Bohr model

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 which 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 naive planetary theory cannot explain why atoms are extremely long-lived.

The naive 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 (that is, electromagnetic radiation), but only at certain discrete frequencies. A naive 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 (or supplied) by a single quantum of light (called a photon) which has an energy equal to the energy difference between the two orbitals.
The allowed orbits depend on quantized (discrete) values of orbital angular momentum, L according to the equation
$\mathbf{L} = n \cdot \hbar = n \cdot {h \over 2\pi}$
Where n = 1,2,3,… and is called the principal quantum number, and h is Planck's constant.

Assumption (4) 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 quantised 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 (4) with an additional radial quantization condition, the Sommerfeld-Wilson quantization condition

\oint p dq = nh

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 lie in the image of $\check{H}^2(M,\mathbb{Z})\to \check{H}^2(M,\mathbb{R})\to H^2_{DR}(M,\mathbb{R})$ , 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.

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

\lambda

, is related to its velocity v by:

\lambda = \frac{h}{m_e v}

where h is Planck's Constant, and

m_e

is the mass of the electron. Bohr did not make this assumption (known as the de Broglie hypothesis) 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:

2 \pi r = n \lambda \,

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:

\frac{kq_e^2}{r^2} = \frac{m_e v^2}{r} \,

where

k = 1 / ({4 \pi \epsilon _0})

, and

q_e

is the charge of the electron.

These are three equations with three unknowns: $\lambda$ , 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:

E \,

=E_{kinetic} + E_{potential} \,

= \begin{matrix} \frac{1}{2} \end{matrix}m_e v^2 - \frac{k q_e^2}{r}

Because of the virial theorem, the total energy simplifies to

E = -\begin{matrix} \frac{1}{2} \end{matrix}m_e v^2

Substituting, one obtains the energy of the different levels of hydrogen:

E _n \,

= -2 \pi^2 k^2 \left( \frac{m_e q_e^4}{h^2} \right) \frac{1}{n^2} \,

= \frac{-m_e q_e^4}{8 h^2 \epsilon_{0}^2} \frac{1}{n^2} \,

Or, after plugging in values for the constants,

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E_n = \frac{-13.6 \ \mathrm{eV}}{n^2} \,

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.

Energy in terms of other constants

Starting with what we found above,

E_n = \frac{-m_e q_e^4}{8 h^2 \epsilon_{0}^2} \frac{1}{n^2} \,

We can multiply top and bottom by

c^2

, and we'll arrive at

E_n = \frac{-m_e c^2 q_e^4}{8 h^2 c^2 \epsilon_{0}^2} \frac{1}{n^2} \,

or re-grouping them to make it more clear:

E_n = -\frac{1}{2} m_e c^2 \left(\frac{q_e^4}{4 h^2 c^2 \epsilon_{0}^2} \right) \frac{1}{n^2}

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

E_n = \frac{-E_r\alpha^2}{2n^2}

where,

E_n \

is the energy level

E_r \

is the rest energy of the electron

\alpha \

is the fine structure constant

n \

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:

E=E_i-E_f=\frac{m_e q_e^4}{8 h^2 \epsilon_{0}^2} \left( \frac{1}{n_{f}^2} - \frac{1}{n_{i}^2} \right) \,

where q_e is the charge of an electron (1.60x10^-19 C),

n_f

is the final energy level, and

n_i

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

E=\frac{hc}{\lambda} \,

the wavelength of the photon given off is

\frac{1}{\lambda}=\frac{m_e q_e^4}{8 c h^3 \epsilon_{0}^2} \left( \frac{1}{n_{f}^2} - \frac{1}{n_{i}^2} \right) \,

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

\mathbf{L} = \hbar

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 (atoms in the lithium group.)
The relative intensities of spectral lines; although in some simple cases, 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.
The Zeeman effect - changes in spectral lines due to external magnetic fields.

References

Historical

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.)

Modern

Atomic physics | Foundational quantum physics | Obsolete scientific theories

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