A possibility for precise Weinberg angle measurement in centrosymmetric crystals with axis

0.19 MB·English

by  T. N. Mukhamedjanov

A possibility for precise Weinberg angle measurement in centrosymmetric crystals
with axis
T. N. Mukhamedjanov, O. P. Sushkov
School of Physics, University of New South Wales,
Sydney 2052, Australia
Wedemonstrate that parity nonconservinginteraction dueto thenuclearweak charge QW leads
6
to nonlinear magnetoelectric effect in centrosymmetric paramagnetic crystals. It is shown that the
0
effectexistsonlyincrystalswithspecialsymmetryaxisk. Kinematically,thecorrelation(correction
0
2 toenergy)hastheformHPNC∝QWE·[B×k](B·k),whereBandEareexternalmagneticandelec-
tricfields. ThisgivesrisetothemagneticinductionMPNC∝QW{k(B·[k×E])+[k×E](B·k)}.
n To be specific, we consider rare-earth trifluorides and, in particular, dysprosium trifluoride which
a looks the most suitable for experiment. We estimate the optimal temperature for the experiment
J
to be of a few kelvin. For the magnetic field B = 1 T and the electric field E = 10 kV/cm, the

8 expected magnetic induction is 4πMPNC ∼ 0.5·10−11 G, six orders of magnitude larger than the
bestsensitivitycurrentlyunderdiscussion. Dysprosiumhasseveralstableisotopes,andso,compar-
] isonoftheeffectsfordifferentisotopesprovidespossibilityforprecisemeasurementoftheWeinberg
r
e angle.
h
t PACSnumbers: 11.30.Er,23.40.Bw,31.15.Md,71.70.Ch
o
.
t
a
I. INTRODUCTION However, the results of the early experiment [12] (1978)
m
performed in Ni-Zn ferrite were not impressive due to
-
d Studies of atomic parity nonconservation (PNC) pro- experimental limitations. In the recent paper [13], Lam-
n videimportanttestsofthestandardmodelofelectroweak oreaux suggested that application of novel experimental
o interactionsandimpose stringentconstraintsonthenew techniques in garnet materials should lead to substan-
c tial increase in sensitivity compared to the previous at-
physics [1]. The effects of PNC in atoms due to the nu-
[
clear weak charge Q have been successfully measured tempt;thissuggestionrenewedtheinterestintheoriginal
W
2 for bismuth [2], lead [3], thallium [4], and cesium [5, 6]. idea. Calculations [14] have shown that improvement of
v The authors of [5, 6] have also performed measurements the statistical sensitivity to the electron EDM of several
3 orders of magnitude was possible compared to the cur-
of the nuclear spin-dependent PNC effect in Cs caused
0
bythe nuclearanapolemoment(e.g.,see text[7]),which rent experimental limit on this value [15]. At the same
0
time, the experimentalsearchwasinitiated—the firstre-
1 remains the only measurement of this kind to date.
sults are alreadyavailable in the literature,[16] (see also
1 Inthepresentworkweconcentrateoneffectscausedby
5 thenuclearweakchargeinsolids. Wepredictanewkind Ref. [17]). The other suggestions for fundamental sym-
0 of (nonlinear) magnetoelectric effect in centrosymmetric metry violation search in solids include proposed search
/ formanifestationsofthenuclearanapolemomentingar-
t crystalsduetotheviolationofparity(P)atfundamental
a nets [18] (violation of parity), and the search for the
level, proportional to Q . Although we show that the
m W
nuclear Schiff moment in the ferroelectric lead titanate
experimental observation of the proposed effect is pos-
- sible at the current level, the uncertainty of theoretical compound [19] (violation of T and P symmetries).
