Transcript Slide 1

Heavy-atom molecules as key objects
to study nonconservation of timereversal symmetry (EDM of electron)
[qchem.pnpi.spb.ru]
PNPI
QChem
Anatoly V. Titov
Group:
A.N. Petrov, L.V. Skripnikov and N.S. Mosyagin
B.P. Konstantinov PNPI RAS,
St.-Petersburg State University,
St.-Petersburg, RUSSIA
Outline
• Why measure electric dipole moment (EDM) of the
electron?
• Milestones in studying PNC effects
• Current status of the electron EDM (eEDM) search
• How to measure eEDM
– Recent experiments with heavy-atom molecules & solids
• Electronic structure modeling for eEDM studies
– Why heavy-atom systems and why relativistic effects are important
– Semi-empirical; RECP / one-center restoration; four-component
• Recent calculations of electronic properties in
heavy-atom molecules for the eEDM experiments
– …
Why measure EDMs?
• EDM is the electric dipole moment of an
elementary particle.
• Dipole moment d is a polar vector
(P-odd, where P is the space parity);
 D
r ~d
 
SS ~ r  p
P
T
[L. Landau, Pis’ma ZhETP 32, 405 (1957)]:
The only vector for a particle at rest is its spin,
therefore, d should be directed along to S,
(moreover, d = deS similar to magnetic moment, where de is a fixed real number
once the coordinate system is chosen)!
• Spin S is a T-odd pseudo-vector (P-even axial vector),
where T is the time-reversal symmetry.
Therefore, for P-even and / or T-even world d should be zero!
Nonzero EDMs can exist only due to the both space Parity, P,
and Time reversal, T, symmetries violated (P,T-odd interactions)
Hamiltonian for an EDM in electric field:
d  de /2 (1)
H EDM ~   E (2)
P :    , P : E   E (3)
T :    , T : E  E (4)
P,T: H EDM  H EDM
(5)
Status
Experimental limit
on the electron EDM:
|de| < 1.610-27 ecm
[B. Regan, E. Commins, C. Schmidt,
D. DeMille, PRL 88, 071805 (2002)]
Physics model
|de| (e·cm)
Standard Model
<10-38
Left-right symmetry
10-26-10-28
Multi-Higgs
10-27-10-28
Technicolor
~10-29
Supersymmetry
~10-25, 10-27 …
Estimated with
the KL0 (T-odd!)
decay exptl data
Basic detection scheme of an EDM
(for a neutral system with S=1/2, magnetic moment  and EDM d)
H    B  d  E
The wavepacket
e -iE t | + e -iE t |
can be prepared using
the electronic spin
resonance method
1
1 
2  B  2dE
Statistical sensitivity:
For S  :
1  2 
2
2 
2  B  2dE
4dE
Single system with coherence time  :
 
1

~ m
:
N uncorrelated systems measured for time TT ~
1
d ~
2 E 2 TN
P,T-odd effects in heavy-atom molecules:
 1965: Sandars suggests to use heavy atoms to search for EDMs
(In the nonrelativistic case Eeff is zero in accord to the Schiff theorem;
relativistic eEDM enhancement Eeff /Eextα2Z3
[V.Flambaum, Sov.J.Nucl.Phys. 24 (1976)]
EDMs of charged particles e-, p etc. can be studied !
)
 1967: Sandars: in polar heavy-atom molecules Emol /Eext >> 1.
He initiated the search for the P,T-odd effects on 205TlF and estimated the
these effects semiempirically (Eeff  20 kV/cm on a valence proton).
 1991: The last series of the 205TlF experiments is finished by Hinds group
at Yale (USA) and the best limitation on the proton EDM,
dp=(-4 ± 6)x10-23 ecm, is obtained.
 2002: Petrov et al. recalculated it with RCC as dp=(-1.7 ± 2.8)x10-23 ecm.
P,T-odd effects in heavy-atom molecules (cont.):
 1978: Labzowsky: ideas to use diatomic radicals CuO, CuS, CuSe due
to additional enhancement of P-odd, because of the closeness of levels of
opposite parity in -doublets having a 2Π1/2 ground state, Emol /Eext 105.
 1978: Sushkov & Flambaum, and in 1979 Gorshkov, Labzowsky &
Moskalev: ideas to use diatomic radicals (-doublets) to search for
P,T-odd effects including EDM of electron due to additional enhancement.
1984: Sushkov, Flambaum & Khriplovich; Flambaum & Khriplovich,
1985 : Kozlov suggest to use diatomics with a 2Σ1/2 ground state.
 Many new molecules, molecular cation and solids are considered up-todate for the eEDM search, mainly by Novosibirsk & SPb groups.
