1 22 Lattice dynamics and elastic properties of α-U at high -temperature and high -pressure by machine learning potential simulations

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Lattice dynamics and elastic properties of α-U at high-temperature and

high-pressure by machine learning potential simulations

Hao Wang1, 2, Xiao-Long Pan1, 2, Yu-Feng Wang2, Xiang-Rong Chen1*, Yi-Xian Wang3, Hua-Yun Geng2, 4

1 College of Physics, Sichuan University, Chengdu 610065, People’s Republic of China

2 National Key Laboratory of Shock Wave and Detonation Physics, Institute of Fluid Physics, CAEP, Mianyang

621900, People’s Republic of China

3 College of Science, Xi’an University of Science and Technology, Xi’an 710054, People’s Republic of China

4HEDPS, Center for Applied Physics and Technology, and College of Engineering, Peking University, Beijing

100871, People's Republic of China

Abstract: Studying the physical properties of materials under high pressure and temperature through

experiments is difficult. Theoretical simulations can compensate for this deficiency. Currently, large-

scale simulations using machine learning force fields are gaining popularity. As an important nuclear

energy material, the evolution of the physical properties of uranium under extreme conditions is still

unclear. Herein, we trained an accurate machine learning force field on α-U and predicted the lattice

dynamics and elastic properties at high pressures and temperatures. The force field agrees well with

the ab initio molecular dynamics (AIMD) and experimental results and it exhibits higher accuracy

than classical potentials. Based on the high-temperature lattice dynamics study, we first present the

temperature-pressure range in which the Kohn anomalous behavior of the Σ4 optical mode exists.

Phonon spectral function analysis showed that the phonon anharmonicity of α-U is very weak. We

predict that the single-crystal elastic constants C44, C55, C66, polycrystalline modulus (E, G), and

polycrystalline sound velocity (CL, CS) have strong heating-induced softening. All the elastic moduli

exhibited compression-induced hardening behavior. The Poisson’s ratio shows that it is difficult to

compress α-U at high pressures and temperatures. Moreover, we observed that the material becomes

substantially more anisotropic at high pressures and temperatures. The accurate predictions of α-U

 Corresponding authors. E-mail: xrchen@scu.edu.cn, s102genghy@caep.cn

 2 /  22 
 
demonstrate the reliability of the method. This versatile method facilitates the study of other complex 
metallic materials. 
Key words: Machine learning potential; lattice dynamics; elastic properties; α-U; high-pressure and 
high-temperature 
 
1. Introduction 
With the increasing shortage of energy today, metallic uranium, as an important nuclear energy 
material, has always attracted much attention[1]. Uranium exists stably in an orthogonal structure (α-
U) at normal  temperature and  pressure[2].  In addition,  it is  the elemental  substance that  can be 
observed the behavior of charge density waves (CDW) at normal pressure. 
  Crummett et al. measured phonon dispersion behavior of α-U at room temperature by neutron 
inelastic scattering experiments, and found that the optical branch in the middle of the [100] direction 
has a huge softening behavior[3]. Smith et al. further experimentally confirmed that this softening 
behavior  increased  further  with  decreasing  temperature,  showing  a  soft-mode-driven  phase 
transition[4].  Bouchet  reproduced  the  phonon  dispersion  behavior  of  α-U  by  using  the  density 
functional perturbation theory (DFPT)[5]. In addition, they found that the pressure effect has a strong 
influence on the softening of the α-U optical branch. This pressure effect was then experimentally 
confirmed by Raymond et al[6]. Bouchet et al.[7, 8] further studied the phonon dispersion behavior 
of  uranium  at  high  temperature  and  high  pressure  by  ab  initio  molecular  dynamics  (AIMD) 
calculation combined with the temperature-dependent effective potential technique (TDEP)[9, 10].   
Both theory and experiments show that α-U has one CDW and multiple Kohn anomalies at qCDW. 
Roy et al.[11] showed that various Kohn anomalies in α-U arise from the combined effect of Fermi 
surface nesting (FSN) and "hidden" nesting, that is, the nesting of electronic states above and below 
the  Fermi  surface.  The  topology  in  favor  of  Fermi  surface  nesting  (FSN)  allows  the  electron 
susceptibility χ0 to diverge and induce a CDW at the wave vector qCDW.   
  At present, the mechanical study of uranium under high-temperature and high-pressure is lacking, 
especially for α-U. Fisher once experimentally measured the variation of the elastic constant of α-U 
with temperature at ambient pressure, and found that the C11 of α-U had anomalous behavior at low 