d
Oneofthemostimportantissuesforanysolidstateex-
n calculationwouldnotallowforanaccurateinterpretation
periment is certainly the existence of various systematic
o in terms of the nuclear weak charge QW itself. Instead,
c the potential importance for fundamental physics lies in effects. It is getting more and more clear that the right
: choice of compound is crucial. In particular, any kind
v the isotope dependence of the effect we suggest. We ar-
of spontaneous magnetic ordering in the crystal (ferro-,
i guethatcomparingthevalueoftheeffectinsampleswith
X antiferro-, or ferrimagentism) should be avoided, thus,
differentisotopecontent,onecan,inprinciple,determine
r the Weinberg angle with high precision. In this respect we consider paramagnetic material—rare-earth trifluo-
a
the idea is similar to the suggestion presented in Ref. [8] ride RF3. The PNC weak interaction of electrons with
the nuclei of rare-earths gives contribution to the total
of an atomic PNC measurements in rare-earth Dy, and
energy of the crystal
the later suggestion of Ref. [9] for a measurement with
atomic Yb. EPNC QW(E [B k])(B k) , (1)
The idea to use solids for studies of fundamental sym- ∝ · × ·
metry violations dates back to 1968, when Shapiro sug- in the applied electric field E and magnetic field B; vec-
gestedthattheexistenceofthepermanentelectricdipole tor k is directed along the crystal symmetry axis. Cor-
moment (EDM) of electron should lead to linear magne- relation similar to Eq. (1) was considered previously by
toelectric effect in solids [10]. (The existence of the per- BouchiatandBouchiat[20]inregardtotheirproposalto
manent EDM of a quantum particle violates both parity measurenuclearanapolemomentusingCsatomstrapped
(P) and time-reversal (T) symmetries, see, e.g., [11].) in solid 4He.
2
Theeffectiveinteraction(1)inducesthefollowingPNC
4f mp ns
macroscopic magnetization which can be measured in a 1/2
magnetometry experiment:
2,4
M= ∂EPNC Q k(B [k E]) ns kp1/2 4f
− ∂B ∝ W · ×
n
+[k E](B k) . (2)
× · FIG. 1: A typical atomic perturbation theory diagram re-
o
sponsibleforthePNCHamiltonian(4). Thewavylineonthe
Naively, one can expect the PNC magnetization of the
diagramrepresentstheresidualCoulombinteractionbetween
form M [B E], however, this kind of magnetization
∝ × electrons, the weak interaction (3) is shown by the cross on
does not lead to any energy shift in the applied mag-
the electron line, and the dashed line denotes interaction of
netic field, and hence, does not appear. Eq. (1) presents electron with theexternal electric field E.
thesimplestenergycorrelationallowedkinematically. To
generate this correlation we need a crystal that allows
nontrivial second rank material tensors, so the crystal the ion from its equilibrium position with displacement
symmetry must be lower than cubic. In addition, we X. The ion has the orbital angular momentum L and
need a necessarily centrosymmetric crystal in order to the spin S, in this section we assume that L and S are
avoid the imitation of the PNC effect by spontaneous decoupled from each other. Certainly, in reality L and
parity breaking in the lattice. S are coupled due to the combined action of the spin-
Rare-earth trifluoride compounds have hexagonal orbit interaction and the non-central crystal field of the
structure described by C3,i space groupand, apparently, latticeattheionsite. Wewillconsidertheseeffectsinthe
satisfy the crystallographic criteria we set. These com- nextsection,butfornowweswitchtheseinteractionsoff.
pounds are widely used as laser materials, and as such, The interaction (3) is a T-even pseudoscalar, therefore,
their electronic structure has been studied extensively theeffectiveHamiltonianwhicharisesaftercalculationof
and is rather well-understood. As it will be seen later, the electronicmatrixelementof(3)mustbe ofthe form
DyF3 is, probably, especially well-suited for the exper-
iment because of the large value of the PNC effect in HPNC E [L S] . (4)
∝ · ×
this compound. Dysprosium has also particularly large
number of stable isotopes which can be used for the de- This is the only structure allowedkinematically, and the
termination of the Weinberg angle as mentioned above. presentsectionis devotedto the calculationofthis effec-
The structure of this paper is as follows. Section II tive Hamiltonian.
