Table Of ContentThermodynamic phase transition of UH : Role of electronic strong correlation
3
Yu-Juan Zhang,1 Bao-Tian Wang,1,2 Yong Lu,1 and Ping Zhang1,3,∗
1LCP, Institute of Applied Physics and Computational Mathematics, Beijing 100088, People’s Republic of China
2Institute of Theoretical Physics and Department of Physics,
Shanxi University, Taiyuan 030006, People’s Republic of China
3Center for Applied Physics and Technology, Peking University, Beijing 100871, People’s Republic of China
The electronic structure and thermodynamical properties of uranium trihydrides (α-UH3 and β-
UH3) have been studied using first-principles density functional theory. We find that inclusion of
strongelectroniccorrelation iscrucialinsuccessfully depictingtheelectronicstructureandthermo-
dynamicphasestabilityofuraniumhydrides. AfterturningontheHubbardparameter,theuranium
1
5f states are divided into well-resolved multiplets and their metallicity is weakened by downward
1
shift in energy, which prominently changes the hydrogen bond and its vibration frequencies in the
0
system. Without Coulomb repulsion, the experimentally observed α→β phase transition cannot
2
be reproduced, whereas, by inclusion of the on-site correlation, we successfully predict a transition
n temperaturevalue of about 332 K,which is close to theexperimental result.
a
J
PACSnumbers: 71.27.+a,71.15.Mb,63.20.D-
7
] Since the report of UH as one ferromagnetic (FM)
l 3
e uraniumcompounds1–3,ithasattractedlotsofattentions
- in nuclear fuel field in the past decades. In solid state,
r
t uraniumhydrideexistsmainlyintheformofcubictrihy-
s
t. dride (Pm3n; No. 223)4 with two allotropes, i.e., α-UH3
a and β-UH . α-UH [Fig. 1(a)] is an intermediate state
3 3
m from orthorhombic α-U saturated with H to β-UH 5–7.
3
- ItisthemetastablelowtemperaturephasewithtwoUH3
d
formulaunitsinoneunitcell. Theuraniumandhydrogen
n
atomsinthisstructureoccupythe 2(a)(0,0,0)and6(c)
o
(1/4, 0, 1/2) sites, respectively. β-UH [Fig. 1(b)] is the
c 3
[ stable high temperature phase with eight UH3 formula
units in one unit cell. The uranium atoms in this struc-
1
ture occupy the 2(a) (0, 0, 0) and the 6(c) (1/4, 0, 1/2)
v sites,andthehydrogenatomsoccupythe24(k)(0,0.156, FIG.1: (Coloronline)Crystalstructuresof(a)α-UH3and(b)
41 0.313)sites. Mulfordetal.6 foundthroughX-raydiffrac- aβn-UdHh3y,drwohgeerneatthoemlsa,rgreespanecdtisvmelayl.l balls denote the uranium
4 tion measurement that the transition temperature of α-
1 UH toβ-UH isbetween373Kand523K,whileGenos-
3 3
1. sar et al.7 reported that α-UH3 transits irreversibly into manyphysicalandchemicalpropertiesofUH areclosely
3
0 β-UH3 aboveroomtemperature. Atevenhighertemper- related to the quantum process of localization and delo-
1 atures(above673K),uraniumhydridereversiblyremove
calizationforpartiallyfilleduranium5f electrons. Mod-
1 the hydrogen. This property makes uranium hydrides
elingoftheelectronlocalization/delocalization,andthus
v: convenient starting materials to create chemically reac- any hydrogenation process involving uranium, is a com-
i tiveuraniumpowderalongwithvariousuraniumcarbide,
X plextask. Conventionaldensityfunctionaltheory(DFT)
nitride, and halide compounds. In contrast to the above
schemes that apply the local density approximation
r mentioned experimental measurements on temperature-
a (LDA)orthegeneralizedgradientapproximation(GGA)
induced α→β phase transition of uranium trihydride, to
underestimate the strong on-site Coulomb repulsion of
date the theoretical exploration of this prominent phase
the uranium 5f electrons and consequently fail to cap-
transition is still totally lacking in the literature. Con-
ture the correlation-driven localization. Therefore, the
sideringthecombinedfactthat(i)UH denotesatypical
3 uranium 5f electrons require special attention in trying
prototype among various kinds of actinide hydrides and
to gain any microscopic insight into the thermodynami-
(ii) the actinide hydrides play extremely important role
cal stability of UH . Up to now several approaches, the
3
in nuclear fuel design (as well as stockpile) in which the
LDA/GGA+U, the hybrid density functional of (Heyd,
thermodynamic stability is a substantial factor, there-
Scuseria, and Enzerhof) HSE, the self-interaction cor-
fore, a revealing theoretical study on the ground-state
rected local spin-density (SIC-LSD), and the Dynami-
properties and α→β phase transition of UH from first-
3 calMean-FieldTheory(DMFT),havebeendevelopedto
principles quantum mechanics is highly needed for the
correctthepureLDA/GGAfailuresincalculationsofac-
relevant important industrial applications.