 3 /  22 
 
temperature[12]. Due to the low symmetry of α-U, to measure the elastic constant at high pressure is 
much more difficult. Density functional theory (DFT) can accurately describe the change of the elastic 
constant of crystal at high pressure [13-15]. In order to take the temperature effect into account, 
however, AIMD simulation must be used. Unfortunately, due to the complexity of the α-U structure, 
the computational cost required to calculate its elastic constant by AIMD is much larger than that of 
materials  with  high  structural  symmetry  such  as  Al[16]  and  TiN[17].  Bouchet  et  al.[8]  gave  a 
relationship between the bulk modulus and shear modulus of uranium with pressure and temperature 
through AIMD+TDEP. However, the variation of mechanical anisotropy and the sound velocity with 
pressure and temperature is unknown. 
It should be pointed out that lattice parameters, phonon properties and elastic constants of α-U 
calculated by using the classical interatomic potential of uranium currently have large discrepancy 
against the experimental data[18-22]. In recent years, there have many different machine learning 
(ML) potential models  been  developed,  such as  Neural  network potentials  (NNP)[23],  Gaussian 
approximation potentials (GAP)[24, 25], Moment tensor potentials (MTP)[26, 27], Deep learning 
potentials  (DPMD)[28-30],  etc.  These  models  have  a  good  performance  for  materials  at  high 
temperature and high pressure, such as Si[31], Fe[32], Ti[33, 34], Zr[35, 36], and alloy compounds[37, 
38]. Existing results indicate that MTP and GAP have higher accuracy, while MTP is more efficient 
than GAP[39].   
So far, investigation on the ML potential of uranium is still few. Although Kruglov et al.[40] 
studied uranium by the moment tensor potential method, they mainly focused on the phase transition, 
and did not report other physical properties. Ladygin et al.[41] mainly used  γ-U to illustrate the 
accuracy of MTP in description of high-temperature phonon dispersion behavior. 
In this paper, we train an accurate machine learning force field based on DFT data. With this 
model, we resolve the lattice dynamics and elastic properties of α-U at high temperature and high 
pressure. Machine learning force field successfully predicts the Kohn anomaly in the lattice dynamics 
of α-U at high temperature and high pressure. We calculate the elastic behavior of α-U, giving its 
dependence on pressure and temperature. Meanwhile, we characterize its elastic anisotropy under 
high temperature and high pressure. 

4 / 22

2. Methods

2.1. Moment tensor potentials

In order to study the physical properties of α-U, we employ a machine learning model called

moment tensor potentials (MTPs)[26]. The MTP training is performed with the software named

“MLIP”[27]. MTP is a local potential. In this model, the total energy of the structure (

mtp

) is

expressed as the sum of atomic contributions V(ni), defined as

MTP

(cfg) ( )

=n

(1)

where N is the total number of atoms in that configuration. The atomic environment neighborhood

(ni) is composed of the i-th atomic type (Zi), the nearest neighbor atomic type (Zj) and the relative

position (rij) within a given cutoff radius.

In MTP, V(ni) can be expressed as a linear combination of a set of basic functions (Bα(ni)), which

is defined as follows

( ) ( )







=

(2)

In order to define the functional form of the basis Bα, the form of moment tensor descriptors is

introduced to describe the local information of atoms, as follows

( ) ( )

times

;

i ij ij

  



= 

crrn

(3)

where

( )



and

ij ij

rr

represents the radial and angular part, respectively. The symbol ⨂

denotes the outer product of vectors. The μ, ν ≥ 0 represents the different descriptors. The ξ and c

constitute the parameters θ(ξ, c) we need to determine during the training process:

( ) ( )

( )

( ) ( )

mtp qm mtp qm

mtp qm

cfg ; cfg cfg ; cfg

cfg ; cfg min

k k i k i k

w E E w

− + −







+ − →











  

(4)

where Nk is the number of atoms in the k-th configuration, E, f, σ represent energy, forces and virial

stress, respectively, we, wf, ws represent the non-negative weights of energy, force, and stress during

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training. For theoretical details of MTPs, such as the specific relationship between Bα and Mμ,ν, please

refer to the Ref. [26, 27, 42]. The training settings can be found in Table I.

Table I. MTP training set size, training parameter settings.

Number of structures

3600

Number of fitting parameters

864

Energy weight (we)

Force weight (wf)

0.1

Stress weight (ws)

0.01

Cut-off radius (Å)

6.0

2.2. Elastic constants

In molecular dynamics, the elastic constants were calculated using the stress-strain method[43].

According to Hooke’s law, the relationship between stress and strain tensor under Voigt symbol is

defined as

i ij j



=

(5)

where σi refers to stress tensor, Cij represents elastic constants, and εj refers to strain tensor. We can

obtain all the elastic constants of a material from Eq. (5), by applying the strain value and calculating

the corresponding stresses

Because α-U belongs to orthorhombic system and has nine independent elastic constants. For

obtaining C11, C12 and C13, we use a strain tensor:

0 0 0







=







(6)

For C22 and C23, a strain tensor follows:

000







=







(7)

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1/22Latticedynamicsandelasticpropertiesofα-Uathigh-temperatureandhigh-pressurebymachinelearningpotentialsimulationsHaoWang1,2,Xiao-LongPan1,2,Yu-FengWang2,Xiang-RongChen1*,Yi-XianWang3,Hua-YunGeng2,41CollegeofPhysics,SichuanUniversity,Chengdu610065,People’sRepublicofChina2NationalKeyLaboratoryofSho...

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