T H E R I G I D I T Y O F T H E E A R T I I ' S I N N E R C O R E 

K . E . B U L L E N 

1. The purpose of tliis paper is to examine and assess, in llie 
light of recent evidence, the theory lliat the Earth's inner core 'ìas 
a significant rigidity. 

The presenee of an inner core in the Earth is revealed from 
observations of the seismie pliase PKP in the « sliadow zone » for 
which the epicentral distance A lies in the range 105" < A < 143". 
Miss I. Lehmann ( r ) in 1936, followed by Gutenberg and Richter ( 2 ) 
in 1938, atlrihuted these observations to tlie presence of an inner 
core; and Jeffreys ( 3 ) in 1939 applied Airy's theory of diffraetion 
near a caustic to sliow that the alternative theory of diffraetion 
round the outer boundary of the centrai core was not capable of 
explaining tlie observations in the shadow zone. The existence of the 
inner core has been f a i r l y generallv accepted sinee tliis ealculation 
of Jeffreys. 

2. The work of Jeffreys ( 4 ) on llie P—-velocity, « say, in the 
centrai core favours tlie view that the inner core is composite, consist-
ing of an outer region F ( 5 ) occupying the range 1250 k m < r < 
1390 km, wliere r denotes tlie distance from the eentre, and an 
inner region G of radius 1250 k m ; (the region E refers to the outer 
part of the core, for wliich 1390 km < r < 3470 k m ; and the region 
D to that part of the mantle for which 3470 km < r < 5400 km, 
approx.). In the solution given by Jeffreys, the value of « in F 
decreases with increase of depth from 10.44 km/sec to 9.5 km/sec 

(the latter figure corresponds to a rate of decrease of a taken arbilra-
rily to he linear by Jeffreys), while, in G, u increases slowly from 
11.16 km/sec at the boundary to 11.3 km/sec at the centre; across 
! he boundary between F and G. tliere is a discontinuous j u m p in <t. 
The existence of the region F in which the velocity gradient is nega-
tive is not accepted by ali authors, for example Gutenberg (°ì; (the 
existence of F rests on a hypothesis of reflection at the inner core 



2 K . E . B U L L E N 

boundary whicli Jeffreys considers may well he true in the liglit of 
various supporting evidence). 

Gutenberg's velocity distribution, on the other band, gives a 
steady increase of a from 8.0 km/see at the outer boundary of the 
centrai core to 10.1 km/sec at r = 1250 k m , tlien a sharp increase 
in the gradient of a , w i t h no discontinuous cliange in a itself, result-
ing in a value of oa eqnal to 11.2 km/sec at about r = 1000 k m ; 
from bere to the centre, Gutenberg fiiicls to he nearly C o n s t a n t . 

3. Following calculations ( 7 ) on the pressure and density distribu-
tions in the Earth, the present writer noted ( 8 ) in 1946 that the 
indicated values of dk/dp, wliere k denotes the incompressi])ility and 
p the pressure, were not sensibly different in the outer part of the 
centrai core from the values in the lower mantle down to a depth 
of 2700 km. (This result is indicated most clearly by consideration 
of the relation 

dk/dp = 1 — g-'d0/dr , 

which applies to a homogeneous material, where g denotes the 
acceleratimi due to gravity, 0 = — 4 fi -/3, and denotes the 
S — velocity). Moreover, the change in k itself was so small as to 
be not necessarily significant. Ali well-established geophysical data 
could in fact be fìtted by assuming that k and dk/dp change smoothly 
between the mantle and the core. At the sanie tiine, it was noted 
that the materials of the mantle and core are widely different in 
respect of the density Q and rigidity [i. Further, in 1946, it was widely 
accepted that the chemical compositions of the mantle and core are 
distinct, that the mantle is coinposed of ultrabasic rock, and the 
outer core of iron or nickel-iron. And, in addition, laboratory investi-
gations at high pressure had sliown that, in general, the differences 
in the compressibilities of different materials diminish with increas-
ing pressure. 

It these circumstances, the writer proposed as a trial working 
hypothesis ("'• 8' n) t h a t : 

a) the compressibility of a m a t e r i a l and its pressure gradient 
are independent of chemical composition at pressures and tempera-
tures of the order of tliose prevailing in the Earth's deep interior 
(say below a deptli of 1000 km). 

