From Elementary Excitations to Microstructures: the Thermodynamics of Metals and Alloys Across Length Scales - CaltechTHESIS

The y-cerium (fcc) has a well-defined crystal field excitation at 17 meV [8]. For a 4f
electron with fcc symmetry this corresponds to a transition from a doublet (17) to quartet (18)
[17]. The f)-cerium (dhcp) data did not show all of crystal field peaks because some were too
weak and broadened to separate from the quasielastic scattering. The details of this problem
are discussed in a separate report on the magnetic scattering in this data set [8]. The basic
problem seems to be a fairly strong hybridization of the f-states with the conduction band.
This makes accurate predictions of thermodynamic quantities from simple crystal field
models difficult without supporting measurements. Fortunately, calorimetry measurements
were made on f)-cerium at low temperatures by Koskimaki, et al. [18], Tsang, et al. [19], and

41
Gschneidner and Pecharsky [20], After subtracting the phonon part using Equation 5 we fit
the remaining heat capacity by assuming three contributions from the electronic degrees of
freedom; (1) crystal field, (2) spin fluctuations, and (3) the usual electronic excitations at the
Fermi energy. The phonon-subtracted specific heat is shown in Figure 3.8.

Y Phonon

-E
---

.0

(5

<5

Electronic

50

100

150

200

T (K)
Figure 3.8. Phonon-subtracted heat capacity of ~-cerium (0). Phonon contribution was
calculated from the ~-cerium phonon DOS shown in Fig. 2. The crystal field (Schottky)
contribution was calculated from the level scheme determined using the inelastic neutron
scattering spectra [8]. The spin fluctuation (Kondo) contribution was calculated with the
Coqblin-Schrieffer model [21]. The thick curve shows the sum of the crystal field, spin
fluctuation, and electronic contributions. The linear electronic contribution was adjusted such
that the sum matched the specific heat at high temperatures. Peak at around 10 K is due to an
antiferromagnetic transition. The heat capacity data are those from measurements of
Koskimaki et al. [18], Tsang et al. [19], and Gschneidner and Pecharsky [20].

42
The ~-cerium dhcp structure has an equal number of sites with cubic local symmetry
and hexagonal local symmetry. The cubic site has the same doublet (r7) to quartet (rg) 17
me V crystal field excitation as the fcc structure [8]. As discussed by McQueeney et al. [8], it
is possible to predict the hexagonal level scheme from the measured cubic levels by
following the assumptions of the Superposition Model [22]. Briefly, the relevant crystal field
scaling parameters depend only on the polar coordinates of the ligands. Because the only
difference between the local environments of the cubic and hexagonal sites is a rtf3 rotation
of the closed-packed plane above the site, the relevant scaling parameters are identical on
both sites. Based on the cubic splitting, the crystal field level scheme on the hexagonal sites
was I± 112> at 0 me V, I±512> at 1.9 me V and I±312> at 9 me V [8]. Thus, if we neglect the
effects of lifetime broadening, the mean crystal field energy can be determined using

[-E J

- -1- E ex __n
Z(T) n n P ksT '

(3.2)

Lexp(-EnJ,
ksT

(3.3)

CF -

where
Z(T) =

and n is summed over all levels (half on cubic sites and half on hexagonal sites). The crystal
field specific heat is then given by

The crystal field specific heat calculated from this level scheme is shown in Figure 3.8. It
should be noted that the lifetime broadening of the crystal field levels is significant [8]. A
justification for using the simple model of the sharp levels is that the specific heat is an

43
integral quantity, so the details of the broadening are smoothed out to a large extent. We do,
however, include the broadening in the ground state separately below.

The interaction of the localized 4f-electrons with the conduction electrons provides an
energy spread for the ground state doublet (I ±1I2> on the hexagonal sites and r7 on the cubic
sites). This can be seen as quasielastic spin fluctuations in the neutron inelastic magnetic
spectra [8]. The spread of these states contributes an additional term to the electronic specific
heat. The simplest way to treat this problem is with the Kondo impurity model as is often
done with heavy fermion systems [23]. The problem is in fact very similar, but with a much
higher Kondo temperature (TK - 40 K [22]) and a much weaker enhancement of the
electronic specific heat at low temperatures. Further support for this approach is that a
resistivity anomally in ~-cerium at around 50 K has been interpreted successfully in terms of
a quenched Kondo scattering mechanism [24]. Raj an calculated an exact expression for the
specific heat using the Coqblin-Schrieffer Model [23]. For the doublet ground states the
specific heat from Kondo spin fluctuations is given by [23]

C (T) = k foo &F(E)(EI2kBT)2 dE
SF
B COsh2(EI2kBT)

(3.5)

-00

where gs~ E) is the spin fluctuation density of states that modifies a standard result for a 2level system. We approximate the spin fluctuation density of states as a lorentzian with a
half-width determined from the neutron quasielastic width extrapolated to zero temperature,
- 4 me V [8]. The calculated specific heat for this contribution is labeled "spin fluctuation" in
Figure 3.8. We did not attempt to fit the specific heat at the lowest temperatures because of
the anti ferromagnetic transition at - 10 K.

44
If we assume temperatures well below the Fermi temperature, Tf , and that the energy

derivatives of the electronic DOS can be neglected, the electronic specific heat in the free
electron model can be expressed in terms of the electronic DOS at the Fermi level, CelT) =
g( Ef )( rr:/3 )k/T. Therefore, with these approximations just one adjustable parameter, the

electronic specific heat constant, r = g( Ef)(n 2/3)kB 2 ,was required to fit the remaining
electronic contribution to the specific heat data. The fit, shown in Figure 3.8, gives r =
(7.0±0.1) mJ/mol-K2. With this it is confirmed that temperatures are well below the Fermi
temperature since Tf = (rr:12)(kEl'r) = 6060 K. Since the narrow 4f-bands result in the largest
derivative, our second assumption depends mainly on the location of the 4f-bands with
respect to the Fermi level. According to Baer and Busch [25] the 4f-bands lie - 900 meV
below the Fermi energy in y-cerium and thus should not affect the derivatives. More recent
results suggest it may be in the 1 to 2 e V range [26]. In any case, if the next term in the
Sommerfeld expansion were large, an additional T3 term would need to be added to the
electronic specific heat. The data do not warrant such a correction.

Despite the simplicity of the models used, the sum of the various electronic
components fit the data surprisingly well above the antiferromagnetic transition as shown in
Figure 3.8. However, it should be pointed out that many assumptions were not quite correct.
In fact, it seems unlikely that crystal field picture is even correct in a strict sense since the

hybridization with conduction electrons is so strong [8]. A more accurate model would
include the hybridization of each crystal field state with the conduction electrons. On the
other hand, since the specific heat is adequately reproduced, the entropy associated with an
effective crystal field can be calculated accurately.

45

The calculated contributions to the specific heat were used to calculate the entropy
difference between y-cerium and rJ-cerium as a function of temperature using

(3.6)
where i indicates the entropy contribution (i = el, vib, and CE). Although the spin fluctuation
part was significant, it made no measurable contribution to the entropy difference since the
quasielastic scattering of y-cerium was nearly identical to rJ-cerium [8]. The crystal field and
vibrational entropy differences are compared in Figure 3.9. At the experimental transition
temperature (420 K), the crystal field contribution is negligible compared to the vibrational
contribution. The latent heat measured at -420 K implies an entropy change of only 0.05
kslatom [27], which is smaller than the vibrational entropy. Thus, by setting the sum of the
entropy differences equal to the latent heat, Figure 3.9 implies an electronic entropy
difference of t1S~-f3 = - (0.04 ± 0.05) ks/atom. This difference is similar to predictions at 420
K using the electronic specific heat constants of Koskimaki et al. [18] for low temperature
calorimetry measurements on y-cerium and rJ-cerium (-0.096 kslatom). However, Koskimaki
et al. [18] noted significant uncertainty in the electronic specific heat constant for rJ-cerium
and attributed it to the low temperature anti ferromagnetic transition (near 11 K in Figure 3.8).
Perhaps more significant was the fact that Koskimaki et al. neglected the spin fluctuation.
(This is not surprising, since at the time of this publication (1974) heavy fermions were
unknown and the calculations of Rajan [23] did not exist.) This latter point probably explains
why they found such an unusually large value for the electronic specific heat constant, y, at
low temperatures (-46 mJ/mol-K2) but could not reconcile it with the high temperature trend.

46
Based on the Kondo impurity model with a TK - 40 K, the low temperature limit of the
specific heat is of order C (T---70)IT = Y = 1.29nkJ6TK - 100 mJ/mol-K2 [23]. Thus, it is not
surprising that Koskimaki, et al. found a significantly enhanced electronic specific heat
constant at low temperatures.

0.2
Latent heat

Vibrational

0.1
..-

0.0

ctS
-co

-0.1

.....

-~

Electronic

c:o.

?-

C/)

-0.2

"- Crystal field

-0.3
-0.4
100

200

300

400

T (K)
Figure 3.9. Crystal field, vibrational and electronic contributions to the entropy difference
between y-cerium and j3-cerium. Thick curve shows the sum of the three components. The
electronic component was adjusted so that the sum equaled the value obtained from the latent
heat measurement of Gschneidner et aI., [27].

