Bohr model

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The Bohr model of the hydrogen atom, where
negatively charged electrons confined to atomic
shells encircle a small positively charged atomic nucleus, and that
an electron jump between orbits must be accompanied by an emitted or absorbed
amount of electromagnetic
energy hν. The orbits that the electrons travel in are shown as
dashed black circles; their radius increases n2, where
n is the principal quantum
number. The 3→2 transition depicted here produces the first line of the Balmer
series, and results in a photon of wavelength 656 nm
(red).

In atomic physics, the
Bohr model depicts the atom as a small, positively charged
nucleus surrounded by electrons that
travel in circular orbits around the nucleus — similar in structure to the solar
system, but with electrostatic forces
providing attraction, rather than gravity. This was an improvement
on the earlier Plum-pudding model
(1904) and the Rutherford model
(1911).

Introduced by Niels Bohr in 1913, the model's key success was
in explaining the Rydberg formula for the
spectral emission lines of atomic
hydrogen;
while the Rydberg formula had been known experimentally, it did not gain a
theoretical underpinning until the Bohr model was introduced. Not only did the
Bohr model explain the reason for the structure of the Rydberg formula, but it
provided a justification for its empirical results in terms of fundamental
physical constants.

The Bohr model is a primitive model of the hydrogen atom. As a theory, it can
be derived as a first-order approximation of the hydrogen atom using the broader
and much more accurate quantum mechanics, and
thus may be considered to be an obsolete
scientific theory. However, because of its simplicity, and its correct
results for selected systems (see below for application), the Bohr model is
still commonly taught to introduce students to quantum mechanics. A similar
model was originally proposed by Arthur Erich Haas in
1910, but was rejected.

Contents

1 History

2 Electron energy levels in
hydrogen

2.1 Energy in terms of
other constants

3 Rydberg formula

4 Moseley's law and
calculation of K-alpha X-ray emission lines

5 Shortcomings

6 Refinements

7 See also

8 References

8.1 Historical

8.2 Further
reading

[edit]
History

In the early 20th century, experiments
by Ernest Rutherford and
others had established that atoms consisted of a diffuse cloud
of negatively charged electrons surrounding a small,
dense, positively charged nucleus. Given this experimental data, it was quite
natural for physicists to consider a planetary model for the atom, the Rutherford model of
1911, with electrons orbiting a sun-like nucleus. However, the planetary model
for the atom has several difficulties. For example, the laws of classical
(Newtonian) mechanics predict that the electron will release electromagnetic
radiation as it orbits a nucleus. Because the electron would be losing
energy, it would be predicted to gradually spiral inwards and collapse into the
nucleus. As this occurred, the emission would change in frequency and would be
predicted to produce a smear, in frequency, of electromagnetic radiation.
However, late 19th century experiments
with electric discharges
through various low-pressure gasses in evacuated glass tubes had
shown that atoms will emit light (that is, electromagnetic radiation), but only
at certain discrete frequencies.

To overcome this and other difficulties in explaining electron motion in an
atom, Niels
Bohr proposed, in 1913, what is now called the
Bohr model of the atom. Two key ideas were:

The electrons travel in orbits and have discrete quantized
momenta, and therefore quantized energies. That is, not every orbit is
possible but only certain specific ones, at certain specific distances from
the nucleus.

The electrons will not slowly lose energy as they travel, and hence will
remain in stable, non-decaying orbits.

The great significance of the model is that it states that the laws of
classical mechanics do not apply to the motion of the electron about the
nucleus. Bohr proposed that a new kind of mechanics, or quantum mechanics,
describes the motion of the electrons around the nucleus. This model of
electrons traveling in quantized orbits around the nucleus, however, was
replaced with a more accurate model of electron motion about ten years later by
the German physicists Erwin Schrödinger and
Werner Heisenberg.

Other points are:

When an electron makes a jump from one orbit to another, the energy
difference is carried away (or supplied) by a single quantum of light (called
a photon) which
has an energy equal to the energy difference between the two orbits.

The allowed orbits depend on quantized (discrete) values of orbital angular momentum,
L according to the equation

Where
n = 1,2,3,… and is called the principal quantum
number, and h is Planck's constant.