4. When the hypothesis (o) was first set down, it had not been 
envisaged that there would soon be a serious challenge to the view 



T H E ItlCIDITY OF T H E E A R T H ' s INNER CORE 3 

that the inantle and outer core are chemically distinct. Following 
work of Kronig, de Boer and Korringa ( I 0 ), however, Ramsey ( n ) 
and the writer (1 2), on the basis of planetary data, have lield that 
there is a probability that the region E consists essentially of a 
pressure modification of ultrabasic rock; (Ramsey, but not the present 
writer, holds that the inner core, also, is caused by pressure). Several 
authors, notably Elsasser (1 3), Urey ( 1 4 ) and Bircb ( l a ) , have put 
forward opposing arguments to the modified-utrabasic-rock core 
theory, but the opposing arguments themselves have been shown ( 
17) to he subject to several weaknesses; the present writer's view is 
that it is stili at least moderately probable that the region E and 
the mantle are not sharply distinct chemically. In these circumstances, 
the originai hypothesis (a) is possibly too broad a generalisation, so 
that it becomes necessary ( I 8 , page 58) to consider the less general 
hypothesis tliat : 

b) a phase transition occurs at the boundary between the 
regions D and E of the Earth and leaves the compressibility and its 
pressure gradient unchanged on the two sides of the boundary. 

At the same tiine it needs to be remarked tliat k, if determined 
by the relation fc/y = dp/do, would be discontinuous at the transition 
point itself, so that there are no a priori physical grounds for expect-
ing (b) to hold. Thus, on the emprical evidence stated in Section 3, 
( a ) may well stili be a f a i r approximation for a class of materials 
whose representative atomic number is close to tliat of the region E. 

5. On the hypothesis (a), it is to be expected tliat the values of 
k and dk/dp in the regions F and G are close to those at the base 
of E. Tliis would, however, be impossible on the results of Jeffreys 
for llie P — velocity a if the inner core is fluid; (we ignore the 
extremely improbable happening that the density decreases witli 
increase of depth in tliis part of the Earth). On the Jeffreys data, 
a increases suddenly at r = 1250 km by 15 per cent, so that if tlie 
inner core were composed of fluid for which k = there would 
be at least a 32 per cent j u m p in k across the boundary. If, on the 
other band, the inner core possesses significant rigidity u , witli 

k -f- 4 n/3 = Qa~ , 

then k will remain continuous across the boundary if u has a value 
equal to about 25 per cent of that of k in the inner core. The hypo-
thesis (a), in conjunction with the data of Jeffreys, thus leads to 
the theory, stated in 1946 and developed in 1949 and 1950 (8, 9, 18), 



4 K . E. B U L L E N 

tliat the Earth's inner core is solid in the sense that description of 
its elastic hehaviour under stresses of periods of a few seconds requires 
the presence of a rigidity parameter of order 3.6 X IO12 dyn cin-. 
Tliis conclusion lias received support in f u r t h e r work of Ramsey ( l t t ). 

The diminution of a with increase of depili found by Jeffreys in 
the region F, in conjunction with tlie relation k = per, incidentally 
entails that, in F, eitlier Q increases r a p i d l y or else k diminishes 
rapidly. On tlie hypothesis (a), the former alternative is preferred, 
the total increase in p in F being 25 per cent. Also tlie relatively 
low gradient of a inside G entails tliat the average density gradient 
in G is more tlian twice as great as the density gradient in the 
region E. 

6. A consequence of rigidity in the inner core is that tlie inner 
core coulcl transmit S seismic waves. On tlie basis of the Jeffreys 
data and the hypothesis («}, theoretical tables for a seismic phase 
PKJKP bave been constructed (2 0), where the symbol J corresponds 
to S waves in the inner core. Companion energy calculation ( 2 1 ) show, 
however, that s u d i a phase could be only 011 the border of obser-
vability. A comprehensive survey by Burke-Gaffney ( 2 2 ) of the whole 
series of seismograms recorded at the Riverview College Observatorv 
between 1909 and 1952 shows that none of these seismograms quite 
conforms to the theoretical requirements for a finn identification of 
the phase PKJKP, thougli a few observations could contribute to a 
wider statistical examination wliich would include data from a num-
ber of observatories. Further, the unknown character of the region 
F leaves it possible that the energy in anv S waves generated in G 
would be even less tlian the energy calculated ( 2 1 ) 011 the assumed 
boundary condilions. It follows that the rigidity of the inner core 
can at best be established by ibis direct seismic metliod only tlirough 
intensive studies of seismograms followed by the use of searching 
statistical significance tests. 

