Nonharmonic contributions to the high-temperature phonon thermodynamics of Cr

PHYSICAL REVIEW B

107

, 054312 (2023)

C. M. Bernal-Choban

†

H. L. Smith

‡

C. N. Saunders

D. S. Kim

L. Mauger,

D. L. Abernathy

and B. Fultz

Applied Physics and Materials Science, California Institute of Technology, Pasadena, California 91125, USA

Physics and Astronomy, Swarthmore College, Swarthmore, Pennsylvania 19081, USA

Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

Jet Propulsion Laboratory, Pasadena, California 91109, USA

Neutron Scattering Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA

(Received 14 June 2022; revised 21 December 2022; accepted 15 February 2023; published 27 February 2023)

Phonon densities of states (DOSs) of body-centered cubic chromium were measured by time-of-flight inelastic

neutron scattering at temperatures up to 1493 K. Density functional theory calculations with both quasiharmonic

(QH) and anharmonic (AH) methods were performed at temperatures above the Néel temperature. Features in the

phonon DOSs decrease in energy (soften) substantially with temperature. A Born–von Kármán analysis using fits

to the experimental DOSs reveals a softening of almost 17% of the high-transverse phonon branch between 330

and 1493 K. The low-transverse branch changes by approximately half this amount. The AH calculations capture

the observed behavior of the two transverse phonon branches, but the QH calculations give some inverted trends.

Vibrational entropies from phonons and electrons are obtained, and their sum is in excellent agreement with the

entropy of chromium obtained by calorimetry, indicating that above 330 K, no explicit temperature-dependent

magnetic contributions are necessary.

DOI:

10.1103/PhysRevB.107.054312

I. INTRODUCTION

Understanding the vibrational, electronic, and magnetic

interactions in condensed matter is fundamental to predicting

the thermodynamic functions of materials such as free energy,

internal energy, and entropy. These thermodynamic functions

are essential for constructing phase diagrams, predicting ther-

mal expansion, and explaining the temperature dependence of

elastic constants, bulk moduli, and magnetization [

]. There

are active investigations into these topics for their own sake,

and for their importance to the structure and properties of

materials [

–

One intriguing system is body-centered cubic (bcc)

chromium, whose vibrational, electronic, and magnetic free

energy contributions result in a transition from an antiferro-

magnet to a paramagnet, and show an apparent anharmonicity

with increasing temperature [

]. Below the Néel transition

temperature,

311 K, a single crystal of Cr is a con-

ventional itinerant antiferromagnet [

]. At

, Cr retains the

bcc structure but becomes paramagnetic [

]. In general,

the free energy of Cr,

, requires three contributions to the

entropy

(

)

−

(

mag

ele

vib

)

(1)

where

is magnetization,

is volume,

is temperature,

is the internal energy,

mag

is the magnetic entropy,

ele

the electronic entropy, and

vib

is the vibrational entropy. The

These authors contributed equally to this work.

†

cmbchoban@gmail.com

‡

hsmith@swarthmore.edu

btf@caltech.edu

vib

gives most of the total entropy at higher temperatures,

even in a harmonic model with fixed phonon frequencies,

{

}

[

]. Phonon frequencies change with volume, and the

“quasiharmonic” (QH) approximation assumes that phonon

frequencies,

(

)

, depend on temperature only through

thermal expansion. The “anharmonic” (AH) approximation

includes an independent change with

, i.e.,

(

The lattice dynamics of Cr show large anharmonic con-

tributions at high temperatures [

]. Transitions from the

antiferromagnetic to the paramagnetic states are not well

understood for Cr [

–

]. It has been suggested that mag-

netic fluctuations exist above 1000 K [

]. These issues

of magnetism and anharmonicity continue to drive work

on the high-temperature thermodynamics of Cr. Calorimet-

ric (JANAF) measurements and third-generation CALPHAD

models provide values for the total entropy of Cr,

tot

(

)

[

], but not the individual components of Eq. (

Here we use time-of-flight (TOF) inelastic neutron scatter-

ing (INS), Born–von Kármán (BvK) analyses, temperature-

dependent effective potential modeling, and non-spin-

polarized density functional theory (DFT) to determine the

individual contributions

vib

ele

, and

mag

, from 330 to

1493 K. Our AH calculations, which include contribu-

tions from electrons and phonons, largely account for the

values of

tot

(

) observed with recent calorimetry mea-

surements and match observed lattice expansion. The QH

approximation also gives a vibrational entropy close to that

observed with TOF INS, but this success is caused by a

canceling effect of individual phonon branches. The BvK

analyses of TOF data support AH phonon branch behav-

ior. A comparison of TOF INS and anharmonic calculations

reveals that AH computations capture most, but not all,

experimentally observed phonon behavior. This additional

2469-9950/2023/107(5)/054312(8)

054312-1

C. M. BERNAL-CHOBAN

et al.