Because of this uncertainty in the electronic specific, the electronic specific heat
constant of j3-cerium (dhcp) was assumed to be equal to that of dhcp-Ianthanum [18]. The
present results make no such assumption, and thus imply independent estimates of the

47
electronic specific heat constants; (7.0±0.1) mJ/mol-K2 for B-cerium from the fit in Figure
3.8 and (6.2±0.8) mJ/mol-K2 for y-cerium accounting for the latent heat and other entropy
terms. Compared with the values used by Koskimaki et al. [18],9.4 mJ/mol-K2 for B-cerium
(which is actually just the lanthanum value) and 7.5 mJ/mol-K2 for y-cerium, our values are
both slightly smaller but their difference is similar.

The crystal field and electronic entropy tend to stabilize the y-phase with respect to
the B-phase at low temperatures. Although the crystal field entropy difference is negligible at
the measured transition temperature (420 K), it becomes important at the lower (true)
transformation temperature, 283 K, determined in a 20 year study by Gschneidner, et al. [27].
Of course in a 20 year study it is not possible to measure the latent heat and, thus, determine
the total entropy change. However, inspection of Figure 3.9 shows that the entropy difference
is essentially the same in magnitude as at 420 K but that the crystal field entropy assumes a
more significant role with respect to the electronic entropy (from continuous excitation of
electrons across the fermi energy.)

3.3.5 Summary
The relative contributions of the vibrational and electronic degrees of freedom to the
entropy of the B- and y-phases of cerium were determined. Many competing sources of
vibrational and electronic entropy need to be included in the equation of state of cerium.
Their different temperature dependencies change their relative importance. However, at the
experimentally observed transition temperature (-420 K) a vibrational entropy difference of
I).S:;13 = (0.09 ± 0.05) kB/atom is dominant followed by the electronic contribution I).S~-13 = -

48
(0.04 ± 0.05) kB/atom and a negligible crystal field contribution. The crystal field entropy
difference dominates at low temperatures and is comparable to the electronic contribution at
the true transition temperature, 283 K. A contribution from quasielastic fluctuations from
Kondo scattering was significant, but showed no difference between the two phases.

References
[1] O. Eriksson, J. M. Wills, and A. J. Boring, Phys. Rev. B 46, 12981 (1992).
[2] P. Soderlind, O. Eriksson, and B. Johansson, Phys. Rev. B 50, 7291 (1994).
[3] P. Soderlind, O. Eriksson, B. Johansson and J. M. Wills, Phys. Rev. B 52, 13169 (1995).
[4] A. Svane, Phys. Rev. Lett. 72, 1248 (1994).
[5] A. Svane, Phys. Rev. B 53,4275 (1996).
[6] T. Charpentier, G. Zerah, and N. Vast, Phys. Rev. B 54, 1427 (1996).
[7] J. L. Robertson, H. N. Frase, P. D. Bogdanoff, M. E. Manley, B. Fultz, and R. J.
McQueeney, Phil. Mag. Lett. 79, 297 (1999).
[8] R. J., McQueeney, M. E. Manley, B. Fultz, G. Kwei, R. Osborn, and P. Bogdanoff,

Dynamic magnetic susceptibility in y-cerium, ~-cerium, and low-density cerium alloys,
Phil. Mag. B., in press (2000).
[9] C. Stassis, T. Gould, O.D. McMasters, K. A. Gschneidner, Jr., and R. M. Nicklow, Phys.
Rev. B 19, 5746 (1979).
[10] G. Beggerow, In Landolt-Bomstein, New Series, Numerical Data and Functional

Relationships in Science and Technology, K. H. Hellwege (editor), Group 4, Vol. 4
(Berlin: Springer-Verlag) p. 9 (1980).

49
[11] I. Barhin and B. Patzki, Thermochemical Data of Pure Substances, Vol. 1 (Weinheim:
VCH), p. 507 (1995).
[12] P. Ravindran, L. Nordstrom, R. Ahuja, 1. M. Wills, B. Johansson, and O. Eriksson,
Phys. Rev. B 57,2091 (1998).
[13] AK. Singh and T.G. Ramesh, The Metallic and Nonmetallic States of Matter, edited by
Edwards, P. P. and Rao, C. N. R. (Taylor and Francis: London and Philadelphia), Ch.
13, p. 359 (1985).
[14] D. C. Koskimaki, K. A Gschneidner, and N. T. Jrand Panousis, J. of Crystal Growth 22,
225 (1974).
[15] N. Nticker, Proc. Int. Con! On Lattice Dynamics, Paris, Flammarion Sci, p. 244 (1977).
[16] F. F. Voronov, L. F. Vereshchagin, and V. A Goncharorva, SOy. Phys. Dokl. 135, 1280
(1960).
[17] K. R. Lea, M. J. M. Leask, and W. P. Wolf, J. Phys. Chern. Solids 23, 1381 (1962).
[18] D. C. Koskimaki and K. A Gschneidner, Jr., Phys. Rev. B 10,2055 (1974).
[19] T. W. E. Tsang, K. A. Gschneidner, Jr., D. C. Koskimaki, and J. O. Moorman, Phys.
Rev. B 14,4447 (1976).
[20] K. A Gschneidner, Jr., and V. K. Pecharsky, J. of Phase Equilibria 20,612 (1999).
[21] V. T. Rajan, Phys. Rev. Lett. 51, 308 (1983).
[22] D. J. Newman and B. Ng, Rep. Prog. Phys. 52, 699 (1989).
[23] A Yatskar, W. P. Beyermann, R. Movsovich, and P. C. Canfield, Phys. Rev. Lett. 77,
3637 (1996).
[24] S. J. Liu, P. Burgardt, K. A Gschneidner, and S. Legvold, J. Phys. F: Metal Phys. 6, L55
(1976).

50

[25] Y. Baer and G. Busch, J. Electron Spectrosc. Related Phenorn. 5, 611 (1974).
[26] J. Rohler, Handbook on the Physics and Chemistry of Rare Earths, Vol. 1, edited by K.
A. Gschneidner, Jr., L. Eyring, and S. Hufner (Elsevier Science, New York, 1987),

Vol. 10, p. 453.
[27] K. A. Gschneidner, Jr., V. K. Pecharsky, Jaephil Cho, and S. W. Martin, Scripta
Materialia 34,1717 (1996).

51

Chapter Four

Heat capacity and microstructure of ordered and
disordered Pd3V

In this Chapter a connection is made between anharmonic atomic scale vibrations and a
microstructural scale contribution to the specific heat. It is also found that the
microstructural contribution to the specific heat is restricted by plasticity.

4.1 Introduction

It is now accepted that crystallographic differences between phases cause differences

in vibrational entropy. It is less clear, however, if microstructure can affect the heat capacity
of a phase, and hence its stability. Nanocrystalline Fe and Ni3Fe exhibit changes in their
phonon DOS that alter somewhat their stability with respect to large grained material [1-3].
A recent experimental study of anharmonic effects in Ni3V [4] suggested that some of the
heat capacity of Ni3V could originate with internal stresses that develop during anisotropic
thermal expansion, but such effects were not shown conclusively. The present study was
designed to test if such microstructural stresses could affect the heat capacity of Pd3V, which
has the same D022 ordering phase transition as Ni3V,

The difference in entropy of two states or phases of a material, a and {3, is found by
measuring differences in heat capacities at constant pressure, I:1C/-a=C/ _Cpa, as a function
of temperature and integrating
I:1Sf3- a =

f I:1Cf3-a
PdT'
T'

(4.1)

52
For measurements of vibrational entropy, the atomic arrangements in the two states, a and {3,
must remain unchanged throughout the differential calorimetry measurements. This is
usually not a problem because most of the harmonic vibrational entropy difference comes
from the low temperature range of the integrand where there is no significant atomic
diffusion.

The standard relationship between the thermal expansion coefficient, a, and the heat
capacity at constant pressure, CP' is with the Grtineisen relation [5]
a=-Y- C
3B V p

(4.2)

where Bs is the adiabatic bulk modulus, V is the volume and Y is the Grtiniesen constant. This
relationship can be derived from the anharmonic contribution to the vibrational specific heat
by assuming a quasiharmonic model (cf. Chapter 2.1 and Chapter 6.1). In this case the
Grtiniesen constant is given by the volume sensitivity of the phonon frequencies

Yi = -~ ~Wi , appropriately averaged over all phonons (typically, Y == 2). For small changes
Wi

between states, the differential thermal expansion coefficient is given by

!:1a f3 - a (T) = _Y_!:1Cf3 - a (T) + C (T)!:1(_Y_Jf33Bs V
3Bs V

(4.3)

where the notation, !:1(xt-a , means the difference in the quantity, x, between the two phases,

a and {3. At low temperatures, where a and Cp vary more strongly than the other parameters,
it seems reasonable to assume that a and Cp are proportional. It is often easier to measure
differential thermal expansion, !:1a, than differential heat capacity, !:1Cp ' and we use both
methods in the present investigation. Thermal expansion measurements were used previously

53
to obtain differences in vibrational entropy [7], but a comparison was not made with the more
direct differential calorimetry measurements.