 2002: The last series of the 205Tl beam experiment is finished at Berkeley
(USA) and the best to-date limitation on de, |de| < 1.610-27 ecm, is obtained
 2002: The first results are obtained by Hinds group on the 174YbF
molecular beam at Sassex (UK) for the electron EDM,
de=(-0.2 ± 3.2)x10-26 ecm;
 2010 (???): some new limitation on de is obtained on YbF.
Experiments on the electron EDM Search
Heavy-atom polar molecules and cations:
 YbF-radical beam
(E.Hinds: Imperial college, London,UK);
 ThO* beam [& PbO* in optic cell ]
(ACME collaboration:
D.DeMille:Yale Uni.; J.Doyle & G.Gabrielse: Harvard);
 PbF radicals in a Stark trap
(N.Shafer-Ray: Oklahoma);
 HfF+ (& ThF+, PtH+ …) trapped cations
(E.Cornell: JILA, Boulder);
 WC (3Δ1 – ground state) molecular beam (A.E.Leanhard: Michigan U.)
Solids:
 Gd-Ga Garnet
(S. Lamoreaux: LANL ; C.-Y. Liu: Indiana)
 Gd-Iron Garnet
(L. Hunter: Amherst),
 Eu0.5Ba0.5TiO3 (perovskite, ferroelectric structure)
(S. Lamoreaux: Yale Uni; J.Haase: Leipzig Uni; O.Sushkov: UNSW).
What should be calculated ?
•
HP,T-odd = Wd de (Je n),
where de=| de |, (Jen)= is projection of the electron
momentum on the molecular axis (n);
Wd || / Elab characterizes the eEDM enhancement.
• The value of Wd || can be considered as some
effective electric field on the electron, Eeff  Wd ||.
It is non-zero only due to the relativistic effects!
• This field is strongly localized near the heavy nuclei, so
the only one-electron-states with small je contribute to Wd:
For point nucleus:
 nlj ~ r
 j 1
,
 j  ( j  1 / 2)  (Z )
2
2
Calculations of PNC effects
in heavy-atom molecules:
 First ab initio nonrelativistic calculations of P,T-parity nonconservation effects in
TlF followed by the relativistic scaling were performed by Hinds & Sandars in
1980 and by Coveney & Sandars in 1983 (Oxford, UK).
 A series of semiempirical calculations was performed since 1978 by Kozlov &
Labzowskii (St.Petersburg); Sushkov, Flambaum & Khriplovich (Novosibirsk) for
many heavy-atom molecules.
 Two-step (RECP / one-center-restoration) relativistic calculations at SPbSU, PNPI:
RECP = Relativistic Effective Core Potential method
without correlations: on PbF & HgF (1985-1991);
with correlations:
on YbF (1996,1998), BaF (1997), TlF (2002),
PbO* (2004), HI+ (2005), liquid Xe & HfF+ (2006+); PtH+(2009)
 First Dirac-Fock calculations on TlF (1997) and YbF (1998) are performed by
Parpia (USA) and by Quiney et al. (EU).
In 2006, correlation four-component calculation of BaF and YbF are performed by
Indian group (Nayak & Chaudhuri).
 … PtH+, ThO & ThF+ (2008) are performed “semi-ab-initio” by Meyer & Bohn
(JILA, Boulder, USA).
Methods of calculations
• Effective Hamiltonian(s):
Generalized RECP / NOCR methods (SPbSU-PNPI):
A.V. Titov & N.S. Mosyagin, IJQC 71, 359 (1999);
A.V. Titov et al., PTCP B15, 253 (2006).
• Correlation Methods:
RCC:
U.Kaldor, E.Eliav, A. Landau, Tel-Aviv Uni., Israel;
SODCI: R.Buenker et al., Uni. of Wuppertal, Germany);
Developments: A.V. Titov et al., IJQC 81, 409 (2001);
T.A. Isaev et al, JPB 33, 5139 (2000);
A.N.Petrov et al., PRA , 72, 022505 (2005).
• Basis Sets: GC-basis:
N.S.Mosyagin et al., JPB, 33 (2000);
T.A. Isaev et al, JPB, 33 (2000);
ANO basis sets for light atoms.
Что делает псевдопотенциал (ПП)
Задачей метода ПП является сведение расчета
электронной структуры системы к явному рассмотрению в
расчете только валентных электронов, т.е.
– исключение химически неактивных (остовных) электронов из расчета
при сохранении достаточно точного описания электронной структуры и
взаимодействий в валентной области;
– обеспечение «ортогональности» (принципа Паули) по отношению к
занятым (но явно исключенным) остовным состояниям, т.е.
предотвращение «провала» валентных электронов в эти состояния;
– эффективный учет релятивистских эффектов (scalar + SO + Breit);
– сглаживание псевдоспиноров для минимизации размеров атомных
базисов и вычислительных издержек в зависимости от задачи:




«large-core» ПП (наиболее экономичные, плохая точность)
«small-core» ПП (менее экономичные, хорошая точность)
корреляционный псевдопотенциал
возможность восстановления электронной структуры в остовах.
