Half-life
A non-system quantity · expressed in SI seconds · a property of the nuclide
In 1900 Ernest Rutherford noticed that the radiation of thorium weakened with time and, what mattered more, weakened not haphazardly but in a perfectly regular way: every minute it lost the same share of its former strength. So was found a quantity that depends neither on temperature, nor on pressure, nor on the chemical state of the substance — the time in which half the nuclei decay.
Its peculiarity is that it belongs to a multitude and not to a single nucleus: one solitary atom has neither age nor term, and whether it will decay in the next second or lie for another million years cannot be foretold even in principle. Nuclei, however, are usually many — a gram holds some ten to the twenty-first of them — and over such a crowd chance turns into an exact law.
01 · Definition
The half-life is the time in which the number of nuclei of a given nuclide falls by half. It is bound unambiguously to the decay constant, which sets the chance that one nucleus decays in unit time, and to the mean lifetime, which is longer than the half-life by about half again.
The exponential law follows from a single assumption: the chance that a nucleus decays does not depend on how long it has already lived. A nucleus does not age, does not wear out and does not approach its term — at every instant it is as ready to decay as at the moment of its birth, and therefore the share of survivors falls by the same factor over equal stretches of time.
The peculiar steadiness of this quantity comes from the decay happening inside the nucleus, where chemistry does not reach. Heat, pressure, a magnetic field and the kind of compound have no effect on the half-life, and only in rare cases, when electrons from the shell nearest the nucleus take part in the decay, can it be shifted by fractions of a percent.
The range of values is vast as for no other quantity in the catalogue: from ten to the minus twenty-second of a second in nuclei that fall apart before the nucleons manage to run round the nucleus, to more than two septillion years for tellurium-128, whose half-life exceeds the age of the Universe a hundred trillion times. And all of it obeys one and the same law of halves.
Lead the time with the slider — it is counted in half-lives, the same for all four nuclides, while the captions turn them into real hours, years or millennia. The orange curve shows the share of surviving nuclei, the blue steps mark a half, a quarter and an eighth, and the bar on the right is what is left of the initial hundred atoms.
The half-life is a property of a multitude and not of a single atom: a nucleus does not age and at any moment has the same chance of decaying as in the first second of its existence. Hence of one atom nothing more definite than a probability can be said, while of a billion almost everything can, and the spread of the prediction falls as the square root of their number.
02 · Conversion
Half-life, constant, activity
Enter a half-life in any units, and the sheet will convert it into the rest, give the decay constant and the mean lifetime, and then show what activity a gram of the pure nuclide yields and how much of it will be left after a given term.
The mass number for computing the specific activity is taken from the nuclide chosen in the interactive part; in the finer settings one can switch to a nominal hundred.
A short half-life means high activity, not harmlessness: the faster a nuclide decays, the more decays fall into a second, and a gram of technetium-99m shines tens of millions of times brighter than a gram of uranium. The danger is therefore set not by the half-life alone but together with it — by the kind of radiation, by what the nuclide settles in and how it enters the body.
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| {u} | {name} | {value} | {note} |
03 · Orders of magnitude
logarithmic scale: fifty-four orders of magnitude04 · Measuring instruments
What measures the thing you cannot wait out
The simplest way to measure a half-life is to count the clicks and watch their rate fall with time. It serves for nuclides living from minutes to a few years: over a shorter term the count has no time to gather, over a longer one the fall is lost in the unavoidable spread of readings.
A semiconductor detector of germanium tells quantum energies apart to thousandths of a percent, and so allows one nuclide to be followed in a mixture of a dozen others. The width of the spectral line itself, meanwhile, betrays the half-life of those states that live shorter than a nanosecond: the shorter the life, the broader the level.
For long-lived nuclides waiting for decays is pointless, and so the atoms are counted one by one, sorted by mass in a magnetic field. Radiocarbon dating rests on precisely this: in the sample one looks not for radiation but for the nuclei of carbon-14 themselves, of which there is one in a trillion ordinary ones.
Half-lives of quintillions of years are measured by taking tonnes of substance and waiting a year: a few decays will happen after all. To tell them apart, the apparatus is hidden under a kilometre of rock, surrounded by ancient lead raised from sunken ships, and the events are counted — a couple a month.
05 · Writing rules
The nuclide first, the unit after
The designation is written with a capital Latin letter and the subscript «one half», while the mass number is placed at the upper left of the element symbol or after its name with a hyphen. The unit is always given and chosen convenient in size; the half-life must not be mixed with the mean lifetime — they differ by a factor of 1.443.
The first writing fails twice over: uranium has several isotopes with different half-lives, and the unit is left out. The second equates the half-life with the mean lifetime, whereas the latter is 1.443 times longer. The third supposes an exhaustion that the exponential does not allow: after two half-lives a quarter remains. The fourth ascribes a term to a single atom, which has none.
06 · Neighbouring units
The half-life stands in a row with those quantities that describe the same decay from other sides: the activity measured in becquerels, the decay constant with the dimension of reciprocal time, and the mean lifetime. It is enough to know any one of them — the rest follow by a factor.
decays per second
reciprocal seconds
longer by a factor of 1.443
Of the Simetrium data sheets nearby stand becquerel and curie, by which the activity itself is measured, gray and sievert for absorbed and equivalent dose, barn from nuclear cross sections and second, in which the half-life is in the end expressed.
07 · Historical section
A clock that can be neither wound nor stopped
Rutherford found that the radiation of the gas given off by thorium falls by the same factor over equal stretches, and introduced a quantity he at first called the time of half-turning. This was the first process in nature whose rate could not be altered by anything whatever.
Kelvin, from the cooling of the terrestrial globe, derived an age of twenty to forty million years, and to the geologists that term was desperately short. The lead accumulated in uranium minerals gave billions at once — and explained along the way why Kelvin's reckoning was doomed: he did not know that the depths are warmed by their own decay.
Willard Libby saw that a living creature exchanges carbon with its surroundings all the time, and that after death the exchange ceases and the clock starts. The method passed its first test on wood from an Egyptian tomb whose age was known from inscriptions, and it differed from that age by only a century and a half.
Repositories of spent fuel are designed from the half-lives of the nuclides they hold, and the terms come out such that the task passes beyond engineering: how to leave a warning for people whose language does not yet exist. The Finnish vault at Onkalo is reckoned for a hundred thousand years — ten times longer than all of written history.
The only clock that cannot be adjusted
Any instrument for measuring time can be put out of order: a pendulum is knocked off by a jolt, quartz drifts with heat, even atomic clocks must be corrected for gravity. Radioactive decay stands apart in this row, since it goes on inside the nucleus, where neither temperature, nor pressure, nor a chemical bond reaches.
Such independence has a reverse side too, and an unpleasant one. Since the process cannot be hurried, waste cannot in principle be rendered harmless: it can only be waited out, and the waiting exceeds the lifetime of any state and any language. A unit that in the laboratory looks like a convenience turns, on the scale of a repository, into a constraint that cannot be argued with.
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