PHYSICAL REVIEW B

107

, 054312 (2023)

nonharmonic behavior is unexplained but is not explicitly

magnetic in origin. In summary, we find that temperature-

broadened electronic and third-order anharmonic contribu-

tions reproduce experimental thermodynamic measurements

well, and no purely magnetic interactions are needed to ex-

plain the thermodynamics of Cr above 330 K.

II. METHODS

A. Inelastic neutron scattering

Inelastic neutron scattering (INS) measurements were

performed on electrochemically deposited plates of polycrys-

talline 99.995% Cr. Two pieces of Cr that gave a large area

for scattering were secured inside a niobium foil sachet sur-

rounded by a frame of boron nitride. All data were taken at

the time-of-flight wide Angular-Range Chopper Spectrometer

(ARCS) at the Spallation Neutron Source (SNS) at Oak Ridge

National Laboratory (ORNL) [

]. The incident energy was

70 meV with the Fermi chopper at 420 Hz and the

chopper

at 90 Hz. Sample temperatures varied from 6–1493 K. Below

330 K, a closed-cycle helium refrigerator was used. For mea-

surements at higher temperatures, samples were transferred to

the high-temperature MICAS furnace [

Data were reduced to phonon density of states (DOS)

curves by subtraction of an empty Nb sachet and were

corrected for multiphonon scattering with Mantid and the

Multiphonon package [

]. Integration to produce the

phonon density of states was performed for values of 3.5 Å

−

10 Å

−

, where magnetic scattering contributes less

than 1.5% of the total scattering. This ensured that the scat-

tering intensity used to obtain the density of states was

vibrational in origin. Additional corrections to account for

sample curvature were performed in MCViNE [

] (see Sup-

plemental Material [

] for more details).

B. Born–von Kármán analysis

Analyses of TOF INS DOSs were performed using the

Born–von Kármán (BvK) model [

]. This model takes a

crystal to be a set of nuclear masses whose interactions act like

springs that provide restoring forces against the displacements

of nuclei. By transforming the forces associated with these

displacements into a dynamical matrix, the BvK model has

often been used to fit phonon dispersions. Fitting phonon DOS

spectra with the BvK model is more involved because the

DOSs are aggregates of all phonon modes in reciprocal space.

To address this challenge, trial force constants were used

to construct a dynamical matrix,

(

)

using the underlying

symmetries of the crystal lattice. A sufficiently dense set of

points in the first Brillouin zone was used to collect the

spectrum of phonon frequencies,

, for each temperature:

(

)

(2)

where

is the mass of the atom and

is the polariza-

tion of the phonon mode corresponding to reciprocal space

vector

. This BvK model was embedded in a genetic al-

gorithm global optimization framework, where trial sets of

force constants were generated randomly according to the

differential evolution algorithm [

]. Each optimization was

repeated several times to ensure convergence. The resulting

DOSs are compared with experimental data. For Cr, a BvK

model including atomic interactions through the second near-

est neighbors (four tensorial force constants) was found to

be sufficient. More details of the fitting process are in the

Supplemental Material [

C. Computation

All density functional theory (DFT) calculations were

performed with the Vienna

Ab Initio

Simulation Package

(VASP) [

–

]. Plane-wave basis sets with a kinetic energy

cutoff of 600 eV and projector-augmented-wave pseudopo-

tentials [

] were used with Perdew-Burke-Ernzerhof

(PBE) exchange-correlation functionals [

]. Each calcu-

lation used a 6

6 supercell consisting of 216 atoms.

Monkhorst-Pack [

]

-point meshes of 4

4 and 8

8 were used for vibrational and electronic supercell cal-

culations, respectively.