tinidecompounds. Amongthesetechniques,theeffective
From basic point of view, it can be visualized that modification of pure DFT by LDA/GGA+U formalisms
2
has been confirmed widely in study of uranium (as well
asits neighbor,plutonium) oxides8–10. By tuning the ef-
fectiveHubbardparameterinareasonablerange,thean-
tiferromagnetic (AFM) Mott insulator features of these
oxides were correctly calculated and the atomic struc-
tural parameters as well as the electronic properties are
well in accord with experiments. Inspired by this series
ofsuccessfulcalculationsonAFMuraniumoxide,aswell
as motivated by the fact that uranium hydride is also a
magnetic ordered (FM instead of AFM) compound and
thus a static mean-field treatment like LDA/GGA+U is
expected to reliably catch its some key features in both
α and β phases, in the present work, we investigate the
ground-state electronic properties and lattice dynamics
of the two allotropes of UH using the LDA+U formal-
3
ism. For comparison the pure LDA calculation is also
performed. The most insightful result of our investiga-
tionisthatitisessentialtotakethe5f on-siteelectronic FIG.2: (Coloronline)Spin-resolvedtotalDOSforthetwoal-
correlationintoaccountfortheoreticallyreproducingthe lotropesofUH3calculatedwithinLDAandLDA+U (Ueff=4
experimentallyobservedα→β phasetransitionofUH at eV) formalisms. Partial DOS of U 5f and H 1s orbitals are
3
also shown. Thefermi level is set to be zero.
finite temperature. Also, the prominent changes in the
hydrogen bond and the consequent hydrogen vibration
(reflected by the optical branches in phonon spectrum)
state with U from 0 to 7 eV, which is well consistent
in UH by the inclusion of on-site Coulomb repulsion of eff
3 with the experiment. The dependence of the lattice pa-
the uranium 5f electrons is for the firsttime highlighted
rameters on U for FM phase of α-UH and β-UH
in this paper. eff 3 3
demonstrates that in α phase, the lattice parameter is
The calculations are performed using the projector-
close to the experimental value when U is in range of
augmented wave (PAW) method of Blo¨chl11, as im- eff
4–6eV,while inβ phase,the rangeis in4–5eV.Consid-
plemented in the Vienna ab initio simulation program
ering the two allotropes, therefore, we choose the value
(VASP)12. For the plane-wave set, a cut-off energy
of U to be 4 eV in our following study for the two
of 500 eV is used. The hydrogen 1s and uranium eff
phases. Our calculated lattice constants for α (β) phase
6s26p66d25f27s2 are treated as valence electrons. The
of UH at U =0 and 4 eV are 4.001 (6.382) and 4.128
strong on-site Coulomb repulsion amongst the localized 3 eff
(6.622) ˚A, respectively. Compared with pure LDA, our
uranium 5f electrons are accounted for by using the for-
LDA+U gives closer values with respect to the experi-
malism formulated by Dudarev et al.13. In this scheme
mentalvaluesof4.1606 and6.63915 ˚Aforαandβ phase,
the total energy functional is of the form
respectively.
ELDA+U =ELDA+ U −2 J X[Trρσ−Tr(ρσρσ)], (1) (DOOuSr) acanldcutlhaetepdarstpiainl-DreOsoSlvfeodr utoratanliudmen5sfityanodfhsytdatreos-
σ
gen 1s states of the two allotropes are presented in Fig.
