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The possibility to study the break of time-reversal invariance in atoms
Volume 142, number 4,5
PHYSICS LETTERS A
II December 1989
THE POSSIBILITY TO STUDY THE BREAK OF TIME-REVERSAL INVARIANCE IN ATOMS M.G. KOZLOV and S.G. PORSEY Leningrad NuclearPhysics Institute, Gatchina, Leningrad district 188350, USSR Received 15 August 1989; revised manuscript received 20 September 1989; accepted for publication 2 October 1989 Communicated by V.M. Agranovich
The possibility to study T-odd electron—nucleon interaction by means of an optical experiment on the 6p1,2—6p3,2 transition in thallium is discussed.
During the last years many experiments were devoted to the investigation of P-odd and P,T-odd interactions in atoms and molecules. Here we want to draw attention to the possibility of studying T-odd electron—nucleus interactions in atomic experiments. Up to flow only the nucleon—nucleon sector was studied, where such interactions were discussed in refs. [1—3],and experiments [4,5] established the upper limits on the constants of the T-odd Hamiltonian. Our proposal is to study the k~Ecorrelation in the refractive index ofthe atomic gas, where k isthe wave vector of the light beam, and Eis the external electric field. On can study this correlation in the interference of two b$ms which cross the cell with the gas in opposite directions. A possible scheme of such an interferometer is shown in fig. 1. If the direction of the field E is changed to the opposite direction, the relative phase of the beams is also changed: L~Ø—_4L~nL 1 w/c,
5
J 3
-._________________
I -
,(
2
1
_.~__jJ~
_________________
U
Fig. 1. Scheme of interferometer for the study of k-E correlation in the refractive index. (1) Laser, (2) semitransparent plate, (3) cell, (4) ~./4plate, (5) detector. The arrow shows one ofthe possible directions ofthe electric field.
(1)
where ~n1 is the T-odd part of the refractive index, and L is the lehgth of the cell. If an additional phase shift ir/2 between two beams is introduced, the intensity on the detector linearly depends on the T-odd phase (1) (see fig. 2). The best sig*al-to-noise ratio is achieved if L = 2L0 (L0 is the absOrption length). The measured quan tity i~Øis governed by the external field. It gives an effectivetest on the systematics. Because of that, one can expect much higher sensitivity, in comparison with the measurements of the rotation of the plane
-_____________
-X/2
_____________
0
iiIZ
Fig. 2. Dependence of the light intensity on the detector from the T-odd phase ~Ø.
0375-9601 /89/$ 03.50 C Elsevier Science Publishers B.V. (North-Holland)
233
Volume 142, number 4,5
PHYSICS LETI’ERS A
11 December 1989
of polarization, where an accuracy 108 is achieved. If we assume ~ 10~’~ [6], then
a4 = /14H_4 + iOr/1’4H~q
L~n1 l0~°c/8L0w.
In this expression d4,
(2)
Below we shall study the limits that can be obtamed from expression (2) for the constants of the T-odd interaction. an interaction electron with the protonSuch includes two termsof (h the = c= I):
H~= h
2~
~
eä ,~(~y~ a,~p).
(3)
Here e and p are the wave functions of the electron and the proton; y,~,y~,a,~are the Dirac matrices; me, m~are the masses of the electron and the proton; h1 and h2 are the dimensionless constants. Assuming that the nucleus has only one valence proton and neglecting nucleus motion, one can reduce expression (3) to the form ~
m~
0E
EaE~,
For the sake ofsimplicity we have dropped the sums over the intermediate states in eq. (5). The z axis is assumed to be parallel to the wave vector k (k~/ 1k I v= ± 1). Taking into account that for the plane-polarized light H_4 = iqvE_4 = ivE/.,,,h and extracting the common phase of the amplitude a,~,we get a4=(a+~) exp(i~), where a=~IE/%J~,
(6) (7)
\ me
(4) where <~>is the proton Pauli matrix, averaged over the nucleus state, r is the electron coordinate, [ ] denotes the commutator. The Hamiltonian (4) has much in common with that of the P-odd weak interaction. So its matrix elements (ME) 3) must rapidly withfor the andgrow must turn to zero allnuclear but the charge Z (cc Z smallest orbital and total electron angular momenta / and j. As far as H~conserves parity and the total angular momentum of the atom F, we conclude that HTcan mix states (n, P1/2, F, mF) with (n’, P3/2, F, mF) (n, s112, mF) with (n’,case d312, F, mF). Theand suitable andF,rather simple is the (6p 112, 6p3,2, F’, m’F) Mi-transition in thallium, F, mF)—’( where calculations can be carried out explicitly for F=OandF’=l.
_O~O’7-d~)vE/J~i.
