Review of the electronic specific heat of B. C. C. and F.
C. C. solid solutions of first long period transition
elements
K.P. Gupta, C.H. Cheng, Paul A. Beck
To cite this version:
K.P. Gupta, C.H. Cheng, Paul A. Beck. Review of the electronic specific heat of B. C. C. and
F. C. C. solid solutions of first long period transition elements. J. Phys. Radium, 1962, 23
(10), pp.721-726. <10.1051/jphysrad:019620023010072100>. <jpa-00236669>
HAL Id: jpa-00236669
https://hal.archives-ouvertes.fr/jpa-00236669
Submitted on 1 Jan 1962
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LE JOURNAL DE
PHYSIQUE
ET LE
TOME
RADIUM
23,
OCTOBRE
1962,
721.
REVIEW OF THE ELECTRONIC SPECIFIC HEAT OF B. C. C. AND F. C. C. SOLID SOLUTIONS
OF FIRST LONG PERIOD TRANSITION ELEMENTS
By
K. P.
GUPTA,
University
of
C. H. CHENG and PAUL A.
Illinois, Urbana, Illinois, U. S.
BECK,
A.
On passe en revue les résultats relatifs aux chaleurs spécifiques des solutions solides
Résumé.
cubiques centrées, et certaines données nouvelles sur les système V-Cr et Ti-V. Les chaleurs spécifiques sont remarquablement bien décrites par un modèle de bandes rigides, même pour les mélanges’
2014
ternaires.
En ce qui
concerne les solutions solides cubiques à faces centrées, les valeurs de 03B3 mesurées ont
interpretation électronique simple seulement pour les alliages ferromagnétiques ou non magnétiques. Pour les alliages où coexistent des interactions ferro- et antiferromagnétiques, la valeur
expérimentale de 03B3 parait comprendre souvent une contribution magnétique. En se limitant aux
alliages non magnétiques ou ferromagnétiques on peut déterminer la forme de la bande d pour la
structure cubique à faces centrées et pour des concentrations allant de 8 à 10 électrons d par atome.
une
A review will be given of the electronic specific heat vs. electron concentration
Abstract.
for b. c. c. solid solutions, including new data for the systems V-Cr and Ti-V. The latter alloys
show a maximum of 03B3 at approximately 60 % V. The electronic specific heat coefficients correlate
well will the superconductive transition temperatures, according to the Bardeen-Cooper-Schriffer
theory, with an interaction coefficient V independent of composition. It was found that the
CsCl-type ordered alloys in the ternary Ti Fe-TiCo and TiCo-TiNi systems possess electronic specific
heat values paralleling those for the b. c. c. Cr-Fe alloys at the same average electron concentrations. The electronic specific heat values for the b. c. c. alloys of transition elements with one
another can be apparently described to a surprising extent in terms of a more or less rigid band,
the degree of filling up of this band with electrons being determined essentially by the average
electron concentration for the alloy. This appears to hold even in cases where the electron concentration is determined by averaging over three different atomic species.
As found also for the
alloy VFe, with a CsCl-type ordered structure, ordering causes little, if any, change in the electronic specific heat values of these alloys.
New low temperature specific heat measurements with f. c. c. solid solutions in the Mn-Fe,
Mn-Ni, Fe-Ni, V-Ni systems gave coefficients for the specific heat term linear in temperature,
which are in some cases quite inconsistent with each other if interpreted as electronic specific
heat coefficients. Consistent 03B3 values were obtained for simple ferromagnetic alloys and for non
magnetic alloys. For alloys where the simultaneous presence of both ferromagnetic and antiferromagnetic interactions is known from magnetic measurements, or may be considered as very likely,
the measured 03B3 values are anomalously high. It has been proposed by Marshall (1) that a temperature-linear contribution to the low temperature specific heat can arise in instances where a
sufficient number of spins are located in a near zero magnetic field. It appears from our results
that this condition is often fulfilled in alloys having both ferromagnetic and antiferromagnetic
interactions. As a result, in such cases the 03B3 value measured at low temperatures comprises, in
addition to the true electronic specific heat coefficient, also a magnetic component. Using only 03B3
values obtained for simple ferromagnetic and for nonmagnetic alloys, the shape of the d-band
for the f. c. c. solid solutions has been nevertheless successfully determined in the electron concentration range from approximately 8 to 10.
2014
In a previous paper [1] electronic specific heat
electron concentration data were published for
b. c. c. solid solutions of first long period transition elements. Measurements made recently [2]
with b. c. c. solid solutions in the Ti-V System
confirm the peak in the electronic specific heat
predicted [1] for these alloys (2). Figure 1 shows
vs.
