Download One Hundred Years of Bohr Model

GENERAL  ARTICLE
One Hundred Years of Bohr Model
Avinash Khare
In this article I shall present a brief review of the
hundred-year young Bohr model of the atom. In
particular, I will ¯rst introduce the Thomson and
the Rutherford models of atoms, their shortcomings and then discuss in some detail the development of the atomic model by Niels Bohr. Further, I will mention its re¯nements at the hand of
Sommerfeld and also its shortcomings. Finally, I
will discuss the implication of this model in the
development of quantum mechanics.
The `Bohr atom' has just completed one hundred years
and it is worth recalling how it emerged, its salient features, its shortcomings as well as the role it played in
the development of quantum mechanics.
Avinash Khare is Raja
Ramanna Fellow at IISER,
Pune. His current interests
are in the areas of low
dimensional field theory,
nonlinear dynamics and
supersymmetric quantum
mechanics. Besides, he is
passionate about teaching
an well as popularizing
Thomson Model of Atoms
Till 1896, the popular view was that the atom was the
basic constituent of matter. The ¯rst important clue
regarding the internal structure of atoms came with the
discovery of spontaneous radiation, ¯rst identi¯ed by
Becquerel in 1896. The very existence of atomic radiation strongly suggested that atoms were not indivisible.
With the discovery of the electron in 1897, J J Thomson
was convinced that electrons must be fundamental constituents of matter and this led to his corpuscular theory
of matter. This model was popularly known as `Plum
Pudding Model'. In this model, the positive charge of
the atom was assumed to be spread throughout the atom
forming a kind of pudding in which negatively charged
electrons were suspended like plums. Thomson showed
that his model had an amazing explanatory power for
the observed periodicity in the elements. Thomson later
applied a modi¯ed version of this model to a variety of
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science at school and
college level.
Keywords
Plum pudding model, Rutherford
atom, Balmer series, Rydberg
formula, star  Puppis, charac-
teristic X-rays, Bohr–Sommerfeld
approach.
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GENERAL  ARTICLE
phenomena such as dispersion of light by dilute gases
and developed methods for estimating the actual number of electrons in an atom. By 1910, experiments had
con¯rmed many of its predictions for the absorption and
scattering of electrons in matter. However, this model
was not suitable for predicting spectral lines which had
already been seen by spectroscopists.
Figure 1. Thomson’s model
of atom. In this ‘plum pudding’ model of atom developed by Thomson and Kelvin
in 1094, the electrons (plums)
are embedded in a sphere of
uniform positive charge (pudding).
Figure 2. Rutherford’s model
of atom. In this model, the
atom consisted of a positively
charged nucleus surrounded
by negatively charged electrons.
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Rutherford Model of Atoms
Rutherford wanted to test this Thomson picture of the
atom. On his advice, Geiger and Marsden carried out
a series of experiments. Using their data, Rutherford
in 1911 provided conclusive proof of the inadequacy of
the Thomson model. In these experiments, a collimated
beam of alpha particles (i.e., He42 nucleus) from a radium source strike a thin gold foil. To their surprise,
they found that few alpha particles were even scattered
at large angles. This would not be expected if the alpha
particle had hit a much lighter particle like the electron. Rutherford argued that these experiments clearly
showed that instead of being spread throughout the atom,
the positive charge is in fact concentrated in a very small
region at the center of the atom. This was one of the
most important developments in atomic physics and was
the foundation of the subject of Nuclear Physics. We go
through the Rutherford argument here.
Let us suppose that a particle of mass M and velocity
v, hits a particle of mass m which is at rest. After the
collision, let us suppose that the particle of mass M
continues along the same line with velocity v0 , giving
the target particle (with mass m) a velocity u (note, we
are using the notation in which positive velocity is in the
same direction as the incident particle of mass M , while
a negative velocity will be in the opposite direction).
Then the energy and momentum conservation equations
are
1
1
Mv = mu + Mv 0 ;
M v2 = (Mv02 + mu2 ) : (1)
2
2
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On eliminating u, we have a quadratic equation for v 0 in
terms of v
m(v2 ¡ v02 ) = M (v ¡ v0 )2 ;
(2)
which has two solutions: either v 0 = v or
µ
m¡M
v = ¡v
m+M
0
¶
:
(3)
The ¯rst solution v 0 = v; u = 0 is a trivial solution.
The interesting solution is the second one given by (3).
It says that v 0 can be negative (i.e., scattered particle
recoils backwards) only if m > M (similarly, somewhat
weaker limits on m can be inferred from scattering at
any large angle).
