G r u n e i s e n ' s g a m m a f u n c t i o n for liquid iron at the E a r t h ' s
outer core conditions
E. B o s c m
( * ) - F . MULARGIA
(*)
Received on July 15 th, 1977
SUMMARY. - WE derive the G r u n e i s e n ' s g a m m a as a f u n c t i o n of temper a t u r e and pressure f o r liquid iron at the t e m p e r a t u r e and pressures of the E a r t h ' s
o u t e r c o r e taking into a c c o u n t the a n h a r m o n i c a n d electronic c o n t r i b u t i o n s in
a self-consistent way. W e obtain n u m e r i c a l values definetely different f r o m those
generally q u o t e d in the geophysical literature, but in a g r e e m e n t with recent
experimental determinations.
RIASSUNTO. - Si ottiene u n ' e s p r e s s i o n e per il g a m m a di G r u n e i s e n in
f u n z i o n e della t e m p e r a t u r a e della pressione per il f e r r o liquido nell'intervallo di
pressioni e t e m p e r a t u r e del n u c l e o esterno della T e r r a t e n e n d o a n c h e conto, in
m a n i e r a consistente, dei c o n t r i b u t i a n a r m o n i c i ed elettronici. Si o t t e n g o n o valori
n u m e r i c i n e t t a m e n t e diversi da quelli g e n e r a l m e n t e riportati nella l e t t e r a t u r a
geofísica, m a in a c c o r d o con i risultati di recenti e s p e r i m e n t i .
It is generally accepted that the E a r t h ' s outer core is mainly
composed by liquid iron. In this w o r k we assume it composed by p u r e
liquid iron and we shall evaluate the G r u n e i s e n ' s g a m m a values in
the outer core range of pressures and t e m p e r a t u r e s . T h e G r u n e i s e n ' s
g a m m a is derived taking into account all the k n o w n possible physical
effects according to recent achievments in solid state theory. F u r t h e r m o r e ,
instead of extrapolating with d u b i o u s rigour values relative to solid
crystals, as it is generally m a d e in the geophysical literature, we apply
our theory to liquid state. T h e p r o b l e m of the determination of the
G r u n e i s e n ' s g a m m a for liquids has had scarce attention (4' " ) .
(*) Istituto di Geofìsica, Università di Bologna.
206
E.
ISOSCHI
-
F.
MULARGIA
By liquid state, at t'he E a r t h ' s core conditions, we m e a n a lattice
with a simple cubic structure in which transverse waves have a transmission coefficient irrelevant in c o m p a r i s o n with the longitudinal transmission coefficient. This liquid model arises f r o m e x p e r i m e n t a l diffraction
data on liquid metals ('• , 5 ) which s h o w a nearest-neighbour pattern
m u c h more randomly located than f o r solids. This gives the reasons to
support for liquids a lower coordination n u m b e r than f o r solids (17) and
a very low transverse « constant » in o r d e r to allow nearly f r e e shear
m o v e m e n t s in the lattice. According to DOMB et al. ( 5 ), by force constants
w e mean the spring constant dF/dr of the intermolecular forces F. W e
believe that at the present stage of k n o w l e d g e s this is the best way
to tackle the p r o b l e m since neither the cell model ( 12 ), nor the distribution
f u n c t i o n model can be profitably used in the lattice calculations (").
An explicit expression f o r the G r u n e i s e n ' s g a m m a can be f o u n d
following the general p r o c e d u r e of considering a solid as a system
of 3 N strongly a n h a r m o n i c oscillators. T h e hamiltonian function of
the system may then be splitted into its h a r m o n i c (in the LeibfriedLudwig sense (") and a n h a r m o n i c parts
H = HH + HA
which can be solved separately. Considering the h a r m o n i c term, w e
find the normal modes of vibration of the system as eigenvalues of
Flu and then, f r o m the energy s p e c t r u m , c o m p u t e the G r u n e i s e n ' s g a m m a
according to the definition
Y =
Zi 0 In v , / 3 In V)
It must be noted that the high t e m p e r a t u r e s of the core allow us
to use a T h i r r i n g ' s series expansion ( l8 ) to express the h a r m o n i c
G r u n e i s e n ' s w i t h o u t passing through the explicit frequency s p e c t r u m ,
but just considering its statistical m o m e n t s . T h e n the h a r m o n i c
G r u n e i s e n ' s g a m m a turns out to be
yH = V (dEvis/dVh
(TCV-EVIB) '
[11
where
Ev,b = E(T)-U
Cv —
(dEvni/dT)\
[2]
CRUNEISEN' S
GAMMA
FUNCTION
FOR
I.IQUID
IRON
ETC.
