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Electron heat capacity in transition metals and alloys. II

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Electron heat capacity in transition metals and alloys. I1
By T.1. K a k u s h a d z e
With 3 figures
Abstract
Group isoenergetic transitions give rise to a layer in one of the overlapping
outer bands of elements or alloys. This layer takes up additional energy,
which accounts for ,,anomalously" large electron heat capacity observed in
transition metals and alloys.
As was indicated in r e f ~ l - ~the
) group transitions occur isoenergetically,
i. e., the interband transition energy of one electron is compensated by the
change in the energy of the other electrons of the group. The isoenergetic
character of the group transition ensures the
conservation of the thermal energy of electron
gas, and thersfore the electron specific heat of
the s a n d d-bands can be evaluated under hhe
conventional methods of Sommerf e l d s theory
of metals (without taking account of the influence of the electron layer in one of the
overlapping outer bands5)).
The pattern is quite different if the effect
of
the
electron layer is taken into account. A
d a
substantial portion of electron heat in this case
d&
is due to group transitions. This portion goes into
Fig. 1
the formation of a layer in one of the overlapping outer bands (fig. 1).Fig. 1 gives a schematic representation of the
overlapping 4s- and 3d-bands of nickel.
Prom the ten electrons, the 4s-band contains 0.6 and the 3d-band 9.4
electron per atom a t 0 OK. The group electron umklapprozesse into the s-band
and back into the d-band lead to the formation of a layer in one of the overlapping bands 6 ) .
T. I. K a k u s c h a d s e , Ann. Physik 7, 3, 352 (1959).
T. I. K a k u s h a d z e , The Role of Group Tra.nsit.ions in the Production of Certain
Satellites. I.
N. F. B l o c h , J. Physique Radoum 4, 486 (1933).
4) W. V. H o u s t o n , Physic. Rev. 65, 1255 (1939).
5 , N. S. A k u l o v , and T. I. K a k u s h a d z e , DAN SSSR 77, 593 (1961).
6 ) For the F e group elements the sojourn of an electron in the layer
10-l' secl).
2)
N
Exited from the d or-s-band into the vacant portion of the s-band, the electron encounters
elastic collision with each head-on ion, i. e., its life is of the order of
sec. Therefore,
the electron in question should descend t o the F e r m i surface with the formation of a
layer before passing back t o the d-band. (Fig. 1).
T . I . Kakushadze: Electron heat cupacity in transition metals and alloys
361
I t is essential that the group umklapprozesse are not confined to the
transition rules, because of a strong electronlattice interaction.
The present paper deals with the electron specific heat in transition metals
and alloys with taking account of group transitions. Below the D e b y e temperature the heat capacity of solids due to the lattice oscillations is known t o
be proportional to T3, whereas the specific heat of electron gas changes linearly
with temperature. At temperatures N 10 OK these heat capacities prove to be
of the same order, while a t lower temperatures the heat capacity of a conductor
is almost entirely due to the electrons, and consequently is proportional to the
temperature.
According to ref.2) the number of electrons in a layer is equal to
An
=
As
= a Ro(no,
nd - nodn,)
T.
Here __
d N ( e )i s the electron density in the layer a t the energy
dF
(1)
E
dsbeing an
(da(&)
approximately constant quantity in the interval AE , AE is the layer energy
)
height, approximately proportional to temperature, a is a constant, R, is
the electric resistance of a metal or alloy a t 0 "CZ) and T is the absolute temperature.
According to ref. l), for the transition energy we obtain the expression
nodnJ2 T2
and for the heat capacity due to the re-settlement of the electrons from one
band to another we have the relation
The validity of the propositions leading t o eq. (3) can be checked as follows : the
coefficient L determined from the electron heat capacity of one substance
must prove suitable for determining the heat capacities C,, of other solids as
well.
Hence the electron heat capacity of transition metals must incorporate :
1. The heat capacity of C,-electrons of the conductivity band. According
t o S o m m e r f e l d , we have
c,
3.26
.lo-5
v2/3T = ys T
f 4)
where n, is the number of s-electrons per atom a t 0 OK and V is the volume
of a gram-atom of the metal. The magnitudes ys evaluated according to (4)
are listed in the table (column 10).
