# What is the correct theoretical formula for the hyperfine splitting of neutral hydrogen (21cm line)

I found two different expressions to express the energy splitting of the hyperfine transition of neutral hydrogen (21 cm line, HI) that are not equivalent, imo.

From Wolfram ScienceWorld:
$$\Delta\ E= \frac{m_e}{m_p} \frac{ 4Z^3 \alpha^3 g_N }{n^3 (2l+1)(j+1))} \frac{1}{2} \begin{Bmatrix} I+\frac{1}{2},\text{ for}\ j\leqslant I \\ \frac{I(j+\frac{1}{2})}{j},\text{ for}\ j\geqslant I \end{Bmatrix} = K_0 .\tag{eq. 1}$$

From the first equation found on 21cm Cosmology, by Tzu-Ching Chang
$$\Delta\ E=\frac{4}{3}\alpha^4\ g_N\ \frac{m_e}{m_p} m_e c^2 = m_e c^2\ K_1 \ \propto m_e , \tag{eq. 2}$$

where the fine structure $\alpha=\frac{1}{4\pi\varepsilon_0} \frac{e^2}{\hbar\ c}$ is a dimensionless constant, and the light speed $c$ as well the ratio $\frac{m_e}{m_p}$ (electron to proton mass ratio) are also measured constants. The other parameters $\alpha,Z,g_N,g_p,n,l,j,I$ are dimensionless constants.

The eq. 1 appears to be incorrect because the LHS is energy and the RHS is dimensionless.

The eq. 2 has energy units in both sides, thus it seems more correct.

I would like to have an independent confirmation of the correct theoretical formula to to see if the splitting energy is proportional to the electron mass.

The question is important to understand if the electron (and atoms) of ancient past have an equal mass as the ones in the local universe (the lab, here and now).

For a simple example of atomic hyperfine splitting from the formal literature, a good place to go is

Hyperfine splitting in the ground state of hydrogen. David J. Griffiths. Am. J. Phys. 50, 698 (1982).

which includes a self-contained derivation. Griffiths gives the splitting, with a full roster of constants (i.e. nothing set to 1) as $$\Delta E_\mathrm{Griffiths}=\frac{\mu_0 \hbar^2 e^2}{6\pi a_0^3}\frac{g_eg_p}{m_em_p}, \tag 1$$ where $a_0$ is the Bohr radius and $g_e\approx 2.0023$ and $g_p\approx 5.5857$ are the gyromagnetic ratios for the electron and the proton, respectively. Substituting in $a_0=\hbar/(m_e c\alpha)$ for the Bohr radius and $$\frac{\mu_0 e^2}{4\pi}=\frac{1}{c^2}\frac{e^2}{4\pi\varepsilon_0} = \frac{\alpha \hbar}{c},$$ Griffith's result $(1)$ can be rephrased as $$\Delta E_\mathrm{Griffiths}=\frac{2g_e}{3}\alpha^4g_p\frac{m_e}{m_p}m_ec^2, \tag 2$$ which is consistent with the OP's second resource via setting $g_e\approx 2$.

On the other hand, the formula from Wolfram ScienceWorld can be traced rather easily to its second reference,

"Hyperfine Structure Splitting", §22 in Quantum Mechanics of One- and Two-Electron Atoms. Hans A. Bethe and Edwin Salpeter. (New York, Plenum, 1977).

where it reads $$\Delta E_\mathrm{Bethe} = \frac{m_e}{m_p} \frac{ 4Z^3 \alpha^2 g_N }{n^3 (2l+1)(j+1)} \mathrm{Ry} \begin{Bmatrix} I+\frac{1}{2}\text{ if}\ j\leq I \\ \frac{I(j+\frac{1}{2})}{j}\text{ if}\ j\geq I \end{Bmatrix} , \tag 3$$ where $$\mathrm{Ry} = hc R_\infty = \frac{m_e e^4}{32\pi^2\varepsilon_0^2 \hbar^2} = \frac{1}{2}\alpha^2m_e c^2 \approx 13.6 \:\mathrm{eV}$$ is the Rydberg unit of energy, equal to $1/2$ in atomic units. Bethe's result coincides almost exactly with the one from Wolfram ScienceWorld, except for the transformation $\alpha^2\mapsto \alpha^3$ (which appears to be a transcription error on the part of Wolfram ScienceWorld).

This explains the substitution $\mathrm{Ry} \to \frac12$ in the Wolfram ScienceWorld result, as well as the observation that puzzled the OP that that result appears to be dimensionless - it is simply stated in atomic units. Here ScienceWorld is at fault for not stating this fact more clearly, but if (within atomic physics) a result that should be an energy is stated as dimensionless, then the convention in the field is simply that it's in atomic units.

Putting in the explicit value of the Rydberg energy in $(3)$ (or, equivalently, translating from atomic units to SI units in the Wolfram ScienceWorld result) transforms it into $$\Delta E_\mathrm{Bethe} =\frac{4}{3}\alpha^4g_p\frac{m_e}{m_p}m_ec^2, \tag 4$$ which is similarly consistent with the Griffiths result, and identical to the OP's second resource.