207
U is the static lattice potential a n d E ( T ) the i n t e r n a l energy of
the
system. Evm can be w r i t t e n as
(
EV,B = 3 NkT
1 -
(
°°
I
£ ( - 1 ) " [B2„/(2n) !] ;íí7„ (h/lc T)2n
«=i
w h e r e B2„ are Bernoulli n u m b e r s a n d fi2n are the static even
of
the
treating
frequency
allow
[31
)
distribution.
The
us to c o n s i d e r
only
very
the
high
first
moments
temperatures
few moments
we
are
since
the
T h i r r i n g ' s series is strongly c o n v e r g e n t u p f r o m the D e b y e t e m p e r a t u r e .
T h e s e m o m e n t s h a v e been calculated
20-th
order
(2'5'
16 13
- )
(BHATIA) a n d
f o r any lattice s t r u c t u r e
HORTON,
1955,
DOMB
et
to
al.,
MONTROLL, 1 9 4 3 ; LEIGHTON, 1 9 4 8 ) . W e s h a l l u s e t h e s e p t e n a r y
the
1959;
partition
m o d e l ( 4 ), w h i c h gives the m o m e n t a of a simple c u b i c 3 d lattice. T h e first
t w o m o m e n t s can be w r i t t e n as
V2 = (K„ + 2K,)Ma-2
[4]
iu = 2 (Kp2 + 2 K,2) Ma~2 + 4 (K„ + 2 K,) MA~2
[51
w h e r e K,, a n d K, are the longitudinal a n d t r a n s v e r s e c o n s t a n t s , respectively. W e e m p l o y an i n t e r m o l e c u l a r potential of R y d b e r g type, n a m e l y
( r „ - r e ) ] e x p [ - f e (r i ; — r e )]
<¡>=-D[\+b
[6]
w h e r e re is the static d i s t a n c e of a p p r o a c h of the molecules, r¡¡ is the
i s t a n t a n e o u s distance. D a n d b are c o n s t a n t s , a n d n a m e l y , D = 7 4 . 6 9 x 10
u
o
ergs, i> = 2 . 0 5 5 4 x 10 M
'. T h i s potential s h o w s a very good a g r e e m e n t
with e x p e r i m e n t a l data for iron a n d nickel c o m p r e s s i o n ( 2I ) (VARSHNI a n d
BLOORE, 1965). T h e force c o n s t a n t s are then given by
K„=Db2
[ 1 -b
(;•„—r,)] e x p [-b
(rfJ—re)]
K, = C Kp
w h e r e C is a r b i t r a r l y taken
10
2
and
10
[7]
[81
3
.
T h i s choice will a p p e a r irrelevant for o u r aims as it will be e v i d e n t
f r o m the f o l l o w i n g . If w e c o n s i d e r in detail e q u a t i o n
[ 2 ] , w e see that
all the terms can be i m m e d i a t e l y c a l c u l a t e d . In p a r t i c u l a r w e note that
the term V (dEvm/Vfr
can be expressed by
oo
V (9 EVIB/dV)r=-3
N kT
X ( - 1 ) " [B2„/(2n)
1] (dft*,/d V) (h/kT)2"
[9]
E.
208
BOSCIII
-
F.
MULAUGIA
w h i c h is easily c o m p u t e d by
V 0 . K p / d V ) = D b2 e x p [ - 6 ( r „ - r . ) ]
[ - ( 2 / 3 ) bm + b2 (r„— r « ) / 3 ]
V 0 K , 2 / 8 V ) = ( 2 / 3 ) D2 b5 r,-/ [ - 2 - f c 2 ( r „ - r e ) 2 + 6 r „ - 6 r , / 2 +
+ 2 6 ( r , - , - r , ) ] e x p [ - 2 6 (r,/— r e ) ] .
