Introduction

By the 19th century, scientific evidence strongly supported the concept of matter being composed of atoms. A key development was J.J. Thomson's experiments on electric discharge through gases in 1897, which led to the discovery of the electron. These experiments showed that atoms of different elements contain identical negatively charged particles (electrons). Since atoms are electrically neutral overall, they must also contain a positive charge to balance the negative charge of the electrons.

The central question then became: How are the positive charge and electrons arranged within an atom? The first attempt to answer this came from J.J. Thomson himself in 1898, with his "plum pudding" model. In this model, the atom was imagined as a sphere of uniformly distributed positive charge, with the negatively charged electrons embedded within it, much like seeds in a watermelon.

However, this initial model did not align with later experimental findings about atomic structure. Furthermore, studies of light emitted by substances provided crucial insights. Condensed matter (solids and liquids) and dense gases emit a continuous spectrum of electromagnetic radiation when heated. In contrast, rarefied gases when heated or electrically excited (like in neon signs) emit radiation only at certain distinct wavelengths, producing a line spectrum (a series of bright lines on a dark background). Each element was found to have its own unique characteristic line spectrum, acting as a "fingerprint".

This relationship between the emitted spectrum and the element's identity suggested a deep connection between the internal structure of an atom and the specific wavelengths of light it can emit. For the simplest element, hydrogen, Johann Jakob Balmer in 1885 found a simple formula that described the wavelengths of a series of lines in its visible spectrum. This empirical formula hinted at an underlying order within the hydrogen atom.

A pivotal experiment to probe the internal structure of the atom was conceived by Ernest Rutherford, a former student of J.J. Thomson. He proposed scattering alpha ($\alpha$)-particles off atoms to investigate their internal arrangement. This experiment, performed by Hans Geiger and Ernest Marsden around 1911, led to significant conclusions about the distribution of mass and charge within the atom, fundamentally altering the view of atomic structure.

Alpha-Particle Scattering And Rutherford’s Nuclear Model Of Atom

To investigate the structure of the atom, Geiger and Marsden, under Rutherford's guidance, conducted experiments involving the scattering of alpha ($\alpha$)-particles by thin metal foils. Alpha particles, known to be positively charged helium nuclei ($^4_2\text{He}^{2+}$), were emitted from a radioactive source (like $^{214}_{83}\text{Bi}$). A beam of these $\alpha$-particles, with kinetic energy around 5.5 MeV, was directed onto a very thin sheet of gold foil (about $2.1 \times 10^{-7}$ m thick) (

Figure 12.2).

The scattered $\alpha$-particles were detected using a rotatable detector comprising a zinc sulfide (ZnS) screen and a microscope. When an $\alpha$-particle struck the ZnS screen, it produced a brief flash of light (scintillation) that could be observed through the microscope. By rotating the detector, they could measure the number of $\alpha$-particles scattered at different angles relative to the original beam direction.

The results of the experiment were surprising and did not fit with the Thomson model:

• Most of the $\alpha$-particles passed straight through the gold foil with little or no deviation. This suggested that atoms are mostly empty space.
• Some $\alpha$-particles were deflected by small angles.
• A small fraction (about 0.14% deflected by more than 1°, and approximately 1 in 8000 deflected by more than 90°) were scattered through large angles, some even bouncing back in the direction they came from (scattering angle close to 180°).

The large-angle scattering was particularly astonishing. Rutherford famously remarked that it was "quite the most incredible event that has ever happened to me in my life. It was almost as incredible as if you fired a 15-inch shell at a piece of tissue paper and it came back and hit you."

Rutherford interpreted these results by proposing a new model of the atom, known as the Rutherford nuclear model or the planetary model. According to this model:

• The atom's entire positive charge and almost all of its mass are concentrated in a tiny, dense central region called the nucleus.
• Electrons, which are negatively charged and much lighter, revolve around the nucleus in orbits, similar to planets orbiting the sun.
• The size of the nucleus is estimated to be very small, on the order of $10^{-15}$ m to $10^{-14}$ m, while the atom's total size is about $10^{-10}$ m. This means the nucleus is about 10,000 to 100,000 times smaller than the atom, leaving most of the atom as empty space.