(1) MARSHALL (W.), Specific heat
of dilute
alloys. Phys.
.Rev., 1960,118,1519.
(2) It was noted [2] that the electronic specific heat
coefficients of these alloys correlate well with their superconductive transition temperatures [2, 3] in accordance
with the B. C. S. theory, and with a value of the interaction
coefficient V independent of composition. This situation
is similar to that found by Goodman et al for the b. c. c.
U-Mo alloys [4].
the combined electronic specific heat data for the
b. c. c. alloys in the entire electron concentration
range where solid solutions with this structure are
formed by the transition elements of the first long
period.
high peak of y corresponds roughly
ferromagnetism in the Cr-Fe alloys ;
the question arises as to whether this peak may be
of magnetic origin. The high temperature specific
heat data of Schrôder [5] clearly indicate (fig. 2),
that the large peak in the term of the low temperature specific heat linear in temperature, which
occurs at around Cr + 19 % Fe is not of magnetic
origin, but that it is indeed an electronic specific
The second
to the onset of
heat coefficient.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphysrad:019620023010072100
722
FIG. 1.
-
Y vs. average electron concentration for b.
c. c.
Ti-V, V-Cr, Cr-Fe and Fe-Co alloys. Filled squares
represent data for close packed structures [2].
average electron concentration for ternary
CsCI type ordered structure. 0 represtands for alloy
sents TiFe-TiCo alloys.
FIG. 3.
-
y
vs.
Ti-alloys with
Dashed line
gives
y values for
b.. c.
c.
Cr-Fe
alloys [1].
nearly the same as that for the disordered
alloy of the same composition.
The density of states vs. energy, calculated from
the data given in figure 1 according to the method
of Hoare [9], is given in figure 4, where the density
to be
FIG. 2.
- Specific
heat
at various
.
composition of
temperatures [5].
vs.
Cr-Fe
alloys
It was recently found [6] that the CsCI-type
ordered alloys in the ternary TiFe-TiCo and TiCoTiNi systems have electronic specific heat values
paralleling very closely those for the b. c. c. Cr-Fe
alloys at the same average electron concentrations
(fig. 3). These , results represent further confirmation of the electronic specific heat nature of the
measured y values. They also underline the surprising extent to which the electronic specific heat
of the body centered cubic alloys of transition
elements with one another can be interpreted in
terms of a more or less rigid d-band, the degree of
filling up of this band with electrons being determined by the average electron concentration of the
alloy. This is now shown for ternary alloys, where
the electron concentration is determined by averaging over three different atomic species’. Ordering apparently makes little, if any, différence in
the electronic specific heat. This is not very surprising in view of the work of Rayne [7] with AuCU3
Also, newer results for VFe with a CsCI-type ordered structure [8] show the electronic specific heat
FIG. 4.
Density of states for each spin orientation per eV
Zero of energy
vs. energy in eV for the d-electrons.
scale corresponds to Cr.
-
72
of states for the two spin directions is plotted separately. The interpretation presented in the figure
makes use of magnetic saturation moment data for
these alloys as shown in the Slater-Pauling
curve [7], and of neutron diffraction data [12].
The combination of these data with the electronic
specific heat results clearly indicates that the first
sub-band, lying in the electron concentration range
between 4 and 6 ( fig. 1) or in the energy range
between -1.2 and 0 eV (fig. 4), represents an
equal number of electrons with positive and with
negative spin direction, since the alloys in this
range have no net permanent moment. The
second sub-band, which extends over the electron
concentration range from about 5.8 to 8.4 ( fig.1)
or in the energy range from - 0. 4 to + 1. 35 eV
(fig. 4), clearly represents electrons in one spin
orientation only. The atomic moments, as measured by saturation magnetization or by neutron
diffraction, correspond to the number of electrons
added to this sub-band. The fractional saturation
moment of Fe (2.2 yB) can now be understood as a
result of the fact that the second sub-band begins
at an electron concentration of approximately 5.8,
and that all the electrons added to this sub-band
contribue to the moment.
Although the above model is consistent with the
known expérimental facts, it certainly does not fit
the usual concept of the structure of the d-band
of Fe. In the usual picture, Fe is considered to
have both + and - spin orientations at the Fermi
surface [13] while the above discussed results
suggest [1] that in Fe there should be d-electrons
at the Fermi surface with one spin orientation only.
Recent measurements by Walmsley [14] of the
change in the contact potential between Fe and Cu
on applying an external magnetic field gave a direct
confirmation of this concept. Walmsley also established that the sign of spin of the d-electrons at
the Fermi surface in Fe is positive.