Point (2) states that the lowest value of n is 1. This corresponds to
a smallest possible radius of 0.0529 nm. This is known as the Bohr
radius. Once an electron is in this lowest orbit, it can get no closer to
the proton.

[edit]
Electron energy levels in hydrogen

The Bohr model is accurate only for one-electron systems such as the hydrogen
atom or singly-ionized helium. It can also be used for
K-line X-ray transition calculations, if other assumptions are introduced (see
Moseley's law below). This section uses the Bohr model to derive the energy
levels of hydrogen.

The derivation starts with three simple assumptions:

1) The energy of an electron in an orbit is the sum of its kinetic and
potential energies:

where k = 1 / (4πε0), and
qe is the charge of the
electron.

2) The angular momentum of the electron can only have certain discrete
values:

where n = 1,2,3,… and is called the principal
quantum number, h is Planck's constant,
and .

3) The electron is held in orbit by the coulomb force. That is,
the coulomb force is equal to the centripetal force:

To begin, multiply both sides of eq (3) by r to see

The term on the left hand side is the potential energy. So the equation for
the energy becomes

Now we just need to figure out what the velocity, v is equal to, so
solve eq (2) for r,

Plug this into eq (4),

Then divide both sides by mev to see

Now we can put in this value for v into the equation for energy, and
then also plug in the values for k and ,
and we'll obtain the energy of the different levels of hydrogen:

Or, after substituting values for the constants,

Thus, the lowest energy level of hydrogen (n = 1) is about -13.6 eV. The
next energy level (n = 2) is -3.4 eV. The third (n = 3) is -1.51
eV, and so on. Note that these energies are less than zero, meaning that the
electron is in a bound state with the proton. Positive energy states correspond
to the ionized atom where the
electron is no longer bound, but is in a scattering state.

[edit]
Energy in terms of other constants

Starting with what we found above,

We can multiply top and bottom by c2, and we'll arrive at

or re-grouping them to make it more clear:

From here we can now write the energy level equation in terms of other
constants to:

where,

is the energy level

is the rest energy of the electron

is the fine structure
constant

is the principal quantum
number.

[edit]
Rydberg formula

The Rydberg formula, which
was known empirically before Bohr's formula, is now in Bohr's theory seen as
describing the energies of transitions or quantum jumps between one
orbital energy level, and another. Bohr's formula gives the numerical value of
the already-known and measured Rydberg's constant,
but now in terms of more fundamental constants of nature, including the
electron's charge and Planck's constant.

When the electron moves from one energy level to another, a photon is emitted. Using the
derived formula for the different 'energy' levels of hydrogen one may determine
the 'wavelengths' of light that a hydrogen atom can emit.

The energy of a photon emitted by a hydrogen atom is given by the difference
of two hydrogen energy levels:

where qe is the charge of an electron
(1.60 × 10−19 C),
nf is the final energy level, and
ni is the initial energy level. It is assumed that
the final energy level is less than the initial energy level.

Since the energy of a photon is

the wavelength of the photon given off is

The above is known as the Rydberg formula. This
formula (with all of the numerical constants lumped into a single empirically
measured Rydberg constant number
R), was known in the nineteenth century to scientists studying spectroscopy, but there was
no theoretical justification for the value of R, or for the form of the
formula itself, until Bohr derived them, more or less along the lines above. In
fact, Bohr's derivation of the empiric Rydberg constant to high accuracy, in
terms of other known physical constants, was an early reason for the acceptance
of Bohr's model as an advance in understanding.

[edit]
Moseley's law and calculation of K-alpha X-ray emission
lines

Niels Bohr said in 1962, "You see actually the Rutherford work [the nuclear
atom] was not taken seriously. We cannot understand today, but it was not taken
seriously at all. There was no mention of it any place. The great change came
from Moseley."