In the present work we find excellent

agreement between the temperature dependence of ~a and ~Cp. This agreement extends to
higher temperatures where significant anharmonicity is observed. We attribute the
anharmonicity to both the conventional thermodynamic source of Equation 4.2 and 4.3, but
also to strain in the microstructure.

4.2 Experimental

Ingots of Pd3V were prepared from elemental Pd (99.9%) and V (99.9%) by induction
melting in an argon atmosphere. For calorimetry, a pair of disks (188.7 mg) were cut from
ingots with a slow speed saw. The edges were ground so that the masses were matched to
within 0.1 mg. Samples for differential thermal expansion measurements were made by coldrolling ingots to about 0.1 mm thickness and cutting out two strips (40 mm by 4 mm). A solid
cylinder (8.63 mm diameter and 8 mm long) was also machined for ultrasonic sound velocity
measurements and for absolute thermal expansion measurements. All five of these pieces
were annealed in evacuated quartz ampoules at 1100 DC for 2 h, and quenched by breaking
the ampoules in iced brine. One of the strips and one disk were used directly in this state,
which was confirmed to be disordered fcc by X-ray diffractometry. The remaining strip and
disk were transformed to the ordered D022 state by annealing in evacuated quartz ampoules
at 780 DC for 10 days. The annealing of the cold-rolled strip also induced recrystallization
and reduced the rolling texture. X-ray diffractometry was performed with an Inel CPS-120
diffractometer using Co Ka radiation with an Al filter to suppress the V Ka fluorescence.

54

Differential heat capacity was measured with a Perkin-Elmer DSC-4 differential
scanning calorimeter (DSC) that had been modified by installing its sample head in a liquidhelium dewar [8]. The sample disks, one disordered and one ordered, were placed in the two
sample pans of the DSC. Heat capacity measurements comprised pairs of runs, with the two
samples interchanged in the sample pans between runs. The difference in heat capacity was
found from the difference of these two sets of runs. Ten matched runs were performed to
ensure reproducibility. To counteract instrumental drift, runs comprised two pairs of scans
over temperature intervals of 30 K, which overlapped by 10K.

Two strips, one disordered and one ordered, were spot-welded to make a bi-Iayer
sample for differential thermal expansion measurements. The spot welds were estimated to
have melted < 5% of the total sample volume. The difference in thermal expansion was
determined by cantilevering the bi-Iayer and measuring the deflection of its free end with an
optical lever. Assuming the elastic constants are similar, the deflection of the laser beam is

8 == ( ~) L !1adis-ord !1T

(4.4)

where D is the distance from the sample to the optical sensor, 2h is the thickness of the bilayer, and L is the length of the sample as shown in Figure 4.1 (see Appendix D for a detailed
derivation of Equation 4.4). The difference in thermal expansion coefficients of the
disordered and ordered strips, !1 adis-ord, causes a difference in their lengths over a
temperature range of !1T that is L!1c/is-ord!1T. The measured deflection, 8, is Dlh times the
expansion difference of isolated strips. Since Dlh == 5000 for our experiment, thermal
expansion differences as small as 2.5x10- 8 could be measured. No attempt was made to

55

measure the precise sample geometry, so absolute thermal expansion coefficient differences
were not measured. We sought to measure the functional form of t1dis-Ord(T), and compare it
to results from calorimetry or t1C/s-ord .

Temperature was controlled by clamping the bi-Iayer to a cold copper finger in a
vacuum chamber. As shown in Figure 4.1, the copper finger was cooled by flowing
cryogenic fluid (liquid N2 or He), and the finger was heated resistively. A steady-state
temperature was maintained for 1 to 2 minutes before recording the deflection. Five runs
were made with the laser reflection from the ordered side and one run was made with the bilayer turned over.

Absolute thermal expansion measurements were performed from 300 K to 770 K with
a Perkin Elmer TMA-7 thermomechanical analyzer. To ensure reproducibility, two heating
and three cooling cycles were measured. Ultrasonic measurements of longitudinal and
transverse wave velocities were performed using 10 MHz transducers of 0.6 cm diameter. A
disordered cylindrical sample (8.63 mm diameter and 8 mm long) was measured first. The
same sample was annealed to develop D0 22 chemical order and measured again.

56
Coolant (liquid N2 or He)

t t
Electrical feed

disordered
,. . . . . . ordered

Turbo Vacuum Pump

Figure 4.1. Schematic of differential dilatometer.

57

Metallography was performed on the ordered calorimetry sample through a procedure
of thermal cycling. The sample was annealed at 780 (C for ten days, after which the surface
was polished and etched with a 3:1 hydrochloric to nitric acid solution. A digital image of the
microstructure was then taken. The sample was subsequently immersed in liquid nitrogen for
about a minute, warmed to room temperature, and a second image was taken. The sample
was cycled ten more times in liquid nitrogen, and an image was taken after each cycle. To
ensure reproducibility, the same experiment was performed again after a second anneal.

4.3 Results

X-ray diffractometry was performed on all samples. Figure 4.2 presents X-ray
diffraction patterns for both the quenched (fcc-disordered) sample and the annealed (D022ordered) sample. The presence of D022 order was shown by the formation of superlattice
diffraction peaks and a slight splitting of the "fcc" peaks owing to the development of some
tetragonality of the unit cell. The tetragonality, cIa = 2.01, was the same as that observed by
Maldonado and Schubert [9] but slightly smaller than that observed by Dwight, Downey, and
Conner [10] (cIa = 2.015). The diffraction data were used to calculate molar volumes of
8.63_10-6 m3/mol for the D022-ordered state and 8.69_10-6 m3/mol for the fcc disordered

state.

Results from an average of ten pairs of differential scanning calorimetry (DSC) runs
are shown in Figure 4.3. Error bars are the standard deviations of the data from the different
runs. The positive sign of the data shows that the disordered state has a larger heat capacity

58
than the ordered state. The magnitude of the difference was less than 1% of the typical heat
capacity of a solid (25 J mol-lK-l) and was a challenge for calorimetric measurement. The
data are qualitatively similar to those observed for Ni3V [4].

Figure 4.2. X-ray powder diffraction pattern of the disordered and ordered Pd3V. Inset shows
peak splitting of the (220) "fcc" fundamental peak. Small unlabled peaks were from a surface
oxide.

Results from an average of six differential thermal expansion (DTE) runs were scaled
to match the calorimetry results and are presented in Figure 4.3. Error bars are the standard
deviations of the data from different runs. The scaled errors were 10 to 30 times smaller than
the DSC results. Note that near 150 K, Cdis-ord ( 0 and (Cp-.!iis-ord_( O. Since Cp must be
greater than zero, Equation 4.3 implies that (((l3BSV)dis-ord is zero. Assuming
(((I3BSV)dis-ord remains zero at all temperatures, the DTE and DSC results should be
proportional to each other. This assumption is supported by the experimental data, since the
DSC results and scaled DTE results have no discern able differences over the full range of
temperature. The consistency of the thermal expansion data and calorimetry data gives more
credibility to both, including the unexpected dip at 150 K.

59

0.15

........

0.10

-0

"E
en

0.05

b-0.00

50

100

150

200

250

300

T (K)
Figure 4.3. Differential heat capacity (_, DSC) and differential thermal expansion coefficient
scaled by 500,000 J mol-1K-1 (., DTE). Curve L1C dis-ord was fit to data using the difference

in two Debye curves. Curve L1(Cp _C v)diS-ord is the anharmonic volume expansion contribution
calculated from measured properties. The curve labeled "fit" is the sum of L1Cv, L1(Cp _C v)diS-Ord
and the microstructural contribution including plasticity (as described in the text).
The high temperature thermal expansion coefficient was obtained by differentiating
the measured displacements in the TMA scans. These results, shown in Figure 4.4, indicate
that the thermal expansion coefficient increases gradually with temperature. The contraction
on cooling was slightly larger than the expansion on heating. This irreversibility is discussed
below.

60

13.0x10

-6

12.5
--.

%'

12.0

11.5

11.0
400

450

500

550

600

650

700

T (K)
Figure 4.4. Thermal expansion coefficient of ordered Pd3V.

The longitudinal sound velocity increased from 5064±5 m/sec in the disordered state
to 5137±5 rn/sec in the ordered state. The transverse sound velocity was more sensitive to
ordering; the transverse velocity was 2547±5 rn/sec in the disordered and 2664±5 rn/sec in
the ordered state. Since the sample length (8 mm) was several times the wavelength, the
adiabatic elastic constants were calculated using the long bar approximation with densities
obtained from diffraction data, and assuming an isotropic polycrystalline average. The
results, presented in Table 4.1, indicate that although elastic constants change by a significant
amount, the bulk modulus is essentially unchanged.