deals with atomic estimates involved in the problem. Rare-earth trifluorides RF3 are ionic crystals contain-
Starting from the electron-nucleus weak interaction, we ing triply ionized rare-earth ions R3+. The general elec-
consideranioninthe latticeenvironmentandderivethe tronic configurationof these ions is 1s2...5s25p64fn with
effective single-ion PNC Hamiltonian. In Section III we the number of4f-electronsrangingfrom1 inCe3+ to 13
averagethisHamiltonianoverthecrystalstructureofthe in Yb3+. So, L and S are carried by the 4f-electrons
compound, which gives the kinematical structure of the which constitute the only open shell. f-electrons practi-
effect and its dependence on the external fields. Section cally do not penetrate to the nucleus and hence, practi-
IV presents our results and discussion of uncertainties cally do not interact with the weak charge directly. In-
involved. stead,theeffectiveinteractionarisesinthethirdorderof
perturbation theory due to many-body effects, a typical
diagram is shown in Fig. 1. The vertical wavy line on
II. SINGLE ION EFFECTIVE HAMILTONIAN the diagramconnecting the two electron lines represents
the residual Coulomb interaction between electrons; the
The weak charge-induced PNC interaction between weakinteraction(3)isshownbythecrossontheelectron
electron and nucleus is of the form (see, e.g., [7]) line; the dashed line denotes interaction of electron with
the externalelectricfield E. This is but one ofanumber
G Q ofdiagramsresponsiblefortheeffect,inordertofindthe
HW =−√2δ(r)γ5 2W . (3) effectiveinteraction(4)onehastoevaluateallofthedia-
grams. A similar atomic calculation has been performed
Here G is the Fermi coupling constant, δ(r) is the Dirac previouslyinthepaper[18]inregardtotheinteractionof
deltafunction forelectronsrepresentingthe nuclearden- atomic electrons with the nuclear anapole moment. The
sity distribution, γ5 is the Dirac γ-matrix. The nuclear effective interaction with the anapole moment appears
weak charge is Q = N +Z(1 4sin2θ ) where θ in the third order of perturbation theory as well, and,
W W W
− −
is the Weinberg angle,N is the number of neutrons, and although it has different kinematical structure, the part
Z is the number of protons. involving atomic radial integrals is practically identical
Letusconsideranionembeddedincrystallattice,and tothe presentcase. Themaindifferenceisthattheweak
let us impose an external electric field E which shifts chargematrixelement H ,Eq.(3),shownbythecross
W
h i
3
in the diagram Fig. 1, has to be replaced with the ma- Kramers ions
trix element of the anapole interaction H . Therefore, Ce Nd Sm Gd Dy Er Yb
a
h i
to estimate the effective Hamiltonian (4), it is not nec- L 3 6 5 0 5 6 3
essary to repeat the atomic calculations—it is sufficient S 1/2 3/2 5/2 7/2 5/2 3/2 1/2
to rescale the results of Ref. [18]. Analytical expressions A 5.0 1.7 1.0 0 −1.0 −1.7 −5.0
forboththeweakcharge H andtheanapolemoment
h Wi non-Kramersions
H interaction matrix elements based on the semiclas-
h ai Pr Pm Eu Tb Ho Tm
sical approximation for electron wave functions can be
L 5 6 3 3 6 5
found in the book [7], and lead to the following ratio:
S 1 2 3 3 2 1
A 2.5 1.3 0.8 −0.8 −1.3 −2.5
hs1/2|HW|p1/2i QW . (5)
s1/2 Ha p1/2 ∼ κa TABLE I: Values of the coefficient A in the effective parity
h | | i
nonconservingHamiltonian(10). Forquickreference,wealso
Here κa is the dimensionless constant characterizing the presentthevaluesoftheorbitalmomentumquantumnumber
nuclear anapole moment, it is of the order κ 0.4. L and thespin S for each rare-earth ion.
a
∼
Rescaling the result of Ref. [18], we find the the single-
electron effective Hamiltonian for 4f-electrons:
−
of fluorine ions F . For the present calculation which
HPNC 2 10−16 (X [l s])QWE0 . (6) concerns the Hamiltonian (10), however, only the spher-
∼ · · ×
ically symmetric part of the electron density from the
Here l and s are the single-electron orbital angular mo- neighboring ions plays role. Since the ions O2− and F−
mentumandspin,andE0 =27.2eVistheatomicenergy have the same electronic configuration, 2p6, and the in-
unit. The ionic displacement X is measured in units of teratomic separation in both compounds is roughly the
the Bohr radius aB. same, we believe the rescaling procedure gives reliable
Simple electrostatic considerations allow to relate the results. At most, we can imagine an error in the coeffi-
displacementXofthetriplyionizedrare-earthioninthe cient A, Eq. (10), of a factor 2 3, which is acceptable
lattice to the strength E of the external electric field: forthepurposesofthepresentes−timate. InthenextSec-
tion,whenweconsidercouplingtotheexternalmagnetic
ǫ 1
P = 3enXa = − E ; (7) field, the anisotropic nature of the environment is very
B
4π
important. There, we treat the ionic environment more
ǫ 1 E
X = − . (8) accuratelyinthe frameworkofthe Coulombcrystalfield
4π 3enaB model.