2. The heat capacity of the Cd-electrons of the d-band. Bearing in mind
that the total number of electron places in the d-band is equal to 10 and that
the energy width of the d-band amounts to about a half of the width of the
s-band'), we have for paramagnetic metals
Cd N 19.06 .
7)
V 2 / 3nz1/3
N. M. K r u t t e r , Physic. Rev. 48, 664 (1935).
T = yd T
(5)
Pt
I
9.1
7.4
7.6
7.03
6.7
6.7
6.7
6.8
7.0
7.32
9.28
10.9
9.12
I
5
10
8
9
10
10.2
10.4
10.6
5
10
5
i
1
0.6
0.9
1.0
0.9
0.8
0.7
0.6
0.6
0.6
0.6
I
I
I
4.1
5.0
6.1
7.2
8.3
9.4
9.6
9.8
10,O
6.0
9.5
4.9
9.4
1
N. S. Akulov, DAN SSSR 106, 935 (1955).
'1
I
~
V
Cr
Mn
Fe
co
Ni
Cu&i,
Cu,Ni,
Cu,JVi0
Nb
Pd
Ta
1
4.1
5.0 I
3.9 I
2.8
1.7
0.6
0.4
0.2
0.0
5.0
0.5
4.9
0.6
13.6
17.0
10.0
17.7
8.8
12.8
13.2
0.8
0.0
3.2
6.4
9.8
12.8
1
19)
'8)
17)
16)
15)
14)
13)
I*)
11)
lo)
1
11.8
27.6
10.7
14.6
11.0
11.8**)
1
i
I
58.8
13.1
83.0
12.0
9.71
11.8
71.8**)
1.2
CUi
0.0
1.37
1.24
1.19
1.11
1.03
0,98
G. D u y c k a e r t s , Physics 6, 401 (1939).
W. H. K e e s o m and B. K u r r e l m e y e r , Physica 6,364, 633 (1939).
1%'.H. K e e s o m and C. W. K l a r k , Physica 2. 230 (1935).
E l s o n , G r a y s o n , S m i t h and W i l h e l m , Can. J. ReRearch 18, 82 (1940).
G. D u y c k a e r t s , Physica 6, 817 (1939).
W. H. K e e s o m and B. K u r r e l m e y e r , Physica ? , l o , 1003 (1940).
J. C. D a u n t and K. Mendelsson, Proc. Roy SOC. London A 160, 127 (]Mi).
G. L. Pickard, Nature 188, 129 (1936).
H. .Jones arid N. F. M o t t , Proc. Roy. Soc. London A 162, 49 (I!lJ7).
.J. A. K o k and W.H. Keesom, Physica 3, 1055 (19%).
9 C. J. K r i e s m a n and H. B. Callen, Physic. Rev. 94, 837 (1954).
8)
41
46
73
78
26
27
28
25
23
24
1
I
5.13
3.63
3.1
2.6
0.0
16.7
6.7
15.9
7.0
if::
6.26
13.13
,
1
I
I
1.3
0
39.6
3,5
5.3
13.2
14.2***)
15.0***)
16.8
44 3
21.2
9.6
11.4
~
3.8
15
42
18.3
18.6
16.8
60.0
28.6
25.6
19.6
1
I
,
1
i
1
15.8 I
15.2
16.2
60
'
31
27
16
i17.8
!: Fi17.4
15.7
13.6
52.2
19
19
16
17
15
15
15
12
13
10,11
14
PD
F
:T
b
10
Q,
w
T . I . Kakushadze: Electron heat capacity in transition metals and alloys
363
and for ferromagnetic metals
Experimental
values for
saturation
Elements
Co
Ni
1
T:;~:i;l
values a t
sufficiently high sufficiently high
temperatures , temperatures
~
I
I
Temperature of
observation
I
1
1.71
0.606
2.36
1.16
~
2.50
1.67
’
1
point
1450 “K
1173 OK
In eqs. (9) and (10) n: is either the number of d-electrons if the d-band
contains less than 5 electrons per atom or the number of holes if the band is
more than half filled (column 7 in the table). For iron both halves of the
d-band (for right and left spins) are partially filled (fig. a), and C d for Fe is
evaluated by the formula
Cd N
+ ni*1/3)T = yd T,
1 2 10” V 2 / 3(nz1I3
(7)
where nz and n:* are the numbers of vacancies in the halves of the d-band for
the right and left spins respectively; the quantities yd evaluated according
to (5), (6) and (7) are listed in the table (column 11).
f
It is clear from the table that the theoretical
values of electron heat capacity factors (column 10
and 11) deviate appreciably from experimental
data (column 4).
