Introduction to Quantum Ideas
51
(iii) inversely proportional to T
2
where
2
1
2
T
mv
=
is the kinetic energy of the
incoming α particles and (iv) inversely porportional to
4
sin 2
θ where θ is the
angle of scattering. These properties were tested by Geiger and Marsden by
varying the thickness and the composition of the foil, the energy of the incident
α particles and the angle of scattering, and were found to be inxcellent
agreement with the experimental observations. For examples, they found in
an experiment with silver foil, that dN was proportional to 111, 680 and 8800
for θ = 150°, 75° and 37.5°, respectively, other variables remaining the same.
For these values, the product (dN) sin
4
(θ/2) is proportional to 96.6, 93.4, 93.9,
respectively. The near-constancy of the product, though dN itself varies by a
large factor, indicates the essential correctness of the θ-dependence of
scattering rates.
It may be noted that the Rutherford formula is the same for attractive and
repulsive Coulomb potentials. It is not valid for values of the impact parameter
b larger than interatomic distances for which an α particle cannot be regarded
as being scattered by a single atom in the metal foil. It is also not valid if the
α particle approaches the nucleus to a distance (for a head-on collision
r min = Ze
2
/πε 0 mv
2
) less than the size of the nucleus, i.e., about 10
–14
m, at which
the nuclear forces become important.
While the Rutherford model of the atom provides a fairly comprehensive
description of the scattering of low-energy α particles by the atoms, there are
some implications of the simple model which are in conflict with the classical
interpretations of experimental observations. The stability of the atom demands
that the electron must revolve around the nucleus. But such an electron, since it
is accelerating, must radiate energy continuously according to the classical theory
of electromagnetism, and ultimately coalesce with the nucleus. Experimentally,
an atom is a highly stable object. Furthermore, it can absorb radiation only of
some well-defined frequencies and then emit radiation again of well-defined
frequencies. The Rutherford model of the atom is unable to explain these
experimental observations. It is with the intention of reconciling the Rutherford
model with the observed stability and spectrum of the atom, that Bohr began his
search for a model of the atom and came up with what is known as the Bohr
model of the atom.
2.7 BOHR MODEL
Bohr (1913) started by assuming that Rutherford’s model of the atom is essentially
correct but that the classical laws of dynamics need to be modified so as to be
51
(iii) inversely proportional to T
2
where
2
1
2
T
mv
=
is the kinetic energy of the
incoming α particles and (iv) inversely porportional to
4
sin 2
θ where θ is the
angle of scattering. These properties were tested by Geiger and Marsden by
varying the thickness and the composition of the foil, the energy of the incident
α particles and the angle of scattering, and were found to be inxcellent
agreement with the experimental observations. For examples, they found in
an experiment with silver foil, that dN was proportional to 111, 680 and 8800
for θ = 150°, 75° and 37.5°, respectively, other variables remaining the same.
For these values, the product (dN) sin
4
(θ/2) is proportional to 96.6, 93.4, 93.9,
respectively. The near-constancy of the product, though dN itself varies by a
large factor, indicates the essential correctness of the θ-dependence of
scattering rates.
It may be noted that the Rutherford formula is the same for attractive and
repulsive Coulomb potentials. It is not valid for values of the impact parameter
b larger than interatomic distances for which an α particle cannot be regarded
as being scattered by a single atom in the metal foil. It is also not valid if the
α particle approaches the nucleus to a distance (for a head-on collision
r min = Ze
2
/πε 0 mv
2
) less than the size of the nucleus, i.e., about 10
–14
m, at which
the nuclear forces become important.
While the Rutherford model of the atom provides a fairly comprehensive
description of the scattering of low-energy α particles by the atoms, there are
some implications of the simple model which are in conflict with the classical
interpretations of experimental observations. The stability of the atom demands
that the electron must revolve around the nucleus. But such an electron, since it
is accelerating, must radiate energy continuously according to the classical theory
of electromagnetism, and ultimately coalesce with the nucleus. Experimentally,
an atom is a highly stable object. Furthermore, it can absorb radiation only of
some well-defined frequencies and then emit radiation again of well-defined
frequencies. The Rutherford model of the atom is unable to explain these
experimental observations. It is with the intention of reconciling the Rutherford
model with the observed stability and spectrum of the atom, that Bohr began his
search for a model of the atom and came up with what is known as the Bohr
model of the atom.
2.7 BOHR MODEL
Bohr (1913) started by assuming that Rutherford’s model of the atom is essentially
correct but that the classical laws of dynamics need to be modified so as to be