of the particular rock. In the latter case that which
flows is heat, measured in so many calories or therms
UNDERGROUND TEMPERATURE AND RADIOACTIVITY. 73
per second. When we know how many calories flow
through a cross-section of one square centimetre for
unit gradient that is, a difference of temperature of
1 C. per centimetre and also know the gradient
prevailing in the rock, we have only to express the
latter as such a fraction of a degree per centimetre,
and then multiply by the conductivity, and we have
the flow in calories per square centimetre. The calorie
here is the quantity of heat required to raise one gram
of cold water through one degree centigrade.
We see, then, that the indications of the gradient
are dependent upon the conductivity of the particular
class of rock in which the observations are made.
Thus there might be cases in which the gradient was
very different, but the flow of heat might be the same.
Again, the gradients might be alike, but the flow of
heat different. These principles will be required in
connexion with various matters later on.
The thermal conductivity is again a very variable
property; and we can only make general use of it
by seeking some mean value. Eocks are what would be
described as bad conductors. Under the conditions of
gradient mentioned i.e., 1 C. per centimetre the flow
in calories per square centimetre per second is in
nearly every case some number in the third place of
The following are the most important determinations
selected from the results published by the British
Association Committee, 1 and in the Landholt-Bornstein
1 Everett, C.G.S. System of Units.
74 RADIOACTIVITY AND GEOLOGY.
Tabellen. The rocks are dry : moisture tends to increase
the conductivity :
Sandstone, , . . . f 0'0055
Micaceous flagstone, . . . . 0*0053
Marbles, Limestone, and Dolomite, . . 0'0051
Caen stone, 0*0043
Chalk, . 0-0025
Clay slate, . 0'0027
Calcareous sandstone, 0'0021
Granite, . . . . . . . 0*0053
Whinstone, Trap rock, and Mica- schist, . 0'0038
Basalt, . . . .... 0-0067
Syenite, . . . * . . . . < . 0'0044
Glass, . .. . . . . . 0-0017
The mean of these values is for the igneous materials
0*0042, and for the sedimentary 0*0041. For approxi-
mate calculations the average 0-004 sufficiently nearly
represents the results on rocks generally.
A very important question is whether the con-
ductivity increases or diminishes at high temperatures.
There is insufficient information on the subject.
K. Weber has found that gneiss, which has a con-
ductivity of 0*00578 at 0, falls in conductivity to
0-00416 at 100 C. ; x and Lees has found, on the other
hand, that the conductivity of glass increases by 0'0025
per cent, per degree between 35 and 60 C. 2 Metals
appear to diminish in conductivity with rise of tem-
perature; but alloys, on the other hand, improve in
1 Landholt-Bornstein, Tabellen.
2 Lees, Phil. Trans., 1898, A, 399.
UNDERGROUND TEMPERATURE AND RADIOACTIVITY. 75
conductivity. 1 Such results as we possess on con-
ductivity, traced through the solid and into the liquid
state, would rather lead to the view that a fluid rock-
magma must possess a lower conductivity still than
the solid. Thus Lees found that ice falls in conductivity
from the temperature of liquid air upwards, and shows
a considerable diminution in the liquid state. Similarly
in the case of aniline the conductivity solid is much
greater than in the liquid state. In the case of
glycerine the change is small, but still one of diminu-
tion. 2 Barus has obtained similar results in the case
of thymol. 3
Knowing the conductivity, it is easy to estimate the
amount of heat which must annually escape at the
We have taken a mean gradient of 32 metres per
degree, and a mean conductivity of O004 for the average
rock. Expressing the gradient as the rise in tempera-
ture in a distance of one centimetre measured down-
wards, we have
x 0-004 = 1-25 x 10-
as the quantity of heat in calories which is conveyed
to the surface per second per square centimetre of the
earth's surface. If we multiply by the number of
seconds in a year (31*5 x 10 6 ), and by the area of the
earth in square centimetres (51 x 10 17 ), we finally arrive
1 Lees, Proc. R. S. 80, A. 143. 2 Ibid.
3 Barus, Am. Journ. Sc. t July, 1892.
76 RADIOACTIVITY AND GEOLOGY.
at the total annual escape of heat from the earth. It
comes out at 2-0 x 10 20 calories.