61

Table 4.1. Measured adiabatic elastic constants.

fcc-disordered Pd3V

2.71±0.01

0.685±0.003

1.797±0.006

D0 22 -ordered Pd3 V

2.81±0.01

0.755±0.003

1.803±0.006

Images of the microstructure showed an unrecovered microstrain of about 3% after
the first liquid nitrogen thermal cycle (see Figure 4.5). The strain was determined by first
removing any rotations by aligning a small part of the microstructure (e.g., grain boundaries
on the left in Figure 4.5b). The remaining bend in the microstructure gave the net
unrecovered microstrain. To test the consistency of the image alignment, additional cycles
were performed and the images were aligned with the uncycled sample. Ten additional cycles
showed no new unrecovered microstrain when aligned with the sample that received one
cycle. These results were repeated for another sample with a full set of images. Further
evidence for plastic strains during thermal excursions was found in X-ray lineshapes, which
broadened slightly from their room temperature widths when the material was heated to
temperatures of less than 400 K. The additional line broadening was difficult to assess
quantitatively, however, since the individual crystallites were too large to provide a good
polycrystalline average in the diffraction pattern.

62

(a)

(b)

Figure 4.5. Metallography of ordered Pd3V. (a) Image taken after annealing. (b) Trace of
grain boundaries in images taken before and after the first liquid nitrogen cooling cycle. The
images were overlain so that the grain boundaries on the left-hand side were aligned.

4.4 Discussion

Following the work of Nagel, Fultz, Robertson and Spooner [4], we explain the
measured shape of the differential heat capacity in Figure 4.3 by assuming three
contributions; (1) harmonic vibrations, (2) the anharmonic C p -

C v term, and (3) a

microstructural strain energy contribution from anisotropic contractions of the D0 22
structure. When the harmonic contribution was approximated as the difference between two
Debye curves with Bdis = 290 K and Bord = 291 K, harmonic vibrations account adequately for
the low temperature, constant volume contribution to the heat capacity difference in Figure
4.3 and give the peak at 60 K. The anharmonic C p - C v term accounts for the energy
expended when thermal expansion works against the bulk modulus of the material. The

63
specific volumes calculated from diffraction data, v, measured thermal expansion
coefficients, a, and the bulk moduli from ultrasonic measurements, B, were used to calculate
this anharmonic contribution to the heat capacity:

Figure 4.3 shows that this anharmonic term accounts for only a small part of the anharmonic
contribution to the heat capacity at 300 K.

The microstructural contribution, first suggested by Nagel, Fultz, Robertson and
Spooner [4], is a consequence of the anisotropy of the thermal expansion of the DOzz-ordered
phase in a polycrystalline sample. As the material is cooled, anisotropies in its contraction
cause a buildup of elastic strain energy. An irregular microstructure of tetragonal grains can
be free of strain at one temperature, To, but changes in temperature cause a buildup of
microstructural strain. We expect no such effect in the disordered phase because the structure
is cubic. When thermal expansion is anisotropic, however, this can be represented as an extra
term in the difference in the Gibbs free energies of the two phases, I1E~:~;ord,
A Gdis-ord

L.l

= /)Jjdis-ord _

T~sdis-ord _

v,b

/1Edis-ord
JlSlr'

(4.6)

The difference in vibrational and "microstructural" entropies is then

~sd,s-ord =
vlb,J,lstr

T ~Cdis-ord

+ ~(C _ C )diS-ord - ~C .

T'

JlSlr dT'

(4.7)

where
~{)rd

U.

I.lslr

~CJlSlr -~

(4.8)

64
A detailed calculation of the microstructural contribution is not practical. However,
assuming linear elasticity (with fixed elastic properties), and assuming that the ordered
microstructure formed in a state of minimum strain energy at temperature To, the
microstuctural strain energy will have the form
(4.9)

where To is a reference temperature where the internal stress is zero, such as the temperature
of the ordering treatment (780 °C). Here a is an averaged thermal expansion coefficient.
The temperature derivative of Equation 4.9 gives
7'0
C}.JStr oc -CiCT)

f Ci(T')dT'·

(4.10)

The negative sign accounts for the fact that the strain energy increases when the temperature
decreases below the temperature To. Thus, Cllstr makes a positive contribution in Figure 4.3.
Assuming that the average thermal expansion coefficient has the same shape as a Debye heat
capacity curve (Grtiniesen's law), C llstr was calculated from Equation 4.10, with the results
shown as "Elastic" in Figure 4.6. This microstructural contribution combined with the
anharmonic and Debye contribution can partially account for the heat capacity data.
However, it cannot account adequately for the dip in the differential heat capacity near 150
K.

65

0.20

0.15

0.10

()~

Elastic-Plastic Model

0.05

0.00

-0.05

Heat dissipated

-0.10

50

100

150

200

250

T (K)
Figure 4.6. Microstructural contribution to heat capacity with and without the effects of
plastic flow. Plastic flow begins to the right of the dashed line in the elastic-plastic model (as
described in text).

The above analysis assumed elastic strains. The microstructural images, Figure 4.5,
showed that plastic strain was imparted to the sample during thermal cycling. The simplest
way to include plastic behavior into the heat capacity is to assume that as the material is
cooled, internal stresses build up to a yield stress and then remain constant (i.e., no work
hardening or softening). The microstructural stress-temperature curve for this process (Figure
4.7) shows how plastic deformation leads to a decrease in the reference temperature, TO-7Tj,
upon cooling, and an increase in the reference temperature on heating, Tj-7T2 • The reference

66
temperature lags T by a characteristic temperature !::.Ty • The !::.Ty is the temperature difference
that provides sufficient thermal expansion to induce local yielding. More realistically,
inhomogeneities in the microstructural stress would lead to a distribution for !::.Ty , but
determining the actual distribution is impractical. As a first approximation we assumed a
single average !::.Ty •

aj.!str

ay

r-Hy
-E:

;>

Figure 4.7. Simplified temperature (n versus microstructural stress (alL'"tr) diagram. The
parameter ay is the microstructural yield stress. The parameter !::. Ty is the change in
temperature from the minimum stress state to the yield stress.

We tested a microstructural heat capacity model that included the elastic and plastic
response of the material induced by thermal expansion. The elastic part was calculated using
Equations 4.9 and 4.10. For the plastic part, we assume that once steady state-p1astic flow is

67

achieved, there is no net generation of defects and dislocations to alter the internal energy of
the material. With this assumption, the work required for plastic flow will equal the heat
evolved by the plastic flow. A microstructural heat capacity was then determined for heating
from 90 K, Figure 4.6. By adjusting I::.Ty , it was found that I::.Ty = 30_gives a dip at 150 K
similar to that observed in the measurements, Figure 4.3. The dip at 150 K corresponds to a
microstructural yielding that is expected outside the range 21::.Ty • At temperatures around 150
K, the plastic response dominates over the elastic response. The transition from initial
yielding at 150 K to steady-state plastic flow at about 250 K was adjusted to match the data.
In this model the "Elastic-Plastic" heat capacity (Figure 4.6) above 150 K is from a net
formation of defects. For temperature changes of more than 160 K, the defect density is
assumed steady-state so our model of a fixed I::.Ty therefore predicts zero microstructural
contribution to the heat capacity of the material.

To check if I::.Ty =30 K is reasonable, a simple comparison was made with the
observed microstrains in the thermal cycle experiments. For a given temperature change, the
average microstrain, Cw is given approximately by a constant, A, times the relevant
temperature change I::.T, as CJJ- = AI::.T. The proportionality constant, A, is a material property
and should be similar for the thermal cycle, the thermal expansion, and the calorimetry
measurements. The yielding micros train in the ordered phase is then estimated from
Cy

=(I::.~{T
)C
/ I::.
cycle

Cycle

=0.004,

(4.11)

where I::.Tcycie is the temperature range of the thermal cycle (223 K), and the observed
unrecovered microstrain (0.03) from Figure 4.5 was used for ccycle' We expect the cycle
microstrain to be greater than the unrecovered microstrain used in the calculation. The

68
temperature dependence of A was neglected. Using the unrecovered strain tends to cause an
underestimate of cY' while neglecting the temperature dependence causes an overestimate of
about a factor of two. Thus, the predicted yield strain at !1Ty = 30 K is at least 0.002, which is
reasonable for a metal.

It is expected that the microstructural contribution to the heat capacity should be

observed in the high temperature thermal expansion measurements, Figure 4.4. The low
temperature range does show a difference in the heating and cooling curves, indicative of a
microstructural elastic response, but the response is not seen at high temperatures. The lack
of response at the high temperatures suggests that the effective yield strength has decreased.
This could be caused by creep. As the temperature is decreased, the increase in yield strength
gives a gradual microstructural elastic response. For this reason it is not possible to
determine an accurate value for !1Ty from these data. However, from the observed difference
between the thermal expansion coefficient on heating and cooling, 0.5x10--{j/K, the magnitude
of the response is found.

When this value is scaled by the same factor relating the

differential thermal expansion to the differential calorimetry (500,000 J/mol), C listr = 0.1 J
mor1K-1, which is similar to the differential calorimetry measurements.