Finally, the third sub-band, extending from
approximately 8.2 electrons per atom to the end
of the range of stability of the b. c. c. structure at
around 8.7 electrons per atom ( fcg. 1) or from
+ 1.06 to + 1. 64 eV (fig. 4), represents electrons
with the opposite spin direction only, as shown by
the decreasing saturation moment with increasing
number of electrons per atom (Slater-Pauling
curve).
separation of the positive (" second ") and
negative (" third ") sub-bands here is approximately 1. 5 eV. This value is much larger than the
exchange energy associated with the Curie temperature. Since Tc 1000 oK for all of the alloys
concerned, the latter exchange energy
The
The observed separation, which is at least an order
of magnitude larger, must correspond instead to
the much larger intra-atomic exchange energy
which gives rise to (unaligned) atomic moments
well above the Curie température.
It is interesting to compare the electronic spécifie
heat results with magnetic susceptibility data for
some of the same alloy systems, published recently
by Taniguchi, Tebble and Williams [15]. Comparison of figure 5 with figure 1 shows the expected
FIG. 5.
Magnetic
tration for b. c. c.
elements [15].
-
susceptibility vs. electron concenalloys of first long period transition
at least in the electron concentration
5 and 6.5. (For the V-Cr alloys
between
range
this was already found earlier by Childs, Gardner
and Penfold [16].) However, the X and the y vs.
composition curves differ from each other very
significantly for the Ti-V alloys. The magnetic
susceptibility of Ti-Nb alloys also varies in a
manner similar to that of the Ti-V alloys in the
same electron concéntration range, as shown by
Jones, Pessall and McQuillan [17]. The reality of
the différence between the variation of x and of y
can not be doubted, but its explanation is, at
parallelism
present, unknown.
Low température specific
heat measurements
solid solutions in the
Mn-Fe, Mn-Ni, Fe-Ni, and V-Ni systems [20].
The resulting coefficients of the low temperature
specific heat term linear in temperature, if interpreted as electronic spécifie heat coefficients, are in
several cases quite inconsistent with the rigid band
model. As seen in figure 6, the y values obtained
by Walling and Bunn [18] for f. c. c. Co-Ni solid
solutions, which are strongly ferromagnetic, are in
reasonably good agreement with the values obtained by Keesom and Kurrelmeyer [19] and in this
laboratory [20] for f. c. c. Fe-Ni alloys in the same
electron concentration range. These Fe-Ni alloys
The value of y
are also strongly ferromagnetic.
for ordered MnNi3, as reported by Goldman [21],
have been made with f.
c. c.
724
is also fairly close to that of the Co-Ni solid solution
same eleetron concentration (fig. 6).
It is
known that ordered MnNi3 is also ferromagnetic,
with a Curie temperature of 475°C [22].
at the
FIG. 6.
y vs. electron concentration for f. c. c. alloys
in the Mn-Fe, Mn-Ni, Fe-Ni, Co-Ni and V-Ni systems.
For MnN’3 y is given fôr the following conditions : disordered (symbol ), short-range ordered (symbol e-), and
long-range ordered (symbol -C). For Fe,N’3 y was
determined after cooling in a magnetic field (symbol
as well as without magnetic field (symbol U) [20].
-
0),
However, the y value reported by Goldman [21]
for disordered MnNi3, is very much higher. This
result was recently confirmed [20]. (See the two
nearly coinciding c symbols in figure 6). Dreesen
concluded from his study of the effect of order on
the Hall constant of MnNi3 [23] that the observed
différence in the y values between ordered and
disordered MnN’3 is not a result of a change in the
band structure upon ordering. This conclusion is
supported by all results obtained to date on the
effect of ordering on the electronic specific heat of
alloys having no magnetic moments [7, 8]. This
raises the question why the coefficient of the low
temperature specific heat term linear in temperature increases upon ordering in the case of MnNi3.
Overhauser first proposed a theory [24] for a low
temperature specific heat term of magnetic origin,
which is linear in temperature. This theory postulates that a large number of spins are located in
a near zero field as a result of the presence of
stationary spin waves in an antiferromagnetic
In view of the fact that disordered
structure.
