Atomic Physics Study Guide

Early Models of the Atom

Background and Classical Context (1897):
* The electron was discovered in 1897 and was observed to be significantly smaller than the entire atom.
* Scientific knowledge at the time established that atoms are electrically neutral.
J.J. Thomson's "Plum Pudding" Model:
    * This was the first modern model of the atom proposed by J.J. Thomson.
    * The model suggests the atom consists of a uniform sphere of positive charge, which accounts for the majority of the atom's mass.
    * Small, negatively charged electrons are scattered throughout this positive mass, similar to raisins in a pudding.
Geiger and Marsden’s Alpha Particle Scattering Experiment:
    * Experiments were conducted to confirm the plum pudding model by observing how alpha particles (helium nuclei) scattered after hitting a thin metal foil target.
    * Results revealed many more large-angle scatters than predicted by the Thomson model.
    * This observation led to the conclusion that the positive charge must be concentrated in a tiny volume rather than being spread uniformly throughout the atom.
Rutherford’s Planetary Model (1911):
    * Ernest Rutherford proposed a model comparable to a planetary system.
    * The Nucleus: Positive charge is concentrated at the center of the atom in a region called the nucleus.
    * Electron Orbits: Electrons orbit the nucleus in the same manner that planets orbit the sun.
    * Atomic Structure: This model reveals that the atom is mostly empty space.
Difficulties with the Rutherford Model:
    * Discrete Frequencies: Atoms are known to emit specific, characteristic discrete frequencies of electromagnetic radiation; Rutherford's model could not explain why these frequencies were discrete rather than continuous.
    * Centripetal Acceleration Paradox: According to classical physics, electrons undergoing centripetal acceleration should radiate electromagnetic waves of the same frequency as their motion.
    * The Spiral Problem: As electrons radiate energy, their orbital radius should steadily decrease, causing the electron to eventually spiral into the nucleus. Observations show that this does not happen, as atoms are stable.

Atomic Spectra

Emission Spectrum Characteristics:
    * When a gas at low pressure has a voltage applied to it, it emits light characteristic of that specific gas.
    * Analyzing this light with a spectrometer reveals a series of discrete bright lines.
    * Each bright line corresponds to a specific wavelength and color.
    * This collection of lines is defined as an emission spectrum.
The Balmer Series for Hydrogen:
    * The wavelengths of hydrogen's spectral lines in the visible range are described by the Balmer series.
    * The Rydberg constant ( $R_H$ ) is used in calculations: $R_H = 1.0973732 \times 10^7\,m^{-1} \approx 1.097 \times 10^7\,m^{-1}$ .
    * The formula for the Balmer series is: $\frac{1}{\lambda} = R_H \left( \frac{1}{2^2} - \frac{1}{n^2} \right)$ .
    * Values of $n$ for the Balmer series are integers: $n = 3, 4, 5, \dots$ .
Standard Spectral Series of Hydrogen:
    * The Rydberg equation applies to any series of spectral lines using the general form:
$\frac{1}{\lambda} = R_H \left( \frac{1}{m^2} - \frac{1}{n^2} \right)$
    * Condition: $m$ and $n$ are positive integers where n > m.
    * Lyman Series: $m = 1$ ; $n = 2, 3, 4, \dots$ (Occurs in the ultraviolet region, $\approx 100\,nm$ to $400\,nm$ ).
    * Balmer Series: $m = 2$ ; $n = 3, 4, 5, \dots$ (Occurs in the visible light region).
    * Paschen Series: $m = 3$ ; $n = 4, 5, 6, \dots$ (Occurs in the infrared region, $\approx 1000\,nm$ ).
    * For the Balmer series, the shortest-wavelength line corresponds to $364.6\,nm$ ; this line is located in the ultraviolet region while the other labeled lines are visible.
Absorption Spectrum:
    * Elements can absorb light at specific wavelengths in addition to emitting it.
    * An absorption spectrum is obtained by passing continuous radiation (containing all wavelengths) through a vapor of the target element.
    * The spectrum appears as a series of dark lines superimposed on a bright, continuous spectrum.
    * The dark absorption lines coincide exactly with the wavelengths of the bright lines in the same element's emission spectrum.
Practical Applications of Spectroscopy:
    * Solar Analysis: Continuous radiation from the Sun passes through cooler gases in the solar atmosphere before reaching Earth.
    * The resulting absorption lines are used to identify elements present in the solar atmosphere.
    * This method led to the initial discovery of helium.