Taking account of heat capacity due t o the
re-settlement of electrons (as a result of group
transitions) changes radically the entire pattern of
this phenomenon.
dN(&)
The heat capacity C,, going into the formation
dE
of the layer in the s-band is given, according to (3),
Fig. 2
by the expression
C,,
N
L Rg(n,,, nd - nodn,)2 T
=Ysd
T‘
(8)
The use of experimental data for one metal yields the value
The quantities y s devaluated according to (8) are listed in the table (column 12).
In the same table (columns 13 and 14) theoretical data are compared with
experimental results. This comparison warrants the conclusion that the group
transitions make a substantial contribution to the electron heat capacity of
metals and alloys.
364
Anmlen der Physik. 7 . Folge. Band 8. 1961
In the light of the problem a t hand concerning the effect of group isoenergetic transitions on heat capacity, the electron heat capacities of the Cu-Ni
alloys are of especial interest. It is supposed that the degree of filling the energy
levels in the Cu-Ni alloy components (fig. 3) varies with changing concentra-
Fig. 3
tion (fig. 3). Therefore : since high electron heat capacities for the transition
cal
metals (say, for Ni, Gel r 174 . lo4 grad mol is entirely due, according to
S l a t e r and M o t t ,to high state densitiesin the upper levels of the filledportions
of the outer d-bands, it should be expected that the electron heat capacity
coefficient y for the Cu-Ni alloy must fall from the value y in pure nickel t o
its minimum value a t 60 per cent content of copper (the d-band being regarded
a s totally filled, fig. 3). As the concentration of copper rises still gihher, the
coefficient y should remain constant and close t o the quantity obtained for
cal
y e 1.2 - 10-4 grad2 mol
-)
--)
The experimental measurements12) diverge rather appreciably from the
values predicted by the S l a t e r and M o t t theory. Instead of a uniform decrease of y down to the magnitude corresponding to 60 per cent of copper content, just the opposite is observed in experiment : the maximum lies approximately a t 60 per cent of copper concentration (cloumn 14). This leads to the
conclusion that ascribing high electron heat capa.city to the d-electrons of metal
cannot be always justified; the d-electrons cannot always account for high
values of C,, . In all probability thermal encrgy which is left out in the theories
of S o m m e r f e l d and M o t t goes into the process (the formation of a layer).
For Cr, the observed electron heat capacity is less than the magnitude
evaluated according to eqs. (4)and (5) ( y s d= 0). Probably the energy widths
of the s- and d-bands of Cr are much larger than their theoretical values, or
else the experimental value for the electron heat capacity of Cr is somewhat
underestimated.
It should be noted that we had incomplete and rather fragmentary experimental material at our disposal (experimental data on the resistance and electron heat capacity were obtained with different samples). Yet the reader can
trace the role of group transitions t o the origin of “anomalously” large electron
heat capacities. I t is essential in this casc that the coeffizient L in eq. ( 3 2)
remain suitable for evaluating y s d of different substances. It should also be
noted that in the present paper all calculations are performed under the assumption of a perfectly free electron (the effective mass equals the free electron
mass). In the approximation as rough as used for evaluating ysd,agreement
between theoretical results and experimental data can be regarded as satisfactory.
To obtain more reliable results it is vital t o measure both electron heat
capacity and resistance as well as the electron distribution in outer bands using
t,he same sample.
T b i l i s s i , Georgian SSR, A. S. Pushkin Pedagogical Institute.
Bei der Redaktion eingegangen am 4.April 1961.
Ann. Phyeik. 7. Folge, Bd.8
26
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