MnNi3 is f erromagnetic [22], it appears that Overhauser’s theory could not be applied to it. On
the other hand,, Marshall’s theory [25]. wich is
also based on spins located in nerr zero field, does
not require antiferromagnetic structure. It appears
very probable that the increment of the obser-
ved y value for disordered MnN’3 over that
of ordered MnN’3 is a result of the mechanism
described by Marshall [25] and that the condition of spins located in near zero average field
is satisfied in the disordered MnN’3 alloy as a
result of its peculiar magnetic structure. It is
known from the work of Kouvel and Graham [26]
that this alloy shows " exchange anisotropy " when
cooled through the magnetic transformation tempe
rature range in a magnetic field. The observed
unusual magnetic effects result from the presence
in this material with an overall ferromagnetic
structure of locally antiferromagnetic spin arrangements. It has been suggested by Kouvel [27]
that in the Cu-Mn and Ag-Mn alloys, which also
show " exchange anisotropy ", the simultaneous
presence of ferromagnetic and antiferromagnetic
interactions on an atomic scale is a result of the
statistical variations in the separation of pairs of
near Mn atoms, and of the strong dependence of
the type of exchange coupling between Mn pairs on
the distance of separation. It is extremely likely
that the same mechanism accounts also for the
exchange anisotropy observed in disordered MnNi3.
Consequently, it is reasonable to assume that many
atomic moments in this alloy are located in a near
zero average local field resulting from the near compensation in these local areas of the ferromagnetic and antiferromagnetic exchange interactions. It may be expected, therefore, that the
condition required by Marshall’s theory is satisfied.
In order to test the above hypothesis, an MnNi3
alloy specimen, which had been disordered by
annealing at 1 000 OC and preserved in this condition by quenching, was re-annealed for two hours
at 485 OC, in order to achieve short range ordering
and thus to decrease the number of nearest neighbor
Mn pairs, which presumably give rise to local antiferromagnetic coupling. This heat treatment was
selected on the basis of results reported by Marcinkowski and Brown [22]. As seen in figure 6, the
measured y value after this heat treatment
(symbol e-) was considerably lower than that in the
disordered condition. In the case of disordered
MnNi. the magnetic component of y approximately
doubles the appearent " electronic specific heat ".
However, the magnitude of this extra specific heat
decreases rapidly with increasing Mn-content.
It may be seen in figure 6 that the curve corresponding to Mn-Fe alloys is crossed by the corresponding curve for the Ni-rich Fe-Ni alloys. The’
measured y values for the latter alloys at high Fecontents become extremely large. The question
should be considered next whether or not these
very high y values may also comprise large magnetic contributions. The magnetic properties of
the Fe-rich f. c. c. Fe-Ni alloys have been investigated by Kondorskii and Fedotov [28] who found
that, superimposed on the overall ferromagnetic
725
alloys we may be dealing with long range
antiferromagnetism with a superimposed " latent "
or short range ferromagnetism.
Again, it may be
structure, a latent antiferromagnetism must be
assumed in order to account for the measured
dered
macroscopic magnetic properties. Consequently,
it may be assumed that, for these disordered alloys, assumed that the average local field in which many
atomic moments are located is near zero, and that
as well as for the disordered f. c. c. Mn-Ni alloys,
the condition required by Marshall’s theory may be the extra large y values observed in the Fe-rich
satisfied in the manner here proposed. In order alloys have a substantial component due to the
effect described above. It should be pointed out
to test this model, the alloy containing 30 % Ni
(and also 5 atomic per cent C to stabilize the f. c. c. however that, in this case, a magnetic contribution
structure) was cooled in a magnetic field of approxi- to y could be also accounted for on the basis of
mately 10 kG from 400 °C to room temperature. the mechanism proposed by Overhauser, since the
After this heat treatment the specimen was again f. c. c. Mn-Fe alloys apparently do have a long
introduced into the calorimeter and its low tempe- range antiferromagnetic structure.
As a further test of the assumption that in
rature specific heat was measured. As seen in
figure 6 (symbol c) the heat treatment increased several of the f. c. c. solid solutions the measured y
the measured y value. The observed effect of ma- values have a large magnetic component, a series
gnetic cooling suggests strongly that this alloy also of non-magnetic V-Ni solid solution alloys were
exhibits exchange anisotropy, and that the mea- also measured. As seen in figure 6, the measured y
sured y value comprises a magnetic contribution. values for the non-magnetic alloys represent a reaThe crossing of the curves for Mn-Ni and Fe-Ni sonable extension of the curve for the " purely "
alloys in figure 6 thus turns out to be of no parti- ferromagnetic Co-Ni and Fe-Ni alloys (no " latent
cular-significance, since there is no reason to expect antiferromagnetism "). The magnetic contribution to Y discussed above, which occurs in
that thé peculiar spin arrangement giving rise to
the large magnetic contribution to y should occur alloys having both ferromagnetic and antiferroin différent alloy systems at the same average elec- magnetic interactions, is absent for both of these
cases.
tron concentration.
