Atomic physics

Atomic Physics

Atomic theory has evolved significantly since John Dalton proposed his model in 1803. Dalton's theory was based on five key postulates:

1. Matter is composed of indivisible and indestructible atoms.
2. All atoms of a given element are identical in mass and properties.
3. Atoms of different elements vary in mass and chemical behavior.
4. Atoms combine in simple whole-number ratios to form compounds.
5. Atoms are neither created nor destroyed in chemical reactions.

While Dalton's model was revolutionary, the discovery of subatomic particles such as the electron (by J.J. Thomson in 1897) revealed that atoms are divisible, disproving his first assumption.

J.J. Thomson's plum pudding model

Thomson proposed the "Plum Pudding Model" in 1904, in which the atom was visualized as a positively charged sphere embedded with negatively charged electrons, like plums in a pudding. However, this model could not explain experimental results from alpha particle scattering experiments.

Rutherford's planetary model

Ernest Rutherford conducted the gold foil experiment in 1911 to investigate atomic structure. Alpha particles (~5 MeV) were directed at a thin gold foil. A phosphor screen detected the scattered particles.

Observations:

• Most alpha particles passed through the foil undeviated.
• Some were deflected at moderate angles $0^\circ < \theta \leq 90^\circ$.
• A few (~1 in 8000) were deflected at large angles $90^\circ < \theta \leq 180^\circ$.

These results suggested that the atom is mostly empty space, with a small, dense, positively charged nucleus at the center, containing most of the atom's mass. Electrons orbit this nucleus, forming what Rutherford called the planetary model.

Limitations of Rutherford's model

1. Did not explain atomic stability Maxwell's theory predicts that accelerating electrons emit radiation and would spiral into the nucleus in $< 10^{-8}$ s.

2. Could not explain discrete spectral lines observed in hydrogen emission spectra. This prediction implies that electrons should not maintain stable orbits, contradicting the observed long-term stability of atoms.

3. Did not account for neutron presence Rutherford hypothesized the neutron in 1926, confirmed by Chadwick in 1932. The model lacked a complete understanding of atomic mass, as it did not include the neutron, which contributes significantly to atomic mass.

4. Lacked a defined mechanism for electron arrangements.

   The model did not describe how electrons are arranged around the nucleus or how their positions determine chemical behavior.

Bohr's atomic model of hydrogen

Niels Bohr improved Rutherford's model in 1913 by incorporating quantum concepts. He introduced the following postulates:

1. Electrons revolve around the nucleus in discrete circular orbits (stationary states) without emitting radiation.

2. Only those orbits are permitted where the electron's angular momentum is quantized:

   $mvr = \dfrac{nh}{2\pi}$

3. Electrons emit or absorb energy only during transitions between energy levels:

   $E_m - E_n = hf$

Bohr's quantized energy levels

The centripetal force is provided by the electrostatic force:

$\dfrac{mv^2}{r} = \dfrac{Ze^2}{4\pi \varepsilon_0 r^2}$

Using the angular momentum quantization condition:

$mvr = \dfrac{nh}{2\pi} \Rightarrow v = \dfrac{nh}{2\pi mr}$

Substituting and simplifying, we obtain the radius of the nth orbit:

$r_n = \dfrac{n^2 h^2 \varepsilon_0}{\pi m Z e^2}$

For hydrogen (Z = 1), the radius of the first orbit (Bohr radius) is:

$a_0 = \dfrac{h^2 \varepsilon_0}{\pi m e^2} \approx 0.529 \times 10^{-10} \ \text{m}$

Total energy in the nth orbit is:

$E_n = -\dfrac{m Z^2 e^4}{8 \varepsilon_0^2 h^2 n^2}$

For hydrogen (Z = 1), this simplifies to:

$E_n = -\dfrac{13.6}{n^2} \ \text{eV}$

Successes of Bohr's theory

Energy levels

Energy levels are quantized. The ground state of hydrogen has $E = -13.6 \ \text{eV}$, and excited states occur at higher (less negative) energies.