where ρσ is the density matrix of f states, and U and 2. The upper panels show the pure LDA results, while
J are the spherically averagedscreened Coulomb energy the loweronesgive the LDA+U results. FromFig. 2 we
andtheexchangeenergy,respectively. HeretheCoulomb can observe evident effects of strong correlation on the
U is treated as a variable, while the exchange energy is f states. For both two phases of UH , after turning on
3
set to be a constant J=0.51 eV. This value of J is in theHubbardU parameter,clearly,theuranium5f states
theballparkofthecommonlyacceptedoneforuranium. aredividedintomorewell-resolvedpeakscomparedwith
SinceonlythedifferencebetweenU andJ issignificant13, pure LDA, and the energy distributions of these multi-
thus we will henceforthlabelthem asone single parame- ple 5f peaks are much localized and separated. As a
ter, for simplicity labeled as U , while keeping in mind result of this multiple peak splitting, for α phase, the
eff
that the non-zero J has been used during calculations. uranium 5f states in the range of −1.0 to 2.0 eV within
TheBrillouin-zone(BZ)integrationsareperformedusing LDA are extended to lie in the range of −2.5 to 5.0 eV
9×9×9 and 7×7×7 Monkhorst-Pack14 special k-points in the LDA+U formalism. For β phase, similar effects
for α-UH and β-UH , respectively. The geometries are of strong correlation can be found. Moreover, the occu-
3 3
optimized until the forces are less than 0.02 eV/˚A, and pation of both spin-up and spin-down electrons at the
the total energy is relaxed until the difference value is Fermi level are loweredafter taking into account the on-
smaller than 10−5 eV. siteCoulombrepulsion. However,noinsulatingbandgap
Through calculating the total energy dependences on are opened by the inclusion of Hubbard parameter and
U ofα-UH andβ-UH innonmagnetic,FM,andAFM UH still remains its metallic nature as observed in the
eff 3 3 3
phases, we find that the FM phase is the most favorable experiment16.
3
TABLE I: Bader effective atomic charges and volumes of α-
UH3 and β-UH3 in the LDA and LDA+U (Ueff= 4 eV) for-
malisms.
Allotrope Methods Q(UI) Q(UII) Q(H) V(UI) V(UII) V(H)
α-UH3 LDA 12.527 1.490 17.913 4.703
LDA+U 12.425 1.525 18.847 5.415
β-UH3 LDA 12.503 12.434 1.503 18.290 17.465 4.904
LDA+U 12.452 12.370 1.535 18.774 19.283 5.706
Duetothedecreaseinmetallicityofuranium5f states
andtheirenergydownwardshifttowardsthehydrogen1s
orbital by taking account of the electronic strong on-site
correlation, the ionicity of the hydrogen bond is promi-
nently enhanced. To analyze the ionicity of the two al-
lotropes of UH , results from the Bader analysis17 are
3
shown in Table I. The charge (Q) enclosed within the
Bader volume (V) is a good approximation to the total
electronicchargeofanatom. Notethatalthoughwehave
included the core charge in charge density calculations,
since we do not expect variations as far as the trends
are concerned, only the valence charge is listed. From
the results we calculated, it is seen that the ionicity of
thetwoallotropeswithinLDA+U aremoreevidentthan
that within pure LDA. For α phase, the electrons trans-
ferredfromeachU atomto H atoms are1.473and1.575
within LDA and LDA+U formalisms, respectively. For
β phase, electrons transferred from two inequipotential
U atoms (U and U ) to H atoms are 1.548 and 1.630,
I II
respectively, in LDA+U formalism. Whereas, in LDA
formalism the values are 1.497 and 1.566 respectively. FIG. 3: (Color online) Phonon dispersion curves (left panel)
and corresponding phonon DOS (right panel) of (a) α-UH3
Toillustratethemainpointofthispaper,i.e,theeffect
and (b) β-UH3. The black curves are computed within the
of electronic strong on-site correlation on the thermody- LDAwhile thered (grey) curves theLDA+U formalism.
namic properties of UH , we havecalculatedthe phonon
3
dispersion and the Helmholtz free energy F of two al-
lotropes with and without including Coulomb repulsion
acoustic branches come from the uranium vibration. As
of 5f electrons. Phononfrequency calculations were car- a result, a large gapbetween optical and acoustic modes
ried out using the Hellmann-Feynman theorem and the
can be observed. The key point revealed in Fig. 3 is
directmethod18. Forαandβ phases,weadopted2×2×2 that the lattice dynamics behaviors are prominently in-
supercells containing 64 and 256 atoms with 3×3×3and
fluenced by electronic strong on-site correlation, which
1×1×1 Monkhorst-Pack k-point meshes in the BZ inte-
has changed the U-H bonding nature. This influence on
gration, respectively. The forces induced by small dis-
latticedynamicsturnsout,asanalyzedbelow,tobe fun-
placementswerecalculatedwithinVASP.Theamplitude
damentalto reproducethe experimentallyobservedrela-
of all the displacements is 0.03 ˚A. tivethermodynamicstabilitysequenceofαandβ phases
The calculated phonon curves along some high- of UH .
