frac{\mu}{m_e}=\frac{1}{1+\frac{m_e}{2m_p+2m_n}}\approx 0.99986396 , which is usually approximated as 1 (in fact, although this number changes for each
isotope of helium, it is approximately constant). A more accurate description may be used with the
Bohr–Sommerfeld model of the atom. The theoretical limit for the wavelength in the Pickering-Fowler is given by: \lambda_\infty^\text{PF} = \frac{4}{R_\infty}, which is approximatedly 364.556 nm, which is the same limit as in the
Balmer series (
hydrogen spectral series for n_2=2). Notice how the transitions in the Pickering-Fowler series for n=6,8,10 (6560Å ,4859Å and 4339Å respectively), are nearly identical to the transitions in the Balmer series for n=3,4,5 (6563Å ,4861Å and 4340Å respectively). The fact that the Pickering-Fowler series has entries inbetween those values, led scientist to believe it was due to hydrogen with half transitions ("half-hydrogen"). However, Niels Bohr showed, using his model, it was due to the singly ionised helium {}_2\text{He}^{+}, a
hydrogen-like atom. This also shows the predictability of Bohr model. ==References==