Rydberg formula from Bohr's model

When an electron jumps from an orbit $m$ to $n$, energy is emitted as:

$hf = E_m - E_n = 13.6 \ \text{eV} \left( \dfrac{1}{n^2} - \dfrac{1}{m^2} \right)$

The corresponding wavelength is given by:

$\dfrac{1}{\lambda} = R_H \left( \dfrac{1}{n^2} - \dfrac{1}{m^2} \right)$

Where $R_H \approx 1.097 \times 10^7 \ \text{m}^{-1}$ is the Rydberg constant. This accurately predicts the Balmer series (visible spectrum) for hydrogen.

Example: Balmer series (n = 2)

Transition (m → 2)	Measured λ (nm)	Predicted λ (nm)	Deviation (nm)
3 → 2	656.29	656.47	-0.18
4 → 2	486.13	486.27	-0.14
5 → 2	434.08	434.17	-0.09

Thus, the Bohr model was the first quantum model to successfully explain atomic spectra and provided a foundation for later quantum mechanics.

Prediction of the Rydberg constant using Bohr's theory

Bohr's third postulate

When an electron jumps from a higher orbit (quantum number $m$) to a lower orbit ($n$), energy is emitted as radiation.

$E_m - E_n = hf = \frac{hc}{\lambda}$

Energy levels of hydrogen atom

Bohr's model gives the energy of the electron in the hydrogen atom as:

$E_n = -\frac{13.6\text{ eV}}{n^2}$

Where:

• $n$ = principal quantum number (1, 2, 3, …)
• Energy is negative (indicating bound states)

Balmer series – visible hydrogen spectrum

• For the Balmer series, the final orbit is n = 2.
• Electrons fall from $m = 3, 4, 5, \dots$ to $n = 2$
• Equation used:

$\frac{1}{\lambda} = \frac{13.6\text{ eV}}{1240\text{ eV·nm}} \left( \frac{1}{2^2} - \frac{1}{m^2} \right)$

Table: Measured and predicted wavelengths for Balmer series

m	Measured λ (nm)	Predicted λ (nm)	Deviation (nm)
3	656.29	656.47	-0.03
4	486.13	486.27	-0.02
5	434.08	434.17	-0.03

General formula for spectral lines

From Bohr's theory:

$\frac{1}{\lambda} = R_H \left( \frac{1}{n^2} - \frac{1}{m^2} \right)$

Where $R_H$ is the Rydberg constant.

Prediction of Rydberg constant

Using:

$hf = 13.6\text{ eV} \left( \frac{1}{n^2} - \frac{1}{m^2} \right) = \frac{hc}{\lambda}$

Solving for $R_H$:

$R_H = \frac{13.6\text{ eV}}{hc} \approx \frac{13.6}{1240 \times 10^{-9}} \approx 1.0968 \times 10^7 \text{ m}^{-1}$

This value agrees closely with experimental results.

Other hydrogen spectral series

Name	Final Level (n)
Lyman	1
Balmer	2
Paschen	3
Brackett	4
Pfund	5
Humphrey	6

Limitations of Bohr's theory

1. Only works for hydrogen and hydrogen-like atoms (one electron) The theory accurately predicts spectral lines only for single-electron systems, failing for atoms with multiple electrons.
2. Cannot explain intensity variations in spectral lines Bohr's model does not address why some spectral lines are brighter or weaker, lacking an explanation for intensity differences.
3. Does not describe electron distribution in multi-electron atoms It fails to account for electron-electron interactions and arrangements in atoms with more than one electron.
4. Cannot explain stability of electron orbits While Bohr introduced quantized orbits, the model does not fully explain why electrons remain stable in these orbits without radiating energy.

Resolution by de Broglie

• Introduced wave-particle duality: electrons as standing waves.
• For stability:

$2\pi r = n\lambda \Rightarrow mvr = n\hbar$

This condition aligns with Bohr's quantization of angular momentum and gives a physical reason for orbit stability.