The large empty space explained why most $\alpha$-particles passed through undeflected. The large deflections and backscattering of the few $\alpha$-particles that did not pass straight through could be explained by a direct collision or close encounter with the small, massive, positively charged nucleus. The strong electrostatic repulsion between the positive $\alpha$-particle and the positive nucleus would cause a large deflection. The light electrons would not significantly affect the trajectory of the much heavier $\alpha$-particles.

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Alpha-Particle Trajectory

The path of an $\alpha$-particle as it interacts with the nucleus can be analyzed using Coulomb's law and Newton's second law. The electrostatic force of repulsion between the $\alpha$-particle (charge $2e$) and the nucleus (charge $Ze$, where Z is the atomic number) at a distance $r$ is given by $F = \frac{1}{4\pi\epsilon_0} \frac{(2e)(Ze)}{r^2}$. Since the gold nucleus is much heavier than the $\alpha$-particle, it is assumed to remain stationary during the interaction.

The trajectory of an $\alpha$-particle is determined by its impact parameter ($b$), which is the perpendicular distance between the initial velocity vector of the $\alpha$-particle and the center of the nucleus (

Figure 12.4). For a given initial kinetic energy, the scattering angle ($\theta$) depends on $b$:

• For a large impact parameter ($b$ large), the $\alpha$-particle passes far from the nucleus and experiences a weak repulsive force. The deflection angle $\theta$ is small.
• For a small impact parameter ($b$ small), the $\alpha$-particle passes close to the nucleus and experiences a strong repulsive force. The deflection angle $\theta$ is large.
• For a head-on collision ($b \approx 0$), the $\alpha$-particle approaches the nucleus directly, momentarily stops at the point of closest approach, and is repelled back along its original path ($\theta \approx 180^\circ$ or $\pi$).

The small number of $\alpha$-particles scattered at large angles or backscattered implies that head-on or near head-on collisions (small $b$) are rare. This is consistent with the nucleus being a very small target compared to the overall size of the atom. By analyzing the number of scattered particles as a function of angle, Rutherford was able to estimate the size of the nucleus and confirm that it is indeed extremely small.

The distance of closest approach ($d$) in a head-on collision ($b \approx 0$) occurs when the $\alpha$-particle momentarily stops. At this point, its initial kinetic energy is completely converted into electrostatic potential energy. $K_{initial} = U_{final} = \frac{1}{4\pi\epsilon_0} \frac{(2e)(Ze)}{d}$. This allows for calculating an upper limit for the size of the nucleus.

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Electron Orbits

In Rutherford's nuclear model, electrons orbit the positively charged nucleus. For a stable orbit, the centripetal force required to keep the electron in a circular path is provided by the electrostatic force of attraction between the negatively charged electron ($e$) and the positively charged nucleus ($Ze$). For a hydrogen atom ($Z=1$), the nucleus is a proton (charge $e$).

Electrostatic force of attraction $F_e = \frac{1}{4\pi\epsilon_0} \frac{e \cdot e}{r^2} = \frac{e^2}{4\pi\epsilon_0 r^2}$, where $r$ is the orbital radius.

Centripetal force $F_c = \frac{mv^2}{r}$, where $m$ is the mass of the electron and $v$ is its orbital speed.

For a dynamically stable orbit: $F_e = F_c$

$$\frac{e^2}{4\pi\epsilon_0 r^2} = \frac{mv^2}{r}$$

This equation relates the orbital radius and velocity: $mv^2 = \frac{e^2}{4\pi\epsilon_0 r}$.

The kinetic energy of the electron is $K = \frac{1}{2}mv^2 = \frac{1}{2} \left(\frac{e^2}{4\pi\epsilon_0 r}\right) = \frac{e^2}{8\pi\epsilon_0 r}$.

The electrostatic potential energy of the electron-nucleus system is $U = \frac{1}{4\pi\epsilon_0} \frac{(-e)(e)}{r} = -\frac{e^2}{4\pi\epsilon_0 r}$. (Potential energy is negative because the force is attractive).

The total mechanical energy ($E$) of the electron in the orbit is the sum of its kinetic and potential energies:

$$E = K + U = \frac{e^2}{8\pi\epsilon_0 r} - \frac{e^2}{4\pi\epsilon_0 r} = \frac{e^2}{8\pi\epsilon_0 r} - \frac{2e^2}{8\pi\epsilon_0 r} = -\frac{e^2}{8\pi\epsilon_0 r}$$

The total energy $E$ is negative. This indicates that the electron is bound to the nucleus. Positive total energy would mean the electron is not bound and would escape.

