**Reference**: John S. Rigden, Hydrogen: *The Essential Element*,
Harvard University Press, 2002

One line in hydrogen’s spectrum is
actually a set of distinct lines. To explain this fine structure, Arnold Sommerfeld,
in 1916, modified Bohr’s theory by including elliptical orbits for electrons.
(to which he also applied Einstein’s theory of relativity). The resulting
energy states of the hydrogen atom were expressed in terms of the dimensionless
constant = 1/137.03599976, and it was named the *fine structure constant*.

It turns out that surfaces in many contexts, in so many, that
along with the speed of light, *c*; *G*; and Planck’s constant, *h*, considered by
many physicists to be one of the fundamental constants of nature[1]. For example,

a is related to the charge of the electron*^{2} by^{}

**Equation(1)** ,

where k =
Coulomb’s constant = 8.9875 X 10^{9} Nm^{2}/C^{2 }

e
= charge of the electron = 1.60217733 X10^{-19} C

h
= Planck’s constant = 6.6260755 X 10^{-34} Js

c
= speed of light = 2.99792458 X 10^{8} m/s,

and the fine structure constant is also related to the mass of the neutron by

**Equation(2)** , where r_{n}
= radius of the neutron= 2.87799 X 10^{-14 }m

m_{n}
= mass of neutron = 1.6749543 X 10 ^{–27} kg.

** Question**: Given

where m_{e}
= electron’s mass = 9.1093897 X 10^{-31} kg,

use basic algebra to express the product of the electron’s mass and radius in terms of the neutron’s mass and radius.

** Answer**: Isolate k

: equation(4).

Substitute equation(4) into eq(2) :

*h ^{2} = 4*

[1] In *Scientific American* June 2005, Barrow and Webb argue that during the
universe’s matter dominated era (we are now in a dark-energy dominated era),
the fine structure constant’s value actually increased. The idea came from
comparing highly accurate measurements of present spectra to those of quasars,
a window into conditions found 7.5
billion years ago.

[2] The fine structure constant is also
expressed as a= e^{2}/(2e_{0}hc), where e_{0} is the vacuum permitivity
constant, which is related to Coulomb’s constant according to e_{0}=
1/(4pk)

*Page
Maintained by Enrico Uva
Technical Asistance: C. Frizzell; and M.Pololos, D.Verrillo and K.Papoulias at
EMSB
Comments: euva@retired.ca
or
euva@prof.emsb.qc.ca *

.

*Copyright
©2006*

[1] In *Scientific American* June 2005, Barrow
and Webb argue that during the universe’s matter dominated era (we are now in a
dark-energy dominated era), the fine structure constant’s value actually
increased. The idea came from comparing highly accurate measurements of present
spectra to those of quasars, a window into conditions found 7.5 billion years ago.