formula gives the angular dependence of
the scattering. In Raman scattering, the
unchanged line (incident frequency) is
called the Rayleigh line.
the Rayleigh-Jeans law
In an 1889 paper on the character of the
complete radiation at the given temperature, Rayleigh defines complete radiation (black-body radiation) as radiation
that, in a steady state, exists in a sealed
enclosure whose walls are impervious to
radiation and are maintained at a given
temperature. Previously, in 1862, Gustav
R. Kirchoff first gave a similar definition
of black-body radiation.
Rayleigh derived his law of black-body radiation by assuming that standing waves in a cavity all have the same
energy; he then divided the spectrum into
small frequency regions and determined
the number of waves with their periods
in a given interval. James Jeans corrected
the missing proportionality constant and
published the results as the Rayleigh-Jeans law. Experimentally it was valid for
long wavelengths—in contrast to Wien’s
law that was valid for short wavelengths.
In 1901, Planck published his law based
on the quantum of energy; this assumption provided a law that was experimentally verified for black-body radiation as
both long and short wavelengths.
Resolving power of a microscope
Rayleigh made seminal contributions
to theoretically defining the resolving
power of gratings, prisms, telescopes and
microscopes. In his 1896 publication
on the theory of optical images applied
to microscopes, Rayleigh cited a paper
of Helmholtz’s (1874) that is based on
an analogous method to that used for
telescopes. Helmholtz concluded that
the smallest resolvable distance, ε, of two
point objects is
Rayleigh’s Work in Early Modern Physics
Rayleigh lived during a time when classical physics
was tested by new discoveries: X-rays, natural
radioactivity, the electron. Novel theories were
also introduced, including relativity, the quantum
of energy and the theory of atoms. He therefore
had plenty of opportunities to solve problems, and
he usually took a mathematical approach. After
all, even though Rayleigh worked in physics, his
degree was in mathematics.
Rayleigh liked to simplify a physical situation
and then generalize the solution he obtained
in higher order terms. Rayleigh used these
approximation or perturbation techniques that were
originally developed earlier by mathematicians
to solve problems in astronomy. Complex physical
applications could be attacked as a perturbation of simpler models. In other
words, Rayleigh sought approximate solutions for problems that did not have
exact solutions by working with related models that did have exact solutions.
Others expanded on these problem-solving methods.
The Rayleigh-Jeans law of black-body radiation correctly gave the frequency
distribution of long wavelength black-body radiation but failed at short
wavelengths. At the same time, Wien derived an empirical law of black-body
radiation that fit the experimental data at short wavelengths but not long ones.
Planck tried to reconcile the empirical relations with the experimental frequency
distribution that resulted in his correct theoretical formulation based on the
quantum hypothesis of energy.
Many of the foundations of wave mechanics are based on the analyses and
equations that Rayleigh derived for the theory of acoustics in his book The
Theory of Sound. Erwin Schrödinger, a pioneer in quantum mechanics,
studied this book and was familiar with the perturbation methods it describes.
He built on Rayleigh’s work to develop the Schrödinger-Rayleigh perturbation
method. Others developed the Rayleigh-Ritz approximation (the variational
principle) and the Wentzel-Kramers-Brillouin (WKB method) of perturbation
theory; the latter is actually a variation by Wentzel of a technique proposed by
Rayleigh in 1912 and earlier by Carlini (1817), Liouville (1837) and Green (1837).
ε = — λ / sin α ,
where α is the divergence angle of the
extreme ray (the semi-angular aperture).
He cites the work of Abbe as defining the
“numerical aperture” as µ sin α where µ
is the refractive index of the medium in
which the object is situated.
Rayleigh stated that this limit can
only be depressed by using incident
light of shorter wavelengths (e.g., that
provided by an ultraviolet or electron
microscope) or by increasing the refractive index of the medium in which the
object is situated. He stated that, for
two point sources of light (i.e., stars),
the light is incoherent, and the intensities of the two lights can be added. But,
for the case of the microscope, there
are permanent phase relations between
two overlapping lights; thus, a different
analysis is required.
Abbe did not treat objects as point
illuminations, but rather as a grating
illuminated with plane waves. Rayleigh
proceeded to develop the diffraction
theory of light by a lens of finite aperture
based on the use of Fourier’s theorem; the
results are consistent with those of Abbe.
Theory of optical image
formation in the microscope
Microscopic resolution was first
approached by Fraunhofer, and then
formulated by Abbe and Helmholtz. In
his 1893 paper on the theory of optical images, with special reference to the
microscope, Rayleigh first discussed the
visibility of a star with a telescope. He
noted that the points of light of stars