Ernest Rutherford.

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spectra of the alkaline earths. Like barium, it is non-volatile at
ordinary temperature. On the other hand, the emanation which
is continually produced from radium is a radio-active and chemi-
cally inert gas which at very low pressures is condensed at a
temperature 150C. A sufficient amount of this gas has been
collected to determine its volume, density, spectrum and boiling
point (see chapter XIII). Both in its spectrum and in the absence
of definite chemical properties, it resembles the argon-helium
group of inert gases. The emanation must be considered to be an
unstable gas of high molecular weight, which breaks up with the
emission of a particles. After the expulsion of helium atoms, the
residual atoms form a new substance called radium A, which
behaves as a solid and is deposited on the surface of bodies. The
chemical and physical properties of this new substance are again
entirely distinct from radium and its emanation. Radium A in
turn gives rise to a long series of .successive products, each of
which has certain distinctive chemical and physical properties
which allow of its separation from a mixture of the others.

The mathematical theory of successive changes and its appli-
cation to the analysis of radio-active substances was put forward
by the writer* in the Bakerian Lecture, to the Royal Society in
1904 and described in a paper entitled "The Succession of Changes
in Radio-active Bodies." The general theory has been discussed
by Stark f, GrunerJ and Bateman. The latter described a simple
method of writing down in a symmetrical form the mathematical
solution for any number of changes.

157. Theory of one transformation. Before considering
the evidence from which these changes are deduced, the general
theory of radio-active changes will be considered. It is very
convenient in discussing mathematically the theory of successive
changes to suppose that the initial matter called A changes
into B, B into (7, C into D, and so on. We shall first consider

* Rutherford, Phil. Trans. Eoy. Soc. A, 204, p. 169, 1904.

f J. Stark, Jahrb. d. Radioakt. 1, p. 1, 1904.

J P. Gruner, Arch. sc. phys. et nat. (4) 23, pp. 5, 113, 329, 1907; (4) 31, p. 509

1911.

H. Bateman, Proc. Camb. Phil. Soc. 15, p. 423, 1910.



XI] THEORY OF SUCCESSIVE TRANSFORMATIONS 419

the transformation of a single substance A. It has been found
experimentally that any single radio-active substance, obtained
by itself, decays according to an exponential law with the time.
We have seen that this can be expressed by the relation
N/N = e~^, where N is the number of atoms initially present
and N the number remaining after an interval t. This general
law of transformation holds universally for all the products, but
X has a different but characteristic value for each product.
Differentiating, dN/dt = \N. or, in other words, the rate of
decrease of the number of atoms is proportional to the number
present. If each atom of A changes into one of B, \N is equal
to the number of atoms of B formed per second. Since the
radiation accompanies the transformation, dN/dt is also propor-
tional to the intensity of the radiation emitted by the product.

We have previously shown in Section 75 that the rate of
transformation of a product is governed by the laws of probability.
The number of atoms breaking up in a given time is subject to
fluctuations round the average value of the magnitude to be
expected from the general probability theory. In this case A,
represents the average fraction of the number of atoms which
break up per second. The value of N given by the relation
NjN Q = e~ xt has been shown by v. Schweidler to represent the
most probable value. The experimental evidence so far obtained
shows that the exponential law of transformation holds universally
for all products, and is completely independent of physical and
chemical conditions. The fraction of a product transformed per
second, is independent, for example, of the age of the product,
and is quite unaffected by the concentration of the active matter
itself.

These conclusions are well illustrated by experiments on the
rate of decay of the radium emanation under different conditions
discussed in Section 131. It has been shown that radium emanation
more than three months old decays at exactly the same rate as
emanation freshly produced from radium. The constancy of the
law of transformation is from the physical point of view very
remarkable. It shows that the chance of any atom breaking up
in a given time is independent of the age of the atom, and is the
same, for example, whether it is produced a second before or has

272



420 THEORY OF SUCCESSIVE TRANSFORMATIONS [CH.

existed independently for more than three months. Such results
show that the decay of the emanation is independent of its past
history. It does not seem possible, for example, to suppose that
each atom at its birth begins to lose energy by radiation, and
that its disintegration is a result of the drain of energy from the
system. On such a view it would be expected that the rate of
decay would increase with the age of the atoms. It would appear
as if the rate of transformation of the atoms depends purely on
the laws of probability, and is uninfluenced by their past history.