Of course this must be regarded as only an approxi-
mation. The gradient beneath the oceans is unknown ;
we assume that it does not differ much from that
which prevails (with variations) over the land surface.
For the purpose of arriving at general views the result
is probably close enough to the truth.
We have now to consider the principles which guide
us in estimating the temperature effects produced by
radium as a widely diffused rock-constituent.
We have already seen that the heat evolved by one
gram of radium and its associated radioactive elements is
closely 5 f 6 x 10" 2 calories per gram of radium per second;
that is to say, one calorie every eighteen seconds. This
is, indeed, a rapid rate of evolution of energy from
so small a quantity of matter. When, however, we
come to deal with billionths of a gram of radium per
gram of rock, of course the absolute quantity of heat
evolved in any small volume of the rock is minute.
On the other hand, in the case of the geological
problems, there is practically unlimited time for its
The accumulation is favoured by the nature of the
circumstances. We are not dealing, it will be noticed,
with an isolated quantity of radium deposited at a
particular depth from the surface, but with materials
containing radium disseminated more or less uniformly
throughout their bulk. Hence the rock whether
igneous or sedimentary is continually producing heat
UNDERGPwOUND TEMPERATURE AND RADIOACTIVITY. 77
throughout every cubic centimetre of its volume.
Thus, at any level beneath the surface, the heat
produced in a particular cubic centimetre of the rock
can only escape in virtue of this little mass of rock
being hotter than the cubic centimetre of material
immediately above it. But in this also heat is being
developed, thus checking the escape of that which is
developed beneath; and this reasoning applies all the
way from base to top of the entire layer of rock : so
that if no heat is escaping below, the maximum
temperature must be at the base, and this basal
temperature gives place to a lesser temperature as we
go upwards at first changing- slowly, and then rather
more rapidly near the surface.
When the question is treated mathematically, the
equilibrium temperature 9, at any depth x from the
surface, is found by the equation 1 :
where D is the total depth of the radioactive layer
supposed neither to gain nor lose heat at the base,
and to retain a constant temperature at the immediate
surface ; K is the conductivity of the particular rock ;
h is the constant of heat-production of radium per
second ; and q the amount of radium contained in one
cubic centimetre of the rock.
At the very base of the layer, x becomes equal
to D, and the basal or maximum temperature is
See Appendix A.
78 RADIOACTIVITY AND GEOLOGY.
The first equation tells us that in order to determine
the temperature at any particular depth in the layer,
we must know the value of D or the total depth of
the radioactive layer. This might indeed be evident
from what we have already said about the temperature
at every level influencing the temperature beneath, 4 ;id,
of course, in sending heat to the surface, influencing
also the temperature above it. The second equation
tells us the very important fact that the maximum
temperature attained increases with the square of the
depth of the radioactive layer. Thus, if equal quantities
of radium were contained in two different radioactive
strata, that stratum which possessed it most diffusedly
distributed that is which had less per cubic centimetre
would have the higher basal temperature ; for the
depth of the stratum is necessarily greater than that
in which the radium is more concentrated. This, if
at first appearing contradictory, will be understood
when it is remembered that the general effect of the
more diffuse distribution is to remove the radioactive
materials further from the surface, and so enable the
heat to be the better retained.
Applying these facts to the problem of interpreting
the radioactivity of the surface-materials in terms of
the gradient of temperature downwards, or, in other
words, of endeavouring to connect the radioactivity
with the underground temperature, we see that our
deepest borings and tunnels are still too high up in the
crust to enable us to pronounce with any certainty on
the effect of radium met with in the rocks. The local
radioactivity in itself can produce little effect on the
temperature of the rocks.
UNDERGROUND TEMPERATURE AND RADIOACTIVITY. 79
An example will make this clear. The Simplon
tunnel runs for a considerable distance at a depth, say,
of 2,000 metres from the surface. The radioactivity of
the rocks would appear to be about 19 x 10" 12 grams per
cubic centimetre, according to the experiments. Now,
if we suppose that this radioactivity extends no further
down than the level of the tunnel, the radium in the
whole 2,000 metres would produce a temperature due
to radioactivity of only 5'3 C. in the tunnel, this being
the basal temperature of the layer.
