Bohr atom

1. Classical model of the atom

After the work of Rutherford on 1911, it is stated that the atom is composed of a nucleus and electrons moving around. In this study, It is considered the simplest atom, hydrogen. The force between the electron and the proton (nucleus) is given by the Coulom's law:

Where:
. F_e is the Coulombian electrostatic force exerted by the proton on the electron.
. - e is the charge of the electron (+ e is the charge of the proton),
. r is the distance between the two particles, and
. e_r is the unit vector in the direction from the proton to the electron.


This F_e needs to be balanced by the centripetal force F_c in order to maitain the electron of mass m moving (rotating) at constant tangential speed v in a circular orbit of radius r.

This phrase is written as: F_e = F_c, that is :
(1/4πε₀)(e²/r²) = m v²/r     (1.2)
We have then:

2. Bohr atom

Bohr's general assumptions (Postulates) in order to derive the Rydberg equation. 1. It exists for the orbiting electrons in atoms some stationary states in which they do not radiate electromagnetic energy.

2. The emission or absorption of electromagnetic radiation for an atom occurs only relatively to the transition of electrons between two stationary states. The frequency ν of the related (absorbed or emitted) radiation obeys the equation: ΔE = (h/2π) ν. ΔE is the energy difference between two stationary states and h is the Planck's constant.

3. The classical laws of Physics govern the dynamical equilibrium of the system in the stationary states, but they are not applicable to transitions between states.

4. For circular motion, the angular momentum of the system (electron-nucleus) in a stationary state is an integral multiple of (h/2π).

For circular motion, the angular momentum L of the electron is: L = r x p. Its magnitude is L = mvr. This angular momentum should be (Assumption 4) as follows: mvr = n (h/2π), or:
n is called the principal quantum number. It quantizes all the related physical values such as velocities, radii and enrgies.

3. The Bohr's Correspondence Principle:

It states:
In the limits where classical and quantum theories should agree, the quantum theory must reduce to the classical result.
Example:
Classical electodynamics gives the frequency of the electron in its orbit around the nucleus:
ν_Classical = 1/T = ω/2π = (1/2π) v/r     (3.1)

Using the new quantized values of r_n and v_n (2.2) and (2.10), we find: v/r = (h/2π)(1/n³ m a₀²). the equation (3.1) becomes: ν_Classical = [h/1/(2π)²][1/n³ m a₀²]     (3.2)

The Bohr's model gives the frequency ν_Bohr of the transition from the state n + 1 to the nearest state n. The equation (2.9) yields:
ν_Bohr = [E₀/h][(1/n²) - (1/(n + 1)²)] = [E₀/h][(2n + 1)/n²)(n + 1)²].     (3.3)

The approximation for large orbits, that is for large values of the quatum number n, we have:
ν_Bohr = 2E₀/hn³. The two frequencies are equal and the Bohr's correspondance is verified.

4. An improvement of the Bohr atom

In fact, the mass of the nucleus is not infinite. the system (nucleus, electron) rotate about the center of mass. The mass in the previous equations would be the reduced mass μ of the system. The more accurate Rydberg constant becomes then , for the hydrogen atom:
R_H= R_∞ μ/m = R_∞ / (1 + m/M)     (5.1)
We have R_H = 1.096776 x 10⁷ m^{- 1}, which fits the measuremements.

The Bohr model may be applied to the hydrogen-like, that is the sigle-electron atom such as H⁺, Li⁺⁺. Because of he Coulomb interaction (between Ze and e), the e² in Bohr's formulas becomes Ze², where Z is the Atomic number of the atom. The Rydberg constant R changes into Z²R and the Rydberg equation becomes:
1/λ = Z²R[(1/n_i²) - (1/n_f²)]     (5.2)

5. Frank & Hertz Experiment

The German physicits James Franck and Gustav Hertz (nephew of Heinrich Hetz), in 1914, wanted to study the ionization phenomenun. They used an electron bombardment of gaseous vapors. The above figure describes the experiment. Electrons extracted from a heated filament (cathode) are accelerated by a varying voltage (0 - 45 V)controlled by a voltmeter. these electrons ionize the monoatomic gas H_g.

The extracted electrons arrive at the grid and then decelerated between the grid and the electrometer (sensitive ammeter). As the potential difference between the electrometer is only 1.5 V, that means only the electrons with energy equal to 1.5 eV can be detected by the electrometer device. The extracted electrons with energy less than 1.5 eV do not pass; the electrons with energy more than 1,5 eV are decelerated and then detected.. The results are as follows:

If the accelerating power supplies is less than 5 V (then the energy of the electrons are 5 eV), a current is detected (all the elctrons within the energy range 1.5 to about 5 eV). The electrons do not lose energy, the collisions are elastic.

When the voltage is icreased to about 5 V (precisely 4.88 eV), the current drops suddenly. The collisions becomes inelastic. That is, all the electrons with 4.88 eV do not go to the grid, they excite the gas atoms. The same process occurs with electrons at energy about 10 eV (precisely about 4.88 x 2 = 9.76 eV). The expected ionization process (wich would occur with very high energy)turns out to be excitation.

The electromagnetic radiation (photon) were not seen because they are not in the visible spectra. Nevertheless, an emission line of wavelength 254 nm (Ultraviolet) was measured. Setting E = 4.88 eV, and used the Planck relationship E = hν Franck and Hertz showed that the Planck constant was in good agreement with the values determined by others. This experiment is another explanation way to verify the quatization of energy.

It exists other highly states in H_g that could be excited ( from L to M , N to L , ...) in enalstic collisions; but the probabilty of that occurrence is more smaller that the first (from L to K , M to K , N to K) excited states.

6. Hydrogen Spectra: related formula

Johann Jakob Balmer, a Swiss mathematician and an honorary physicist, in 1885, heated a Hydrogen tube and analyzed the lines emitted. He established an empirical formula for the spectral lines of the Hydrogen atom and obtained:

λ = hn²/(n² - 2²)

In 1888, Johannes Robert Rydberg, Swedish physicistinverted both sides of Balmer's formula and gave:

1/λ = R_H[(1/2²) - (1/n²)]

7. Comments:

Due its circular motion about the nucleus, the electron is accelerated. An accelerating charged particle looses (gives off or radiates)energy in the form of electromagnetic radiation ( as light, xray, ...). If energy of the electron decreases continusly, then its radius deccreases continusly, spiralling inward until it crashes into the nucleus, so the atome collapses!

Bohr strived to build an ad hoc theory to account for the hydrogen spectral lines experimented by Balmer. He said himself: " As soon as I saw the Rydberg formula, everything became clear to me". The set hypothesis mvr = nh/2π, matched the predictions and became then the found key formula; that is the angular momentum is quantized. The quatization of energy was already thought before by Max Planck. The introduction of the integral number n gives the discrete values for the related values such as radius, velocities and energies. This hidden quantization feature appears in the experimental (empirical) Rydberg formula.

Whilst the Bohr model was a major step toward understanding the quantum theory of the atom, considering a circular orbiting motion of the electron (as a planet) around the nucleus is just approximative; setting definite values for the radius and momentum violates the uncertainty principle in which position and momentum cannot be determined simultaneously.

The correct motion of the electron and the explanation of the brightness of the spectral lines will be described by the probability of presence (charge distribution) and the transition probabilities (spectra) concepts derived from Quantum Mechanics.