HerePisthedielectricpolarization,e= e istheelemen-
tary charge, n 1.8 1022 cm−3 is the|n|umber density
≈ · III. CRYSTAL STRUCTURE AND INDUCED
of rare-earth ions in the compound, and ǫ 14 is the
MAGNETIZATION
≈
dielectric constant. Evaluating Eq. (8) we get
X[a ]=2.5 10−8 E[V/cm] . (9) In the present section we take calculation of the PNC
B
· interaction Hamiltonian (10) one step further. The idea
Finally, we have to average the single-electron effec- is that the PNC part is exceptionally small comparedto
tive Hamiltonian (6) over the ground state of the 4fn the other terms in the full Hamiltonian for the ion in
electronic configuration of the R3+ ion with the total external magnetic field:
momenta L=l1+...+ln and S=s1+...+sn (see also
the remark [21]). The averagingwas performed with the H =ALS(L S)+Hcf+µBB (L+2S)+HPNC . (11)
· ·
help of the usual fractional parentage coefficients, see,
e.g., [22]. The result reads: (HereALS istheconstantforfine-structuresplitting,Hcf
denotesthecrystalfieldinteraction,µ istheBohrmag-
B
HPNC A 10−24 E [L S]QW E0 , (10) neton.) So, it is sufficient to consider the HPNC in the
∼ · · ×
first order of perturbation theory only,
where the electric field E is measured in units of V/cm.
Values of the coefficient A in the above equation for dif- ∆EPNC = g HPNC g . (12)
h | | i
ferentrare-earthionsarelistedinTableI. Note,thatthe
effect vanishes for Gd3+ ion which has the ground state The ground state g is determined by the first three
| i
with L=0. terms in the Hamiltonian (11).
Eqs.(6), (10)were obtainedbyrescalingthe resultsof For the case of isotropic environment, the direction
our previous calculation [18], where the rare-earth ions of L and S is completely determined by the external
were considered in the garnet environment consisting of mahgnieticfiehldi: L , S B;thespin-orbitcouplingcon-
eight oxygen O2− ions. In the present case the lattice serves J = L+hS,isho tih∝at L , S J , and J B.
structure is different, and the environment is composed Intuitively, it is then clearhthiath tihe∝ehxpiressionh Li ∝ S
h × i
4
in (10) must vanish; the result is rather easy to prove Cs trapped in solid 4He. The nature of the second term
rigorouslyif one considers the groundstate g = J,J . is not as obvious. An interesting feature of this term is
z
So, the PNC effect (10) vanishes for enviro|nmient|s witih that the vector bk is a pseudovector: while a true vector
spherical symmetry, and the same result is expected for isforbiddenincentrosymmetriclattice,apseudovectoris
cubic lattices. The situation is different, however, for an generally allowed. We have confirmed numerically that
ion in the crystal environment with an axis k. This is the contribution proportional to bk appears only due to
especially obvious in the case of zero spin-orbit coupling 4th and 6th multipolarities of the crystal field—indeed,
when L S = L S . Indeed, the directionof L is one can construct a pseudovector from the irreducible
h × i h i×h i h i
then determined not only by the external magnetic field tensors of 4th and 6th rank, but not from a 2nd rank
B, but also by the lattice crystal field: in the simplest tensor. For example, one can construct a pseudovector
casewecanwrite L B+(B k)k. Thetotalspin S from the 4th rank irreducible tensor tˆ, which represents
h i∝ · h i
is oriented along the magnetic field. Then, the value of the crystal field, in the following way:
the cross product L S (B k)[B k] is, generally,
not zero. Existenche of×theiL∝S-co·upling×certainly compli- bki ǫijk tjlmn tlmnp trsuv tuvpq tqrsk . (15)
∝
cates the analysis, but the general picture remains the
same. Here ǫijk is the usual totally antisymmetric tensor, and
The crystalfield HamiltonianHcf inEq.(11)wascho- summation over the repeated indices is implied. At zero
sen in the form suggested in paper [23]: temperature, the combination (15) does not appear ex-
plicitlyintheexpressionforthecoefficientb—inthiscase
4π thecoefficientdoesnothaveaperturbationtheorystruc-
Hcf = ρnAnm Ynm(ri) , (13) ture, and, hence, the answer cannot be presented as a
2n+1
nm i r simple series in powers of crystal field. However,we also
X X
considered a nonzero temperature case. At high tem-
where r is the radius-vector of the 4f-electron, and i
rduonnsootvedirepalelnedleocntrotnhseiinont,hebu4tfasrheelslp.ePciafircamtoettehrseAhonsmt preeprraetsuernet,ekdTas≫a sHercife,sAiLnSp,owtheersposfeuHdcof/vkecTt.orInbkthcisancabsee
numerics clearly indicate that, in agreement with (15),
compound(localenvironmentofthe ion). Togetherwith
the leading contribution to bk appears in the fifth order
theion-dependentparametersρ theydescribetheeffect
n in crystal field.