Figure 4.3 shows the final calculated curve labeled "fit," which combines all terms
for the differential heat capacity -

the elastic-plastic Clistr (from the ordered alloy only), the

harmonic Debye curves (for both phases), and the anharmonic Cp - C v terms (for both
phases). The agreement with experimental data is reasonable, but the high temperature
anharmonic effects are underestimated. This extra heat capacity difference could not

69
originate with the ordering of the disordered phase because this would provide a contribution
of the wrong sign. The deviation at high temperatures is likely caused by our
oversimplification of the microstructural plastic flow. Deformation processes with ATy > 30
K probably contribute to the heat capacity at higher temperatures. The important point is that
it was necessary to include plastic flow to account for the dip in the data at 150 K in Figure
4.3.

The microstructural contribution appears in the scaled thermal expansion data without
the use of any new scale factor at high temperatures, Figure 4.3. The standard expression
relating thermal expansion and specific heat, Equation 4.2, is derived without considering the
effects of microstructure [5]. From classical thermodynamics the linear thermal expansion
that results from energy stored in the microstructure can be expressed by
1(

aJ1..llr

d(dE J J

(4.12)

J-l'lr

= 3V aT -;;;- T p

For clarity we assume a simple expression for the energy stored in the microstructure:
(4.13)

(8

where c is the average compliance coefficient and

is the mean square strain stored in

the microstructure (this energy is considered in more detail in Chapter 5). Substituting
Equation 4.13 into Equation 4.12, a relationship is found between the thermal expansion
coefficient and the specific heat due to strain energy stored in the microstructure:

(de] .!.c(d(8aT
3VC dp 2

= _1

ex
J-l'lr

= _1

)]

(de] C

3VC dp T

J-l'lr

(4.14)

70

The pressure derivative in Equation 5.3 can be written in terms of a volume derivative using
the bulk modulus, B, resulting in a familiar form,

_r
}LIlr -

)lslr

3BV

}LIlr·

(4.15)

This is exactly the same form as the standard Grtiniesen relation (Equation 4.2) only with a
Grtiniesen constant given by

vac

r = - c (IV .
)lslr

(4.16)

This gives the volume sensitivity of the elastic constants rather than the usual volume
sensitivity of the phonon frequencies. Of course, these quantities are directly related and thus
it is expected that r }LIlr "'" r . Therefore, it is not surprising that only one scale factor was
needed to relate the thermal expansion and heat capacity over the entire temperature range.

Evaluating the three terms in the integrand of Equation 4.7, we obtain three
contributions to the entropy difference between ordered and disordered Pd3V. The harmonic
and anharmonic vibrational entropy differences are ~S:~~:~~l~C = O.OIka/atom and
tlS:~~~::!oniC = 0.025 kalatom at 300 K. It is expected that at the ordering temperature (l088 K)

the anharmonic effects will be larger. The third part of the heat capacity comes from defect
formation, which provides an effective entropy of ~S:~;~.~rd = -O.OIka/atom at 300 K. This
plastic microstructural contribution to the heat capacity is highly temperature-dependent, and
becomes zero at high temperature. These three contributions to the entropy are all expected
to be smaller than the configurational entropy of the order-disorder transformation, which
could be as large as 0.56 kalatom if there is no short-range order in the fcc phase.

71
Since a significant portion of the entropy comes from the microstructure, it is
expected that the stability of the D0 22 ordered phase will be different in a single crystal. Both
theoretical [11] and experimental [12] studies have suggested that coherency stresses can
have a significant effect on the phase diagram. In the present study a temperature-dependent
strain energy within the microstructure of the ordered phase was observed over a range of
temperature. The type of strain energies observed here are expected to be at least as large as
the coherency strain energies since the elastic limit is exceeded. A consequence of including
plasticity, as was done here, is that the effects on phase stability are expected to be dependent
on thermal history.

4.5 Conclusion

A polycrystalline alloy of Pd3 V was prepared in states of D0 22 chemical order and as
a disordered fcc solid solution. Differences in the heat capacity and thermal expansion of
these two materials were measured, as were the density, bulk moduli, and thermal expansion
coefficients that are needed to assess anharmonic contributions to the heat capacity. When
scaled by a positive constant factor, the differential thermal expansion coefficient and the
differential heat capacity were the same over the temperature range from 80 - 300 K. The
heat capacity was larger for the chemically disordered Pd3V at both the lowest and highest
temperatures in this range. The differential heat capacity curve and ancillary measurements
were used to assess the harmonic and the anharmonic contributions to the vibrational
entropy, which at 300 K was Sdis - sord = (+0.035± 0.001) kiatom, with 70% of this coming
from the anharmonic contribution.

72

The conventional anharmonic contribution, f).(Cp-Cvts.ord, was too small to account
for the measured differential heat capacity. This anharmonic contribution was also unable to
account for a peculiar behavior in the differential heat capacity and differential thermal
expansion at 150 K, where both differential curves were nearly zero. We argue that this dip
originates with the elastic/plastic response of the polycrystalline microstructure of the
ordered alloy during thermal expansion. We propose a model where elastic energy is stored
in the polycrystalline microstructure owing to anisotropies in thermal expansion, but this
energy is limited by plastic flow of the material.

References
[1] B. Fultz, J. L. Robertson, T. A. Stephens, L. J. Nagel, and S. Spooner, J. Appl. Phys. 79,
8318 (1996).
[2] B. Fultz, C. C. Ahn, E. E. Alp, W. Sturhahn, and T. S Toellner, Phys. Rev. Lett. 79,937
(1997).
[3] H. Frase, B. Fultz, and J. L. Robertson, Phys. Rev. B 57, 898 (1998).
[4] L. J. Nagel, B. Fultz, J. L. Robertson, and S. Spooner, Phys. Rev. B 55, 2903 (1997).
[5] T. H. K. Barron, J. G. Collins and G. K. White, Adv. Phys. 29,609 (1980).
[7] G. D. Mukherjee, C. Bansal, and A. Chatterjee, Phys. Rev. Lett. 76, 1876 (1996).
[8] L. Anthony, J. K. Okamoto, and B. Fultz, Phys. Rev. Lett. 70, 1128 (1993).
[9] A. Maldonado and K. Schubert, Z. Metallkde 55,619 in German (1964).
[10] A. E. Dwight, J. W. Downey, and R. A. Conner, Jr., Acta Crystallogr., 14, 75 (1961).
[11] M. J. Pfeifer and P. W. Voorhees, Metall. Trans. A 22, 1991 (1991).

73
[12] M. J. Pfeifer, P. W. Voorhees, and F. S. Biancaniello, Scripta Metall. Mater 30, 743

(1994).

74

Chapter Five

Microstructural strain energy of a-uranium
determined by calorimetry and neutron
diffractometry

In this Chapter we take what was learned about the microstructural contribution to the
specific heat in Pd3 V and design better experiments to isolate the contribution in uranium.
This Chapter focuses on the largest length scale in this thesis.

5.1 Introduction

We report the first direct measurement of the microstructural effects on the heat
capacity of a-uranium. Results from prior work imply that microstructure can affect the
charge density wave (CDW) transitions in a-uranium. Distinct CDW transitions at 23 K and
37 K can clearly be seen in calorimetry measurements on single crystal uranium, but these
transitions are broadened severely in measurements on polycrystalline samples [1]. Work by
Hall [2] shows that the constraints on anisotropic thermal expansion in uranium polycrystals
either partially inhibit or prevent the CDW transformations, as evidenced by the diminished
effects on thermal expansion, specific heat, and electron transport properties.

Recently it has been proposed that an anomaly in the specific heat of Ni3 V and Pd3V
alloys was the result of a microstructural contribution [3,4]. However, in these experiments,
the microstructural contribution was mixed with contributions from both harmonic phonons
and anharmonic volume expansion. In this chapter the microstructural contribution in
uranium is isolated by subtracting a single crystal specific heat directly from a mass matched
polycrystal. In addition, neutron powder diffraction experiments are used to measure the

75
distribution of elastic strains in the polycrystalline material along various crystallographic
directions. The strain distribution data were used to calculate the microstructural strain
energy. We find good agreement between the elastic strains that develop in polycrystalline auranium during thermal expansion and the reduction in the measured heat capacity as this
energy is released. The temperature and energy scales of these phenomena are consistent
with measured distortions of the CDW transitions in polycrystalline a-uranium.

5.2 Experimental

Uranium crystals were grown by electro-transport through a molten salt bath of LiCIKCI eutectic containing on the order of 3 wt. % UCl 3 [5]. The uranium was deposited onto a
stainless steel cathode as dendrites in the form of parallelogram-edged platelets. The
individual platelets are high purity single crystals of a-uranium. The residual resistivity ratio
(RRR) of 115 was about three times higher than any RRR reported previously [1]. Because
the uranium was deposited below the a-B transformation temperature, single crystals are
strain-free. Strips were cut by spark-erosion cutting, and were cleaned in concentrated HN0 3
and electropolished in H3P04 .

Uranium polycrystals were prepared by induction melting the dendritic electrorefined product described above in a BeO crucible under an inert atmosphere. The ingot was
melted only once to minimize the risk of contamination from the crucible or the atmosphere.
The samples were sectioned directly from the cast ingot with a diamond saw.