However, the Rutherford model faced fundamental problems when viewed from the perspective of classical electromagnetic theory:

1. Atomic Stability: According to classical electromagnetism, an accelerating charged particle, like an electron moving in a circular orbit, must continuously emit electromagnetic radiation and lose energy. As the electron loses energy, its orbit radius would decrease, causing it to spiral inwards and eventually fall into the nucleus. This would lead to the atom's collapse, contradicting the observed stability of atoms.

2. Atomic Spectra: If the electron were to continuously spiral into the nucleus, its orbital frequency would change continuously. According to classical theory, the frequency of the emitted radiation should equal the orbital frequency. Thus, the atom should emit radiation with a continuous range of frequencies (a continuous spectrum). This directly contradicts the observed line spectra, where atoms emit light only at specific, discrete wavelengths/frequencies.

Clearly, classical physics was insufficient to explain the observed properties of atoms.

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Atomic Spectra

As discussed earlier, when a rarefied atomic gas is excited (e.g., by heating or electric discharge), it emits light at specific, discrete wavelengths. Observing this light through a spectrometer produces an emission line spectrum, which appears as a series of bright lines against a dark background. Each element has a unique set of spectral lines.

Conversely, when white light (containing a continuous range of wavelengths) passes through a cold sample of a gas, the gas atoms absorb light at those same specific wavelengths that they would normally emit when excited. Analyzing the transmitted light reveals a continuous spectrum with dark lines at the wavelengths that were absorbed. This is called the absorption line spectrum.

The fact that atoms emit or absorb light only at discrete wavelengths strongly suggests that the energy states within an atom are also discrete or quantized. This provided a crucial clue to understanding the internal structure and behavior of atoms, particularly how they interact with electromagnetic radiation.

Bohr Model Of The Hydrogen Atom

Niels Bohr, working in Rutherford's laboratory, recognized the limitations of the classical Rutherford model, particularly its inability to explain atomic stability and the existence of line spectra. In 1913, Bohr proposed a model for the hydrogen atom by incorporating concepts from the emerging quantum theory. His model is based on three postulates:

(i) Bohr's First Postulate (Postulate of Stable Orbits): An electron in an atom can revolve around the nucleus in certain specific, discrete orbits, called stationary orbits or stable states, without emitting any electromagnetic radiation. Each such stationary state has a definite, fixed total energy. This directly contradicts the classical prediction of continuous energy loss by an orbiting electron.

(ii) Bohr's Second Postulate (Postulate of Quantisation of Angular Momentum): The electron revolves around the nucleus only in those stationary orbits for which its angular momentum ($L$) is an integer multiple of $h/2\pi$.

$$L_n = mvr_n = n \frac{h}{2\pi} \quad \text{where } n = 1, 2, 3, \dots$$

Here, $m$ is the electron mass, $v$ is its speed, $r_n$ is the radius of the $n$-th allowed orbit, $h$ is Planck's constant, and $n$ is an integer called the principal quantum number. This postulate introduces the concept of quantization directly into the atomic model.

(iii) Bohr's Third Postulate (Postulate of Energy Transitions): An electron can transition (jump) from one stationary orbit of higher energy ($E_i$) to another stationary orbit of lower energy ($E_f$). When it does so, the energy difference is emitted as a single photon of frequency $\nu$.

$$h\nu = E_i - E_f \quad \text{where } E_i > E_f$$

Conversely, an atom can absorb a photon of precisely the same energy $h\nu = E_f - E_i$ to cause an electron to jump from a lower energy state ($E_i$) to a higher energy state ($E_f$).

Bohr used these postulates, along with classical mechanics for the forces, to derive expressions for the radii of the allowed orbits and the energies of the stationary states in a hydrogen atom ($Z=1$).

From the centripetal force condition $\frac{mv_n^2}{r_n} = \frac{1}{4\pi\epsilon_0} \frac{e^2}{r_n^2}$ and the angular momentum quantization $mv_n r_n = n \frac{h}{2\pi}$, he solved for $r_n$ and $v_n$. The radius of the $n$-th stationary orbit is found to be:

$$r_n = \frac{n^2 h^2 \epsilon_0}{\pi m e^2}$$

Substituting the fundamental constants, the radius of the first Bohr orbit ($n=1$) is $r_1 = a_0 \approx 5.29 \times 10^{-11}$ m, known as the Bohr radius. The radii of higher orbits are $r_n = n^2 a_0$. Thus, the allowed orbits are discrete and depend on the square of the principal quantum number $n$.