При универсальности метода ПП он является
наиболее гибким в расчетах электронной структуры.
Radial parts of large components of spinors 5s1/2 and 6s1/2
and of corresponding pseudospinors for the Thallium atom.
«Согласованные-по-форме» ПП
Наиболее важные особенности СФ ПП являются следствием
двух естественных ограничений при его построении:
 требования «жесткости» ПП в остове (r<Rc)
(с учетом свойства жесткости исходного атомного потенциала по
сравнению с амплитудами взаимодействий в валентной области и
валентными (орбитальными) энергиями);
 требования «физичности» ПП в валентной области (r>Rc)
(т.е. взаимодействия, смоделированные посредством ПП должны с
высокой точностью отслеживать исходные атомные потенциалы в
валентной области).
 как валентные, так и виртуальные псевдоорбитали будут с высокой
точность отслеживать исходные атомные орбитали в валентной
области вместе с их орбитальными энергиями даже при введении
возмущения в валентной области (хим.связь, внешние поля и т.п.);
 точность расчетов с ПП становится прогнозируемой и управляемой,
а процедура восстановления орбиталей в остове – обоснованной.
Radial parts of the 7s1/2 spinor (all-electron Dirac-Fock) and pseudospinor
32-electron GRECP/SCF) of Uranium for the state averaged over the
nonrelativistic 5f26d17s2 configuration and their difference multiplied by 1000.
Nonvariational One-Center Restoration (NOCR)
of electronic structure in cores of heavy-atoms in a molecule:
Advantages & disadvantages of GRECP / NOCR scheme:
 Удается естественным образом разделить задачу на две части –
атомную (с большим числом электронов и численными функциями) и
молекулярную (с минимальным числом электронов и гауссовыми функциями);
 «Естественное» представление в расчете остовных функций как спиноров, а
валентных – как спин-орбиталей – за счет «приближенного» учета их ортогональности в ПП-расчете, что невозможно в полноэлектронном расчете;
 Выполнение молекулярного расчета в спин-орбитальном базисе дает очень
большую экономию ресурсов, позволяет существенно повысить точность;
 Хотя молекулярные псевдоорбитали не ортогональны (строго!) к остовным, но
при их восстановлении также восстанавливается и их точная
ортогональность; при этом восстановленные валентные функции уже
являются не спин-орбиталями, но спинорами!
 Спин-орбитальным взаимодействием в валентном расчете с ПП часто
можно пренебречь (или учесть приближенно), и «включить» его только при
восстановлении в остовах, что очень важно в расчетах сложных соединений;
 Не учитывается поляризация (релаксация) остова, кот. обычно невелика; она
может быть учтена в «вариационной» схеме восстановления.
First two-step calculations of 199HgF and 207PbF
HI+ model:
Iodine (Z=53): [Kr] 4s24p64d10 5s25p5 + H+: 1s0
[ outer core ] [valence]
[valence]
HI+ ground state: 23/2;
configuration: […] 2 1/22 3/21 (derived from 5p5)
Highest doubly occupied -orbital is bonding and
most “mixed”:   5p0(I) +1s(H)
 is not highest-by-energy among the occupied
orbitals, but it gives 77% to the molecule-frame
dipole moment.
HI+:
<Basis sets>:
I:[5s5p3d2f] + H:[4s3p2d]
HI+:
<Basis sets>:
I:[5s5p3d2f] + H:[4s3p2d]
Calculations of PNC effects
in heavy-atom molecules (continued):
(1 GV/cm = 0.2421024 Hz/e∙cm)
1
Old (2006)
(2008)
[См. постер К. Бакланова]
[См. постер А. Петрова]
??
60 (2010)
PtH+
28 (2009)
73
Неэмпирический расчёт Eu++
во внешнем электрическом поле
[См. постер Л.Скрипникова]
Eu++: 4s24p6 4d10 5s2 5p64f7
= -4.6
Вклады в K от матричных элементов s-p, p-d, d-f :
s
p
d
f
s
p
d
f
-
-3.3
0
0
-
0.3
0
-
-1.6
-
Thanks to:
 L.Labzowskii – initiator & supervisor of the PNC study
at SPbSU & PNPI
 M. Kozlov (PNPI)
 Yu.Yu. Dmitriev, A. Mitrushchenkov (SPbSU)
 I. Khriplovich, O. Sushkov & V. Flambaum
(Novosibirsk & Sydney, Australia)
 D. DeMille (Yale, USA)
 E. Cornell (Boulder, USA)
 E. Eliav & U. Kaldor (Tel Aviv, Israel)
 R. Buenker & A. Alekseyev (Wuppertal, Germany)
 A. Zaitsevskii (Kurchatovskii institute, Moscow)
Concluding remarks:
 The eEDM experiments on heavy-atom molecules (and solids ?) are
of key importance for modern theory of fundamental interactions and
symmetries – window for a new physics beyond the Standard model.
