Principles of radioactive dating
Francis Albarède
The energy binding protons and neutrons in the nucleus is about a factor 106 larger than the
energy of chemical reactions: radioactivity is therefore independent of the chemical bonding
of atoms (in minerals or in solutions) as it is of the temperature and pressure inside the planets
and even the Sun. In addition, radioactivity is a memoryless process (atoms do not age) and is
therefore a nuclear event whose probability of occurrence per unit of time, which we note 
is independent of time. This probability, termed the decay constant, is specific to each
radioactive nuclide. Radioactive decay, like incoming calls at a telephone exchange, is a
prime example of a Poisson process, in which the number of events is proportional to the time
over which the observation is made. In the absence of any other loss or gain, the proportion of
parent atoms (or radioactive nuclides) disappearing per unit of time t is constant:
dP
 λ
Pdt
(1)
For a number of parent atoms P = P0 at time t = 0, this equation integrates as:
P  P0e  t
(2)
To determine the age of a system from the measurement of the number of parent atoms at the
present time, we must know P0 and therefore, in this form, this equation is not a chronometer
(a notable exception is the method 14C). For each parent atom, a daughter atom (or radiogenic
nuclide) is created, usually of a single element, whose number can be noted D. In a closed
system and for a stable daughter nuclide D, the number of parent and daughter atoms is
constant, therefore:


D  D0  P0  P  D0  P e t  1

(3)

The term P e t  1 is a measure of the accumulation of the radiogenic nuclide during time t
and therefore:
t
1 
D  D0 
ln 1 

 
P 
(4)
Even if D and P are measured, this equation is no more a timing device than equation (2)
unless we know the number of daughter atoms D0 at time t = 0. A first class of chronometers
arises when the condition D0 << D applies as for the U-Pb dating of zircons, for example:
206

Pb 
ln 1  238 t 
 238U 
Ut 
and the K-Ar dating method. This age dates the isolation of the host mineral.
t
1
For a second class of chronometers, the condition D0 << D does not apply and we replace it by
the principle of isotopic homogenization. We need not go into the existence of isotopic
fractionation (both natural and analytical), a phenomenon which is of little importance in
heavy elements and which, for the needs of geochronology, is simply eliminated by internal
normalization against some arbitrary reference ratio of stable isotopes, e.g., 86Sr/88Sr = 0.1194
for Sr. When carbonates precipitate out from seawater, the 87Sr/86Sr ratio is exactly the same
in calcite as in the seawater from which it precipitates; as the mantle melts, 143Nd/144Nd is the
same in the molten liquid as in the residue. Let us therefore divide Equation (3) by the number
D’ of atoms of a stable isotope of the same element as the radioactive nuclide represented by
D. As the system is closed, D’ remains constant and therefore:


 D   D   P  t
        e 1
 D' t  D' 0  D' t
Example:
(5)


 87 Sr   87 Sr   87 Rb   t

 
 
 e 8 7 Rb  1
 86 Sr   86 Sr   86 Sr 

t 
0 
t
This is the standard ‘isochron’ equation. D/D’ represents the ratio of the radiogenic nuclide to
its stable isotope (e.g., 87Sr/86Sr) and P/D’ is the ‘parent/daughter’ ratio, in most cases
proportional to an elemental ratio (e.g., 87Rb/86Sr). If two samples 1 and 2 formed at the same
time from an isotopically homogeneous m edium (ocean, magma), they share the same
(D/D’)0 and, taking the 87Rb-87Sr system as an example, the time is obtained from:


 
 


87

Sr/ 86 Sr 2  87 Sr/ 86 Sr 1 

t
ln 1  87
87Rb 
Rb/ 86 Sr 2  87 Rb/ 86 Sr 1 


1
This age dates the time at which the two samples last shared a same 87Sr/86Sr ratio. This
method is commonly used for parent-daughter systems with a long half-life, typically 143Nd144
Nd, 176Lu-176Hf, 187Re-187Os. A particular application combines the two chronometers 238U206
Pb and 235U-207Pb, in which the two parent nuclides on the one hand, and the two daughter
nuclides on the other hand: as the U isotopes are not explicitly considered, this method is
known as the Pb-Pb method.
A third class of chronometers relevant to astrobiology is that of extinct radioactivities. These
have a short half-life (and therefore a large ). For large values of t, P becomes negligible
and therefore the closed system condition reads:
Dtoday  D  Pt
(6)
Let us write this equation for a sample (spl) and divide it by D’:
spl
spl  SN
D
D
 
 
 D' today  D' t
spl  SN
P
 
 P' t
spl
 P' 
 
 D' today
(7)
in which we emphasize that the sample was born in isotopic equilibrium with the solar nebula
(SN) (P’ is a stable isotope of the parent nuclide P). This equation is equivalent to:
spl
sple SN 
SN
D
D
P
 
 
 
 D' today  D' today  P' t
spl
SN 
 P' 
 P' 
 

 
 D' today  D' today 
(8)
The isotopic properties of the solar nebula are those of chondrites. For the example of the
26
Al-26Mg chronometer, equation (7) reads:
spl
spl  SN
 26 Mg 
 26 Mg 





 24 Mg 
 24 Mg 

today 
t
spl  SN
 26 Al 
  27 
 Al 

t
spl
 27 Al 


 24 Mg 

today
This is the equation of the extinct radioactivity isochron (the conventional isochron cannot be
used when all the parent nuclide has decayed away). When 26Al/24Mg is plotted against the
27
Al/24Mg (note that the two nuclides of the latter ratio are stable), samples formed at the same
moment in isotopic equilibrium with the solar nebula form a straight line with a slope
indicative of the 26Al/27Al ratio of the gas at the time the system formed. The 26Al/27Al ratio of
the solar nebula at the time the sample formed is obtained from:
spl  SS
 26 Al 


 27 Al 

t






spl
SN
SS
26
24
26
24
 26 Al 
Mg/
Mg

Mg/
Mg
today
today
  27  et 
spl
SN
 Al 
27

0
Al/ 24 Mg today  27 Al/ 24 Mg today


If the 26Al/27Al ratio of the solar nebula at the reference time t = 0 is assumed, the result can
be converted into an age. This age dates the time at which the dated sample shared the same
26
Al/27Al ratio as the solar nebula. If no history of the 26Al/27Al ratio is assumed for the solar
nebula, dividing this equation for one sample by the same equation for a second sample gives
the age difference between the two samples. This method is employed for a number of
‘extinct’ short-lived nuclides, such as 41K-41Ca, 60Fe-60Ni, 53Mn-53Cr, 146Sm-142Nd, etc.