Using the
gamma
above
equations
we
determine
the
for various compressions. The volume
harmonic
dependence
taken into a c c o u n t by c h a n g i n g the v a l u e of r„ in the
potential e q u a t i o n .
equation
[8|
Results are s h o w n
in Fig.
Gruneisen's
is
directly
intermolecular
1. T h e values of C of
t u r n o u t to be i m m a t e r i a l f o r the values of
yH.
T h e a n h a r m o n i c p a r t of the h a m i l t o n i a n f u n c t i o n is then
written
in the f r a m e w o r k of the self-consistent p h o n o n theory ( ,6 ) as
t
4 = + .088
1
!06
2« 106
3«tO6
4»IB6
P bars
Fig. I — T h e G r u n e i s e n ' s h a r m o n i c g a m m a a n d the total g a m m a f u n c t i o n a r e
p l o t t e d versus p r e s s u r e . T h e total g a m m a is t a k e n along an isotherm at 5800° K.
Ha=
12 k2 h2 ZKK-
4>K. -K.K-.~K- (cjk OJK-)1-24
1(QK.K-.K-)2 + (3/2)
ÎK.-JC.-JC"] (OJK
k2 h-3
CJK- CJK••)-'
Zkk-K[10]
GRUNEISEN' S GAMMA
FUNCTION
FOR
LIQUID
IRON
209
ETC.
w h e r e &k.k\... is the total lattice p o t e n t i a l i n t e n d e d as a p a i r
and
G, K', ...
accounting
for
the
summation
pairs
(").
potential
cjk a r e
the
e i g e n v a l u e s of the d y n a m i c a l m a t r i c e s .
Since w e are t r e a t i n g a m e t a l w e m u s t also a c c o u n t f o r the e l e c t r o n i c
c o n t r i b u t i o n s to the f r e e e n e r g y of the system. Let us w r i t e t h e m as
•I
[11]
Hel = rT2/2
where r
is the e l e c t r o n i c specific h e a t d i v i d e d by T.
We
incorporate
all the e l e c t r o n i c effects in t h e a n h a r m o n i c h a m i l t o n i a n f u n c t i o n .
electronic
contributions
for
liquid
iron
can
be
evaluated
The
from
the
d e n s i t y of the states relatively to the n o n - f e r r o m a g n e t i c p h a s e b e c a u s e of
t h e h i g h t e m p e r a t u r e s w e are c o n s i d e r i n g . U p to n o w , n o e x p e r i m e n t a l
d a t a a r e a v a i l a b l e f o r these q u a n t i t i e s , b u t t h e r e is physical
evidence
t h a t liquid iron d e n s i t y of the states v a l u e s s h o u l d slightly d i f f e r f r o m
the solid
iron o n e s
(')• F u r t h e r m o r e ,
a physically
reliable
theoretical
a p p r o a c h , b a s e d o n the c l u s t e r m o d e l ( 10 ), s h o w s t h a t e l e c t r o n i c
f o r liquid iron d i f f e r f o r n o m o r e t h a n 5 %
solid state. W e shall use v a l u e s c o m p u t e d
f r o m those relative to the
f o r solid
non-ferromagnetic
i r o n w h i c h h a v e been tested w i t h good a c c o r d a n c e w i t h
data
( 23 ). T h e
effects of
pressure
on
this
bands
term
were
experimental
computed
HENRY ( 9 ) by u s i n g a F e r m i - T h o m a s a t o m m o d e l , w i t h the r e s u l t
by
that
e l e c t r o n i c specific h e a t d e c r e a s e s by a f a c t o r s m a l l e r t h a n 3 f o r p r e s s u r e s
of 10 2 M e g a b a r s . T h e r e f o r e w e shall s u p p o s e t h a t the e l e c t r o n i c contrib u t i o n is not affected by the p r e s s u r e s w e c o n s i d e r .