Consequently, it seems justifiable to proA similar lack of consistency may be observed pose that the true electronic specific heat of the
in figure 6 between the y values for Mn-rich Mn-Ni f. c. c. alloys in the electron concentration range
alloys and Fe-rich Mn-Fe alloys. The fact that y between 8 and 10 may be represented approxiis pnusually large for a f. c. c. Fe +11 % Mn alloy mately by the heavy line in figure 6. The points
has been reported earlier by Goldman [29], and for the two Mn-Ni alloys near the equiatomic comthe much lower y value for Fe + 45 % Mn was also position are also near this curve.
known [30]. Corliss and Hastings found [31] that
Acknowledgements. - This review is largely
f. c. c. Mn-Fe alloys possess long range antiferro- based on experimental work performed during the
magnetism, and Kimball, Gerber and Arrott last six years with the following colleagues and
found [32] that the observed magnetic moments graduate students : C. T. Wei, K. Shroeder, E. A.
Starke Jr. and E. C, van Reuth. The work is
must be largely associated with the Mn-atoms.
It seems very probable that, here too, the exchange supported by a grant from the National Science
interaction between Mn-atoms with a somewhat Foundation. In various earlier phases it was
larger distance of separation than that of nearest supported by the U. S. Air Force and by the U. S.
neighbors is ferromagnetic. Thus, in these disor- Army Research Office (Durham).
.
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BECK (Paul A.), Phys. Rev. , 1962, 126, 2030.
[3] HULM (J. K.) and BLAUGHER (R. D.), Phys. Rev., 1961,
128, 1569.
[4] GOODMAN (B. B.), HILLAIRET (J.), VEYSSIE (J. J.) and
WEIL (L.), C. R. Acad. Sc.,1960, 250, 542.
[5] SCHRÖDER (K.), Phys. Rev., 1962, 125, 1209.
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published.
[9] HOARE (F. E.), MATTHEWS (J. C.) and WALLING (J. C.),
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RADIUM
TOME
23,
OCTOBRE
1962,
FERROMAGNETIC PROPERTIES OF Fe AND Ni IN RELATION TO THEIR BAND STRUCTURE
By R. GERSDORF,
Natuurkundig Laboratorium
der Universiteit
van
Amsterdam.
Résumé. 2014 De nombreuses propriétés de Fe et Ni peuvent être reliées les unes aux autres au
moyen d’une théorie empirique supposant que le moment atomique magnétique n’est pas constant
dans les métaux de transition mais est supposé dépendre dans ces cas de la température, du
volume et du champ magnétique externe.
Les propriétés en question sont, parmi d’autres, la magnétostriction en volume dépendant du
champ, la variation de magnétisation et du point de Curie avec la pression et les déviations par
rapport aux courbes de Brillouin de la magnétisation à saturation en fonction de la température.
Les variations du moment atomique magnétique sont estimées en incorporant les moments
magnétiques localisés dans la théorie des bandes pour les électrons 3d.
Several properties of the elements Fe and Ni can be correlated with each other
of an empirical theory, which assumes that the atomic magnetic moment does not have
a constant value in transition metals, but is thought to depend in these cases on temperature,
volume, and external magnetic field.
The properties in question are, among others, the field-dependent volume magnetostriction,
the change of magnetization and Curie point with pressure, and the deviations of the temperature
dependence of the saturation magnetization from the appropriate Brillouin curves.
Variations of the atomic magnetic moment are estimated by incorporating localized magnetic
moments into the electron band theory of the 3d-electrons.
Abstract.
2014
by means
Some of the physical properties of iron metal rature, volume, etc. For a certain value of a,
display unexplained anomalies. We found [1] that the energy will be a minimum ; we assume a quathe field-dependent volume magnetostriction, h’O, dratic dependence of the energy upon a in the
does not disappear at low temperatures, but re- neighbourhood of this equilibrium value.
If we define a as the magnetic moment per atom
mains approximately constant. Moreover, the
temperature dependence of the saturation magne- expressed in Bohr magnetons, and m as a relative
tization deviates more than 10 % from the theore- change in volume of the lâttice, then the terms of
tically expected Brillouin curve [2], while the value the free energy per atom, F, which depend on a
for the exchange energy as found from spin wave or w, can be written as :
theory, differs considerably from the value, calculated from the experirnentally determined Curie point
[3]. In other ferromagnetics these properties give
a better fit with theory.
These difficulties can be removed by considering
the localized atomic magnetic moment, 6, of the
The eff ects of lattice vibrations, causing the
atoms in a magnetic metal as a variable. This is
in contradistinction to the moments in magnetic thermal expansion are omitted here. The first
salts, which have fixed values that have been calcu- term gives the quadratic dependence of F on a,
lated. We assume that a is a function of tempe- which we have already mentioned ; the second