Page 131 - Physical Principles of Sedimentary Basin Analysis

6.3 Heat generation 113

where C i is the mass fraction of each isotope i. (The index i and the corresponding heat
production H i are defined by Table 6.1.) Uranium is at present composed of 99.27%
238 U and 0.72% 235 U by weight. Thorium is 100% 232 Th, and potassium is composed
of 0.0128% 40 K. It is customary to give the mass fractions of isotopes in terms of the
fractions of U, Th and K. The total heat production then becomes

H total = 0.72%C U H 1 + 99.27%C U H 2 + C Th H 3 + 0.0128%C K H 4 (6.38)
where C U , C Th and C K are the mass fractions of uranium, thorium and potassium,
respectively. The total heat production can then be expressed in terms of C U , C Th and C K as
(6.39)
H total = H U C U + H Th C Th + H K C K
where H U = (0.72% H 1 + 99.27% H 2 ) · 10 −6 = 9.8 · 10 −5 μWkg −1 , H Th = H 3 · 10 −6 =
2.6 · 10 −5 μWkg −1 and H K = 0.0128% H 4 · 10 −2 = 3.4 · 10 −5 μWkg −1 . Notice that the
factors 10 −6 and 10 −2 have been added because the mass fractions of U and Th are given
in ppm (parts per million) and K in %. The expression of heat production per unit mass of
rock can be multiplied with a typical crust density = 2700 kg m −3 , and it becomes an
expression for heat production per unit volume:

S total = S U C U + S Th C Th + S K C K (6.40)
where S U = H U = 0.267 μWm −3 , S Th = H Th = 0.070 μWm −3 and S K = H K =
0.091 μWm −3 . Table 6.2 lists the heat production for some rock samples. We see that
granite is the most heat-producing rock, and that mantle rocks (undepleted mantle and peri-
dotite) have almost no heat production. Anther observation is that there is a considerable
scatter in the heat production among rocks, even for rocks with similar types of minerals.
We also observe the ratios C Th /C U ≈ 4 and C K /C U ≈ 1.2, whichareoftenusedtoesti-
mate the content of thorium and potassium once the content of uranium is known. These
ratios can also be used to invert a measurement of heat production to an approximation for
the contents of the elements U, Th and K. The concentrations are then

C U ≈ 1.5S total , C Th ≈ 6S total and C K ≈ 1.8S total (6.41)

where the mass fractions C U and C Th are in ppm and the mass fraction C K is in percent,
when S total has the units μWm −3 . (See Exercise 6.4.)
The heat generation from radioactive decay is not constant through time, because the
decay to stable elements leaves less and less radioactive atoms. The number of decays per
unit time is proportional to the number of atoms N of the radioactive isotope

dN
=−λN (6.42)
dt
where the disintegration constant λ tells how fast the decay is. We will see that a large λ
leads to a short half-life. The decay constant can also be interpreted as a decay probability