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Soddy-Fajans Displacement Laws: Alpha and Beta Shifts

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When a nucleus emits an alpha or beta particle, it transforms into a new element that sits at a different place in the periodic table. In 1913, Soddy and Fajans independently stated simple rules — the displacement laws — that predict exactly where the new element lands.

These laws matter because decay chains in uranium ore, reactor waste, and medical isotope production involve many steps. With the displacement laws, we can trace an entire chain on paper.

Key Concepts

  • Alpha displacement: Z decreases by 2, A decreases by 4 — the product shifts two places left in the periodic table.
  • Beta-minus displacement: Z increases by 1, A unchanged — the product shifts one place right, and is an isobar of the parent.
  • Beta-plus displacement: Z decreases by 1, A unchanged — the product shifts one place left.
  • Gamma: no displacement, since Z and A do not change.

How It Works

  1. Write the parent as ᴬZX.
  2. For alpha decay, subtract 4 from A and 2 from Z; look up the new Z to name the element.
  3. For beta-minus, keep A, add 1 to Z. For beta-plus, keep A, subtract 1 from Z.
  4. Repeat for each step in a chain; total A and charge are conserved at every step when emitted particles are included.

Worked Example

Radium-226 (Z = 88) emits an alpha particle, then the daughter emits a beta-minus particle. Identify the final element.

  1. Alpha: A = 226 − 4 = 222, Z = 88 − 2 = 86 → radon-222 (Rn).
  2. Beta-minus: A stays 222, Z = 86 + 1 = 87 → francium-222 (Fr).

Common Misconceptions

  • Students apply the alpha shift as Z−4. It is A−4 but Z−2, because a helium nucleus carries only 2 protons.
  • Many think beta decay changes A because an electron has mass. The electron mass is tiny and A counts nucleons only, so A stays constant.
  • Some forget that beta-minus products are isobars of the parent — same A, different element.

Real-World Applications

Displacement laws map the uranium-238 decay series ending at stable lead-206, predict radon gas generation in mines, design production of medical isotopes, and identify unknown radioactive contaminants from their decay products.

A nice check is the uranium-238 series: it takes 8 alpha decays and 6 beta-minus decays to reach lead-206, and applying the displacement laws step by step predicts exactly the intermediate elements that miners and geologists actually observe in uranium ores.

Remember: alpha moves the element two places left (A−4, Z−2), beta-minus one place right (Z+1), beta-plus one place left (Z−1) — the Soddy-Fajans rules trace any decay chain.

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