of the host crystallattice on the rare-earthion. Authors
ofpaper [23]performedconsistentanalysisofthe optical The systemof electronspins inRF3 at sufficiently low
temperatureundergoestransitiontotheorderedstate. It
spectra of nine different rare-earth ions when they dope
is clear that the lowesttemperature in the experiment is
lanthanum trifluoride by substituting lanthanum ions,
limited to the transition temperature. However, the ex-
and deduced the values of parametersA for this com-
pound. Inouranalysis,weneglecttheponsmsibledifference istingexperimentaldataonRF3is,unfortunately,scarce.
betweenthelocalfieldinlanthanumtrifluoridedopedby It is known that the electron spins in TbF3 order ferro-
rare-earthions and the crystal field in generic rare-earth magnetically below 3.95 K [24], and the spins in HoF3
order antiferromagnetically at 0.53 K [25]. From the re-
trifluoride compounds, and use the crystal field parame-
ters derived in paper [23]. sults of Ref. [26] we also know that (Ce,Nd,Pr)F3 com-
pounds remain paramagnetic at least down to 2 K. It is
AccordingtoEq.(10),inordertocalculate g HPNC g
we need to find the expectation value of [Lh | S]. |Toi reasonable to conclude that other rare-earth trifluorides
× have transition temperature of < 1 K as well, so, in our
be specific, let us consider non-Kramers ions, i.e., ions
estimates, we assume the tempe∼rature of 1 K. For non-
that contain even number of electrons. In this case the
Kramers ions where the ground state is not degenerate
ground state is non-degenerate and, as the direct calcu-
the effect is, actually, temperature independent as soon
lation shows, the expectation value [L S] vanishes in
h × i as the temperature in the compound is above the tem-
the zero magnetic field. So, HPNC = 0, if B = 0, and,
h i peratureofcollectivespinordering 1K,andbelowthe
certainly,thisagreeswiththegeneralstatementthatthe ∼
first energy interval in the crystal field splitting ( 10 K
usualweakinteractioncannotgeneratelinearStarkeffect ∼
for compounds where effect is the largest). However,the
[7]. In the nonzero magnetic field the expectation value
temperature dependence is very important for Kramers
is of the form
ions where the groundstate is degenerate,andthe effect
there scales as 1/T. In the calculation we set T =1 K.
g [L S]g =a[B k](B k)
h | × | i × · If we consider the external magnetic field of the order
+b kB2 B(B k) , (14) B < 1 T (tesla), the system is in the regime µ B
B
− · kT∼ ∆ǫcf < ∆ǫLS. Numerical analysis of the Ham≪il-
where k is the unit vector ort(cid:2)hogonal to the (cid:3)plane of toni≪an (11) ∼in this case gives the following parametric
hexagonalsymmetry,andaandbaresomeconstantsde- dependencies for the coefficients a and b describing the
rivedfromcalculation. Thefirsttermin(14)corresponds effect (14):
tothesimplestructurethatfollowsfrommostgeneralar-
guments, as discussed in the beginning of this Section; a 1 1
non-Kramers ions: a , b ; (16)
similar structure was consideredin the literature [20] for ∝ HcfALS ∝ A2LS
5
Kramers ions
Ce Nd Sm Gd Dy Er Yb
a 0.30 ·10−4 0.40 ·10−3 0.21 ·10−3 0 0.56 ·10−2 0.87 ·10−3 −0.76 ·10−4
b −0.25 ·10−4 −0.29 ·10−4 0.23 ·10−5 0 −0.15 ·10−4 −0.21 ·10−5 0.60 ·10−7
non-Kramersions
Pr Pm Eu Tb Ho Tm
a 0.41 ·10−5 0.57 ·10−5 0.46 ·10−5 0.24 ·10−2 0.77 ·10−3 −0.57 ·10−5
b 0.36 ·10−6 0.27 ·10−7 0.93 ·10−9 0.13 ·10−5 0.24 ·10−5 0.12 ·10−7
TABLE II: Values of the coefficients a and b, Eq. (14), for different rare-earth ions. We assume that the magnetic field B in
Eq. (14) is expressed in units of T (tesla). The values of a and b for Kramers ions correspond to thetemperature T =1 K.