We performed spin-polarized DFT calculations at 1000 K

from initial paramagnetic, ferromagnetic, and antiferromag-

netic spin configurations. Upon convergence, the magnetic

polarizations in all cases were less than 0.08

on individual

atoms and less than 0.05

in the orientational averages of

spins. Calculations were performed for positive and negative

dilations of the lattice, with no notable effect on the converged

magnetic polarization. To balance computational cost with

supercell size and complexity, non-spin-polarized calculations

were used for the phonon dynamics.

1. Quasiharmonic

Phonon calculations within the quasiharmonic approxi-

mation were conducted with Phonopy [

]. A finite atomic

displacement was introduced into each supercell of a grid

of minimized 0 K supercells scaled by

volume. Static calculations of each were converged to within

−

eV for accurate force constant determination. The har-

monic approximation from

0–1500 K was applied to

each volume, and a grid of these free energy curves was

fitted to a Birch-Murnaghan equation of state. The mini-

mized volumes at 330, 1000, and 1500 K were used to

create corresponding dynamical matrices and predict phonon

properties. A

-point mesh of 70

70 was necessary

for proper convergence, and the calculated phonon DOSs

were convoluted with a Gaussian of 1.0 meV to approximate

the broadening at higher phonon energies from the in-

strumental resolution. Lattice and thermodynamic properties

calculated within this approximation depend on tempera-

ture only through a volume mapping,

(

)

.More

details on this process are provided in the Supplemental

Material [

2. Anharmonic

Anharmonic contributions to thermodynamic properties

were calculated using the stochastic temperature-dependent

effective potential method (sTDEP) [

]. In this procedure,

the Born-Oppenheimer surface of a material at a given

temperature is represented using a collection of static cal-

culations on supercells of thermally displaced atoms. These

displacements were generated by a stochastic sampling of a

054312-2

NONHARMONIC CONTRIBUTIONS TO THE ...

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, 054312 (2023)

canonical ensemble at the temperature of interest. The ener-

gies, forces, and displacements of each configuration were

tabulated and used to generate force constants with a least-

squares fit to a model Hamiltonian,

∑

αβ

∑

ijk

αβγ

ijk

(3)

where

{

}

is the displacement of atom

{

}

and

α, β, γ

are the Cartesian components. The temperature-dependent

is a fit parameter for the baseline of the potential energy sur-

face. The sum containing

, the quadratic force constants,

captures some anharmonic and electron-phonon effects at a

given temperature, and the final sum includes phonon-phonon

interactions through the cubic force constants,

ijk

. The latter

two terms in the model Hamiltonian are used to calculate

anharmonic shifts and broadenings of phonon modes with

respect to temperature. The force constants computed with

this method include explicit temperature and volume depen-

dencies.

A grid of 36 volumes and temperatures was created. For

each point on the grid, an ensemble of ten supercells was

generated with stochastically displaced atomic positions. DFT

calculations were performed on each supercell to obtain

energy-force-displacement data. The resulting energies were

fitted to the Birch-Murnaghan equation of state to find the

optimized volume for a given temperature. This volume and

the calculated force constants were used to create another set

of configurations on the temperature volume grid, and the

minimization process was repeated. In this way, the force con-

stants are numerically converged with respect to the number

of configurations and supercell size. The final minimized free

energies were utilized to calculate the equilibrium volume

at each temperature, and phonon properties were evaluated

at these conditions. Renormalization of phonon frequencies

due to anharmonicity was included in these evaluations. Us-

ing phonon self-energy corrections from many-body theory

[

∑

(

)

(

)

(

), shifts and broadenings of

the phonon DOS were calculated. Further details are provided

in the Supplemental Material [

III. RESULTS

Phonon densities of states (DOSs) from TOF spectra of

Cr are shown in Fig.

. Shifts of the phonon DOSs from

6–1493 K follow the expected trend of “softening,” or reduc-

tion of energy, with an increase in temperature. Interestingly,

above 600 K, the three defined mode peaks (low transverse,

, high transverse, T

, and longitudinal, L) appear as two

peaks. This is traced to a large thermal softening of approx-

imately 8 meV of the high-transverse mode between 6 and

1493 K.

Figure

shows the temperature dependence of the phonon

dispersions between 6–1493 K, obtained by fitting the ex-

perimental DOSs to a BvK model [

]. We performed fits

iteratively using a genetic algorithm optimization [

](see

Sec.

II B

). Along the

→

pathway, the trans-

FIG. 1. Cr phonon DOSs from 6–149 K measured by TOF.