7. In the meantime, it is desirable to examine the sources of 
evidence for a solid inner core in some f u r t h e r detail. Points on 
wliich further investigation can usefully be iliade at present i n c l u d e : 

£) a comparison of Ihe effeets of using Gulenberg's and Jef-
freys' observational data on « in the inner core; 

ii) questions arising from the forms that the compressibility-
pressure hypothesis miglit l a k e : 



T H E ItlCIDITY OF T H E E A R T H ' s I N N E R CORE 5 

iii) the hearing on the problem of quantum-mechanical (lata 
at extreinely high pressures. 

8. In considering (i), it will he sufficient to malte an approxiinate 
calculation hased on Gutenberg's velocity data quoted in Section 2. 

We shall represent tliis data by 

a 2 = 11.22 — a(r—1000)2 , [ 1 ] 
wliere 

a = (11.22 — 10.12)/2502 , 
taking cla/dr to be zero at r = 1000 km, and taking the k m and sec 
as units. The equation [ 1 ] yields 0.19 km/sec2 for the magnitude 
of da2/dr wlien r is just less tlian 1250 km, as against 0.019 km/sec2 

when r is just greater than 1250 k m , and implies that the magnitude 
of d J (/c — 4 [i/3)/p{ dr suddenly increases (as the depth increases) 
by a factor of ten at r = 1250 km. On the hypothesis (a), it would 
follow that, while il and its gradient are zero in the fluid region 
above r = 1250 km, |d(4 jx/3p)/o!r| clianges to the value 0.17 km/sec2 

just below; i.e. di u j oìdr suddenly rises from zero to 0.13 km/sec2. 
Assuming for (32 ( = u/o) a forni similar to [ 1 ] , we tlien obtain for 
1000 k m < r < 1250 km 

P2 = Po2 — b (r — 1000)2, 
wliere 

b = 0.13/500, 

and |30 is the S — velocity at r = 1000 km. Since is bere being 
taken as zero at r = ]250 km, we tlien have (30 evi 4.0 km/sec at 
r = 1000 km. 

Previous work using the data of Jeffreys baci yielded p0 = 4.8 
km/sec. The Jeffreys data had involved an increase in q from 12 to 15 
g/cin:i in the region F (see Section 5). If we ignore the possibility 
of an abnormal increase in density in the range 1000 k m < r < 1250 
km ( s u d i increase is neiIber indicated nor precluded on Gutenberg's 
data), tlien the relation = p (32 would, on the hypothesis (a) and 
Gutenberg® data, y i e l d a value of u at r = 1000 km equal to 2.0 X IO12 

dyn/cm2, approximately, i.e. 0.6 of the value yielded on the Jeffreys 
data. If an abnormal density increase were present, the yielded 
rigidity would of course he greater. Thus, on the hypothesis (a) and 
Gutenberg's data, the minimum rigidity in the innermost 1000 k m 
of the Earth appears to be about 2.0 X IO12 dyn/cm2. Tliis value is 
stili comparable witli the rigidity of the mantle, and is appreciably 
greater than that of surface rocks. 



6 K . E. BULLEN 

Between r = 1000 km and tlie centre, Gutenberg's data would, 
on the hypothesis (a), y i e l d a slightly higher density gradient tlian 
the Jeffreys data because of the less velocity gradient, but the density 
reaclied at tlie centre need not be much greater than 15 g/cm3. This 
is closer to the figure preferred by Birch (1 5), altliough Gutenberg's 
data do not preclude an appreciably higher value. 

A further point is that whereas the inain change in ti would take 
place suddenly on the Jeffreys data, it is only the gradient of ;i that 
would change suddenly on the Gutenberg data, the change in u being 
spread over 250 km. The boundary conditions of the inner core would 
tlien be very different from tliose assumed in the energy calcula-
tions ( 2 1 ) for PKJKP, and the energy transformed into S waves in the 
inner core would be very much less. Tlius, if Gutenberg's velocity 
distribution is approximately correct, the phase PKJKP is quite 
unobservable, and it would not be possible to establisli the r i g i d i t y 
of the inner core directly by observations of tliese waves. 

9. The point ( « ) of Section 7 on the possible limitations of 
the hypothesis ( « ) of Section 3 w i l l now be considered in relation 
to the inner core. 