 High-accuracy calculations of prospective heavy-atom systems are of
increasing interest for the eEDM experiment.
 The two-step method – RECP / one-center-restoration – has better
flexibility than the four-component approaches and good prospects for
further improvement of accuracy [A.V.Titov et al., PTCP B15, 253 (2006)]
Accuracy is limited by present possibilities of correlation methods
rather than by basis set limitations, RECP and other approximations.
Extension of the method to study more complicated systems (solids
etc.) is simple (in contrast to four-component ones); applicability to
study other physical-chemical properties is straightforward.
 Further development of accurate effective Hamiltonians, correlation
methods and new basis sets is highly desirable for actinides/lanthanides!
The end.
Gadolinium Gallium Garnet (Gd3Ga5O12)
• Gd3+ in GGG - 4f75d06s0
(7 unpaired electrons)
• Atomic enhancement factor =
-4.91.6
• Langevin paramagnet
• Dielectric constant ~ 12
• Low electrical conductivity
and high dielectric strength
• Volume resistivity = 1016 -cm
• Dielectric strength = 10 MV/cm
for amorphous sample
Garnet Structure:
{A3}[B2](C3)O12
–A {dodecahedron}: M3
•Ca, Mn, Fe, R (La,..Gd,..Lu)
–B [octahedron],C (tetrahedron):
•Fe, Ga, …
Methods of calculations: GRECP & NOCR
Why use molecular ions?
[R.Stutz & E.Cornell, Bull.Am.Phys.Soc. 49, 46 (2004)]
• Ions are easy to trap (in RF quadruple trap);
• Potential for long spin coherence times (ion-ion repulsion);
• Can get Eeff/Elab= 109 (for Ω>1/2 have closely spaced
levels of opposite parity fully polarized with E ~ 10 V/cm);
• Rotating external electric field can be used for eEDM
measurements keeping the cold ions in the trap.
Mass-spectrometry:
HfF+ model
Proposal: HfH+: [L.Sinclair et al., Bull.Am.Phys.Soc. 450, 134 (2005)];
HfF+ & ThF+: [E.Cornell & A.Leanhardt, private communication].
Calculation: HfF+: [A.N.Petrov et al., PRA 76, 030501(R) (2007)+ …]
HfF+ working state - 31; config.:
[…] 12 21 1,
,
2
1
Hf 2+: […4f14 ]5s25p6 5d1 6s1 + F– : 1s2 2s2 2p6
[outer core] [ valence ]
[core] [ valence ]
1st question: which state is the ground one,
3
1 (config.: […]  2  2 )?!
or
1
1
1
2
(and if 31 is not the ground one, how to populate it?)
2nd question: which is effective field on e-, Eeff ?
3rd question: which transitions to excited states
(3, 1) can be used to measure the EDM signals?
Our SODCI
calculations
with HfF+
HfF+
14
2
Hf: [Xe 4f ] 5s 5p
2
F:
1s
[inner core]
6
[outer core]
2
2
5d 6s
2
5
2s 2p
[ valence ]
+
HfF valence (4e) configuration:
2
2 1 +
σ1 σ2 ( Σ )
2 1 1 3
3
3
1
σ1 σ2 δ ( Δ1, Δ2, Δ3, Δ2)
2 1 1 3
3
3
3
1
σ1 σ2 π ( Π2, Π1, Π0+, Π0ˉ, Π1)
σ1 ≈ 2pz
σ2 ≈ 6s
δ, π ≈ 5d
Our SODCI calculations with HfF+
The GRECP (with 60 e- in core) and basis set for Hf (Z=72) is
generated and used in 10e- & 20e-SODCI calculations;
basis sets: Hf: (12s,16p,16d,10f,10g) / [6s,5p,5d,3f,1g] our GC basis;
F: (14s,9p,4d,3f) / [4s,3p,2d,1f] ANO basis set.
Up to 12×106 selected SAFs are used in SODCI calculations.
• Effective electric field on e- : Eint = 2.51010 V/cm (5.81024 Hz/e∙cm);
• Hyperfine constants:
A|| [177Hf] = -1239 MHz ; A|| [19F] = -58 MHz ;
• The ground state is 11 and 31 is the long-lived (1/2~0.5 sec)
lying only about 2000 cm-1 higher [calc-n by Skripnikov L., Dec.2008];
• Spectroscopic constants and curves, electric dipole moments (moleculeframe and transition), radiative lifetimes are calculated for ten lowest
states. Errors for energetic properties are about 500 cm-1;
• 4f14 relaxation is shown to be not important for these studies.