The law of transformation shows that theoretically any atom
may exist unchanged for any time from zero to infinity. In
practice, however, it is often convenient to speak of the average
life of a large number of atoms. This has a definite value which
can be simply calculated. Suppose that N atoms of a substance
are initially present. After a time t, the number which change
in the time dt is equal to \Ndt or \N e~ M dt. Each of these
atoms has a life t, so 'that the average life of the whole number

is given by I \te~ M dt or 1/X. The average life of an atom is

Jo

consequently measured by the reciprocal of the radio-active
constant.

158. Theory of successive transformation. We shall
now consider a number of important cases where the active
matter passes through a succession of changes.

Suppose that P, Q, R ... represent the number of particles
of the matter A, B, and C ... respectively at any time t. Let
Xj, X 2 > ^3 be the constants of change of the matter A, B, and
C ... respectively.

Each atom of the matter A is supposed to give rise to one
atom of the matter B } one atom of B to one of (7, and so on.

The expelled "rays" or particles are non-radio-active, and so do
not enter into the calculation.

It is not difficult to deduce mathematically the number of
atoms of P, Q, R) . . . of the matter A, B, C, . . . existing at any time t
after this matter is set aside, if the initial values of P, Q, R, ...
are given. In practice, however, it is generally only necessary to
employ three special cases of the theory which correspond, for



XI] THEOKY OF SUCCESSIVE TRANSFORMATIONS 421

example, to the changes in the active deposit, produced on a plate
exposed to a constant amount of radium emanation and then
removed, (1) when the time of exposure is extremely short
compared with the period of the changes, (2) when the time of
exposure is so long that the amount of each of the products has
reached a steady limiting value, and (3) for any time of exposure.

There is also another case of importance which is practically
a converse of case 3, viz. when the matter A is supplied at a
constant rate from a primary source and the amounts of A, B, C
are required at any subsequent time. The solution of this can,
however, be deduced immediately from case 3 without analysis.

159. CASE 1. Suppose that the matter initially considered
is all of one kind A. It is required to find the number of
particles P, Q, R, S ... of the matter A, B, (7, D ... respectively
present after any time t.

Then P = P e~ Al *, if P is the number of particles of A initially
present. Now dQ, the increase of the number of particles of the
matter B per unit time, is the number supplied by the change of
the matter A, less the number due to the change of B into C,

thus

dP/dt = -\ l P ........................ (1),

dQ/dt=*\P-\Q ..................... (2).

Similarly dR/dt = \ 2 Q -\ S R ..................... (3).

Substituting in (2) the value of P in terms of P ,

dQ/dt = \P0-W-\iQ.
The solution of this equation is of the form

Q = P (ae-^+be-*) .................. (4).

By substitution it is found that a Xi/(X 2 ^i).
Since Q = when t = 0, b = - \ a (A, - \).



Thus Q = _L. (e -M_ e -A^ .................. (5).

A, 2 A!

Substituting this value of Q in (3), it can readily be shown that
R=P (ae~^ + be-^ + ce-^) ............ (6),



422 THEORY OF SUCCESSIVE TRANSFORMATIONS

where



[CH.



a



(x, -



- x 2 ) (x 3 - x 2 ) '






(X 1 -X 3 )(X 2 -X 3 )'
Similarly it can be shown that

S = P (ae-^ + be~^ + ce~^
where



.(7),



(X 2 -X 1 )(X 3 -X 1 )(X 4 -X 1 )'



(X 1 -X 2 )(X 3 -X 2 )(X 4 -X 2 ) >



(X 1 -X 3 )(X 2 -X 3 )(X 4 -X 3 )'



(X 1 -X 4 )(X 2 -X 4 )(X 3 -X 4 )



.1 I I

Relative number of atoms of A, B, C, D
present at any time (Case i)




20



40



60 80

Time in minutes
Fig. 99.



TOO



120 14O



The method of solution of the general case of n products has
been given in a symmetrical form by Bateman*. The amount of
the nth product N (t) at the time t is given by

N (t) = w-W + c#-** ...c n e~*J (8),



* Bateman, Proc. Camb. Phil. Soc. 15, p. 423, 1910.



Xl] THEORY OF SUCCESSIVE TRANSFORMATIONS 423

where GI = -TT >. / r-r TT -r ,

(Xa Xi ) ^A-3 A/j ) . . . \h n A,i)
\ ~P

... ^71 1 -L Q



.etc.