76
Differential heat capacity measurements were performed with a Perkin-Elmer DSC-4
differential scanning calorimeter (DSC) that had been modified by installing its sample head
in a liquid-helium dewar [5]. Mass matched -200 mg samples, one single crystalline and one
polycrystalline, were placed in the two sample pans of the DSC. Heat capacity measurements
comprised pairs of runs, with the two samples interchanged in the sample pans between runs.
The difference in heat capacity was found from the difference of these two sets of runs. Four
matched runs were performed to ensure reproducibility. To counteract instrumental drift,
runs comprised two pairs of scans over temperature intervals of 30 K, which overlapped by
10K.

Neutron diffraction patterns were obtained on the Neutron Powder Diffractometer
(NPD) at the Lujan Center, Los Alamos National Laboratory. To reproduce the thermal
history of the calorimetry measurements, the -150 g sample was first cooled to 77 K.
Diffraction patterns were then obtained at 77 K, 90 K and in steps of 10 K up to 290 K. The
sample was equilibrated at each temperature for 10 minutes before acquiring each diffraction
pattern. Each diffraction pattern was acquired for 20 minutes. The sample was re-cooled to
check for irreversibilities, and measurements were then performed at lower temperatures (40
K, 30 K, and 20 K).

5.3 Results and analysis

The shape of the microstructural contribution showed the basic form expected from
the "Elastic-Plastic Model" described by Manley, et al. [4], Figure 5.1. The material showed
the release of microstructural strain energy upon heating from liquid nitrogen temperature.

77

The minimum in strain energy appears to be at about To = 280 K. The temperature change
associated with yielding, as defined by Manley, et al. [4], is about !1Ty = 150 K (see Figure
4.7). Both of these values are much larger than for Pd3V [4], reflecting the higher yield strain
of uranium metal. The low temperature range, however, showed a significant difference from
the "Elastic-Plastic Model." With constant thermal and elastic properties, the model predicts
linear behavior until about half the Debye temperature (-150 K for uranium), where the
thermal expansion coefficients decrease owing to a depopulation of phonons. The data,
however, show nonlinear behavior at much higher temperatures. This is probably a result of
the strong temperature dependence of the elastic properties of uranium. A more detailed
analysis, including the temperature dependence of the thermal and elastic properties, is
discussed below.

78

40
Q'
..!..

-J

20

O~----------------------------~~~~~--------~

('(j

....

::s
.()
::s
....
.en

-20

....
.~

-40x10

-3

Elastic-Plastic Model

100

150

200

250

300

T (K)

Figure 5.1. Microstructural contribution to the specific heat of uranium. The "Elastic-Plastic
Model," described by Manley et al. [3], was scaled arbitrarily and a Debye temperature of
250 K was used. The error bars come from the standard deviation between an average of 4
pairs of runs.

The strain energy per unit volume in an arbitrary stress state is given in matrix
notation by [7]
(5.1)

79
where cij is the compliance matrix and Cj is the strain matrix. The total energy per unit volume
in a polycrystal is obtained by averaging over the entire volume. Neutron diffraction provides
a sampling of the strain distribution in crystallites in specific orientations determined by
Bragg's law. We assume that Ef in the set of all crystallites in selected by Bragg's law, have a
gaussian distribution characterized by
(5.2)
where 8j~ = a 2 ( E;). the variance of Ej. By multiplying Equation 5.2 by cij , considering the
appropriate sums, and rearranging terms, a general expression for the average strain energy
(Equation 5.1) of this set of crystallites is given by
(5.3)
where the relation between the stress and strain matrix, a j = CijEj' has been used. The first
term in Equation 5.3 originates with the average distortion, and the second term originates
with deviations from the average. For a polycrystal with random crystallite orientations the
average of this set of crystallites is equivalent to any other orientation and hence it is
equivalent to a volume average. Experimentally, each component of Equation 5.3 is
determined by an average over a different set of crystallites.

We now show that for an unconstrained polycrystal, the first term in Equation 5.3
vanishes. Consider a plane normal to the Xl axis cut through an arbitrary polycrystal as shown
in Figure 5.2. The force normal to the plane on an area dx 2dx 3 cut by the plane at position Xl

equilibrium requires

80

F;(x,) =

ff a,(X"X2,X3)dx2 dx3= o.

(5.4)

Since this condition must hold true for all x, in the polycrystal, it follows that

(aJ = ~f f f a,(x"x2,x3)dx,dx2dx3 = 0,

(5.5)

where V is the total volume of the polycrystal. Similar arguments can be made for all of the
stress matrix elements, so in general =O. Thus, the first term in Equation 5.3 is zero in
the case of an unconstrained polycrystal. The microstructural strain energy can then be
reduced to
(5.6)

Like the compliance tensor, the strain-broadening tensor is fourth rank because it connects
two second-rank tensors.

Figure 5.2. Planar cut through an arbitrary unconstrained polycrystal.

Since we need to know only the deviations from the average strain to determine the
microstructural strain energy, we have to consider only the strain broadening in the neutron

81

diffraction data. This is an important simplification because the average strains can be
determined accurately only with precise knowledge of the free crystal lattice parameters,
which are often sensitive to impurities, defect concentrations, etc.

A measure of the strain-broadening matrix can be extracted from neutron diffraction
data using a formalism developed by P. W. Stephens [8]. In this formalism the variance of
the diffraction peak widths are fit using [8]
(J2(hkl) = LSHKLhHkK1L,
HKL

(5.7)

where the coefficients SHKL are restricted by symmetry (6 for orthorhombic) and h, k, and 1
are the Miller indices. In the following discussion we convert this into strain using
S(hkl) =

~(JZChkl)
d ( hkl) Cdiffraction

(5.8)

where d(hkl) is the spacing of (hkl) planes and Cdiffraction is the diffraction constant (converts
time-of-flight to A). The Stephens formalism has been incorporated into the well-known
GSAS (General Structure and Analysis Software) Rietveld refinement package [9].
Refinements were fit to all neutron diffraction data in the Le Bail mode [10]. In this mode the
diffraction peak intensities are treated as free parameters. Only the peak positions and
profiles are fit. A typical fit is shown in Figure 5.3. Using profile function 4 in the GSAS
software, the appropriate Stephens strain broadening parameters (6 unique SHKL) were
extracted at each temperature. From these parameters the microstrain broadening was
calculated as a function of direction using the Mathcad file in Appendix E. The function,
S(hkl), fits the strain broadening in all of the peaks simultaneously. To check for consistency,

the peak broadening from this function was compared to single peak fits for some easily-

82

separated peaks, Figure 5.4. The fair agreement was reassuring, although it fell short of what
Stevens obtained using much higher resolution synchrotron radiation [8]. The temperature
trend of a single peak, shown in the inset of Figure 5.4, was much better.

Three-dimensional representations of the strain broadening function at several
temperatures are shown in Figure 5.5. The strain broadening is strongly anisotropic as is
expected from the anisotropic elastic and thermal expansion properties of a-uranium [1]. The
largest strain broadening is in the [010] direction. This is expected since the Young's
modulus is lowest in this direction. Strain broadening increases in the [100] direction at low
temperatures, consistent with the softening of ell below about 250 K [11]. The magnitude of
the strain broadening decreases with increasing temperature. This agrees with the
microstructural specific heat. The microstructural specific heat is negative at low
temperatures, implying a decrease in strain energy with increasing temperature.

83

Uranium 290K
Bank 1, 2-Theta

148.0, L-S cycle

Hist
352 Obsd. and Diff. Profiles

""
(J)
U)

...±

+,

*ll ~tl

- -. . . . . . ,11 111111,11 11 ,11111 111111

•..:11:,
III

111111

"-

.w
I=!
::1

~ I

I I

,. ,.L t

s ""I

,,~

.5
1.0
D-spacing, A

1.5

2.0

Figure 5.3. GSAS refinement of a uranium diffraction pattern at 290 K.

2.5

3.0

84

800
r-

'--'

.. ..
.........

'c.....o

[131 ]

600

.... ' .. ' ..

... ..

Ul

.....

200

V)

~.

o ~~__~~__~~=
' . .. .....

~. ~~m r~tOu~e
pe

. ........... .

.~

........

..

(K)

. .'.

[112]

400

.c
........

[110]

Cf)

[132]

[200]

200

1.4

1.6

1.8

2.0

2.2

d-spacing (A)

Figure 5.4. Comparison of strain broadening at 190 K calculated from refinement of the
entire diffraction pattern, open symbols CO), and from single peak fits, filled symbols ce).
Inset shows the temperature dependence of one peak. Units are in rnicrostrain (1000 = 0.1 %
strain).

85

290 K

...----

[010]
-1000

- 1000

- 500

200 K

-1000

- 1000

77 K

-1000

-1000
-500

Figure 5.5. The shapes represent the anisotropic microstrain broadening in uranium at several
temperatures. Units are in microstrain (1000 = 0.1 % strain) and all axes are on the same
scale.