In a n y case
the
e r r o r is m u c h less t h a n the u n c e r t a i n t y in t h e a n h a r m o n i c t e r m . T h e n
the G r u n e i s e n ' s g a m m a f u n c t i o n is e v a l u a t e d by the
equation
Y = YH+Y'i
[!2]
w h i c h leads to
Y = y„{
1 + (1 / 1 2 ) [{hojJkT>2-
Yn~x ((huK/kT)2)]
X { ( r ~ 2 A2)-2
} - (3NnkT~ T
Yh~' V [ d ( r - 2 A)/dV]
w h e r e the a v e r a g e s are p e r f o r m e d o v e r the n o r m a l
}
X
[13]
m o d e s , A2
a n h a r m o n i c l e a d i n g t e r m of t h e f r e e energy ( " ) , a n d n is the
of p a r t i c l e s in a cell. E q u a t i o n
1
is the
number
[ 1 3 ] , in the high t e m p e r a t u r e c o n d i t i o n s
of t h e E a r t h ' s c o r e , simplifies to
r
=
Yll-(3NknT-
> l
y
{ ( r - 2 A2)-2
yh-1
V
[ d ( f - 2 A2)/dV]
}
[14]
14
210
E.
ISOSCHI
-
F.
MULARGIA
Concluding, by m e a n s of the above e q u a t i o n s , w e c o m p u t e d the
G e u n e i s e n ' s g a m m a f u n c t i o n f o r liquid iron versus p r e s s u r e along
an isotherm at 5800° K, t e m p e r a t u r e that most recent theories ( 7 ), q u o t e
as the inner core outer core b o u n d a r y t e m p e r a t u r e in the a s s u m p t i o n of
a p u r e iron core. T h e c o r r s e p o n d i n g results are plotted in Fig. 1. W e
c o m p u t e d also the isobars at 1.4 a n d 3.1 M e g a b a r s , i. e. at the pressures
of the mantle-core a n d the i n n e r core-outer core b o u n d a t i e s . T h e uncertainty on the a n h a r m o n i c g a m m a leads to an average u n c e r t a i n t y
on the values of the g a m m a f u n c t i o n of 0.09. T h e results are plotted
in Fig. 2, w h e r e the zone c o r r e s p o n d i n g to the o u t e r core is s h a d e d .
G r u n e i s e n ' s g a m m a ranges f r o m 1.0, at the mantle-core b o u n d a r y , to
0.5, at the inner core-outer core b o u n d a r y .
BOHELER,
RAMAKRISHNAN,
and
KENNEDY
(3)
have
recently
mea-
sured the G r u n e i s e n ' s g a m m a at high pressures f o r iron and o u r results
are in agreement with theirs. T h e r e f o r e the values of the G r u n e i s s e n ' s
g a m m a generally assumed f o r iron at core conditions, 1.7, are definitely
y
2.0-
1.5-
r
P = 0 Mbar
1.0-
L
T
P ' = 1.38 Mbar
0.5-
r
±
P = 3.10 Mbar
t
1500
3000
4500
Fig. 2 — T h e total G r u n e i s e n ' s g a m m a f u n c t i o n is p l o t t e d
along the isobars at 0, 1.4, a n d 3.1 M e g a b a r s .
6000
versus
T K°
temperature
GRUNE1 S E N ' S G A M M A
unacceptable.
Finally w e
wish
FUNCTION
FOR L I Q U I D
to e m p h a s i z e
211
IRON ETC.
the fact that,
for the
first
t i m e , t h e t e m p e r a t u r e e f f e c t s h a v e b e e n a c c o u n t e d f o r in a s e l f - c o n s i s t e n t
way.
ACKNOWLEDGMENTS.
We
w i s h to t h a n k
publication.
their kind
Thanks
are
due
also
to
V.
LELLI
and
G.
MOLINARI
to
for
assistance.
This work
Nazionale
R . BOEHLER f o r l e t t i n g u s u s e h i s d a t a p r i o r
has been performed
delle
with a contribution
from
Consiglio
Ricerche.
REFERENCES
(') BAER, Y., cl al., 1970. Band structure
Phys. Scripta, 1, p. 55.
of transition
metals
studied
by
ESCA.