Curves are offset for clarity, and the 333 K dataset is overlaid with

the 1493 K curve to show the magnitude of the shift between 333

and 1493 K. Experimental error bars (based on counting statistics)

are not shown because their height is approximately the width of the

line used to connect data points.

FIG. 2. Phonon dispersion relations from force constant opti-

mization of BvK fits of the experimental phonon DOSs (333–

1493 K). Two nearest neighbors (four variables) were considered in

the fit.

054312-3

C. M. BERNAL-CHOBAN

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PHYSICAL REVIEW B

107

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verse modes are degenerate and exhibit a softening similar

to the longitudinal mode. From N

→

, there is a branch-

dependent decrease in energy with respect to temperature. We

illustrate these shifts relative to 330 K at each high-symmetry

point in Fig.

. The largest softening (of approximately 17%)

occurs in the high-transverse mode between 330–1493 K. This

behavior is consistent with the DOSs of Fig.

Figure

shows the calculated and experimental DOSs

at 330, 1000, and 1500 K. Colored bars representing mode

peaks fitted to three Lorentzians show the average shift of

each feature. Densities of states calculated with sTDEP show

similar peak location, shape, and softening to the experimental

spectra at 330 and 1500 K. Our quasiharmonic predictions

do not accurately reproduce the features in the DOS at these

temperatures, indicating that anharmonicity is important for

the thermal trends of phonons in Cr.

Our experimental and calculated phonon dispersions at

330, 1000, and 1500 K are plotted in Fig.

. Quasiharmonic

and anharmonic calculations agree well with experimental

data along most of the high-symmetry reciprocal space path-

ways. A notable exception is the N

→

direction, where the

QH low-transverse modes show an anomaly at the N point,

and the high-transverse modes do not shift as strongly as

expected. The AH and QH calculations also give different

magnitudes of the longitudinal mode along the

→

H path.

A way to assess the thermodynamic consequences of an-

harmonic phonon behavior is by calculating entropy. Figure

shows the electronic, vibrational, and electron-phonon com-

ponents of entropy and their respective contributions to the

total entropy as determined by JANAF [

The calculated lattice expansions are compared to exper-

imental results in Fig.

. Both QH and sTDEP

ab initio

calculations are in agreement with lattice parameters obtained

from the elastic region of our neutron scattering measure-

ments, and with previous experimental results that are labeled

in the figure.

IV. DISCUSSION

A. Phonons

The experimental phonon DOSs of bcc Cr show significant

softening at elevated temperatures, and the apparent disap-

pearance of one of the Van Hove singularities in Fig.

a result of this temperature dependence. Phonon dispersions

calculated with fitted force constants show that the largest

softening occurs in the high-transverse mode. At the high-

symmetry points,

, H, and P, the two transverse branches are

degenerate. At the N point near 1000 K, Fig.

shows a soft-

ening of the high-transverse (T

) mode and little softening of

the low-transverse (T

) mode. The shift of the high-transverse

mode from

∼

32 to

∼

27 meV confirms that features in the

phonon DOS with the largest thermal softening are asso-

ciated with the T

[

ξ,ξ,

0] phonon branch. At the N point,

this T

mode involves the opposing displacements of two

neighboring (110) planes along the [1

10] direction [

]. The

relatively small thermal softening of the low-transverse mode,

[

ξ,ξ,

0], up to 1000 K also may originate from interactions

beyond the harmonic approximation.

FIG. 3. Thermal shifts, relative to ambient temperature, of the

phonons in Cr at each non-

high-symmetry point for (a) our TOF

measurements, (b) previous triple-axis experiments [

], (c) AH

calculations, and (d) QH calculations. A comparison of the thermal

shifts for each method at 1500 K for the T

and T

modes at the N

point is shown in (e).

At 330 K, the high-transverse and longitudinal modes in

the DOS have higher energies in the QH calculations than

from TOF measurements and anharmonic sTDEP calcula-

tions. This overestimation of peak locations continues to

054312-4

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FIG. 4. Phonon densities of states at 330 (bottom), 1000 (mid-

dle), and 1500 K (top). Each panel compares the experimental

phonon DOS (dark blue) to the calculated DOSs from quasiharmonic

(purple) and anharmonic (light blue) approximations. The colored

markers indicate peak locations from a Lorentzian fit to the features

of the DOSs.