We see from this how unlikely it is that volcanic
temperatures can be brought very near the surface
by the radioactivity of the rocks in the upper crust;
for this involves a temperature of some 1200 being
so produced. To bring about such a temperature, as
a basal temperature, at a depth of, say, 7 kilometres
from the surface, there must be a radioactivity through-
out the 7 kilometres of over 340 x 10~ 12 grams per cubic
centimetre. This is quite in excess of what prevails in
any known considerable rock- mass. And as lavas, so far
as examined, show no very excessive radioactivity, there
can be no support for this theory of volcanic origin.
The gradients met with in deep borings, as already
observed, vary considerably. The variations are doubt-
less in most cases due to variations in the rock-con-
ductivity. But in some very erratic cases this may
not be so, and it may be found possible to establish
some connexion with the radioactivity of the local rocks.
The majority of researches in this direction must, how-
ever, probably prove unfruitful. Consider in this
connexion some radium measurements made in the
RADIOACTIVITY AND GEOLOGY.
case of the Iklfour Bore in Fifeshire, which is carried
through Carboniferous shales and sandstones to a depth
of 4534-5 feet (1383 metres). The boring passes through
a succession of beds changing frequently in character ;
the most continuous rock being a coarse, diabase-
like layer some 500 feet thick. The shales contain
vegetable fossils, and are probably of fresh-water
origin. Throughout these materials, wherever tested,
save in one c|se, there is a notable sameness in
Black argillaceous rock, .
Argillaceous, fossilif erous,
Argillaceous rock, . ,,
Arkose sandstone, .
It is probable that the varying gradients which prevail
at the levels from which the samples are taken are
referable almost entirely to the differing conductivi-
ties of the beds : the radioactivity only affecting the
temperatures in quite a secondary manner.
In the case of the two great tunnels, the Simplon and
the St. Gothard, there is evidence of the intervention of
UNDERGROUND TEMPERATURE AND RADIOACTIVITY. 81
radioactivity in affecting the temperatures encountered
in making the tunnels. This evidence is, however, only
to be obtained by assuming and the assumption is
doubtless quite legitimate the downward continuation
of the radioactivity observed at the levels of the tunnels.
Tht, temperatures, which are so essential to any scientific
discussion of the subject, were read in the rocks when
the headings were being made. It has been found that
the means taken to keep down the temperature in the
progress of the works soon result in a fall of the
rock-temperatures, so that the data to be of any real
value must be obtained before this artificial source of
cooling becomes effective. Nothing can exceed the
pains devoted in both cases to this difficult work.
Its value is incalculable, not only from the scientific
but from the practical point of view.
The high temperatures met with in the Simplon, as
all know, nearly put an end to the project, although
the actual difficulty was more that arising from the
quantity of heat which entered in the torrents of hot
water than from the temperature of the rocks. The
facts in outline are these : For a stretch of some
seven or eight kilometres the mean depth of the
tunnel below the surface of the mountain is about
1700 metres. At the north end of this stretch, the
rocks attain the temperature of about 55, and this at
the south end sinks to 35. The temperature 5 5 '4 was
the highest encountered. It was not at the deepest part
of the tunnel, but at a point somewhat to the north of
How unexpected such temperatures were, may be
82 KADIO ACTIVITY AND GEOLOGY.
shown by the predictions of the best authorities when
the tunnel was still unattempted. Stapff distinguished
at once as engineer and as geologist had in the case
of the St. Gothard made a scientific study of the dis-
tribution of rock-temperature, and would rightly be
regarded as the most competent authority of the time.
His prediction for the Simplon was a maximum tem-
perature of 47 C. He was, however, alone in making
so high a prediction. Stockalper, also of St. Gothard
fame, foretold 37 at a depth of 2000 metres from
the surface. Heim thought 38 to 39 the probable
These predictions were, doubtless, influenced by the
accepted views as to gradients in mountains. The
gradient actually attained at the hottest part of the
tunnel would not be excessive for a boring made in the
plains. Thus, if we take it that 50 C. was reached at
1600 metres below the mountain surface at C., the
gradient is no more than the average value of 32 metres
per degree. However, previous experience of gradients
in steep mountains would lead to the belief that the
gradients in such cases are influenced by the convexity
and steepness of the ground. In the case of the
St. Gothard, the gradient for the major part of the
tunnel was 46*6 metres, 2 and in the case of the Mont
Cenis tunnel it has been estimated at 427 metres. 3 In
1 Schardt, Verhandl Schweizerischen Naturf. Oeselkch. 1904. 87,
Jahresversammlung, p. 204.