Thus it was clear to Rutherford that the experimental
results of Geiger and Marsden could not be explained in
terms of multiple encounters with a positively charged
sphere of atomic dimensions, as was Thomson's view.
The fact that a few alpha particles were observed to be
scattered at large angles clearly showed that the alpha
particles must be hitting something in the gold atom
which is much heavier than the electron. As Rutherford
later explained, \it was quite the most incredible event
that has ever happened to me in my life. It was almost
as incredible as if you ¯red a 15-inch shell at a piece of
paper, and it comes back and hits you". Hence, Rutherford concluded that the positive charge of the atom is
concentrated in a small heavy nucleus at the center of
the atom, around which the much lighter, negatively
charged electrons circulate in orbits, like planets around
the sun. This is why the Rutherford nuclear model is
often referred to as the planetary model of the atom.
This model, despite being successful, had one major
problem. It predicted that even light atoms like hydrogen were unstable. The point was, according to the
classical electromagnetic theory, an electron revolving
around a nucleus will radiate electromagnetic waves and
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Rutherford
concluded that the
positive charge of
atom is concentrated in a small
heavy nucleus at
the centre of the
atom, around which
the much lighter,
negatively charged
electrons circulate
in orbits.
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GENERAL  ARTICLE
hence will deplete the electron's energy and it will eventually spiral inwards towards the nucleus. Thus an atom
would rapidly collapse to nuclear dimensions (the collapse time can be computed to be of order 10¡12 sec!).
Further, the continuous spectrum of the radiation that
would be emitted in this process was not in agreement
with the observed line spectrum.
Bohr Model
A key feature of the
Bohr model was the
prediction of the
experimentally
observed line
spectrum by atoms.
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In the autumn of 1911, Bohr went to England for his
post-doctoral research. He had already done interesting work on the electron theory of metals during his
PhD thesis { it was so advanced that no one in Denmark could evaluate it fully. Bohr ¯rst went to Cambridge University and worked for about a year with J
J Thomson before being invited by Rutherford to work
with him at University of Manchester. Using the data
from the absorption of alpha-rays, and using Rutherford's model, Bohr showed that the hydrogen atom has
only one electron outside the positively charged nucleus
while the helium atom has two electrons outside the positively charged nucleus. It is worth noting here that till
1912, physicists were not sure about the number of electrons in the helium atom or even in the hydrogen atom.
All this time, the problem which was really troubling
Bohr was, however, the stability of the Rutherford atom
and ¯nally he came up with a simple model of atomic
structure. A key feature of this very successful model,
proposed by Bohr in 1913, was the prediction of the
line spectrum of radiation by atoms. So we shall digress
here and describe what was known experimentally about
atomic spectra at that time.
By 1900, the amount of information available about
atomic spectra was enormous. Spectroscopists had noticed that an atom can only absorb certain energies of
light (the absorption spectra) and once excited can only
release certain energies (the emission spectra), and these
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GENERAL  ARTICLE
energies happen to be the same. Further, the spectra
coming from di®erent atoms showed that each atom has
its own characteristic spectrum, i.e., a characteristic set
of wavelengths at which the lines of the spectrum are
found. Amongst all the atoms, the spectrum of hydrogen is relatively simple. Since most of the universe
consists of isolated hydrogen atoms, the hydrogen atom
spectrum is of considerable importance. It was found
that the hydrogen atom spectrum had a great regularity. This tempted several people to look for an empirical
formula which would represent the wavelengths of the
lines. Such a formula was discovered by Balmer, a Swiss
school teacher, in 1885. He found the simple relation
¸ = 3646
n2
; n = 3; 4; 5; ::: ;
n2 ¡ 4
(4)
where ¸ is the wavelength. Using this formula, he was
able to predict the wavelengths of the ¯rst nine lines of
the series to better than one part in 1000. This discovery initiated a search for similar empirical formulas that
would apply to other series. Most of this work around
1890 was done by Rydberg, who found it convenient
to deal with the reciprocals of the wavelengths of the
lines, instead of their wavelengths. In terms of reciprocal wavelength º, the Balmer formula can be written
as
1
1
1
º = = RH ( 2 ¡ 2 ) ; n = 3; 4; 5; :::
(5)
¸
2
n
where RH is the so-called Rydberg constant for hydrogen. It might be noted here that Balmer had already
accurately calculated RH to one part in 10000.
In 1913, in 3 seminal papers [1] Bohr, who was then
just 27 years old, presented his model for the atom and
was successful in accurately explaining some of the spectroscopy data. Bohr's model can be said to be based on
the following four postulates.