The variations in the values of P, Q, R, S, with time t after
removal of the source are shown graphically in Fig. 99, curves
A, B, C and D respectively. The curves are drawn for the
practical and important case of the first four products of the
active deposit of radium, known as radium A, B, C, and D. The
matter is supposed to consist initially only of radium A. This
corresponds to the case of a body exposed for a few seconds in the
presence of the radium emanation. The values of X 1? X 2 , X 3 are
taken as 3'85 x 10~ 3 , 4'33 x 10~ 4 , 5'93 x 10~ 4 (sec.)- 1 respectively
corresponding to the half value periods of A, B and C of 3,
26*8 and 19'5 minutes respectively. The half value period of
radium D is about 16 '5 years. Over the small interval under
consideration in the figure, one may suppose that the atoms of
radium D suffer no' appreciable decrease by transformation. The
equation (7) can then be simplified for calculation by putting
X 4 = 0.

The ordinates of the curves represent the relative number of
atoms of the matter A, B, C, and D existing at any time, and the
value of P , the original number of atoms of the matter A
deposited, is taken as 100. The amount of matter B is initially
zero, and in this particular case passes through a maximum about
10 minutes later, and then diminishes with the time. In a
similar way, the amount of C passes through a maximum about
35 minutes after removal. After an interval of several hours the
amount of both B and G diminishes very approximately according
to an exponential law with the time, falling to half value in both
cases in 2 6 '8 minutes.

Over the interval considered, the amount of D increases
steadily with time, although very slowly at first. A maximum
is reached when B and C have disappeared. Finally the amount
of D would decrease exponentially with the time with a period
of 16*5 years.



424 THEORY OF SUCCESSIVE TRANSFORMATIONS [CH.

160. CASE 2. A primary source supplies the matter A at a
constant rate and the process has continued so long that the amount
of the products A, B, C, .,. has reached a steady limiting value.
The primary source is then suddenly removed. It is required to
find the amounts of A, B, C, ... remaining at any subsequent time t.

In this case, the number n of particles of A, deposited
per second from the source, is equal to the number of particles
of A which change into B per second, and of B into C, and so
on. This requires the relation

M =^o = X a Q = \ s Bo .................. (9),

where P , Qo> ^o are the maximum numbers of particles of the
matter A, B } and C when a steady state is reached.

The values of Q, R, and S at any time t after removal of
the source are given by equations of the same form as (4), (6)
and (7) for a short exposure. Remembering the condition that
initially



it can readily be shown that

P = %-^

A-!



R = n (ae-^+be-^ + ce-^) ............ (12),

where

a= (X 8 -X 1 )(X,-X 1 )' = (X 1 -X 2 )(X 3 -X 2 )'



Similarly for four changes it can be shown that

S = n (ae-M + be-** + ce-^+de"^) ...... (13),

where

A.iA.3






! - X 3 ) (X, - X 3 ) (X 4 - X,) ' (X* -X 4 ) (X, - X 4 ) (X 3 - X 4 )



XI]



THEORY OF SUCCESSIVE TRANSFORMATIONS



425



Bateman has pointed out that the solutions for case 2 can be
derived from case 1. This is obvious when it is remembered
that the amount of G, for example, remaining after a definite
interval t is made up of (1) supply from A through B, (2) supply
from B, (3) part of G remaining. Remembering that the amounts
of A, B, G initially present are n /\ 1} n /\^ n /\ 3 respectively,
the solution can be written down with the aid of equations (5)
and (6).

The relative numbers of atoms of P, Q, R existing at any
time are shown graphically in Fig. 100 curves A, B, G respectively.



100



80



6'0



40



20




I I I I I

Relative number of atoms of matter A, B, C

present at any instant (Case 2)






20



40



60 80

Time in minutes
Fig. 100.



100



120



140



The number of atoms Q is taken as 100 for comparison, and the
values of X l5 Xj, X 3 are taken corresponding to the 3, 26'8, and
19 '5 minute changes in the active deposit of radium. It should
be pointed out that the curves show the variation of A , B and G
on removal of a body which has been exposed for a long interval
to a constant amount of emanation. The curves are slightly
different for the practical case where the body is exposed in a
decaying source of emanation.



426 THEORY OF SUCCESSIVE TRANSFORMATIONS [CH.

A comparison with Fig. 99 for a short exposure brings out
very clearly the variation in the relative amounts of P, Q, R
corresponding to cases 1 and 2. In case 2 the amount of R
decreases at first very slowly. This is a result of the fact that
the supply of C due to the breaking up of B at first nearly com-
pensates for the breaking up of G. The values of Q and R
after several hours decrease exponentially, falling to half value
in 26*8 minutes.