The Bohr Model

Bohr’s Assumptions for Hydrogen (1913):
    1. Circular Orbits: Electrons move in circular orbits around the proton, held by the Coulomb force of attraction, which provides the necessary centripetal acceleration.
    2. Stable Orbits: Only specific electron orbits are stable and allowed. In these orbits, the atom does not emit electromagnetic radiation, and the energy of the atom remains constant.
    3. Quantized Radiation: Radiation is emitted only when an electron "jumps" from a higher energy initial state ( $E_i$ ) to a lower energy final state ( $E_f$ ). This jump is non-classical.
    4. Frequency Relation: The frequency ( $f$ ) of the emitted radiation is independent of the orbital frequency and is determined by the change in energy:
$E_i - E_f = hf$
    5. Standing Matter Waves: The circumference of an electron's orbit must contain an integral number of de Broglie wavelengths ( $\lambda$ ):
$2\pi r = n\lambda$
    where $n = 1, 2, 3, \dots$
Bohr Radius and Quantized Radii:
    * The radii of orbits are quantized, meaning they can only exist at certain allowed distances determined by the integer $n$ .
    * Bohr Radius ( $a_0$ ): The smallest radius ( $n=1$ ) is calculated as:
$a_0 = \frac{\hbar^2}{m k e^2} = 0.0529\,nm$
    * General Radius Formula: The radius for any orbit $n$ is:
$r_n = n^2 a_0 = n^2(0.0529\,nm)$
Energy Levels of the Hydrogen Atom:
    * Atoms exist in stationary states called energy levels.
    * General Energy Formula: $E_n = -\frac{13.6}{n^2}\,eV$ .
    * Ground State: The lowest energy state occurs at $n=1$ . The energy is $-13.6\,eV$ .
    * The second energy level ( $n=2$ ) has an energy of $-3.40\,eV$ .
    * Excited States: Represented by n > 1. The uppermost level corresponds to $E = 0$ as $n \rightarrow \infty$ .
    * Excitation vs. Ionization:
        * Excitation: Electrons absorb photons to jump to higher, more energetic, and more "unstable" levels.
        * Ionization Energy: The energy required to completely remove an electron from the atom. For hydrogen, the ionization energy is $13.6\,eV$ .
Generalized Wavelength Equation for Transitions:
* Whenever a transition occurs between an initial state ( $n_i$ ) and a final state ( $n_f$ ), a photon is emitted with a wavelength determined by:
$\frac{1}{\lambda} = R_H \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)$
* Requirement: n_i > n_f where $n_f$ determines the series (e.g., $n_f = 2$ for Balmer).

Example 28-1: Balmer Series Calculations

Problem Statement: The Balmer series for hydrogen corresponds to transitions terminating at $n = 2$ .
(a) Calculation for the Longest-Wavelength Photon:
* The longest wavelength corresponds to the smallest energy transition, which occurs from $n_i = 3$ to $n_f = 2$ .
(b) Calculation for the Shortest-Wavelength Photon:
* The shortest wavelength corresponds to the highest energy transition, which occurs from $n_i \rightarrow \infty$ to $n_f = 2$ .
* Calculation steps:
$\frac{1}{\lambda} = R_H \left( \frac{1}{2^2} - \frac{1}{\infty} \right) = \frac{R_H}{4}$
$\lambda = \frac{4}{R_H} = \frac{4}{1.097 \times 10^7\,m^{-1}} = 3.646 \times 10^{-7}\,m = 364.6\,nm$

Characteristic X-Rays

Definition and Generation:
* X-rays are electromagnetic radiation emitted when highly energetic electrons strike a metal target.
Production Mechanism:
* If the incident electrons possess sufficient energy, they can knock a K-shell ( $n=1$ ) electron out of a multielectron atom.
* Because of the large positive charge of the nucleus in heavy metals, this process requires energies in the range of tens of thousands of electron volts ( $eV$ ).
Properties and Applications:
* They are highly penetrating.
* A primary application includes medical imaging.

Chapter Summary

Evolution of Atomic Models:
* Progressed from the Thomson "plum pudding" model (positive pudding with embedded electrons) to the Rutherford model (planetary system with a dense nucleus and empty space).
Hydrogen Emission: Excited hydrogen atoms emit light at specific quantized wavelengths.
Key Bohr Model Points:
    * Electrons exist in circular orbits.
    * Only specific angular momentum values and orbits are allowed.
    * Electrons do not radiate while in these allowed stationary orbits.
    * Radiation is only emitted or absorbed during jumps between levels.
    * Allowed orbits correspond to standing matter waves of electrons.
X-Ray Production: Occurs when an inner-shell electron is displaced by a high-energy collision.