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Balmer had
already accurately
calculated the
Rydberg constant
to one part in
1000.
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GENERAL  ARTICLE
The four postulates
of Bohr are an
unusual mixture of
classical and
nonclassical physics.
1. Atomic electrons move in circular orbits about a
massive nucleus under the in°uence of the Coulomb
attraction between the electron and the nucleus,
obeying the laws of classical mechanics.
2. Instead of the in¯nity of orbits which would be
possible in classical mechanics, an electron can in
fact only move in an orbit for which its angular
momentum L is quantized, i.e., it is an integral
h
multiple of ¹h = 2¼
, h being the Planck constant.
3. Even though it is constantly accelerating, an electron moving in such an allowed orbit does not radiate electromagnetic energy.
4. Electrons can only gain or lose energy by jumping from one allowed orbit to another, absorbing or emitting electromagnetic radiation with a
frequency º, determined by the energy di®erence
of the levels according to the Planck's relation
¢E = Ef ¡ Ei = hº.
The ¯rst postulate of Bohr's model is based on the existence of the atomic nucleus. The second postulate introduces quantization. On the other hand, the third
postulate removes the problem of stability of an electron
moving in a circular orbit, due to the emission of electromagnetic radiation required of the electron by classical
theory, by simply postulating that this feature of the
classical theory is not valid for the case of an atomic
electron. The fourth postulate is just Einstein's postulate that the frequency of a photon of electromagnetic
radiation is equal to the energy carried by the photon
divided by Planck's constant [2].
Clearly, these four postulates are an unusual mixture of
classical and nonclassical physics. The electron moving
in a circular orbit is assumed to obey classical physics,
and yet the nonclassical idea of quantization of angular momentum is included. Further, while the electron
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is assumed to obey Coulomb's law of classical electromagnetic theory, yet it is assumed that it does not obey
the other feature of emission of radiation by an accelerated charged particle. The postulate about transition
between allowed orbits was in obvious con°ict with classical electrodynamics, but to Bohr it appeared to be
necessary in order to account for the experimental data.
Of course, Bohr was clear from the beginning that his
model is in con°ict with the classical theory. That is
why, he had already said that no attempt will be made
at a mechanical formulation as it seems hopeless. I think
this clearly shows Bohr's ability to entertain several con°icting ideas. Blazing courage is one thing but tolerance
of ambiguities is quite another.
Bohr was clear
from the beginning
that his model was
in conflict with the
classical theory.
Predictions of Bohr Model
Let us now derive the predictions that Bohr obtained
using these postulates. Consider an atom with nuclear
charge Ze and mass M and a single electron with mass
m and charge ¡e. Note, for hydrogen Z = 1. Following
Bohr, we initially assume that the mass of the electron
is completely negligible compared to the mass of the
nucleus and consequently that the nucleus remains ¯xed
in space. The condition of the mechanical stability of
the electron (following from classical mechanics) comes
by balancing the Coulomb force acting on the electron
and the centripetal acceleration keeping the electron in
a circular orbit, i.e.,
Ze2
mv2
=
;
r2
r
(6)
where v is the speed of the electron in its orbit of radius r. Now applying the quantization condition on the
angular momentum of the electron, L = mvr, we have
mvr = n¹h ; n = 1; 2; 3; :::
(7)
Using (6) and (7), we can solve for the radius of the
circular orbit and the velocity of the electron in this
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Blazing courage is
one thing but
tolerance of
ambiguities is
quite another.
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GENERAL  ARTICLE
Box 1.
orbit. We get
In the Rydberg formula, the
transitions to the ground
state (i.e., ni = 1, nf = 2, 3,
4, ...) are known as the
Lyman series. All lines in
this series are in ultraviolet
region with wavelengths
ranging from 1216 to 912
Angstroms. On the other
hand, the transitions to the
r=
n2h
¹2
n¹h
Ze2
;
v
=
=
:
mZe2
mr
n¹h
The kinetic energy of the electron is then
1 2 Ze2
K:E: = mv =
;
2
2r
E = K:E: + P:E: = ¡
tute the Balmer series. Four
of these lines are in the
visible region with wavelengths ranging from 6562
to 4101 Angstroms while
the other lines are in the
ultraviolet region. Transitions to the second excited
state (i.e., ni = 3, nf = 4, 5,
lengths are in the infrared
region.
Bohr was able to
derive the famous
Rydberg formula
and hence the
Balmer formula
which is a special
case of the
Rydberg formula.
