Encyclopedia
In atomic physics, the
Bohr model depicts the
atom as a small, positively charged
nucleus surrounded by waves of
electrons in orbit — similar in structure to the
solar system, but with
electrostatic forces providing attraction, rather than
gravity, and with waves spread over entire orbit instead of localized planets.
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.
The Bohr model is a primitive model of the hydrogen atom that cannot explain the fine structure of the hydrogen atom nor any of the heavier atoms. As a theory, it can be derived as a first-order approximation of the hydrogen atom in the broader and much more accurate
quantum mechanics, and thus may be considered to be an obsolete scientific theory. However, because of its simplicity, the Bohr model is still commonly taught to introduce students to quantum mechanics.
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 is quite natural to consider a planetary model for the atom, with electrons orbiting a sun-like nucleus. However, a naive planetary model has several difficulties, the most serious of which is the loss of energy by
synchrotron radiation.That is, an accelerating electric charge emits
electromagnetic waves which carry
energy; thus, with each orbit around the nucleus, the electron would radiate away a bit of its orbital energy, gradually spiralling inwards to the nucleus until the atom was no more. A quick calculation shows that this would happen almost instantly; thus, the planetary theory cannot explain why atoms are extremely long-lived.
The planetary model also failed to explain
atomic spectra, the observed discrete spectrum of light emitted by electrically excited atoms. Late
19th century experiments with
electric discharges through various low-pressure gasses in evacuated glass tubes had shown that atoms will emit light , but only at certain discrete frequencies. A planetary model cannot explain this.
To overcome these difficulties,
Niels Bohr proposed, in 1913, what is now called the
Bohr model of the atom. The key ideas were:
- The orbiting electrons existed in orbits that had discrete quantized energies. That is, not every orbit is possible but only certain specific ones.
- The laws of classical mechanics do not apply when electrons make the jump from one allowed orbit to another.
- When an electron makes a jump from one orbit to another, the energy difference is carried off by a single quantum of light which has an energy equal to the energy difference between the two orbitals.
- The allowed orbits depend on quantized 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.
Assumption 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.
The Bohr model is sometimes known as the
semiclassical model of the atom, as it adds some primitive quantization conditions to what is otherwise a
classical mechanics treatment. The Bohr model is certainly not a full quantum mechanical description of the atom. Assumption 2) states that the laws of classical mechanics don't apply during a quantum jump, but it doesn't state what laws should replace classical mechanics. Assumption 4) states that angular momentum is quantized but does not explain why.
Refinements
Several enhancements to the Bohr model were proposed; most notably the
Sommerfeld model or
Bohr-Sommerfeld model, which attempted to add support for elliptical orbits to the Bohr model's circular orbits. This model supplemented condition with an additional radial quantization condition, the
Sommerfeld-Wilson quantization conditionwhere
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 Cech cohomology groups induced by the inclusion of the integers in the reals, and the second map is the natural isomorphism between the Cech 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.
Electron energy levels in hydrogen
The Bohr model is accurate only for one-electron systems such as the
hydrogen atom or singly-ionized
helium. This section uses the Bohr model to derive the energy levels of hydrogen.
The derivation starts with three simple assumptions:
- 1) All particles are wavelike, and an electron's wavelength , is related to its velocity v by:
where
h is
Planck's Constant, and is the mass of the electron. Bohr did not make this assumption in his original derivation, because it hadn't been proposed at the time. However it allows the following intuitive statement.
- 2) The circumference of the electron's orbit must be an integer multiple of its wavelength:
where
r is the radius of the electron's orbit, and
n is a positive integer.
- 3) The electron is held in orbit by the coulomb force. That is, the coulomb force is equal to the centripetal force:
where , and is the charge of the electron.
These are three equations with three unknowns: ,
r,
v. After solving this system of equations to find an equation for just
v, it is placed into the equation for the total energy of the electron:
Because of the virial theorem, the total energy simplifies to
Substituting, one obtains the energy of the different levels of hydrogen:
Or, after substituting values for the constants,
Thus, the lowest energy level of hydrogen is about -13.6 eV. The next energy level is -3.4 eV. The third 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.
Energy in terms of other constants
Starting with what we found above,
We can multiply top and bottom by , 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.
Rydberg formula
The
Rydberg formula describes the transitions or quantum jumps between one energy level and another. When the electron moves from one energy level to another, a
photon is given off. Using the derived formula for the different 'energy' levels of hydrogen one may determine the 'wavelengths' of light that a hydrogen atom can give off.
The energy of photons that a hydrogen atom can give off are given by the difference of two hydrogen energy levels:
- where qe is the charge of an electron , is the final energy level, and 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 was known in the nineteenth century to scientists studying
spectroscopy, but there was no theoretical justification for the formula until Bohr derived it, more or less along the lines above.
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.
The Bohr model also has difficulty with or fails to explain:
- The spectra of larger atoms. At best, it can make some approximate predictions about the emission spectra for atoms with a single outer-shell electron
- The relative intensities of spectral lines; although in some simple cases, it was able to provide reasonable estimates .
- The existence of fine structure and hyperfine structure in spectral lines.
- The Zeeman effect - changes in spectral lines due to external magnetic fields.
See also
References
Historical
- Reprinted in The Collected Papers of Albert Einstein, A. Engel translator, Princeton University Press, Princeton. 6 p.434. '
Modern