If the view of Elsasser, Urey and Birch that the region E consists 
of iron is substantiated, then the hypothesis (a) remains, as in 1946, 
an important working hypothesis. It is l i k e l y to be reliable at least 
to the extent that, for atomic numbers between about 18 and 29, 
k and dk/dp at pressures of 1.3 million atmosplieres differ by not 
more than the uncertainties in k and dk/dp in the writer's Earth 
model A ( 5 i 18). In Model A, the formai solution gave a reduction 
(not necessarily significanti of 5 per cent in k at the boundary 
between the regions D and E. Ramsey ( 1 2 , l f t) prefers a greater change 
in k at this boufidary, but his argument involves deviation from 
homogeneity in the region D to an extent that Birch ( 1 5 ) regards as 
improbable. Ramsey (1 B) finds dk/dp for the mantle and core to be 
about the same, namely 3.7 and 3.5-3.8, respectively, wliile the writer's 
estimates y i e l d 3.2-3.3 near the base of the mantle, and the sanie or 
a little less at the top of the core. Birch (1 5), on independent grounds, 
finds the value in the mantle to be 3.3. In these circumstances, appli-
cation of the hypothesis (a) to the inner core probably does not lead 
to errors in k and dk/dp of more than the order of 5 per cent. 

On the assumption that the region E consists of iron, and in the 



T H E I t l C I D I T Y OF T H E E A R T H ' s I N N E R CORE 7 

light of cosmic abundance data for elements, it is strongly probable 
that any deviation from the hypothesis (a) will involve changes of k 
between E and G smaller than between D and E, and therefore much 
smaller than would be required to bring the value of ;i obtained in 
Section 5 down to zero. A 5 per cent j u m p in k near the boundary 
of G would stili require the rigidity |i in the inner core to be nearly 
3.1 X IO12 dyn/cm2 on the Jeffreys data. 

If, on the other band, the region E consists of modified ultrabasic 
rock, the remarks in Section 4 become relevant. On the view of Ramsey 
that the region G is not chemically distinct from either E or D, the 
expected change in k between E and G would not be greater than 
between D and E, and the argument favouring rigidity in G is essen-
tially unchanged. But the case favoured by the writer on present evi-
dence ( ' " • 2 3 , 2 4 ) , in which G consists of iron or denser materials 
while E lias the same chemical composition as the ultrabasic-rock ma-
terial of D, is more difficult. The remark i n the last paragraph of 
Section 4 shows that there is stili some probability that the value 
of k in G is very close to that in the lower part of E, but a less ge-
neral hypothesis than (a), closer to (b) of Section 4, would not lead 
of itself to any inference on the rigidity of G, and furtlier conside-
rations are desirable. 

10. Such further considerations come from the point (iii) of Sec-
tion 7; tliis supplements the argument in Section 9 in cases in which 
an estimate of the rigidity is yielded, and also goes some distance 
towards filling the gap in the more difficult case. 

Several authors, including Jensen (2 5), Feynman, Metropolis and 
Teller (2fi), have considered the compressibility-pressure relations for 
Thomas-Fermi-Dirac gas models. Jensen, Elsasser ( 1 3 ) and the writer 
( 1 7 ) have discussed the hearing of relevant parts of the results on the 
composition of the Earth's core. Elsasser quotes data at pressures of 
the order of 10 to 30 million atmosplieres, according to which k in-
creases by 8 per cent when the atomic number Z changes from 18 
to 36. The pressure, pi say, at the boundary of the region G is ap-
proximately 3.2 million atmospheres, and Elsasser's comparison of 
experimental results of Bridgman witli the tlieoretical high-pressure 
results indicates about the same proportionate change of k witli Z at 
the pressure pi as at the liiglier pressures. The writer has amended 
certain parts of Elsasser's work, and fìnds that Bridgman's results, 



8 K . E. B U L L E N 

geophysical data and tlie Feynman data are w.ell fitted together by 
taking smaller values of k at the pressure pi than those assumed by 
Elsasser; this procedure raises the proportionate change in k when Z 
changes from 18 to 36 to about 12 per cent at the pressure pi. Most 
theories of the composition of the Earth's core require the difference 
in Z between the outer part of the region G and the base of E to be 
nearly zero; the writer's theory, referred to in Section 9, gives the 
biggest difference, of order seven units. TIius the quantuin-mechanical 
calculations indicate that any j u m p in k between E and G is of tlie 
order of 5 per cent or less, and that, on the Jeffreys velocity data, the 
rigidity of the inner core is stili not less than the minimum value 
arrived at in Section 9. This argument is of course subject to uncer-
tainties in the interpolations between Bridgman's data and the quan-
tum-mechanical data, and again falls well short of deductive proof, 
but, in combination with the earlier discussion, leaves it most pro-
bable tliat the inner core is significantly rigid. 