86

5.4 Microstructural strain energy

To determine the strain energy stored in the microstructure we need to determine the
strain broadening matrix elements, 8i~' corresponding to the nine compliance constants, cij'
allowed by orthorhombic symmetry. The first six components correspond to the variance of
the three pure normal strains and the three pure shear strains. They can be written down
directly as

8;1 = S2(100)
8;2 = S2(01O)

8;3 = S2(001)
8~ = S2(011)

(5.9)

8;5 = S2(101)
8~6 = S2(110)

where the numbers in parentheses correspond to the crystallographic hkl indices. The
remaining non-zero components (corresponding to CI2,CI3,C23) contain subscripts with i:f:. j
and thus represent connected variances between strain components (Equation 5.2). The six
coefficients obtained in the Stephens formalism cannot be used to determine these strain
components. However, these strain components can be estimated using some additional
information.

The connected variance terms have the individual strain components subtracted away
leaving only the strains in different directions that occur together. Thus, it is a measure of the
correlation between strain components in different directions averaged over the volume of
the sample. For random intergranular stresses, the strain in direction 1 due to the stress in
direction 1 is uncorrelated with the strain in direction 2 due to the stress in direction 2.

87
However, the strain in direction 1 due to the stress in direction 2 (determined by the Poisson
ratio V 12 ) is correlated with the strain in direction 2 due to the stress in direction 2 and thus
contributes to 8122 , Therefore, assuming the only correlations between strains in different
directions comes from the Poisson effect, the remaining components can be written
(5.10)

Fisher and McSkimin [12] measured all the single crystal elastic constants at room
temperature. Some of the resulting properties are most unusual. For example, there is an
extraordinarily strong coupling between strains in the [010] direction and the [001] direction
with V32= 0.548. On the other hand, the strains along the [100] direction and the [001]
direction are almost uncoupled with V31 = -0.017. This results in the unusual property that
[010] is most compressible in uniaxial compression, while [100] is most compressible under
hydrostatic compression. Thus, it is important that the coupling terms, Equation 5.8, be
included in the strain energy calculation. Fisher [11] measured the temperature dependence
of the shear and normal compliance coefficients. The temperature dependence of the offdiagonal components (C I2 , C 13 , and C 23 ) is unknown and was thus neglected. Substituting
Equation 5.9 and 5.10 into Equation 5.6 with the temperature dependent elastic constants, the
strain energy stored in the microstructure was calculated as a function of temperature, Figure
5.6.

For comparison, the microstructural specific heat shown in Figure 5.1 was integrated
to give a measure of the strain energy using
(5.9)

88

.-

--0

>-

0)
....

a.>

'ct;

....

( J)

nl
....

-....
-....
::J

(J

::J

C/)

(J

100

150

200

250

300

Temperature (K)

Figure 5.6. Strain energy stored in the microstructure: The points (e) were calculated from
strain broadening in neutron diffraction patterns and the solid line comes from integrating the
microstructural specific shown in Figure 5.1 and adding an arbitrary constant.

89
where Eo is an arbitrary constant set to match the strain energy determined from the strain
broadening in the diffraction pattern, Figure 5.6. The origin of Eo is probably from residual
strain fields. The agreement between the neutron diffraction and calorimetry results gives
more credibility to both measurements. Errors in the total energy integral are not shown in
Figure 5.6 because they depend on the uncertainty in Eo and the systematic accumulation of
errors through integration. For example, if Eo is known exactly at some temperature, say To,
then the error in energy by integrating away from To is given by I1EJ.lSlr = I1CJ.lSlr (T- To) where
I1CJ.lSlr = 0.01 J/mol-K the error in the specific heat measurement. Thus, assuming Eo = 1

J/mol at 280 K, then the energy at 77 K would be (4±2) J/mol-K. The strain energy
calculated from the diffraction measurements give a more precise measure at 77 K of
(3.7±0.5) J/mol-K.

Attempts were made to measure how the strain energy stored in the microstructure
changed below the CDW transitions by measuring the diffraction pattern at 40 K, 30 K and
20 K. However, there was no clear indication that the CDW transition had occurred.
Specifically, Barrett et al. [13] observed a sudden increase in the a (0.2%) and b (0.05%)
lattice parameters and a decrease in c (-0.09%) in single crystals. Our measurements, on the
other hand, showed no significant changes (other than the usual continuous thermal
contractions). Our results, therefore, agree with earlier measurements suggesting that the
CDW transformations are either suppressed or completely smeared out in temperature in the
presence

of

the

microstructural

expansions/contractions [1,2].

constraints

on

the

anisotropic

thermal

90

5.5 Elastic strains and the charge density wave transitions

The magnitude of the strain energy stored in the microstructure, (3.7±0.5) J/mol-K at
77 K, is comparable to the latent heats of the CDW transitions. The latent heats of these 37 K
and 22 K transitions are 2.08 and 1.38 Jlmol, respectively [14]. Since these transitions are
accompanied by lattice strains, it is therefore not surprising that the transitions are either
smeared out in temperature or suppressed by constraints imposed by the microstructure.

The expansion of the a-axis and b-axis during the CDW transformations [13] would
tend to undo some of the strains that built up on cooling because the strains that build up on
cooling originate with contractions along these directions. Specifically, in single crystals the
CDW transformation strains cause the lattice parameters to recover to their values at about
180 K for the a-axis and about 300 K for the b-axis [13]. Therefore, one would expect that in
regions dominated by thermal strains along [100] and [010], the strain energy would favor
the transformations. Therefore, these regions may in fact transform at a higher temperature
than in the single crystal. On the other hand, the contraction of the c lattice parameter during
the transformation would tend to further increase the strain energy. Therefore, the opposite
affect would be expected in grains dominated by strains along [001]. Of course, the actual
strain energy depends on all of the strain components in a given region. In a polycrystal with
randomly oriented grains we would expect a distribution of strain energies either gained or
lost in the transitions and therefore a distribution of transition temperatures. This has the
effect of smearing out the effective CDW transition temperature seen in the polycrystal.

91

The spread of the transition temperatures depends on the strain energy gained or lost
in the transition and the free energy change in the free crystal. To predict the range of this
spread, we would need to add the strain energy gained or lost in a region to the free energy
difference between the phases in the single crystal and then recalculate a new local
equilibrium temperature (if there is one). Presently, we do not have enough information to do
this because we do not know how the free energy difference of the single crystal scales with
temperature. For example, if the entropy difference was primarily electronic, then the
difference would scale linearly with temperature. If it where vibrational, the entropy
difference would scale roughly as T 3 in the lower temperature range.

The elastic energy in the polycrystalline microstructure depends on the thermal
history of the material. We have made the above arguments based on the strains induced by
our particular thermal path. It is conceivable that different results could be found with
different heat treatments. In particular, it would make sense to try and find a way to minimize
the strain energy stored in the microstructure near the phase transition. This may allow these
transitions to be studied in more detail in polycrystalline samples.

5.6 Concluding remarks

The results presented in this Chapter not only confirm the mechanism for
microstructural specific heat described in Chapter 4, but also show that it is possible to
deduce the same results from neutron diffraction. Estimated strain energies stored in the
microstructure showed good agreement between calculations from the diffractometry and

92
calorimetry. It is now clear that the strain energies stored in the microstructure of uranium
have a significant affect on the temperatures of the low temperature charge density wave
transitions.

References
[1] G. H. Lander, E. S. Fisher, and S. D. Bader, Adv. Phys. 43, 1-111 (1994).
[2] R. O. A. Hall, Inst. Phys. Conf. Series (Lond.) 27, 60 (1978).
[3] L. J. Nagel, B. Fultz, J. L. Robertson, and S. Spooner, Phys. Rev. B 55, 2903 (1997).
[4] M. E. Manley, B. Fultz, and L. J. Nagel, Phil. Mag. B. 80, 1167 (2000).
[5] C. C. McPheeters, E. C. Gay, P. J. Karell, and J. P. Ackerman, J. Met. 49, p. N7 (1997).
[6] L. Anthony, J. K. Okamoto, and B. Fultz, Phys. Rev. Lett. 70, 1128 (1993).
[7] J. F. Nye, Physical Properties of Crystals: Their Representation by Tensors and Matrices
(Oxford Science Publications, 1989) p. 137.
[8] P. W. Stephens, J. Appl. Cryst. 32, 281-289 (1999).
[9] A. C. Larson and R. B. Von Dreele, GSAS - General Structure Analysis System. Los
Alamos National Laboratory Report LAUR 86-748.
[10] A. Le Bail, H. Duroy, and J. L. Fourquet, Mater. Res. Bull. 23,447-452 (1988).
[11] E. S. Fisher, J. Nucl. Mater. 18,39 (1966).
[12] E. S. Fisher and H. J. McSkimin, J. Appl. Phys. 29, 1473 (1958); Phys. Rev. 124,67
(1961).
[13] C. S. Barrett, M. H. Mueller, and R. L. Hittermann, Phys. Rev. 129,625.
[14] J. Crangle and J. Temporal, J. Phys. F 3, 1097 (1973).