The total energy of the electron in the $n$-th stationary state is given by substituting $r_n$ into the energy expression $E_n = -\frac{e^2}{8\pi\epsilon_0 r_n}$:

$$E_n = -\frac{m e^4}{8 \epsilon_0^2 h^2} \frac{1}{n^2}$$

Substituting the values of the constants, the energy levels for the hydrogen atom are approximately:

$$E_n = -\frac{2.18 \times 10^{-18}}{n^2} \text{ J}$$

Converting to electron volts ($1 \text{ eV} \approx 1.602 \times 10^{-19} \text{ J}$):

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

The negative sign signifies that the electron is bound to the nucleus. The state with $n=1$ has the lowest (most negative) energy and is called the ground state ($E_1 = -13.6$ eV). States with $n > 1$ are called excited states, having progressively higher (less negative) energies ($E_2 = -3.4$ eV, $E_3 = -1.51$ eV, etc.). The energy levels become closer together as $n$ increases.

The state with $n=\infty$ corresponds to $E_\infty = 0$ eV, which represents a free electron infinitely far from the nucleus. The energy required to remove an electron from the ground state of a hydrogen atom (ionise it) is $0 - (-13.6 \text{ eV}) = 13.6 \text{ eV}$. This theoretical prediction matches the experimental value extremely well.

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Energy Levels

The possible discrete energies that an electron in a hydrogen atom can have are called its energy levels. These levels are determined by the principal quantum number $n=1, 2, 3, \dots$. The energy level diagram for hydrogen visualizes these allowed energies as horizontal lines (

Figure 12.7).

• The $n=1$ level is the ground state, with the lowest energy (-13.6 eV). The electron is in the smallest allowed orbit (Bohr radius $a_0$).
• Levels $n=2, 3, 4, \dots$ are excited states with energies -3.4 eV, -1.51 eV, -0.85 eV, and so on. As $n$ increases, the energy approaches 0 eV.
• The energy $E=0$ eV and all positive energies correspond to an unbound or free electron, forming a continuum of energy states above $E=0$.

An atom in the ground state can be excited to a higher energy state by absorbing energy, typically through collisions with other particles or by absorbing a photon with energy equal to the energy difference between the initial and final states. Excited states are unstable, and the electron quickly transitions back to a lower energy state, emitting a photon in the process.

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The Line Spectra Of The Hydrogen Atom

The key achievement of Bohr's model was its ability to explain the discrete line spectrum of hydrogen. According to Bohr's third postulate, spectral lines are produced when an electron makes a transition from a higher energy stationary state (with principal quantum number $n_i$) to a lower energy stationary state (with principal quantum number $n_f$, where $n_f < n_i$). The energy difference between these two states is emitted as a single photon with frequency $\nu$ given by:

$$h\nu = E_{n_i} - E_{n_f}$$

Substituting the expression for the energy levels $E_n = -\frac{m e^4}{8 \epsilon_0^2 h^2} \frac{1}{n^2}$, the frequency of the emitted photon is:

$$\nu = \frac{1}{h} \left(E_{n_i} - E_{n_f}\right) = \frac{1}{h} \left(-\frac{m e^4}{8 \epsilon_0^2 h^2 n_i^2} - \left(-\frac{m e^4}{8 \epsilon_0^2 h^2 n_f^2}\right)\right)$$

$$\nu = \frac{m e^4}{8 \epsilon_0^2 h^3} \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$$

The speed of light $c = \nu\lambda$, so the wavelength $\lambda = c/\nu$.

$$\frac{1}{\lambda} = \frac{\nu}{c} = \frac{m e^4}{8 \epsilon_0^2 h^3 c} \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right)$$

The constant term $\frac{m e^4}{8 \epsilon_0^2 h^3 c}$ is called the Rydberg constant (R) for hydrogen. Its theoretical value is approximately $1.097 \times 10^7 \text{ m}^{-1}$. So, the formula for the wavelength of spectral lines in hydrogen is:

$$\frac{1}{\lambda} = R \left(\frac{1}{n_f^2} - \frac{1}{n_i^2}\right) \quad \text{where } n_f < n_i \text{ are integers.}$$