In 1913 Henry Moseley found an
empirical relationship between the strongest X-ray line emitted by atoms under
electron bombardment (then known as the K-alpha line), and their atomic
number Z. Moseley's empiric formula was found to be derivable from Rydberg and
Bohr's formula (Moseley actually mentions only Earnest Rutherford
and Antonius Van den
Broek in terms of models). The two additional assumptions that [1]
this X-ray line came from a transition between energy levels with quantum
numbers 1 and 2, and [2], that the atomic number Z when used in the
formula for atoms heavier than hydrogen, should be diminished by 1, to
(Z-1)2. Moseley wrote to Bohr in puzzlement about this fact, but Bohr
at that time was not able to help, thinking that the postulated inner or "K"
shell of electrons in elements should have at least four electrons, not two.
Thus, Moseley left the matter without an attempt at theoretical explanation.
Much latter, the effect was realized to be in compensation for the screening
effect of the single electron remaining in the lowest atomic energy level, which
actually only has two electrons. As noted, Moseley did not choose to speculate
on the reason for this even more general screening effect (which was numerically
much higher for the L-alpha transition between levels 2 and 3), and the basic
mechanism for it which would only become clear later, after further
understanding of the correct atomic electron shell structure.

In Bohr's formula for hydrogen above, the charge q4 is a
product of the electron charge q2 and the nuclear charge
(Zq)2 = q2 Z2. The nuclear
charge Z2 may then be factored out as a pure number.

Moseley's law for K-alpha
lines is given by the following changes in Bohr's formula:

or

This latter relationship had been empirically derived by Moseley, in a simple
plot of the square root of X-ray frequency against atomic number. Moseley's law
not only established the objective meaning of atomic number (see Henry
Moseley for detail) but, as Bohr noted, it also did more than the Rydberg
derivation to establish the validity of the Rutherford/Van den Broek/Bohr
nuclear model of the atom, with atomic number as nuclear charge.

The K-alpha
line of Moseley's time is now known to be a pair of close lines, written as
(Kα1 and Kα2) in Siegbahn notation.

[edit]
Shortcomings

The Bohr model gives an incorrect value
for the ground state orbital angular momentum. The angular momentum in the true
ground state is known to be zero. Although mental pictures fail somewhat at
these levels of scale, an electron in the lowest modern "orbital" with no
orbital momentum, may be thought of as not to rotate "around" the nucleus at
all, but mearly to pass back and forth straight "through" it, in a spherical
cloud of probability which grows more dense as the nucleus is approached, but
which is also characterized by a larger distance where the electron is most
likely to be found (due to its slowing as it travels outward, and also
volume-shells increase with distance). This radius in hydrogen is equal to the
Bohr radius, but since one model is based on the electron having a minimal
angular momentum and other on its having none at all, the fact that both return
the same distance number, is in some sense a coincidence of nature.

The Bohr model also has difficulty with, or else fails to explain:

Much of the spectra of larger atoms. At best, it can make predictions
about the K-alpha and some L-alpha X-ray
emission spectra for larger atoms, if two additional ad hoc assumptions
are made (see Moseley's law above).
Emission spectra for atoms with a single outer-shell electron (atoms in the lithium group)
can also be approximately predicted. Also, if the empiric electron-nuclear
screening factors for many atoms are known, many other spectral lines can be
deduced from the information, in similar atoms of differing elements, via the
Ritz-Rydberg combination principles (see Rydberg formula). All
these techniques essentially make use of Bohr's Newtonian energy-potential
picture of the atom.

The relative intensities of spectral lines; although in some simple cases,
Bohr's formula or modifications of it, was able to provide reasonable
estimates (for example, calculations by Kramers for the Stark
effect).

The existence of fine structure and hyperfine
structure in spectral lines, which are known to be due to a variety of
relativistic and subtle effects, as well as complications from electron spin.

The Zeeman effect - changes
in spectral lines due to external magnetic fields; these
are also due to more complicated quantum principles interacting with electron
spin and orbital magnetic fields.

[edit]
Refinements

Several enhancements to the Bohr model were proposed; most notably the
Sommerfeld model or Bohr-Sommerfeld model, which suggested that
electrons travel in elliptical orbits around a nucleus instead of the Bohr
model's circular orbits. This model supplemented the quantized angular momentum
condition of the Bohr model with an additional radial quantization condition,
the Sommerfeld-Wilson quantization condition

where p is the generalized
momentum conjugate to the angular generalized
coordinate q; the integral is the action of action-angle
coordinates.