Some final remarks:

• The OP's question of whether the hyperfine splitting is "proportional to $m_e$" is essentially meaningless, so long as it is not specified what other constants are held constant. If$m_e/m_p$ is held constant, then $\Delta E\propto m_e$, but if $m_p$ is held constant instead then $\Delta E \propto m_e^2$. The distinction is essentially meaningless, and if the OP wants to push their personal theories they can do so elsewhere.

• While Wolfram ScienceWorld is at fault for not specifying the use of atomic units, for someone in the field this is a very obvious candidate solution to the dimensionlessness observation, and the OP is remiss for not drawing that inference, since they are presumably already aware of the concepts of natural and atomic units if they want to challenge the accepted wisdom in related areas.

• The OP is certainly remiss in not following up on the references from the resources they quote, which are plenty to resolve the observed contradiction (and easy to find in open forms). The OP's reliance on an unsubstantiated statement in an online encyclopaedia and a set of slides is a red flag regarding resource management, and the claim that a formula "is used in Universities", referring to further slide sets, is also worrying. The golden standard is peer-reviewed literature (for all its flaws), and in this topic it is readily available and easy to find.

• About your remark: is "proportional to $m_e$" is essentially meaningless, so long as it is not specified what other constants are held constant. Please, it is stated in the OP: “where the fine structure α.. is a dimensionless constant, and the light speed c as well the ratio $m_e/m_p$ (electron to proton mass ratio) are also measured constants. The other parameters $\alpha,Z,g_N,g_p,n,l,j,I$ are dimensionless constants.” Your words: " If $m_e/m_p$ is held constant, then $\Delta\,E\propto\,m_e$”, imply that there is no way to distinguish the spectrum of a scaled atom and the one in motion. – Helder Velez Jun 6 '16 at 16:11
• @HelderVelez I will not address the content of the linked question beyond that remark. It is inconsistent, on the other hand, to regard $m_e/m_p$ as a "measured constant" but exclude $m_p$ and $m_e$ from that status, since both of the latter can be and have been measured. If you want to speculate on what would happen if they changed, please do so elsewhere - here I am only addressing the current question as originally posed. – Emilio Pisanty Jun 6 '16 at 16:19
• In atomic physics when dealing with the actual atoms in the lab it is useful to use the operative value of 13.6 eV as the Rydberg constant, but one should not use 'atomic units' - set to 1 the electron's mass - when investigating the parametric evolution of the relations. How can one measure anything else than the atom constancy if we are using itself as a standard? We are blinded to variations. You are correct saying that $m_p$ and/or $m_e$ are measured constants but only irt the local universe (lab). Provide a reference if you are able to assert the same irt the past (theory or measures). – Helder Velez Jun 6 '16 at 17:15
• I've accepted your answer but your final remarks and comments are not correct. $m_p$ and $m_e$ were indeed measured as you wrote but one can say noting about their constancy, and in the literature you can not find a reference stating or investigating their constancy. You can only 'presume' and jump to conclusions as you did or investigate as I did. The mass unit $kg$ was defined as a definite set of atoms, say $N$ (in S.I. is the prototype in Paris) and then we concluded the atomic unit is $1/N$ . This is a tautology which adds nothing about the constancy. – Helder Velez May 8 '18 at 15:34
• If the atom and the particles changed then we can not discover that with our way of measuring (if you measure your size with your palm you will find that you have never grown in your life, right?). The universe is in motion at the expense of the energy of the fields that expand from them. That energy comes from the particles themselves. To avoid a free lunch they must decrease their internal energy. My formal proof that the atoms shrink is in A self-similar model of the Universe unveils the nature of dark energy – Helder Velez May 8 '18 at 16:09

The following refs show that the eq.2 is used in Universities :
IMO the eq.1 is not reducible to the eq.2 .

$$\Delta\ E= \frac{ m_e c^2}{2} \left (\frac{Z\alpha}{n}\right )^4 n\times \frac{8}{3}\ g_N \frac{m_e}{m_p} \frac{1}{2}\begin{Bmatrix} I\\ -I-1 \end{Bmatrix} \propto m_e$$

Splitting of the Hydrogen Ground State in quantum mechanics at ucsd.edu $$\Delta\ E= \frac{4}{3} (Z\alpha)^4 \frac{m_e}{m_p} m_ec^2 g_N \frac{ 1 }{n^3} \propto m_e;$$

$Z,n,c,m_e/m_p,\alpha,g_N$ are measured constants.

Conclusion: The hyperfine splitting of Hydrogen (21 cm wavelength - HI line) is proportional to the electron mass.

The relevant part of this result is that the attribution of the red-shift of the light we receive from the past can be attributed to larger atoms (corresponding to a larger electron), as compared with the ones we can measure in the laboratory.

The evolution of the universe can be described with varying atoms in static space (see my profile).

In this way I show that the answer PSE - how to distinguish between the spectrum of an atom in motion and the one of a scaled atom is NOT correct.

I've searched for HI surveys to see if the observations are inconsistent with my viewpoint and found no contradiction.