1 Hcf responds to the macroscopically averaged quantity, and
Kramers ions: a , b . (17)
∝ kTALS ∝ kT A2LS as such, is consistent with the combined C6,i symmetry
(the local C2, plus C3,i which describes the site orienta-
The aboveexpressionsdescribe variationofthe effectfor
tions). ThevaluespresentedinTableII,also,correspond
allrare-earthions,withtheexceptionofEu3+ andGd3+.
to the kinematics averagedover the lattice.
Eu3+ ion has the ground state with J = 0, so that the
From equations (10) and (14) we derive the energy
first energy interval is determined by the fine structure
density induced by the weak interaction
splitting rather than crystal field; in this case the effect
iAnH5Lcc3Sflu,daensdasdod,iitsiosunpaplroersdseerds. oTfhHecGf/dA3+LSio:naha∝saAHh3LcaSfl,f-fiblle∝d EPNC =nhg|H=PNaCn|AgEi 0QW(E·[B×k])(B·k) , (18)
4f-orbitalwith L=0, so the expression [L S] , as we
h × i the number density of rare-earth ions in the compound
havealreadymentioned,vanishescompletelyforthision.
is about n 1.8 1022 cm−3. In the above equation
Table II lists the valuesof a andb for the rare-earthions ≈ ·
we neglect the second term on the right-hand side part
Ce through Yb derived from numerical calculations.
of Eq. (14): although appearance of the pseudovector
Thegeneralfeaturesofthe variationofaandbfordif-
bk is itself an interesting point, in practice, as it follows
ferent ions can be understood with the help of Eqs. (16)
fromTableII,thecontributionoftheb-termisnegligible
and(17). Inparticular,thelargestvaluesofaamongthe
compared to the contribution of the a-term.
non-Kramersions correspondto the ions with the small-
The macroscopic magnetization corresponding to
estfirstenergyintervalinthecrystalfieldsplitting. This
Eq. (18) is
agreeswellwiththeformula(16). Tb3+ andHo3+ which
have particularly large values of a, have very small first
deniffeergreyncineteinrvathlse∼cry5sctaml−fi1e,ldwhspereecatsrutmheisavera1g0e0ecnmer−g1y. M=−∂E∂PBNC =−naAE0QW k(B·[k×E])
Another general tendency observed for all∼ions is that +n[k E](B k) . (19)
× ·
the a-effect is larger than the b-effect by approximately
o
the factor Hcf/ALS, just as it should be according to NumericalvaluesofthecoefficientsaandAarepresented
Eqs. (16), (17). Although the expressions (16) and (17) in Tables I, II; we assume that the electric field is ex-
describe most general trends observed in Table II, there pressed in units of V/cm, and the magnetic field B and
is still considerable dispersion in the values. This is be- magnetizationMareexpressedinunitsofT(tesla). The
cause the many-electron states of the different ions have temperature is set to T = 1 K, although, this is im-
significantlydifferentstructure,andthiskindofvariation portant for Kramers ions only. From Tables I, II and
cannot be accounted for in a simple way. the Eq. (19) we see that the effect is maximal for Dy3+,
The rare earth ions in RF3 occupy the sites with local whichisaKramersion,andHo3+ whichisnon-Kramers.
C2 symmetry. In addition, there is the C3,i symmetry Underfavorableexperimentalconditions,itisreasonable
which describes different ionic sites in the lattice: for to assume the applied magnetic field of B = 1 T and
any arbitrary site, there are also sites rotated on 2π/3 the electric field of E = 10 kV/cm. Then, the magnetic
and 2π/3 (rotation axis is the same as the local C2 induction in the sample is
−
axis), as well as the sites with the invertedenvironment.