3
symmetrydirectionsintheBZtogetherwiththe phonon For the sake of determining phase transition tempera-
DOS are plotted in Fig. 3. Clearly, the effect of elec- ture ofα-UH andβ-UH atambientcondition, wehave
3 3
tronic strong correlation on phonon dispersion can be calculated the Helmholtz free energy F in the LDA and
observed by comparing results from LDA and LDA+U LDA+U formalisms. This quantity at volume V and
approaches. Compared with pure LDA calculation, the temperature T can be expressed as
optical branches calculated within LDA+U shift down
by around 2.5 THz and 2.3 THz for α-UH3 and β-UH3, F(V,T)=E(V)+Fvib(V,T)+Fele(V,T), (2)
respectively. However,the acousticbrancheshaveno ev-
ident changes for both phases. Due to the fact that ura- where E(V) is the 0 K band energy, F (V,T) is the
vib
niumatomismuchheavierthanhydrogenatom,thenthe phonon vibrational free energy, and F is the thermal
ele
opticalbranchesdenotethehydrogenvibrationwhilethe electronic contribution to the free energy. The phonon
4
temperature range of 0 to 1000 K, as shown in Fig.
4. This is contrary to the experimental observations.
Whereas, after including the electronic strong correla-
tion, the calculated Helmhotz free energy curves of the
two allotropes cross at the temperature of 332 K, which
is well consistent with the experimental value for α→β
phase transition. Therefore, we arrive at that while the
pureLDAcalculationtotallyfailstoreproducetheexper-
imentallydeterminedthermodynamicphasetransitionof
UH ,theLDA+U calculationcanwelldepictit. Thisisa
3
good news for the science of actinide hydrides. Needless
to say, more advanced many-body techniques, such as
DMFT, which can account for some spin fluctuations at
finite temperatures,willfurtherimprovethecalculations
FIG. 4: (Color online) Temperature dependences of the and we would like to leave it for future consideration.
Helmholtz free energies within LDA (the black curves) and
In conclusion, the ground-state properties of two al-
LDA+U (the red/grey curves) formalisms. The inset gives
lotropes of UH have been comparatively studied within
closer view of the crossing point within the LDA+U. 3
the LDA and LDA+U formalisms. After switching on
the on-site Coulomb interaction, the uranium 5f states
vibrational free energy in the quasiharmonic approx- split into the multiple well-resolved peaks that shift up-
imation can be calculated from phonon DOS g(ω,V) ward or downward with respect to the Fermi energy,
as F (V,T)=k T ∞g(ω,V)ln[2sinh( ~ω )]dω. The which reduces the metallicity of the 5f states and en-
thermviabl electroBnicR0contribution can2kBbTe described hances the ionicity of the hydrogen bond in the system.
as Fele(V,T)=Eele−TSele with electronic entropy Asaresult,thecalculatedopticalbranchesinthephonon
Sele(V,T)=−kBR n(ε,V)[flnf+(1−f)ln(1−f)]dε, dispersionoftwoallotropeshavebeenfoundtoshiftdown
where f is Fermi-Dirac distribution and n is the elec- by about 2.5 and 2.3 THz for α-UH3 and β-UH3, re-
tronic DOS. The chemical potential at temperature T is spectively, by turning on the Hubbard parameter. Our
fixed by conservation of total valence electron number theoreticalHelmhotz free energycurveshaveshownthat
during thermal excitation. It turns out that the value of whilethepureLDAfailstodescribetheα→βphasetran-
the thermal electronic contribution is very small, only sition, the LDA+U calculation gives the transition tem-
one-tenth of the phonon vibrational free energy. perature of 332 K, which well lies within the experimen-
We have calculated and plotted the temperature de- tally observed range. These results clearly indicate that
pendences of the Helmhotz free energy for the two UH the electronic strong on-site correlation plays an impor-
3
phases within LDA and LDA+U formalisms (Fig. 4). A tant role in the uranium hydride systems.
significantdifferencecanbeclearlyobservedbetweenthe This work was supported by NSFC under Grant No.
two formalisms. Without including the on-site Coulomb 51071032, by the Foundation for Development of Sci-
interaction,theHelmhotzfreeenergiesoftwoUH phases ence and Technology of China Academy of Engineering
3
have no crossing point, and the α phase is always more PhysicsunderGrantNo. 2009A0102005,andby the Na-
thermodynamically stable than the β phase in a wide tional Basic Security Research Programof China.
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