Stapff, Trans. North of England Mining and Mech. Engineers,
xxxiii., 19 et seq.
3 Prestwich, Proc. E. S., xli., p. 1 et seg*
UNDERGROUND TEMPERATURE AND RADIOACTIVITY. 83
the last case the neglect of observations during the
progress of the headings deprives the estimate of much
of its reliability. Osmond Fisher, indeed, contends
with much force that the convexity of the ground
cannot exercise a marked influence on the gradients. 1
However, the fact remains that the temperatures
observed must be regarded as above those generally
experienced in previous cases, and, therefore, as
abnormal on the evidence of the, perhaps, small
number of observations of reliable character.
In explanation of the observed facts, it was pointed
out that the north end of the tunnel, where the
temperature was high, is exceptionally dry. The
conductivity of the rock is, therefore, less, and there is
no circulating water to carry off the heat ; while the
opposite conditions prevail at the south end. This is,
of course, a physical explanation of why the gradients
at the south end are low, and gives no reason for the
high temperatures at the north end. The circulation of
water is, in fact, a condition tending to disguise the
true flow of heat at the south end if the average
conductivity of rock is assumed where in truth it does
not apply. The question really turns on the con-
ductivity of the rock where the temperatures were
highest ; we want to know whether these temperatures
indicate an abnormal heat-flow or not. The next
explanation is, therefore, more to the point: the
direction of the schistosity is appealed to. It is well
known that considerable differences exist in the
conductivity of certain rocks according as the heat-flow
1 Phyric* of the Earth'* Crust, 2nd ed., p. 222.
84 ItADIOACTIVITY AND GEOLOGY.
takes place across the schistose cleavage or along the
cleavage. The explanation would be a good one if the
schistosity was actually horizontal. In point of fact,
the sections show it as pointing at a high angle to
the horizontal at stretches near the north end where
the temperature already nearly reached its maximum. 1
It cannot be doubted that the whole explanation of the
very remarkable difference between the gradients in
the central reaches of the Simplon and the St. Gothard
tunnels is not fully explained in the angle of schistosity
of the rocks in the former, nor yet, probably, in the
conductivities, as we shall presently see.
In the case of the St. Gothard, the temperatures
observed were of the most interesting character. In
the central parts of the tunnel the mean gradient
comes out as 46*6 metres per degree. These rocks are
gneisses and schists of varying character. At the north
end, where the tunnel passes through granite the
same granite that enters into the massif of the
Finsteraarhorn the gradient steepens to 20*9 metres.
This last gradient, exceptionally steep even for a level
plain, will again claim our attention. It is desirable
to first confine our considerations to the temperature
conditions where they are apparently least affected by
special circumstances in the two mountains. Failing
actual comparative measurements of the conductivities,
we must assume that in the central St. Gothard there
is actually a lesser flow of heat than prevails in the
Simplon massif. We assume that in those stretches
of both tunnels where there is no circulation of water
1 Schardt, loc. cit., and Schmidt, Eclogie Geol. Helvetia, ix., No. 484.
UNDERGROUND TEMPERATURE AND RADIOACTIVITY. 85
in the rocks, and both rocks are of much the same
character, the differing gradient must represent a real
difference iii the quantity of heat escaping to the
surface. In support of this assumption, it is to be
observed that there appears no good reason to explain
a difference in gradients of 32 and 46'6 metres, or
nearly 50 per cent., on any quality of the rocks.
Both are rocks of schistose and gneissic character,
highly compressed, and not cooled by circulating water.
The schistosity is more vertical, apparently, in the
St. Gothard than in the Simplon; but this cannot
create so great a difference in conductivity. Thus
Zollinger 1 gives the conductivities in kilogram-calories
per metre per hour of the schists and gneiss of the
Simplon tunnel, taken across and with the cleavage,
as 2'28 and 3*02 respectively ; in the cases of calcareous
schists and limestone, as 2 '10 and 3 '26 ; and in the case
of granite 1'94 across the stratification, and 2'49 in the
line of stratification. These reduced to the units
previously used the centimetre, gram, and second
Gneiss and schists across cleavage, . . 0'0063
along . 0-0084
Calcareous schists 0'0058
Granite across stratification, .... 0'0054
,, along .... 0-0069
As schistosity is oriented in the tunnel, only a fraction
of the difference of conductivity in the several cases
1 Discussion on Mr. Fox's paper : Minutes, Proceedings Institute Civil
Engineers, 168, p. 53.