A consideration of the formulae for cases 1 and 2 brings
out the interesting point that the amount of G ultimately de-
creases according to an exponential law with the period of
26*8 minutes, which is characteristic of radium B and not of
radium C. This is an expression of a general result that the
product of longest period ultimately governs the decay curve in
all cases.

161. CASE 3. Suppose that a primary source has supplied
the matter A at a constant rate for any time T and is then
suddenly removed. Required the amounts of A, B, C at any
subsequent time.

Suppose that n particles of the matter A are deposited each
second. After a time of exposure T, the number of particles PT
of the matter A present is given by



At any time t, after removal of the source, the number of
particles P of the matter A is given by

P = P r e-V = ^(1-6-^)6 -A,* ............... (14).



Consider the number of particles n Q dt of the matter A produced
during the interval dt. At any later time t, the number of
particles dQ of the matter B, which result from the change in A,
is given (see equation 5) by



(15).

-i A, 2



XI] THEORY OF SUCCESSIVE TRANSFORMATIONS 427

After a time of exposure T, the number of particles Q T of the
matter B present is given by

Q T =n [f(T)dt+f(T-dt)dt + ......



If the body is removed from the emanation after an exposure
T, at any later time t the number of particles of B is in the same
way given by



It will be noted that the method of deduction of Q r and Q is
independent of the particular form of the function /().

Substituting the particular value off(t) given in equation (15)
and integrating, it can readily be deduced that
Q ae~^ be~* lt



Q T ~ a -b < 16 >'



where



a =



In a similar way, the number of particles R of the matter C
present at any time can be deduced by substitution of the value
of/() in equation (6).

R ae ~ AI * + be ~ *** 4- ce ~ A ^






In a similar way the amount of any product may be written
down.

162. CASE 4. The matter A is supplied at a constant rate
from a primary source. Required to find the number of particles
of A, B, C at any subsequent time t, when initially A, B, C are
absent.



428 THEORY OF SUCCESSIVE TRANSFORMATIONS [CH.

The solution can be simply obtained in the following way.
Suppose that the conditions of case 2 are fulfilled. The products
A, B, G are in radio-active equilibrium and let P , Qo, R Q -be the
number of particles of each present. Suppose the source is
removed. The values of P, Q, R at any subsequent time are given
by equations (10), (11) and (12) respectively. Now suppose the
source, which has been removed, still continues to supply A at
the same constant rate and let P lt Q l} R be the number of
particles of A, B, C again present with the source at any
subsequent time. Now we have seen, that the rate of change of
any individual product, considered by itself, is independent of
conditions and is the same whether the matter is mixed with the
parent substance or removed from it. Since the values of P ,Q , R
represent a steady state where the rate of supply of each kind
of matter is equal to its rate of change, the sum of the number
of particles A, B, C present at any time with the source, and in
the matter from which it was removed, must at all times be equal
to P , QQ, R 0> ..., that is



This must obvio.usly be the case, for otherwise there would be a
destruction or creation of matter by the mere process of separation
of the source from its products ; but, by hypothesis, neither the
rate of supply from the source, nor the law of change of the
products, has been in any way altered by removal.

Substituting the values of P, Q, R from equations (10), (11),
and (12), we obtain

= ! - * ..................................... (18),

*



(19),
(20),



etc.,

where a, b, and c have the values given after equation (12). The
curves representing the increase of P, Q, R are thus, in all cases



XI] THEORY OF SUCCESSIVE TRANSFORMATIONS 429

complementary to the curves shown in Fig. 100. The sum of
the ordinates of the two curves of rise and decay at any time is
equal to 100. We have already seen examples of this in the case
of the decay and recovery curves of Ur X and Th X.

163. Secular and transient equilibria. The theory of
cases 2 and 4 have been worked out on the assumption that there
is a permanent equilibrium between the successive products of
transformation. This is impossible to realise completely in practice
since the amount of every radio-active substance is always de-
creasing with time. No sensible error, however, is introduced
when the primary source is transformed so slowly that there is no
appreciable change in its amount in an interval of time required
for the later products to attain approximate equilibrium with the
primary source. This condition is very nearly fulfilled, for example,
in the case of radium and the radium emanation, where the period
of the former is 2000 years, and of the latter 3*85 days. The latter
approaches its equilibrium value very closely after the emanation
has been supplied continuously from the radium for an interval of
2 months. During this time, the fraction of the radium trans-
formed is only about 6/100,000, so that for the interval under
consideration, it may be regarded as a constant source without
sensible error. It is convenient to apply the term " secular "
equilibrium to this and similar cases.