This formula successfully explains the various series of lines observed in the hydrogen spectrum. Different series correspond to transitions ending in specific lower energy levels ($n_f$):

• Lyman series: Transitions ending in the ground state ($n_f = 1$), $n_i = 2, 3, 4, \dots$. These lines are in the ultraviolet region.
• Balmer series: Transitions ending in the first excited state ($n_f = 2$), $n_i = 3, 4, 5, \dots$. These lines are in the visible region (
  Figure 12.5), which is the series Balmer originally described.
• Paschen series: Transitions ending in the second excited state ($n_f = 3$), $n_i = 4, 5, 6, \dots$. These lines are in the infrared region.
• Brackett series: Transitions ending in the third excited state ($n_f = 4$), $n_i = 5, 6, 7, \dots$. These lines are in the infrared region.
• Pfund series: Transitions ending in the fourth excited state ($n_f = 5$), $n_i = 6, 7, 8, \dots$. These lines are in the infrared region.

Each integer value of $n_f$ defines a series, and within each series, different values of $n_i$ produce distinct lines. Since $n_f$ and $n_i$ must be integers, the possible energy differences ($E_{n_i} - E_{n_f}$) are discrete, resulting in the emission of photons with only discrete frequencies (or wavelengths), thus explaining the observed line spectra. Bohr's explanation of the hydrogen spectrum was a monumental achievement in physics.

Absorption spectra occur when an atom in a lower energy state absorbs a photon with energy equal to the difference needed to reach a higher energy state. Only photons with these specific energies (wavelengths) can be absorbed, resulting in dark lines at those wavelengths in a continuous spectrum.

De Broglie’s Explanation Of Bohr’s Second Postulate Of Quantisation

While Bohr's postulates successfully explained the hydrogen spectrum, the second postulate – the quantization of angular momentum ($L = n \frac{h}{2\pi}$) – seemed arbitrary from a classical viewpoint. A physical justification for this quantization rule was provided by Louis de Broglie in 1923, building upon his hypothesis of matter waves.

De Broglie proposed that material particles, like electrons, also possess wave-like properties, with a wavelength $\lambda = h/p$, where $p$ is the particle's momentum. De Broglie suggested that the electron in a Bohr orbit should be thought of as a standing matter wave. For a standing wave to form in a circular orbit, the circumference of the orbit must contain an integer number of wavelengths. If this condition is not met, the wave would destructively interfere with itself over successive orbits, and its amplitude would quickly diminish.

So, for the electron wave to form a stable standing wave in the $n$-th orbit of radius $r_n$, the circumference ($2\pi r_n$) must be equal to an integer multiple of the electron's de Broglie wavelength ($\lambda_n$):

$$2\pi r_n = n \lambda_n \quad \text{where } n = 1, 2, 3, \dots$$

Substituting de Broglie's relation $\lambda_n = h/p_n = h/(mv_n)$ (assuming non-relativistic speed):

$$2\pi r_n = n \frac{h}{mv_n}$$

Rearranging this equation:

$$mv_n r_n = n \frac{h}{2\pi}$$

Since $mv_n r_n$ is the magnitude of the angular momentum ($L_n$) of the electron in the $n$-th orbit, this equation becomes:

$$L_n = n \frac{h}{2\pi}$$

This is precisely Bohr's second postulate. Thus, de Broglie's hypothesis provided a natural and intuitive explanation for why only certain orbits are allowed in the atom: they are the orbits where the electron's matter wave forms a stable standing wave pattern (

Figure 12.8).

De Broglie's explanation linked the wave nature of the electron to the quantization of its angular momentum and energy levels, reinforcing the idea of wave-particle duality and paving the way for modern quantum mechanics.

Despite its success for hydrogen, Bohr's model has limitations:

• It is strictly applicable only to hydrogenic atoms (atoms with a single electron, like H, He$^+$, Li$^{2+}$). It cannot accurately explain the spectra of atoms with multiple electrons because it does not account for the complex electrostatic interactions between the electrons themselves.
• While it predicts the wavelengths/frequencies of spectral lines for hydrogenic atoms, it cannot predict the relative intensities of these lines (i.e., why some lines are brighter or weaker than others).

Although superseded by the more complete theory of quantum mechanics, the Bohr model remains historically important as it successfully introduced quantization into atomic structure and provided a simple, intuitive picture that correctly predicted many features of the hydrogen atom.

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