VIBRATIONAL PROPERTIES OF
URANIUM AND PLUTONIUM
JOHANN BOUCHET,
FRANÇOIS BOTTIN,BORIS DORADO,ALOIS CASTELLANO
CEA, DAM, DIF, F-91297 ARPAJON, FRANCE

ACTINIDES 2013 KARLSRUHE| 21-26 July2013| PAGE 1

DENSITY FUNCTIONAL THEORY, T= 0 K
DFT (GGA, +U, +DMFT…) has been a successful tool to understand the ground state
properties of the actinides and their compounds : Structures, Equilibrium volume, Bulkmodulus, elastic constants, phase transitions in pressure…

itinerant

localized[A. Lindbaum et al. J. Phys Cond Matt15, S2297 (2003)]
R. C. Albers, Nature 410, 759-761 (2001)

T ≠ 0 K ???

PuU
Fisher and McSkimin 1961
[Los Alamos Science, number 26, 2000]
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Comparison with experiments at room temperature.
Low melting points.
Dynamical instability of the bcc structure.
Elastic constants of uranium at low T.
CDW in uranium
Thermal conductivity of nuclear fuels
Thermal dilation (uranium, plutonium)
Softening of the bulk modulus of Pu
Phase transitions (low symmetry vs high symmetry)…

UA. Migliori, Phys. Rev. B73, 052101 (2006)

PHONON SPECTRUM
a-U Phonon Spectrum

Soft modes,structuralstability

𝑉𝑠 =𝐶𝑖𝑗𝜌

PDOS g(w) w j  q   Fph V , T   k BT  ln  2sinh  2k T  q, jB U, Svib, CV…
| PAGE 4

ATOMIC MOTIONS AND PHONON SPECTRA IN DFT
Density functional perturbation theory (DFPT) T= 0 K
Harmonic approximation : no thermal expansion, no phase transitions (melting)
Quasi harmonic approximation : phonon frequencies are volume dependent

F (V , T )  E (V )  Fph w , T   Fe T Structures dynamically stable at 0 K
Weak anharmonicity

 w j  q   Fph V , T   k BT  ln  2sinh  
2kTq, jB Bcc unstable at 0 K
Low melting point, phase transitions

HARMONIC-ANHARMONIC : Al VS Pu

bcc Pufcc AlOUTLINE

 Introduction. DFT, a ground state theory (T=0 K)

a, orthog, bcc

 T≠0 K : DFPT and Quasi Harmonic approximation

 Failure of the QHA for uranium at low T.
Introduction of a new method : TDEP Phase transitions in uranium The case of plutonium.

d, fcc U-Mo alloyse, bcc| PAGE 7

URANIUM METALUranium is the only element discovered so far to exhibit CDW phase transitions atambient pressure.
Evolution of the soft mode in temperature shows a phase transition and a doublingof the unit cell in the [100] direction.

Ta-U

a1-U
[Smith et al., Phys. Rev. Lett. 1980]Uranium-Phonon spectrum with DFPT (T=0 K)
Pressure[ W.P. Crummett et al. Phys. Rev. B 19, 6028 (1979)][J. Bouchet Phys Rev B, 77 (2008)]

Pressure behavior confirmed by IXS[S. Raymond, J. Bouchet, G. H. Lander et al., Phys. Rev. Lett. 107,136401 (2011).]

FAILURE OF THE QHA (T≠ 0 K)

 QHA only takes into account the thermal dilatation
w(T)= w(V)
 Inadequate for uranium because of the soft modes a-U is NOT the correct structure at 0 K=A. Dewaele, J. Bouchet, F. Occelli, M. Hanfland, andG. Garbarino, Phys. Rev. B 88, 134202 (2013)

The phonon frequencies have to be explicitly dependent of the temperature

HOW TO TAKE INTO ACCOUNT THE TEMPERATURE?
AB INITIO MOLECULAR DYNAMICS
Ab-initio Molecular Dynamics (AIMD)?
FTAt each time step 𝜏 :𝐹𝑖 𝜏 =𝜱𝒊𝒋 𝑢𝑗 𝜏𝑗Equation of motion

Forces are related to displacements by
the interatomic force constants (IFC)𝜱𝒊𝒋 and then w will be temperature dependent
Temperature-dependent effective potential (TDEP)O. Hellman et al. PRB 84 180301 (2011)

TDEP METHOD
At each time step of the AIMD, we have the forces and the displacements :FTSecond Order : Phonon frequenciesw (T)Third Order : Grüneisen parameter
SiO. Hellman et al. PRB 84 180301 (2011)| PAGE 12

NEW METHODS TO TREAT ANHARMONICITY
BEYOND THE QHA
 Self-Consistent Ab-Initio Lattice Dynamics (SCAILD) [P. Souvatzis et al. 2008, P. Souvatzis etal. 2009, W. Luo et al. 2010],
 Stochastic Self-Consistent Harmonic Approximation (SSCHA) [I. Errea et al. 2014, I. Errea etal. 2014, L. Paulatto et al. 2015, M. Borinaga et al. 2016],
 Temperature Dependent Effective Potential (TDEP) [O. Hellman et al. 2011, O. Hellman
2013, P. Steneteg et al. 2013, J. Bouchet et al. 2015],
 Anharmonic LAttice MODEl (ALAMODE) [Tadano et al. 2014, Tadano et al. 2015],
 Compressive Sensing Lattice Dynamics [L. J. Nelson et al. 2013, F. Zhou et al. 2014].
 DynaPhopy [A. Carreras, A. Togo, and I. Tanaka, 2017, T. Sun, D. Zhang D., R. Wentzcovitch
2014] Other methods obtain anharmonic contributions via a derivation of the Gibbs energy [A. Glensket al. 2015],