The local site symmetry describes the geometry of the DyF3: 4πM 0.5 10−11 G ,
∼ · (20)
single site environment: it poses constraints on the crys- HoF3: 4πM 0.1 10−11 G .
tal field parameter values, and hence, on any kinematics ∼ ·
thatappearlocally. Theexistenceoflocalionicsiteswith Quadratic dependence of EPNC on the value of mag-
differentorientationsinthelatticecellandtheassociated netic fieldseen in Eq.(18) is valid only if µ B kT for
B
≪
symmetry pose limitations on the form of any material Kramersions,andµB ∆ǫcf fornon-Kramersions. For
≪
(macroscopic)tensor. The kinematical relation(14) cor- higher values of magnetic field the dependence turns to
6
linear. This means that for Dy and Ho compounds the is measured in atomic experiments. Electronic matrix
PNC magnetization (19) increases linearly with B up to element of the operator (3) of weak charge interaction
B 1 T and then stops to grow. contains the expression ρ(r)Q = ρ(r)( N + Z(1
W
∼ 4sin2θ )) averaged over the electron radi−al wave func−-
W
tions, ρ(r) here is the nuclear density which replaces the
IV. DISCUSSION Dirac delta function in Eq. (3). It is known, however,
that the density distribution is different for neutrons
and protons, so the correct expression to be averaged
Theparity-violatingweakinteractionofelectronswith
is ( Nρ (r)+Zρ (r)(1 4sin2θ )), where ρ (r) and
the weak charge of rare-earth nuclei in rare-earth triflu- − n p − W n
ρ (r) are the neutron and the proton nuclear density re-
orides leads to the parity nonconserving magnetization p
spectively. Thus, in a real experiment one probes the
presented in Eq. (19). The effect depends on the exter-
effective weak charge
nal electric and magnetic fields. According to our esti-
mates, the effect is maximal in Ho and Dy trifluorides. Q = Nq +Zq (1 4sin2θ ) (21)
W n p W
− −
Assuming that the external magnetic field is B = 1 T,
which contains the neutron and proton form factors q
theexternalelectricfieldisE =10kV/cm,andthetem- e n
and q . The form factor q can, at least in principle,
peratureisT =1K,wefindthatthePNCmagnetization p p
isabout10−11G. Wecancomparethisestimatewiththe be measured reliably with electron scattering, in muonic
atoms, etc. It is expressed in terms of the proton distri-
current experimental sensitivity that is under discussion
butionroot-mean-square(RMS) radiusR , whichisalso
in the literature. According to Ref. [13], it is possible to p
achieve the sensitivity of 3 10−16 G for 10 days of aver- the RMS radius of nuclear electric charge distribution.
· We assume that the uncertainty in the value of R is
aging with SQUID magnetometry, and it even might be p
negligible—the neutronRMSradiusR andthe neutron
possible to do 2 orders of magnitude better with appli- n
form factor q are known with much lower precision.
cation of different techniques. This leads to conclusion n
Let us estimate the uncertainty in sin2θ due to un-
that the value ofthe PNC magnetizationcan,at leastin W
certainty in the neutron form factor q . To be specific,
principle, be measured to very good precision. n
let us consider the suggested experiment with dyspro-
However, an accurate theoretical treatment of the
sium trifluoride. There are seven stable isotopes of Dy:
problem with precision of even severalpercents does not
156, 158, 160, 161, 162, 163, 164. The odd and the even
seempossible. Atbest,wecanexpecttheaccuracyatthe
isotopes should not be considered together, because the
level of 20-50%. Thus, extraction of the precise value of
odd isotopes then would give rise to nonmonotonic de-
the weak charge directly from the results of experimen-
pendences due to the presence of the unpaired neutron.
talmeasurementsdoesnotseemaviablepossibility. Itis
Clearly,it is bestto analyzethe evenisotopes only. This
possible, however, to look for the variation of the effect
leaves dysprosium-156,158, 160,162 and 164—these are
in a chain of nuclear isotopes in the same compound.
completely paired nuclei with practically the same de-
In this respect, dysprosium, which has seven naturally
formations, so the dependence of q on the number of
occurring isotopes, looks most promising. This opens a n
neutrons must be smooth and monotonic.