11. Gutenberg's data show appreciably smaller changes in a than 
Jeffreys' data in the vicinity of the boundary of the inner core, and 
may be reasonably taken as indicating the least allowable changes 
in a . Taking tlien the lowest estimate of [i in Section 8, diminished 
by a possible 0.5 X IO12 dyn/cm2 in accordance with the discussions 
in Sections 9 and 10, we arrive at the value 1.5 X IO12 dyn/cm2 as 
the least value of the rigidity in the inner core that is reasonably 
compatible with ali present evidence. This value is less one-half the 
value 3.6 X IO12 dvn/em2 involved in earlier calculations of the writer 
( 1 S i 20, 21), but is stili about twice the rigidity of stecl at zero pressure. 

As in earlier papers, temperature effects bave been disregarded 
in ali the foregoing discussions. Birch's recent work ( 1 5 ) gives further 
confirmation that this would not lead to serious error. 

If the rigidity of the inner core were in actual fact as low as 
the above suggested m i n i m u m value of 1.5 X IO12 dyn/cm2, there 
would be repercussions on the calculations of the travel-times of the 
phase PKJKP, and the time of the J rays would need to be increased 
by a factor of 1.3, approximately. This would entail increases ranging 
from zero u p to about 2% minutes in the calculated times. If the 
phase PKJKP arrived as late as this, however, the theoretical travel-
time table ( 2 0 l would be devoid of practical value, since, 011 Guten-
berg's data (see Section 8), the phase would not then be observable. 
But from the discussion in Sections 9 and 10 it follows tliat, even on 



9 

the leffreys data, the limes of the j rays may need to he increased 

fly 11 fad or of 1.08, approximately, This would entail increases in the 
('.ilculalcd times for PK]KP, reaching nearly 40 seconds at J. = 180". 

It follows that, in seeking evidence on the occurrence of the 

phase I'KJKP, it is desirahle to consider readings on seismograms 
rangin;! up 10 perhaps a minute later than the limes in the calculated 

table. TIH~ existence of the phase could he estahlished only if sufficient 
of sllch readinp:s fitted to adc(Iuale precision 11 travel-time curve po~­

;<e~~in;! the rip:ht p:radienL This cousideration emphasises further that 

the cxi~lenee of the phase PKjKP can he estahlished only hy statio 
stical exatllination of 11 large numher of readin:rs. ' 

SUMMARY 

The I'tJidf'nce for the existence of significant rigidity in the Earth's 

il!l!f'r ("orl' is Slllll11lf1risf'CI (IIul discussed in the lif!,ht of recent work. 

Quantum-mf'chanical calculfltions, based on a Thomfls-Fermi·Dirac 

model, ,m{!,w'st that (Ill f'flrlif'r f'stimatl' of 3.6 X ltJl~ dyn/cm~ for this 

ri{!,idi/.I" mfly lIeed to bl' reduced by 0.5 X 1012 dyn/cm~. TIU' tlwore-

tical travl'i·ti/IH'S of the phase PKJKP would then need to be increased 
b.\· fl1llOu/lIs rrmg':'!{!. from zero to 40 sf'conds. It is shown that tlm use 

of GII/I'1I1H'rg's seismic da/a in place of that of leflreys would reduce 

thl' I,s/imfl/f'd righlity of the inner .core fry a furthl'r 1.6 X lOI~ dynicm 2 , 

approximatl'ly. It is su{!,w~sted that GlItenfJerg's data lead to the lowest 
likl'ly lm[m' for tIll' rip:idity of the inner core, nallH,lv 1.';XI01~ dyn " Cill~ , 

so tfw/, 011 all prr'sent elJidence, it is improbable that tlw inner core 

is [('ss Ihall about tldce as rigid as steel is at zero pressure. 

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G"nphy.,. SUI'PI., ·l, 1938,1'1'.363·372; 0" .• "i .• ",ic ,,·"l·e.'. G"rland" I)dlr. Geophy • ., 
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