Atomic energy levels Knowledge-Based Study

Study notes

1. Quantization of Atomic Energy Levels

   Classical electromagnetic theory predicted that an orbiting electron would continuously accelerate, radiate electromagnetic energy, and spiral rapidly into the nucleus. To resolve this instability, early quantum models postulated that electrons inside an isolated atom can only occupy discrete, stable orbits corresponding to well-defined, quantized energy values.

   For a hydrogen-like atom with atomic number Z, the allowed energy levels are given by:

   $E_n = -R_H \frac{Z^2}{n^2}$

   where $R_H \approx 13.6\text{ eV}$ is the Rydberg energy unit and n is the principal quantum number ($n = 1, 2, 3, \dots$). Key features of this energy structure include:

   • Ground State ($n = 1$): The lowest, most stable bound state of the atom. For neutral hydrogen ($Z = 1$), $E_1 = -13.6\text{ eV}$.
   • Excited States ($n > 1$): Higher energy configurations where the electron is less tightly bound (e.g., $E_2 = -3.4\text{ eV}$, $E_3 = -1.51\text{ eV}$).
   • Ionization Limit ($n \to \infty$): The state where the electron is completely liberated from the electrostatic pull of the nucleus, corresponding by convention to $E_\infty = 0\text{ eV}$.

   Transitions: Absorption and Emission of Photons

   An atom changes its internal energy state through transitions between two discrete levels. Because energy is strictly conserved, any upward or downward transition must account for the exact energy difference:

   $\Delta E = |E_f - E_i| = hf = \frac{hc}{\lambda}$

   1. Photon Absorption

   An electron in a lower level $E_i$ transitions to a higher level $E_f$ only when it absorbs a photon whose energy precisely matches the level separation: $E_{\text{photon}} = E_f - E_i$. If the incident photon energy is even slightly different from any allowed gap (and less than the ionization threshold), the photon cannot be absorbed.

   2. Photon Emission

   An excited electron spontaneously or stimulatedly drops from an initial upper level $E_i$ to a lower level $E_f$, releasing the excess potential energy as a single emitted photon with frequency $f = \frac{E_i - E_f}{h}$. The collection of discrete emitted wavelengths constitutes the characteristic atomic emission spectrum of the element.

   Excitation Mechanisms: Photons versus Colliding Electrons

   A crucial distinction in atomic physics lies in how an atom absorbs energy from incoming particles versus incident electromagnetic radiation:

   • Excitation by Photons: Photons are absorbed as indivisible single quanta. Therefore, an un-ionized atom absorbs a photon if and only if $E_{\text{photon}}$ matches an exact transition energy $E_n - E_1$. An exception occurs only when $E_{\text{photon}} \ge E_{\text{ion}}$, in which case the photon is fully absorbed, ionizing the atom and transferring any surplus energy as kinetic energy to the ejected electron: $E_k = E_{\text{photon}} - E_{\text{ion}}$.
   • Excitation by Inelastic Electron Collisions: An incident free electron does not disappear upon collision. It can transfer a portion of its kinetic energy to an atomic electron as long as its initial kinetic energy $E_k$ is greater than or equal to the energy gap to an excited state: $E_k \ge E_n - E_1$. The colliding electron then leaves the atom with the remaining kinetic energy: $E_k' = E_k - (E_n - E_1)$.

   Hydrogen Spectral Series

   When hydrogen atoms de-excite, the emitted photons are classified into spectral series according to the final state $n_f$ of the transition:

   • Lyman Series ($n_f = 1$): Transitions from $n_i = 2, 3, 4, \dots$ directly to the ground state. These involve the largest energy gaps, producing radiation strictly in the ultraviolet (UV) region.
   • Balmer Series ($n_f = 2$): Transitions from $n_i = 3, 4, 5, \dots$ down to the first excited state. Four of these transitions fall within the visible spectrum (responsible for hydrogen\'s characteristic spectral lines), while higher-order transitions reach the near UV.
   • Paschen Series ($n_f = 3$): Transitions from $n_i = 4, 5, 6, \dots$ down to $n = 3$, emitting in the infrared (IR) region.

   The total number of possible distinct emission lines from a gas of atoms excited to state n undergoing all possible cascade transitions down to the ground state is given by $N = \frac{n(n - 1)}{2}$.