possibility for the precise measurementsof the Weinberg
Strictly speaking, in order to find the neutron form
angle. There are several issues related to the hypotheti-
factor q and the corresponding uncertainty one has to
cal measurements. The first is the normalization of the n
perform calculations of q for deformed nucleus in the
effect. Imagine, there are two samples made of different n
rotational s-wave state. However, to estimate the un-
isotopes. Theywillbeofslightlydifferentshapesandvol-
certainty, it is sufficient to use the known formula for
ume, and then one has to perform precise measurements
sphericalnucleusinthe “sharp-edge”approximation[27]
ofthe usualelectricandmagnetic susceptibilities to nor-
malize the PNC effect per unit volume. Another issue is
2
related to the possible systematics. At the moment, we 3 R
q =1 (αZ)2 1+5 n . (22)
dimointaottesteheeaPnyNmCamcraogsnceotpizicatsiyosnte(m19a)t.icAelfftheoctusgthhEatq.w(o1u9l)d, n − 70 " (cid:18)Rp(cid:19) #
inessence,describesthenonlinearmagnetoelectriceffect, The difference ∆Rnp = Rn Rp was found in Ref. [29]
−
the crucialfeatureisthatitviolatesparityatfundamen- from analysis of experiments with antiprotonic atoms.
tallevel. Inthelatticewithcenterofinversion,theusual The empirical fit in Ref. [29] reads:
magnetoelectric effects cannot generate this correlation.
N Z
A possible source of systematics can be due to the small ∆Rnp[fm]=( 0.04 0.03)+(1.01 0.15) − . (23)
− ± ± N +Z
localimperfectionsinthelatticewhichdestroytheinver-
According to this expression the uncertainty in R is
sion symmetry. However such effects are of local nature n
δR 0.058 fm. From Eq.(22), uncertainty in the neu-
andshouldaverageouttozerointhemacroscopicsample n
≈
with about 1023 rare-earthsites. tron form factor is related to δRn in the following way
Another serious issue ([27, 28]) is the unknown value
of nuclear neutron form factor which appears in the for- δq = 3(αZ)2 Rn δR . (24)
mulaforeffectivenuclearweakcharge—thequantitythat n 7 R2 n
p
7
The experimentwith the chainofisotopesinthe same δs2 in the value of sin2θ according to the formula:
W
compound would be measuring the ratio of Q (21) for
W
′ ′
twodifferent isotopeswith A=Z+N andA =Z+N :
N2 δQ δQ
e δs2 √2 W =6 W . (27)
∼ 4Z∆N QW ! QW !
Q Nq +Zq (1 4s2) e e
W n p
= − − , (25)
QW′ −N′qn′ +Zqp′(1−4s2) This corresponds to the meeasurement weith two dyspro-
e sium isotopes, 156Dy and 164Dy, ∆N = 8. In order to
and thus, it would allow to eliminate the solid-state pre-
e match the uncertainty (26), the experiment should be
factorandtheassociatedtheoreticaluncertainty. Assum-
carried out with the precision exceeding δQ /Q
ing there is no contributionfrom experimentalerror,the 8 10−5. According to the estimates, statisticWal coWnsid∼-
uncertaintyins2 ≡sin2θW thatcorrespondstotheerror er·ations allow for precision of the order 3e 10e−5, or
bounds in the Eq. (23) is: ∼ ·
better, depending on the magnetometry technique.
δs2 5 10−4 . (26) The current experimental precision for the value of
∼ · sin2θ measured at Z-peak, according to Ref. [1], is
W
Note, that the uncertainties in q and q′ in Eq. (25) δs2 1.5 10−4. The solid-state experiment proposed
n n ∼ ·
are correlated according to Eqs. (24), (23). Moreover, in the present paper could achieve comparable level of
main contribution to δs2 comes from the second term precision, and, what is more important, would measure
in Eq. (23), while contribution of the constant term is the value of the Weinberg angle in the domain of low
negligible. energies (effective momentum transfer 30 MeV).
∼
There is also the experimental uncertainty δQW/QW. The authors would like to thank J. M. Cadogan and
In the possible experiment that measures the ratio A.I. Milstein forstimulating discussions,andD. Budker
′
Q /Q ,thisuncertaintytranslatesintotheunecertaeinty and V. A. Dzuba for important comments.
W W
e e
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