WORKSHOP CECAM : “Anharmonicity and thermal properties of solids”
January, 10-12th 2018, PARIS| PAGE 13TEST CASE: Al| PAGE 14

FORCES : TDEP VS AIMD

| PAGE 15

CALCULATIONS DETAILS OF AIMD FOR U• Supercell : 4x2x3 of a-U = 96 atoms of uranium (up to 11th shell of nearestneighbors)• 32 kpoints• Experimental parameters (Llyod, Barrett J. Nucl. Mater. 1966)• 50, 300 and 900 K starting with the ideal positions• Around 3 000 time steps
Around 1-2 millions CPU hours
All the calculations have been performed using the ABINIT
package, PAW (14 valence electrons), GGA.

URANIUM : AVERAGE POSITIONS AT 300 AND 50 K300 K50 K[110][011]No change in the [011] plane, theatoms stay in the ideal positions

At 50 K, the atoms adopt new equilibrium positions
with a small displacement in the x direction

URANIUM : TDEP (T≠ 0K) VS DFPT(T=0K)
Comparison TDEP-Exp at 300 K

Comparison TDEP-DFPTExp

 At V(900 K), the a-U structure is unstable with DFPT At V(300 K), TDEP gives results comparable to exp
while DFPT still predict a destabilization of a-U
 At V(50 K), TDEP predicts the phase transition towards
the CDW state

J. Bouchet & F. Bottin., Phys. Rev. B 92, 174108 (2015)

URANIUM : PHASE DIAGRAM
To find the transition line between two structures we need tocompare their Gibbs energies :G(P,T) = F(P,T)+PV(P,T)
With

F(T,V)=E(0,V)+Fvib(V,T)+Fel(V,T)

bcc g phase[J. Bouchet & F. Bottin., Phys. Rev. B 95, 054113 (2017)]

URANIUM : PHASE DIAGRAM
F(T,V)=E(0,V)+Fvib(V,T)+Fel(V,T)

CS Yoo et al, Phys. Rev. B 57 10359 (1998)

Bulk & Shearbcc g phasea phase

PHONONS IN d-PU
Exp: J. Wong et al., Science 301, 1078 (2003)

 All the unusual features are reproduced at 600 K
 The d phase is unstable at 300 K (a-Pu) and at 900 K (e-Pu)
 At 300 K, the d phase is stabilized by a small amount of Ga[B. Dorado, F. Bottin & J. Bouchet , Phys. Rev. B 95, 104303 (2017)]

NEGATIVE THERMAL EXPANSION Experimentally, d-Pu has a NTE. Until now, no theory has ever been able to capture it.
 Grüneisen and thermal expansion coefficients in d-Pu:

Influence of volume change on
phonon frequencies

Volume variation as a function of T

D. C. Wallace
Phys. Rev. B 58, 15433 (1998)

 d-Pu NTE also correctly reproduced (though larger than experiments).
 Analysis shows the soft mode in G-L is responsible for the NTE.

Plutonium: bcc ε phase stabilization
DMFT (T=0K)

bcc e phasebcc is unstable at 0 K even with DMFT or LDA+UAIMD with LDA shows a disordered structure
AIMD with LDA+U gives a gradual stabilization of the bcc structure around 900 K
Calculated transition temperature = 1000K (exp=750K)
See also P. Söderlind, Scientific Reports 7, 1116 (2017)
B. Dorado, J. Bouchet & F. Bottin., Phys. Rev. B 95, 104303 (2017)| PAGE 23

Uranium-Molybdenum Alloys
 Motivations
Uranium metals are promising nuclear fuels
Pure uranium has three allotropes :α-U orthorombic, β-U tetragonal, γ-U body centered cubic
The γ-U phase is a good option for nuclear fuel, but it's unstable at low temperature (T<1000K)

Stabilize the γ phase by alloying uranium with a bcc metal such as Mo

 Goals
Construct the phase diagram of the bcc U-Mo system
Study the γ-stabilization effect of molybdenum

Steiner et al, J Nucl. Mater. 500 (2018) 184

29/03/2019| PAGE 24

Ab-initio computation

 Ab-initio Molecular Dynamics (AIMD) in the NVT ensemble
 GGA functional with the PAW formalism as implemented in Abinit 4x4x4 supercells with 128 atoms
 Random alloys are modeled by Special Quasirandom Structures (SQS)

Zunger et al. Phys. Rev. Lett. 65, 353 (1990)

29/03/2019| PAGEγ-stabilization effect in UMo

Stabilization of the bcc phase in UMo
29/03/2019| PAGE 26

MIXING FREE ENERGY| PAGE 27

CONCLUSIONS The standard methods (DFPT, QHA) have limited applications for the actinides.

 AIMD and TDEP give phonon frequencies with an explicit temperature dependence.
 The CDW phase transition is well predicted as the transition line between a and g-U
 The high temperature phases of Pu are found stable with TDEP Stabilization of bcc U by Mo Phase transitions mechanisms between a and d plutonium
 Phase diagram of Pu
 Higher orders terms (phonon lifetime, thermal conductivity…)

Arigatou gozaimasu
Thank you for your attention