1500 K. Consequently, the shape of the DOS is skewed in

the QH approximation, with more features appearing at higher

energies than those found with measurements and sTDEP

calculations. The agreement of the DOS calculated with the

AH approximation and TOF measurements is excellent at

330 and 1500 K, despite some excessive broadening of the

longitudinal Van Hove singularity.

There is a general agreement of phonon dispersions calcu-

lated with each model and experimental data at 330, 1000, and

1500 K along

→

(Fig.

). However, the QH

approximation inverts the energies of the T

and T

modes

at the N point. This is accentuated by the anomaly in the

low-transverse branch behavior at N in the QH dispersions.

This discrepancy is better seen with Fig.

3(e)

.IntheQH

approximation, the low-transverse branch shows the largest

thermal softening at N. This disagrees with the general trend

seen in the experimental data and AH calculations: the high-

transverse phonon branch has the larger thermal softening.

Quasiharmonic models sometimes predict accurate macro-

scopic properties without capturing the underlying phonon

physics [

FIG. 5. Comparison of experimental and calculated phonon dis-

persion relations at 330 K (bottom), 1000 K (middle), and 1500 K

(top). Symbols to distinguish the low transverse (diamonds), high

transverse (circles), and longitudinal (triangles) phonon branches are

also shown. Anharmonic (light blue) and quasiharmonic (purple) dis-

persions are in good agreement with a BvK fit to experimental data

(dark blue) and existing triple-axis data (gray) [

]. The anharmonic

calculations capture behavior along the H to P high-symmetry path

better than the quasiharmonic simulations.

We attribute the failure of the QH approximation in Cr to

the underlying assumption of noninteracting phonons. This

assumption is known to fail at higher temperatures, where

temperature-dependent phonon-phonon interactions begin to

dominate [

–

]. The QH model cannot capture these

effects because the frequencies within this model,

(

)

, incorporate temperature by shifting harmonic fre-

quencies with changes in volume. This approach ignores

terms beyond the quadratic phonon self-energy, which are

needed for lifetime broadening and purely temperature-

dependent (AH) shifts. Our sTDEP calculations include

cubic order corrections of the phonon self-energy, so

(

The QH approximation did not successfully predict the

thermal softenings of individual phonons, but there is rea-

sonable agreement between the vibrational entropy calculated

with QH and AH methods. The change in phonon frequency

at the N point gives insight into why. Summing the fractional

thermal shifts (T

L) from the QH calculations at the

N zone boundary gives a change of about 22%. A similar

sum for our AH calculations gives a change of 26%, which is

054312-5

C. M. BERNAL-CHOBAN

et al.

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FIG. 6. Components of entropy compared to the total JANAF

entropy [

]. (Bottom) Electronic entropy including temperature-

dependent electron-phonon coupling. (Middle) Vibrational entropy

calculated from DOSs using TOF INS, a quasiharmonic approxima-

tion, and an anharmonic approximation. (Top) Sum of the electronic

and vibrational components of entropy from experiment and compu-

tations versus the calorimetric total entropy.

FIG. 7. Lattice expansion of Cr from 6–1500 K with respect

to ambient temperature. Previous lattice expansions are reproduced

from [

comparable. This shows some cancellation of errors in the av-

erage behavior of phonon branches from the QH calculations,

giving them better success with the vibrational entropy.

Likewise, a good prediction of thermal expansion does

not validate the QH approximation for predictions of phonon

physics. Using

∂

=−

(4)

and

(

)

(

)

−

(

), it is evident that thermal

expansion,

, and bulk modulus,

, are explicitly dependent

on both volume and temperature. Any cancellation of errors

introduced in calculated entropy will be present in predictions

of thermal expansion (and bulk modulus). It is therefore plau-

sible that the QH approximation can successfully reproduce

the thermal expansion of Cr (Fig.

), even with the wrong

thermal trends of individual phonons.

There is an underestimation of 4% in the sTDEP vibra-

tional entropy at 1500 K, illustrated in the middle panel of

Fig.

. This is caused by an overbinding in the generalized

gradient approximation (GGA) for Cr [

], which also affects

our ground state lattice parameter (

845 Å). This is

consistent with the

∼

4% average overstiffening of the phonon

dispersions (Fig.

), and the slight underestimation of the

lattice expansion (Fig.

). A similar effect occurs in our QH

calculations.