( 2 ) BHATIA, A. B., a n d G . K. HORTON, 1955. Vibration
spectra and specific
o/ cubic metals, Applications
to silver. Phys. Rev., 98, p. 1715.
heats
(3)
gamma
BOEHLER, R . ,
for
J. RAMAKRISHNAN, a n d
metals
at high
pressures.
( 4 ) BRADELY, J. C., 1961. Vibrational
A n n . Phys., 15, 411.
G.
C . KENNEDY,
1977.
Gruneisen's
J. A p p l . Phys..
frequency
spectrum
of random
crystal
( 5 ) CHAN, T . , H . A SPETZLER, a n d M . D . MEYR, 1976. Equation-oj-state
for
p.
liquid
metalls
and
energetics
oj
the
Earth's
core.
lattices.
parameters
T e c t o n o p h y s . , 35,
271.
6
( ) DOMIS, C., et al., 1959. Vibrational
115, p. 18.
( 7 ) FAZIO, D . , F . MULARGIA, a n d
distribution
in the
Earth's
( 8 ) FAUER, T . E., 1972. Introduction
versity Press.
spectra
E . BOSCHI,
outer
core.
to theory
of disordered
1977. Melting
lattices.
of
iron
temperature
N u o v o C i m e n t o B. 40, p. 206.
of liquid
metals.
( 9 ) HENRY, J .F., 1962. Equation
of state and conduction
pressures.
J. G e o p h y s . Res., 67, p. 4843.
bands
( 10 ) KELLER, J., and R. [ONES, 1973. The density
J. Phys. F, 45, p . 753.
Phys, Rev.,
of states of liquid
Cambridge
Uni-
of iron
high
transition
at
metals.
212
E. ISOSCHI -
( " ) KIRKWOOD, J. G., 1935. Statistical
Phys., 3, p. 300.
F.
MULARGIA
mechanich
of
fluid
C 2 ) KNOPOFF, L., and J. N. SHAPIRO, 1970. Pseudo-Cruneisen
Phys. Rev., B 1, p. 3893.
C 3 ) LEIBFRIED, G., and LUDWIG, W., 1961. Theory
crystals. Solid State Phys., 12, p. 275.
of
mixtures.
J.
Chem.
parameter
in
liquids.
anharmonic
( I4 ) LEIGHTON, R. B., 1948. The vibrational
spectrum and specific
centered-cubic
crystal. Rev. Mod. Phys., 20, p. 165.
0 5 ) LENNARD-JONES,
gases.
J.
E.,
and
A.
F.
DEVONSHIRE,
1937.
Critical
effects
heal
of a
in
face-
phenomena
in
Proe. Roy. Soc., 163, p. 58.
( 16 ) LUDWIG, W., 1958. The effects of the anharmonicity
crystals. ). Phis. Chem. Solids, 4, p. 283.
on
( " ) MADDIN, R., et al., 1958. Liquid metals and salification,
American
Society
for Metals. N e w Y o r k .
( 18 ) MONTROI.L, E. W., 1943. Frequency
and applications
to simple
( 19 ) STISHOV, S. M „ 1969. Melting
spectrum
cubic
of crystallic
lattices.
at high pressures.
F . J. BLOORE, 1965. Rydberg
for metals.
Proceedings
solids
of
of
general
the
theory
Sov. Phys. Usp. 11, p. 816.
( ) THIRRING, H., 1913. Zur tlieorie der rauingitterschuingeiger
warne fester koper. Phys. Z., 14, p. 867.
potential
properties
J. Chem. Phys. 11, p. 481.
20
( 2 1 ) VARSHNI, Y . P., a n d
the
function
und der
speziftschen
as an
interatomic
Phys. Rev., 129. p. 115.
i22) WALLACE, D. C., 1965. Thermal expansion
crystals. Phys. Rev., 139. p. 877.
( A ) WOOD, ]. H., 1962. Energy bands
Phys. Rev., 126, p. 517.
aiul other
anharmonic
in iron via augmented
plane
properties
waves
of
method.