The Bohr-Sommerfeld model proved to be extremely difficult and unwieldy when
its mathematical treatment was further fleshed out. In particular, the
application of traditional perturbation theory
from classical planetary mechanics
led to further confusions and difficulties. In the end, the model was abandoned
in favour of the full quantum mechanical
treatment of the hydrogen atom, in 1925, using Schrödinger's wave
mechanics.

However, this is not to say that the Bohr model was without its successes.
Calculations based on the Bohr-Sommerfeld model were able to accurately explain
a number of more complex atomic spectral effects. For example, up to first-order
perturbation, the
Bohr model and quantum mechanics make the same predictions for the spectral line
splitting in the Stark effect. At
higher-order perturbations, however, the Bohr model and quantum mechanics
differ, and measurements of the Stark effect under high field strengths helped
confirm the correctness of quantum mechanics over the Bohr model.

The Bohr-Sommerfeld quantization condition as first formulated can be viewed
as a rough early draft of the more sophisticated condition that the symplectic form of a
classical phase space M be
integral; that is, that it lies in the image of ,
where the first map is the homomorphism of Čech cohomology groups
induced by the inclusion of the integers in the reals, and the second map is the
natural isomorphism between the Čech cohomology and the de Rham cohomology
groups. This condition guarantees that the symplectic form arise as the curvature form of a connection of a
Hermitian
line
bundle. This line bundle is then called a prequantization in the theory of
geometric
quantization.

[edit]
See also

Franck-Hertz
experiment provided early support for the Bohr model.

Moseley's law provided
early support for the Bohr model. See also Henry Moseley

Inert pair effect is
adequately explained by means of the Bohr model.

Lyman series

Schrödinger
equation

Theoretical
and experimental justification for the Schrödinger equation

Balmer's Constant

[edit]
References

[edit]
Historical

Niels Bohr (1913). "On the Constitution of Atoms and Molecules (Part 1 of 3)".
Philosophical Magazine 26: 1-25.

Niels Bohr (1913). "On the Constitution
of Atoms and Molecules, Part II Systems Containing Only a Single Nucleus".
Philosophical Magazine 26: 476-502.

Niels Bohr (1913). "On the Constitution
of Atoms and Molecules, Part III". Philosophical Magazine 26:
857-875.

Niels Bohr (1914). "The spectra of helium
and hydrogen". Nature 92: 231-232.

Niels Bohr (1921). "Atomic Structure". Nature.

A. Einstein (1917). "Zum Quantensatz von
Sommerfeld und Epstein". Verhandlungen der Deutschen Physikalischen
Gesellschaft 19: 82-92. Reprinted in The Collected
Papers of Albert Einstein, A. Engel translator, (1997) Princeton University
Press, Princeton. 6 p.434. (Provides an elegant reformulation of
the Bohr-Sommerfeld quantization conditions, as well as an important insight
into the quantization of non-integrable (chaotic) dynamical systems.)

[edit]
Further reading

Linus Pauling (1985).
General Chemistry, Chapter 3 (3rd ed). Dover Publications.
A great explainer of Chemistry describes the Bohr model, appropriate for
High School and College students.

George Gamow (1985). Thirty
years that shook Physics, Chapter 2. Dover Publications.
A popularizer of physics explains the Bohr model in the context of the
development of quantum mechanics, appropriate for High School and College
students

Walter J. Lehmann (1972).
Atomic and Molecular Structure: the development of our concepts, chapter
18. John Wiley and Sons.
Great explanations, appropriate for High School and College students

Paul Tipler and Ralph
Llewellyn (2002). Modern Physics (4th ed.). W. H. Freeman. ISBN
0-7167-4345-0.

Retrieved from "http://en.wikipedia.org/wiki/Bohr_model"

Categories: Atomic
physics | Foundational
quantum physics | Obsolete
scientific theories | Hydrogen
physics | Atomic
Cloud Theory

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