93

Chapter Six

Future work

At this point it is clear that the thermodynamics of metals and alloys depend on
phenomena across a wide range of length scales. We have found many interesting
contributions to phase stability from electronic excitations to strain energy produced at the
microstructural scale (from the forces that grains exert on one another). There is still
considerable work remaining. In this Chapter some future work is outlined.

6.1 Phonon softening in the absence of anharmonicity

In Chapter 2 the search for the nature of the vibrational softening in uranium was
motivated in part by the failure of the quasiharmonic approximation to explain the phonon
softening from thermal expansion. Specifically, there was about an order of magnitude more
entropy from phonon softening than needed to account for the elastic energy generated from
volume expansion. This argument can be reformulated in terms of the specific heat. First,
express the entropy as
(6.1)

Then take the temperature derivative at constant pressure,

(6.2)

where

94

Dp = L __
1 (d{J)j)

ill

ar

(6.3)

Considering only the vibrational contribution, the constant pressure specific heat can be
written,
(6.4)
Comparing Equation 6.4 and Equation 6.2 it is clear that under the assumptions of the
quasiharmonic approximation 9Bva 2 = 3k B 8p • These terms were calculated for several
different materials and are shown in Table 6.1. The first three elements (Pd, Ni, and Cu) were
calculated using the temperature dependence of the phonon DOS published in the LandoltBomstein series [1]. The remaining Pu phases and a-V were calculated using the temperature
dependence of the Debye temperature extracted from neutron diffraction data [2]. For both
Pu and V the discrepancy is quite large, suggesting further study of the phonon contribution
to the equation of state of both materials.

Table 6.1. Excess specific heat above 3kB from volume expansion (column 1) and from
phonon softening (column 2).
Material

9Bvd (10-4k B /atom-K)

3kB 8p (10-4k B /atom-K)

Pd
Ni
Cu
a-Pu
8-PUo.9sAlo.os
a-V

2.4
2.3

3.2
2.7
3.2
11.8
10.6
23.4

2.8
6.2
2.5

In the case of V, the effects of pressure on the vibrational spectrum could be
explored. At ambient pressure the vibrational and electronic entropy have been shown to be
significant. However, the origin of the electronic entropy is not understood. The origin of

95
the electronic contribution could be explored by measuring the partial electronic density of
states using photoelectron spectroscopy (PES). For Pu the temperature dependence of the
vibrational spectra in all of its phases should be measured. Measured vibrational spectra
measurements could be used to determine the vibrational entropy, and this could be
compared with calorimetry measurements in the literature. The electronic contribution to the
entropy will be the difference between the vibrational entropy and the total entropy
determined by calorimetry. It is expected that a significant electronic contribution will also
be found in Pu and could be explored using the same techniques as with U and Ceo

This work will help lay the groundwork for more accurate first principles calculations
of the behavior of these metals. For example, PES may give the information needed to take
into account the temperature-dependence of the force constants. The understanding of such
complex materials will also lead to a more general understanding of other materials.

6.2 Crystal-field splitting and Kondo spin-fluctuations

We have shown that both the crystal-field splitting and Kondo spin-fluctuations
make a significant contribution to both B-cerium and y-cerium at low temperatures.
However, both were small at the transition temperature so the difference made only a minor
contribution to the transition. In the case of a-cerium, however, the transition is at a much
lower temperature (100 K). Also, a-cerium has no localized f-electron and thus both the
spin fluctuation and the crystal field contribution should be zero in this phase. Therefore, it

96
seems likely that a detailed study of the entropy change of these transitions (a- to ~-cerium
or a- to y-cerium) will reveal a significant contribution from these degrees of freedom.

6.3 Strain energy stored in microstructures

The strain energy stored in the microstructure of uranium was large enough to affect
the low temperature CDW transitions. However, the range of transition temperatures could
not be worked out without prior knowledge of the components of the free energy differences
between the low temperature phases. Unfortunately, it is the spread in the transition
temperatures seen in polycrystals that makes it so difficult to study these transitions. We
could not, for example, perform the kind of detailed analysis that was performed for cerium
in Chapter 3.2 because in polycrystalline uranium we are likely to have some mixture of
transformed and untransformed uranium well below the transition temperature. In cerium the
strain problem was overcome by thermal cycling the sample (although it is not entirely clear
why this works). A more detailed study including microstructural modeling and modeling of
the electronic and phonon contributions is likely to contribute to an understanding of, and
possibly allow the control of, the CDW transitions in uranium.

References
[1] P. H. Dederichs, H. Schober, and D. J. Sellmyer, in Numerical Data and Functional
Relationships in Science and Technology, edited by K. -H. Hellwege and J. L. Olsen,
Landolt-Bomstein, New Series, Group III, Vol. IIU13a (Springer-Verlag, Berlin, 1981).
[2] A. C. Lawson, B. Martinez, J. A. Roberts, and B. I. Bennett, Phil. Mag. B 80, 53 (2000).

97

Mathcad files used to determine multiphonon
scattering

Appendix A

1 Phonon Scattering Calculation
Z :=REAOPRN(UOOS)
Se :=REAOPRN(U300KS)
r :=0, 1.. rows(Z) - 1

inn,,' DOS
data to be compared with

rows(z) - 2
Zr.1 :=

]_1
Zm,l!
. Zm+l.O - Zm,O)

·Zr,l

normalizes the DOS

m=O

h := 1

M :=238

E := 3.55

meV

atomic mass

incident energy

AE(W) :=[(1- 0·0.0753·h.w) + 0·0.00195· (h.w)2]
kT :=25
81 :=0.61

meV
to

92 :=2.32

insturment resolution function

detector angle range (radians)

rows(Z) - 2 Zm 1
[Z
m 0
-'-·coth h·-'- . Z
- Z
2.kT
m+ 1,0
m,O
m= 0
m,O

msd :=

meV

temperature

(mean square displacement) *2M/h1\2

b,-~

time-dependent correlation function

ott) := rows(Z)

i:

m = 0

:m, 1
m,O

[i .

sin!Zm, o·t) + cos!Zm. O· t)'COth[h.

~:;]l !Zm+ 1,0 - Zm.O)

n :=0, 1.. 1214
tn :=b· (n - 606.5)

single phonon dynamic structure factor convoluted with resolution function

0(8,00) :=_1_. exp[-1.008 E.[2 2·,,·h

h.~2.J1- h'~'COS(2.9)].msd]

m:=10
SI(w)

:=82~91'

i:
n= 0

Output data

r :=0, 1 .. rows (Se) - 1

____ ______
~n

,- !Zl,O - Zo, 0)' 606.5

S[81+n. (92:81)

,w} (82:81)

98

Total Scattering Calculation
inout DOS
data to be compared with

Z :=READPRN(UDOS)
Se :=READPRN(U300KS)
r :=0, 1.. rows(Z) - 1
rows(Z)- 2
Z"I :=

]_1
Zm.1 . (m+I.O
- Zm.O)

·z,.1

normalizes the DOS

m=O
M :=238

E := 3.55

meV

atomic mass

incident energy

~E(w) :=[(1- 0.0.0753.h.w) + 0·0.00195· (h.w)2]
kT :=25

meV

SI := 0.6110

meV

temperature

S2:= 2.32 detector angle range (radians)

rows(z) - 2 Z

[Z]

~ ~,colh h.~ .(z +1 o-z 0)

msd:=

insturment resolution function

m= 0

m.O

2.kT

m.

(mean square displacement) '2M/h"2

m.

time-dependent correlation function

G(I):=

rowS(Z)~2

:m.I.[i

m= 0

m.O

.sin(Zm.o.tl+cos(Zm.o'll.colh[h'~~~;]l(Zm+l.o-Zm.ol

n :=0, 1.. 1214
In :=b· (n- 606.5)
G n:=G(lnl

total dynamic structure factor convoluted with resolution function

D(S,w) :=_I_. exp[-l.008 'E.[2~n·h

m:=lO

Output data
r :=0,1.. rows(Se)- 1
E, :=Se,.O
SI, :=St(E,l

h'~2.Jl- h'~'COS(2'S)].msd]

99

Appendix B

Anisotropy error in phonon DOS measurement

The only error from anisotropy in experiments on isotropic polycrystals comes from a
Debye-Waller factor weighting of the measured phonon DOS. It is well-known, however,
that an isotropic Debye-Waller factor is a good approximation [1], especially at low
temperatures and modest Q where the Debye-Waller factor is close to 1 for all directions of

Q.

To settle definitively this point about the anisotropy of the Debye-Waller factor, we
have obtained results from our lattice dynamics calculation. The anisotropic Debye-Waller
factor second rank tensor in orthorhombic symmetry has three independent coefficients with
principal axes parallel to the crystallographic a, b, and c axes. The three diagonal components

«u/>, , and along each of the principal directions. Using the force constant model of Crummett et al. [2]
the partial DOS projected along each of the principal axes was calculated and the resulting
components are = 0.015 A2, < u/> = 0.0047 A2, and < u/> = 0.0056 A2 at room
temperature. The Q-range varies with energy and angle, but has a range from about Q = 1-3