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Ze2
mZ 2 e4
=¡ 2 2 :
2r
2n h
¹
(10)
We thus see that the quantization of the angular momentum of the electron leads to a quantization of its total energy. Hence the frequency of the electromagnetic
radiation emitted when the electron makes a transition
from the quantum state nf to a state ni is given by
º = R1 Z 2(
6, ...) constitute the Paschen
series and all the wave-
(9)
which is half of the potential energy, and hence the total
energy E is given by
first excited state (i.e., ni =
2, nf = 3, 4, 5, ...) consti-
(8)
1
1
¡ 2);
2
ni
nf
(11)
where ni ; nf are integers with nf > ni and
R1 =
me4
:
4¼¹h3 c
(12)
Equation (11) is the famous Rydberg formula obtained
by him in 1890 (see Box 1). The essential predictions
of the Bohr model are contained in (10) and (11). It is
worth pointing out that Bohr in fact derived this spectrum in three di®erent ways in his classic 1913 paper.
Further, in a footnote, Bohr mentioned that his results
might Ralso be obtained if one assumes that the line integral p dl is an integral multiple of Planck's constant,
i.e.,
Z
pdl = nh ;
(13)
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where p is the electron's momentum and dl is an element
of length.
In one of his papers
Bohr had also put
It is interesting to note here that in his ¯rst paper, Bohr
had also put forward the celebrated `Correspondence
Principle' which states that the laws of quantum physics
must reduce to those of classical physics when quantum
numbers such as n, as de¯ned above, become large. In
fact one of the derivations given by Bohr for (10) and
(13) was by using this principle. The incredible thing
was that while the proof was given for large n, the ¯nal
result as given by (10) and (13) is claimed to be true for
any value of n! What luck that it could work!
forward the
celebrated
For hydrogen (Z = 1), in case ni = 2, Bohr immediately recovered the Balmer formula provided R1 = RH .
Bohr evaluated R1 by using (12) and found that the
resulting value was in quite good agreement with the
experimental value of RH . Later on, when Bohr made a
correction for ¯nite nuclear mass, i.e., substituted m by
the reduced mass ¹ = MmM
, he found that the theoreti+m
cal and experimental values of RH agree to within three
parts in 100,000! Another success of the Bohr model was
about star ³ Puppis. Before Bohr, it had been wrongly
interpreted as a new series of lines of hydrogen. It was
another triumph for the Bohr model that it could explain these lines as those belonging to the spectra of
ionized Helium. It may be noted here that till that time
the spectral lines of ionized helium had not yet been observed in the laboratory. But as soon as this was done,
the Bohr model was regarded as a great success. Yet another triumph was the explanation due to Mosley about
the characteristic K® lines of X-rays using Bohr's theory.
I might add here that while the most compelling of
Bohr's results was the derivation of the Balmer formula
of 1885, Bohr claimed throughout his life that he was
unaware of the formula until he was already well along
in the development of his theory. This appears rather
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Correspondence
‘Principle’ and using
it was able to derive
the famous Bohr
formula (10).
Bohr successfully
explained the
spectral lines from
the star  Puppis
as due to ionized
Helium.
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GENERAL  ARTICLE
Bohr was awarded
the Nobel Prize in
Physics in 1922.
strange given the fact that Balmer's work was extensively discussed during major international physics conferences in 1890 as well as in subsequent years.
How was the Bohr model received by the physics community? Sommerfeld immediately wrote a letter to Bohr,
complimenting him for calculating the empirical Rydberg constant in terms of the more fundamental constants, though he was skeptical about the atomic model
in general. Once there was an explanation of the spectrum of star ³ Puppis, Bohr's theory received wide attention. Einstein too immediately recognized the importance of Bohr's theory saying it was a major development. It is fair to say that by and large people
were unhappy with the two postulates of the Bohr model
but were very impressed with its unprecedented success,
which eventually led to its widespread adoption. And
of course the crowning glory came when, for this work,
Bohr was awarded the Nobel Prize in Physics in 1922.
Bohr{Sommerfeld Approach
Sommerfeld
generalized the Bohr
quantization condition
and calculated the
energies of electrons
in elliptical orbits.
