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Hydrogen Spectral Series

Energy levels of hydrogen, $E_n = -13.6\,\text{eV}/n^2$ , drawn to scale with the transitions of the Lyman ( $n \to 1$ , ultraviolet), Balmer ( $n \to 2$ , visible) and Paschen ( $n \to 3$ , infrared) series. Each emitted photon has wavelength $1/\lambda = R_\infty (1/n_f^2 - 1/n_i^2)$ given by the Rydberg formula, so the four visible Balmer lines H $\alpha$ (656 nm), H $\beta$ (486 nm), H $\gamma$ (434 nm) and H $\delta$ (410 nm) form the characteristic emission spectrum shown below the arrows. Within each series the lines crowd together toward a series limit set by the ionization threshold $n_i \to \infty$ (dashed line).

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hydrogen-spectral-series.typ (101 lines)

#import "@preview/cetz:0.5.3": canvas, draw
#import draw: content, line, rect

#set page(width: auto, height: auto, margin: 8pt, fill: none)
#let ink = rgb("#19324f")
#set text(font: "Avenir Next", size: 22pt, fill: ink)

// Hydrogen levels E_n = -13.6 eV / n^2 on a linear energy scale; n = oo is the ionization limit.
#let y-span = 11.5 // canvas height from n = 1 to the ionization limit
#let y-of(n) = y-span * (1 - 1 / (n * n))
#let (x-left, x-right) = (1.2, 10.4)

#let lyman-color = rgb("#6a3fc8")
#let paschen-color = rgb("#9b2335")
// Balmer lines (upper level, wavelength in nm, color, arrow x): H-delta, H-gamma, H-beta, H-alpha.
// Arrows are ordered by wavelength like the spectrum below.
#let balmer = (
  (6, 410, rgb("#8f2bd6"), 4.6),
  (5, 434, rgb("#2f55e0"), 5.25),
  (4, 486, rgb("#0098b8"), 5.9),
  (3, 656, rgb("#e3261b"), 6.55),
)
// Visible-spectrum gradient for the Balmer series label
#let rainbow = gradient.linear(
  ..balmer.slice(0, 3).map(line => line.at(2)),
  rgb("#2e9e3e"),
  rgb("#e09a00"),
  balmer.last().at(2),
)

#let arrow(x, n-from, n-to, color) = draw.line(
  (x, y-of(n-from)),
  (x, y-of(n-to) + 0.03),
  stroke: (paint: color, thickness: 2.6pt, cap: "round"),
  mark: (end: "stealth", fill: color, scale: 0.9),
)

#canvas({
  // Energy levels, the ionization limit dashed
  for n in (1, 2, 3, 4, 5, 6, calc.inf) {
    let dash = if n == calc.inf { "dashed" }
    line((x-left, y-of(n)), (x-right, y-of(n)), stroke: (paint: ink, thickness: 1.4pt, dash: dash))
  }

  // Level labels: quantum number on the left, energy on the right
  for (n, label) in ((1, $n = 1$), (2, $n = 2$), (3, $n = 3$), (calc.inf, $n = oo$)) {
    content((x-left - 0.15, y-of(n)), label, anchor: "east")
  }
  for (n, label) in ((1, [−13.6 eV]), (2, [−3.4]), (3, [−1.5]), (calc.inf, [0])) {
    content((x-right + 0.15, y-of(n)), label, anchor: "west")
  }

  // Lyman series: n >= 2 -> 1 (ultraviolet)
  let lyman-x(idx) = 1.75 + 0.48 * idx
  for (idx, n) in range(2, 7).enumerate() {
    arrow(lyman-x(idx), n, 1, lyman-color)
  }
  content((lyman-x(2), -0.2), text(fill: lyman-color, weight: "bold")[Lyman], anchor: "north")
  content((lyman-x(2), -1.05), text(fill: lyman-color, size: 18pt)[UV], anchor: "north")

  // Balmer series: n >= 3 -> 2 (visible)
  for (n, _, color, x) in balmer {
    arrow(x, n, 2, color)
  }

  // Visible emission spectrum of the Balmer lines
  let (s-left, s-right, s-bottom, s-top) = (4.3, 10.1, 1.5, 3.6)
  let (lam-min, lam-max) = (390, 680)
  let x-of-lam(lam) = s-left + (s-right - s-left) * (lam - lam-min) / (lam-max - lam-min)
  rect((s-left, s-bottom), (s-right, s-top), fill: rgb("#cdd3da"), stroke: none, radius: 0.12)
  for (_, lam, color, x) in balmer {
    let xl = x-of-lam(lam)
    line(
      (x, y-of(2) - 0.12),
      (xl, s-top + 0.08),
      stroke: (paint: color, thickness: 1pt, dash: "densely-dashed"),
    )
    line((xl, s-bottom + 0.08), (xl, s-top - 0.08), stroke: color + 3.2pt)
  }
  for (lam, label) in ((410, [410]), (486, [486]), (656, [656 nm])) {
    content((x-of-lam(lam), s-bottom - 0.15), text(size: 18pt, label), anchor: "north")
  }
  content((9.5, 6.3), text(weight: "bold", fill: rainbow)[Balmer], anchor: "south")
  content((9.5, 6.25), text(size: 18pt)[visible], anchor: "north")

  // Paschen series: n >= 4 -> 3 (infrared)
  let paschen-x(idx) = 8.4 + 0.5 * idx
  for (idx, n) in range(4, 7).enumerate() {
    arrow(paschen-x(idx), n, 3, paschen-color)
  }
  content(
    (paschen-x(1), y-of(3) - 0.15),
    text(fill: paschen-color, weight: "bold")[Paschen],
    anchor: "north",
  )
  content(
    (paschen-x(1), y-of(3) - 0.95),
    text(fill: paschen-color, size: 18pt)[IR],
    anchor: "north",
  )
})