86 RADIOACTIVITY AND GEOLOGY.
can be supposed to affect the flow of heat. It is also
evident that there is nothing in the absolute values
of the conductivities to account for the high tempera-
tures in the Simplon ; the conductivities are rather
above the average values.
It would appear, however, that in the radium-content
of the Simplon, contrasted with that of the St. Gothard,
a possible explanation is forthcoming. The following
summary embodies the results of experiments on the
radioactivities of the Simplon rocks. A detailed
statement of the individual results on the 49 rocks
investigated is not necessary, and would hardly be
desirable in view of the fact that a revision of the
averages arrived at may be necessary (see p. 44).
Jurassic and Triassic altered sediments, . . 6*4
Crystalline schists, partly Jurassic and Triassic,
partly Archaean, . . . . . . 7'3
Monte Leone gneiss and primitive gneiss, . .6*3
Schistose gneiss a fold from beneath, . .6*5
Antigorio gneiss, . . . . . .6*8
This subdivision of the Simplon rocks is that of
The mean of the individual experiments is 7*1 x 10~ 12
grams per gram, and if some exceptionally high readings
be substituted for low ones obtained in the same rock,
the mean rises to 97 x 10~ 12 .
Now it will be presently seen that the experiments
on the St. Gothard rocks do not give such high values
unless in the granite at the north end of the tunnel.
The gradient, according to Stapff, 1 declines from the
1 Stapff, loc. eit.
" s ^i
UNDERGROUND TEMPERATURE AND RADIOACTIVITY. 87
point 3550 metres measured from the north end to
4050 metres, after which it keeps a very low value
to the south end of the tunnel. Water-circulation
appears, however, to affect the gradients at the south
extremity, giving rise to the lowest gradients anywhere
observed in the tunnel. This applies probably to the
last 3 kilometres. If we take the radium determin-
ations from the beginning of the true St. Gothard
massif to the distance 11,000 metres from the north
end, we have the radium readings corresponding to the
central low gradients. A glance at the accompanying
chart, Plate IV., where the radium values are plotted
as a full line, and the gradients as a dotted line, will
more fully explain. The mean of these radium readings
is 3*3. In this mean we have apparently a record
of local radioactivity which we may compare, with
security against causes of special disturbance, with
the Simplon results. We find, in fact, the low radio-
activity associated with the low gradients, and the
high radioactivity with the high gradients.
We have already seen that without making assump-
tions as to the downward extension of the materials
examined so near the surface, we cannot expect to
connect our observations with the gradients. It is of
interest to estimate what downward extension of the
two rock-masses must be assumed to exist, always
supposing them to maintain the observed mean radio-
activities, in order to account for the temperature
observations. Such an estimate is simple, if we suppose
the conductivities alike throughout, and that therefore
the differing gradients represent a different heat-flow
88 EADIOACTIVITY AND GEOLOGY.
to the surface. If now this difference of heat-flow is
due to the different radioactivities of the rocks, then
obviously there must be some distance to which if both
masses extend the superior heat-flow from the Simplon
becomes accounted for. This will be evident when it
is remembered that each metre downward increases
the superiority of the Simplon in respect of the total
radium transmitting heat to the surface.
It has been explained that the product of the
gradient and the conductivity is a quantity of heat
flowing to the surface ; accordingly, it follows that if
J2 and r be the radium content per cubic centimetre
of the Simplon and the St. Gothard rocks respectively,
and G and g the gradients in each case, and K the
conductivity, then when the rocks extend to a depth
sufficient to produce the difference of heat-flow by the
difference of total radium, we have X(E - r)h = K(G-g),
where X is the required depth, and li the heat-constant
of radium. Inserting the values of the radioactivities per
cubic centimetre (19 x 10~ 12 , and 8 x 10~ 12 ), and taking