|San José State University|
& Tornado Alley
of a Muonic Hydrogen Atom
and an Electronic One
Experimental physicists have found an interesting puzzle. Protons can form hydrogen atoms with muons as well as electrons. The puzzle is that the measured dimensions of the proton in a muonic hydrogen atom are smaller than than those of a proton in an electronic hydrogen atom.
First note that a hydrogen atom revolves at such a rapid rate, many millions of times per second, that all that can be observed about the proton is its dynamic appearance. In the case of the proton in a hydrogen atom that may be an oblate spheroid or a torus dpending upon the orbit radius relative to the radius of the particle. .
An electron has a mass only 1/1836 of the mass of a proton so the center of mass of a proton-electron atom is close to the center of the proton and hence the proton has relatively little motion compared to that of the electron. The muon, on the other hand, has a mass 208 times that of an electron so its mass is approximately 1/9 of the mass of a proton. The center of mass of the muon-proton system is distinctly outside of the proton and the proton circles that center of mass just as the muon does.
The subatomic particles execute periodic motion so rapidly that all that can be observed of them is their dynamic appearance. What follows here is an analysis which is essentially an extension of Niels Bohr's analysis of a hydrogen atom. That analysis is marvelously successful in explaing the spectrum of hydrogen. However the analysis is put in most general terms so it applies to positronium as well as electronic and muonic hydrogen atoms.
Let the characteristics of the two particles be labeled 0 and 1. The masses, orbit radii and magnitudes of the charges are given as m, r and q, respectively. It is assumed that the charges of the particles are opposite. The force of attraction between them is given by
The balances of that attractive force with the so-called centrifugal forces are given by
where ω is the rate of rotation of the system.
The second equality above implies
This is just the condition for determining the center of mass of the system.
The angular momentum of the two particle system is quantized as
where h is Planck's constant divided by 2π and n is a positive integer.
This condition may be expressed more succinctly as
where J is the moment of inertia of the system and J=m0r0² + m1r1².
Note that, since m1r1=m0r0
The kinetic energies of the particles are
The total kinetic energy of the system K is given by
The potential energy of the system is kq1q0/(r0+r1). This can be reduced to
Since ω is equal to L/J
The total energy of the system can then be expressed as
If this equation is divided by m0 the result is
This is a quadratic equation in 1/(m0r0). The expression (1/m0 + 1/m1) can be represented as 1/μ, where μ is called the reduced mass of the system. The equation determining (1/m0r0) is then
The equation determing (1/m1r1) is the same, as it should be since m1r1 = m0r0.
The solutions to this equation are
The analysis can be limited to the minimum case of n=1 and the + of the ±. For a hydrogen atom q1q0=e² where e is
the electrostatic charge. Furthermore ke² can be represented as α
hc where α is the fine structure constant,
h is Planck's constant divided by 2π and c is the speed of light. These specializations reduce the equation to
This may be rearranged to
Now let r1e and r1m be the values of r1 for the radii of the proton's orbit for the electron and the muon in a hydrogen atom. From the above formula
Let particle 1 be the proton. When particle 0 is an electron the reciprocal of the reduced mass of the atom is proportional to 1+1/1836. When it is a muon the recirocal of reduced mass is proportional to 1/9+1/1836. This is a reduction to roughly 1/9. Thus
As with intuition the proton has a much larger orbit radius with a muon than does the proton with an electron. So the puzzle is even more extreme.
(To be continued.)
Dedicated to my long term friend Lydia
for alerting me to this puzzle
but with more graditude for her perfecting
the department she was chairman of.
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