Consider next the important case of the emanation and
its products, radium A, B and C. A stage of equilibrium
between the emanation and its products is reached after the
emanation has been stored about 5 hours, and the amount of
each of the products then decays exponentially with the period of
the emanation. This is a case of " transient " equilibrium, for the
amounts of the products are changing comparatively rapidly. The
amounts of radium A, B and C at any subsequent time are always
appreciably greater than the amounts for secular equilibrium if
the supply of emanation were kept constant.

Consider, for example, the case of radium C. Using the
notation of Section 159, the number of atoms S of radium C at
any time t is given by

S = ae'^t + be~^ + ce~^ + de~^



430 THEORY OF SUCCESSIVE TRANSFORMATIONS [CH.

where \, X 2 , X 3 , X 4 are the constants of transformation of the
emanation, radium A, B and C respectively. Since the value of
X l is much smaller than \2, X 3 or X 4 , by making t very large only
the first term becomes important, i.e. when t is large



_ 123 o _g-M (21)

(X 2 -X I )(X 3 -X 1 )(X 4 -X 1 )

For the instant t, the amount S of radium C which would be in
secular equilibrium with the emanation is given by

Xi-t Q 6 = X 4 o .

Consequently g-

when the values of X are substituted in the equation.

This shows that the amount of radium C present is 0*89 per
cent, greater than corresponds to secular equilibrium. For example,
consider a certain quantity of emanation existing alone by itself
and an equal quantity of emanation associated with the amount
of radium with which it is in equilibrium. The amount of
radium C in transient equilibrium with the emanation in the
first case is *89 per cent, greater than in the latter. Now the 7
radiation (after passing through 2 cms. of lead) from radium in
equilibrium or from a tube filled with radium emanation arises
mainly from radium C. If the 7 radiation from the emanation
tube is compared with that due to a standard radium preparation
through 2 cms. of lead, the amount of emanation actually present
in the tube at the time of observation is obviously '89 per cent.
less than that deduced by direct measurements of the 7 ray effects.

In a similar way, it can be shown that the amount of radium
A and radium B present are '054 and '54 per cent, greater
respectively than the true equilibrium amount.

It is clear from these considerations that the decay curve of
the active deposit after a long exposure to a source of radium
emanation stored by itself does not follow exactly the theory given
in Section 160, for the relative ratios of radium A, B and C above
the true equilibrium are as T00054 : T0054 : 1-0089. It is obvious,
however, that the differences from the theoretical curves given in
Fig. 100 will be small.



XI] THEOKY OF SUCCESSIVE TRANSFORMATIONS 431

The general question of the ratios which the amounts of
substances in radio-active equilibrium bear to one another has
been examined by Mitchell* and Lotkaf for any number of
successive changes. If the successive products have periods
jTj, T z ...T n of which T^ has a period long compared with any of
the others, then it can be simply shown from equations analogous
to (21) that the ratio of the amount of the nth product to the
first



This question is of interest in considering the relative quan-
tities of various radio-active substances in very old uranium
minerals. These arise from uranium which is half transformed in
about 6 x 10 9 years. The period of radium (2000 years) and of
ionium (possibly 10 6 years) are so short compared with uranium/
that secular equilibria should be reached in minerals about 10
million years old, and the ratio of the amount of the nth to the
first product is practically T n \T^. The case of uranium itself is
of special interest and will be considered later in Section 171.

164. Application to some practical cases. In the analysis
of radio-active transformations and in radio-active problems gene-
rally, frequent use has been made of the calculations previously
discussed. It may be of value and of interest to consider the
application of the general theory to several of the great number of
problems to which it is applicable.

Growth of radium D and polonium (radium F) from radium.
Suppose that radium initially deprived of all its products is placed
in a closed vessel. The successive transformations occurring are
shown below with the half value period of each product added.

Radium Emanation Rad. A B C D E F

2000 3-85 3 26'8 19'5 16'5 5 136

years days min. min. min. years days days

* Mitchell, Phil. Mag. 21, p. 40, 1911.



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