Decay Chains and Daughter Nuclides

Introduction

One radioactive transformation can produce a daughter that is itself unstable. The resulting sequence is a decay chain. To follow one, calculate the daughter after each arrow, use its new atomic number to name the element, and then treat that daughter as the parent of the next arrow. A shortcut based only on the first and last symbols can hide an incorrect intermediate step.

Core explanation

Every link in a decay chain obeys the same conservation checks as a single nuclear equation. For alpha emission, subtract four from mass number A and two from atomic number Z. For beta-minus emission, keep A and add one to Z. For beta-plus emission or electron capture, keep A and subtract one from Z. Gamma emission alone leaves A and Z unchanged. These are bookkeeping consequences of the emitted or captured particles, not a promise that every proposed route occurs for a given nuclide.

Consider the simplified sequence uranium-238 → thorium-234 → protactinium-234. The first step is alpha emission: ²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He. The thorium-234 daughter then undergoes beta-minus decay: ²³⁴₉₀Th → ²³⁴₉₁Pa + ⁰₋₁e + ν̅ₑ. The final nuclide has A = 234 and Z = 91. If you try to name the final element before working the thorium step, it is easy to overlook that beta-minus decay raises Z by one after alpha emission lowered it by two.

The word daughter is relative to a particular arrow. Thorium-234 is the daughter of uranium-238 in the first step and the parent of protactinium-234 in the second. A chain may continue through many nuclides before reaching a stable end product. Do not assume an intermediate is stable simply because one question stops its calculation there; the stated scope of a problem may end before the physical chain does.

Net arithmetic can be useful as a check. If a chain contains one alpha emission and one beta-minus emission, total A changes by −4 and total Z by −2 + 1 = −1. Starting at uranium-238, the two-step net result is A = 234 and Z = 91, matching protactinium-234. But this net result does not tell the order of steps or the identity of the intermediate. An alpha-then-beta path and a beta-then-alpha path can have the same net A and Z while involving different intermediate nuclei, and one order may be physically unavailable.

Real chains can branch. A nuclide may have more than one decay route with different probabilities, producing different daughters. A diagram can therefore be a network rather than one unbroken line. When a school exercise supplies an explicit sequence, follow that sequence. When it asks for a natural decay chain without specifying a branch, use reliable isotope data rather than inventing a route from neutron-proton balance alone.

The time spent at each link also differs. A parent and its intermediate daughters can have very different half-lives. The amount of each nuclide present at a given time depends on both production from the preceding link and loss through its own decay. It is not generally correct to apply the first parent's half-life to every member of the chain. Later half-life lessons handle the simplest single-nuclide calculation; more complicated chain populations require additional equations.

Gamma photons can appear within a chain after a daughter is produced in an excited state. A gamma step changes energy state but not the nuclide's A or Z. Drawing it as a separate arrow keeps the chemical element ledger clear. Similarly, positron annihilation after beta-plus emission is not another nuclear daughter transformation.

Step-by-step reasoning

1. Write the first parent with A and Z, and note the specified first emission. 2. Calculate and name the first daughter; check the equation's A and charge sums. 3. Promote that daughter to parent for the next stated emission. 4. Continue one arrow at a time, recording intermediate nuclides. 5. Use net A and Z changes only as a final check, and check whether branching or gamma steps are stated.

Visual explanation

Draw three boxes in a row: ²³⁸₉₂U, ²³⁴₉₀Th and ²³⁴₉₁Pa. Label the first arrow “α: A − 4, Z − 2” and the second “β⁻: A same, Z + 1.” Beneath the chain draw a net arrow from first to last labelled “A − 4, Z − 1,” while preserving the middle box so the actual path remains visible.

Real-world analogy

A parcel can move through several sorting centres. Its final destination alone does not reveal every stop or the order of travel. Likewise, net A and Z changes may identify the final nuclide but not the intermediate path. The analogy does not imply that nuclei choose routes intentionally; decay probabilities come from nuclear physics.

Real-world example

Naturally occurring radioactive families involve sequences of daughter nuclides. In environmental measurement, detecting one daughter can provide information about a parent source, but the conclusion depends on the daughter's own half-life and whether material has moved between locations. A decay-chain diagram helps identify which nuclide a detector might be observing.

Why?

Why calculate each step separately if the net change is easy? The intermediate nuclide has its own identity, possible half-life and possible branch. Net arithmetic cannot show which intermediate was formed or whether the stated order is consistent with an observed pathway.

Common misconception

“A radioactive daughter has to stay the daughter forever.” A daughter of one event can be the parent of the next. The labels parent and daughter describe positions in one transformation, not permanent categories of elements.

Worked example

A hypothetical stated sequence begins with ²²⁶₈₈Ra, emits alpha radiation and then the resulting daughter emits alpha radiation again. First: A = 226 − 4 = 222 and Z = 88 − 2 = 86, so ²²²₈₆Rn forms. Second: A = 222 − 4 = 218 and Z = 86 − 2 = 84, so ²¹⁸₈₄Po forms. The equations are ²²⁶₈₈Ra → ²²²₈₆Rn + ⁴₂He and ²²²₈₆Rn → ²¹⁸₈₄Po + ⁴₂He. Net A falls eight and Z falls four, matching two alpha emissions. Radon-222 is an intermediate, not the final nuclide of this two-step question.

Quick check

1. A parent has A = 210 and Z = 84. After alpha then beta-minus decay, what are the final A and Z? Answer: After alpha, A = 206 and Z = 82; after beta-minus, A = 206 and Z = 83.

Exam focus

Write an equation or clear A/Z ledger for each arrow. Name every daughter using its new Z. Check the final net change but do not let it replace intermediate identities, and do not apply one half-life to every chain member.

Advanced insight

In a branching chain, each route has a branching fraction: the proportion of a parent's decays that follow that route. Production and loss of an intermediate can sometimes balance approximately, a situation called secular equilibrium when a long-lived parent feeds a much shorter-lived daughter under suitable conditions. This is more involved than repeatedly applying a single half-life.

Summary

A decay chain is a sequence in which a daughter may decay again. Follow each arrow using its own alpha, beta or gamma rule and name intermediate nuclides from atomic number. Net changes provide a useful check but do not establish the path, branches or timing.

Practice questions

1. Complete the two-step path ²³⁸₉₂U —α→ ? —β⁻→ ?. Answer: The intermediate is ²³⁴₉₀Th and the next daughter is ²³⁴₉₁Pa. 2. After two alpha emissions from ²²⁶₈₈Ra, what is the final nuclide? Answer: A = 218 and Z = 84, so the final nuclide is ²¹⁸₈₄Po. 3. Can the same intermediate be called both daughter and parent? Answer: Yes. It is the daughter of the preceding step and the parent of its own subsequent decay.