B. Entropy and free energy

The top panel in Fig.

shows that for temperatures up

to 600 K, the phonon entropy from TOF INS experiments

accounts for nearly all of the total entropy [

]. At 1500 K,

the

vib

from TOF measurements or calculations accounts for

over 89% of the total entropy of Cr. The remainder originates

primarily from the occupancy of electronic states, the change

of these states with temperature, and perhaps magnetic en-

tropy [

Figure

shows that an electronic entropy

ele

, calculated

first by including the temperature dependence of the electronic

entropy through the Fermi-Dirac distribution with the ground

state electronic states, brings the sum of entropies

tot

closer

to the JANAF thermodynamic data, where

tot

vib

ele

mag

(5)

Thermal motions of atoms broaden the electronic states

through electron-phonon interactions. This temperature de-

pendence was calculated with supercells consisting of ther-

mally displaced atoms obtained by sTDEP at 330, 1000, and

1500 K, and is shown in Fig.

. The electronic entropy was

also calculated by the simpler process following Thiessen [

]

and Grimvall [

], where the ground state electronic DOS,

gnd

(

), is convoluted with a function of Lorentzian form

(

)

gnd

(

)

∗

)

(6)

to approximate the effects of electron-phonon interactions

on electronic DOS,

(

). The amount of broadening, 2

πλ

, used a value of

]. Figure

shows very

good agreement between these two approaches.

Adding the thermally broadened

ele

to the experimental

and computational

vib

gives our best estimates of the total

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entropy of bcc chromium. Figure

shows that the sum of the

measured TOF vibrational and electronic components gives

excellent agreement with the JANAF data from calorimetric

measurements. There is, however, something missing from the

anharmonic phonon calculations, as seen by the inset at the top

of Fig.

. A similar discrepancy was found in previous anhar-

monic calculations [

], which proposed that the extra entropy

required for agreement with calorimetry was magnetic in

origin. However, the experimental phonon entropy is larger

than these anharmonic calculations, and with the electronic

entropy, the total entropy agrees well with calorimetry. The

discrepancy is in the anharmonic calculations of the phonon

DOS, which are stiffer and give a lower phonon entropy than

the measurements. The additional nonharmonic contribution

to the phonon self-energy is unexplained, but an effect from

phonon-paramagnon interactions could be a candidate. Para-

magnon energies were calculated recently by time-dependent

density functional theory at 0 K [

]. These energies were

found to be large, and may not change strongly with tempera-

ture. Showing the effects of paramagnons on phonon energies

at high temperatures may require new computations. Never-

theless, a large, explicit contribution from magnetic entropy is

not needed to account for the thermodynamic entropy of Cr at

high temperatures.

V. CONCLUSIONS

The phonon DOS was measured on bcc chromium from 6

to 1493 K by TOF INS, and calculations of the phonons were

performed with QH and AH approximations using Phonopy

and sTDEP. To obtain detail on individual phonon branches,

the experimental DOSs were fitted to a Born–von Kármán

model using force constants adjusted with a global minimizer.

Both measurements and computations showed significant

thermal softening of the phonons, and a similar average

phonon softening. However, the QH approximation predicted

that the low-transverse branch would soften faster than the

high-transverse branch, whereas the opposite trend is found by

AH sTDEP calculations, and by TOF results from the present

work and a previous study. The thermodynamic entropy of

chromium was obtained from the experimental phonon DOS,

and the electronic entropy from

ab initio

calculations. Their

sum gave an entropy for chromium that was in excellent

agreement with JANAF results obtained by assessing calori-

metric data. An explicit magnetic entropy contribution is not

needed for temperatures above 330 K, but a paramagnon sus-

ceptibility may perturb phonon energies beyond the known

effects of quasiharmonic, anharmonic, and electron-phonon

interactions.

ACKNOWLEDGMENTS

We thank Jiao Y. Y. Lin for his discussions and assis-

tance with MCViNE modeling. This research used resources

at the Spallation Neutron Source, a DOE Office of Science

User Facility operated by the Oak Ridge National Laboratory.

This work used resources from National Energy Research

Scientific Computing Center (NERSC), a DOE Office of Sci-

ence User Facility supported by the Office of Science of the

U.S. Department of Energy under Contract No. DE-AC02-

05CH11231. This work was supported by the DOE Office of

Science, BES, under Contract No. DE-FG02-03ER46055.

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