894
The Bohr model was developed for circular orbits, but
just as in the solar system, the generic orbit of a particle
in a Coulomb ¯eld is not a circle but an ellipse. A generalization of the Bohr quantization condition (7) was
proposed by Sommerfeld and used to calculate energies
of electrons in elliptical orbits. In particular, in addition to the Bohr quantization of the azimuthal motion
(angular momentum) of an electron around the nucleus,
Sommerfeld quantized phase integrals for the radial motion (allowing for elliptical orbits), and the orientation
of the orbital plane (spatial quantization). He replaced
equation (7) of Bohr by a system of three conditions
Z
pr dr = n1 h ;
Z
pÁ dÁ = n2h ;
Z
pµ dµ = n3h : (14)
In this way, Sommerfeld made signi¯cant progress over
the Bohr model. Further, Sommerfeld removed the degeneracy in the hydrogen atom spectrum by treating
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GENERAL  ARTICLE
the problem relativistically thereby explaining the ¯ne
structure observed experimentally (see Box 2). Note
that the Bohr model was unable to explain this ¯ne
structure observed in the spectrum of the hydrogen atom.
Further, Sommerfeld's approach could explain the Stark
and the Zeeman e®ects in hydrogen. After reading these
papers, Bohr wrote a letter to Sommerfeld saying \I do
not believe ever to have read anything with more joy
than your beautiful work".
What were the eventual failures of the Bohr{Sommerfeld
approach? While this approach was reasonably successful for atoms with one valence electron, it failed to
explain much of the spectra of atoms containing more
than one electron. Even for the hydrogen atom, the
Bohr model gives incorrect value for the orbital angular momentum of the ground state. Broadly speaking
the Bohr{Sommerfeld approach was fundamentally inconsistent and led to many paradoxes. The framework
they proposed, a classical description of atoms to which
quantization rules were added, was ¯nally rendered untenable. I might add here that Bohr himself had realized
that his model was not the ¯nal answer and he believed
that a deeper revision of physics was required.
Box 2.
Whereas the ground state
energy of the hydrogen
atom is 13.6 eV, the fine
splitting between the 2P3/2
and 2P1/2 states due to spin–
orbit interaction is 4.5 
10 –5 eV. On the other hand
the Lamb shift (occurring
mainly due to continual
emission and reabsorption
of photons by the electron)
between the 2S1/2 and 2P1/2
states is 4.35  10–6 eV.
Finally, the hyperfine splitting (occurring due to the
interaction between electron and proton spin) between the 1S3/2 and 1S1/2
states is 5.87  10–6 eV.
The radiative transition
between these two states is
the famous 21 centimeter
line in the radio spectrum
of hydrogen.
Bohr Model and Development of Quantum Mechanics
Finally, what role did the Bohr atom play in the eventual development of quantum mechanics? It is fair to
say that Bohr's model of the atom is simple, elegant,
revolutionary but rather preliminary. It was undoubtedly an essential step towards a correct theory of the
atomic spectra. Since the Bohr model could well explain the spectra of atoms with one valence electron, it
had a domain of applicability. So it could not be entirely
wrong and would have to correspond in some way with
another, possibly more successful theory. It is fair to
say that the Bohr model has a special place in history,
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Since Bohr atom
could successfully
explain the spectra
of atoms with one
valence electron, it
could not be
completely wrong.
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GENERAL  ARTICLE
Bohr model has a
special place in
history, a bridge
between the older
classical thinking
and the newer
quantum thinking. It
was important for
being a tangible
break from the
priciples of classical
mechanics.
Address for Correspondence
a bridge between the older classical thinking and the
newer quantum thinking. It was a bridge over a chasm
but one that collapsed after it was crossed. One may
wonder, how it could hold up so long and reach so far.
The reason probably is that with our present knowledge
of quantum mechanics, we can look across the chasm
from the other side and see that the semi-classical situation works for two problems one of which is the hydrogen atom! The Bohr model was pathbreaking for
physics because it marked the transition between classical and quantum thinking. The model, though could
not evolve continuously to modern quantum mechanics. The Bohr model was important for being a tangible
break from the principles of classical mechanics which
were useless at explaining the quantum mechanical effects in the atoms. Thanks to the Bohr model, physicists recognized this and insisted on building on what
they had [3].
Suggested Reading
Avinash Khare
Raja Ramanna Fellow
[1]
N Bohr, Philos. Mag. Vol.26, pp.1–25, pp.476–50, pp.857–875, 1913.
Indian Institute of Science
[2]
There are several simple accounts of the Bohr model; see for example,
R Eisberg and R Resnick, Quantum Physics of Atoms, Molecules, Solids,
Education and Research
Nuclei and Particles, Chapter 4, Wiley India, 2006.
(IISER)
Pune 411 008, India.
Email:[email protected]
[3]
There are some recent books exclusively discussing Bohr model of atom;
see for example, H Kragh, Niels Bohr and the Quantum Atom: The Bohr
Model of Atomic Structure, pp.1913–1925, Oxford Univ. Press, 2012.
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