-
-
(8)
‘~= (OEOrdI~/~uI
In these expressions we assume that OEdl <
[g(z) —if(z)] (9) TD is the Doppler where N0 is the concentration, width, and functions g and ~(—iz)}, fdescribe the line profile: f+ig=.,/~exp(—z2)[l— =
1
—
N0
4it
,
(10)
~
External electric fields admix to this transition El transitions 6p—7s and 6p—6d, while T-odd interaction results in corrections to both Ml and El -transitions. As the ME ofHT are purely imaginary, these corrections to the amplitude have an additional phase 3t/2. Finally the amplitude can be written in the form 234
(5) the ME of the electric
and magnetic dipole moments, q= ±1 denotes spherical components of vectors, H4, E4 are magnetic 0E define and electric fields of the by light ô~-and the admixture of states thewave, operator H~and the external electric field,
1ö~(ë~y5a~~e)jYy,4y5p +
ji~are
h
mpme
+ OEd4E_4 + iO’~O~d’4E_4.
0
The complex detuning parameter z is (11) where us the collision width. The absorption length
Volume 142, number 4,5
PHYSICS LETTERS A
ii December 1989
is determined by the imaginary part of expression (9),
So, if our treatment of the experimental conditions is realistic, thallium experiments provide a rather sensitive test on the time-reversal invariance
L ~‘ = 2wn2,
in electron—nucleon interaction. Another source of the break of time-reversal invariance in atoms are the T-odd nuclear toroidal moments (first of them is the quadrupole toroidal moment) [10]. Because of the small spin of the thallium nucleus, these moments must turn to zero, and so thallium experiments cannot give any information about nuclear Todd forces. Of course, there must be some other possibilities of measuring constants of the T-odd interaction. It is clear from the above that T-odd effects must be larger of levels in the atoms same with parity largeand polarizabilities total angularand momenclose
(12) and we can rewrite formula (1) for L = 2L0 in the form = —8 (n/a)
(g/J).
(13) The Stark ME which enter the expression (8) can be extracted from the oscillator strengths of the conesponding transitions [7]. The largest terms are connected with the 6p—7s and 6p—6d transitions. The result of the calculation is a =
\ 6P3/2 112iH~ RyI 6p1,2 1> (6.1 X l0~ <
—4.4 x ~o—3 <7s 1,2 1121H~ E RyI 6d3,2 i > ~ 1 0~V/cm~ (14)
)
The ME of the operator H~can be calculated in the quasi-classical approximation [8,9], 6P3/2 11 H~I 6p < 1/2 1> 8[h = —6.2x10 1 +(me/mp)h2](2i Ry) (15) ,
<7s112 11 HT I 6d3,2 1> 8[h =0.5x 10 1 (me/mp)h2] (2i Ry) —
.
(16)
The second term in (14) can be neglected, and we obtain -~
E = —3.8 x 10— ‘°[h1 + (m~/m~)h2] l0~V/cm~ (17)
In the discussion of the experimental conditions we shall follow ref. [9]. If the temperature in the cell 3, r=o.l8x isl09c’, 1200°C,then N0=0.66x 1018 cm r’D= l.7x l09c~,and if we choose the detuning parameter x= (w w 0) /ID =5, then the absorption length L0 1 m and g/f~500. The electric intensity E in expression (15) is limited by the gas discharge. It can be increased by adding buffer gas. Taking a moderate value(17) E= that l0~the V/cm, we obtain from eqs. (2), (13) and following limit on the constants h 1 and h2 can be set:
turn F. Theseconditions are fulfilled for the rare-earth atoms. But it seems impossible to make calculations for them. In fact, the amplitude (5) is determined by the wave functions of the second order ofthe perturbation theory. In our case there are few leading terms in the sums over intermediate states, but in general it is not so. Moreover, even for other hyperfine components of the same electron transitions in thallium, calculations are more complicated, and within the accuracy of our approach the result is undertermined because of the large cancellations of the leading terms. Summing up, we think, that the transition in thalhum discussed above is the most clear case to study the break of time-reversal invariance in atoms by means of optical experiment. Now we shall dwell on the spurious effects. As was pointed out in ref. [11], macroscopic dissipation processes break time-reversal invariance and lead to the same correlation k-E in the refractive index. In fact the electric field willthis cause ionmust flowbeincompenthe cell. In Stationary conditions flow sated by the flow of neutral atoms. That will result
—
Ih1 +(m~/m~)h2l
(18)
in a Doppler shift of different sign for two opposite beams. The corresponding phase shift will be N~ (19) 1’D dg/dx f where CNN0 and v are the concentration and speed of
~
ions. It follows from expression (19) that this shift is an iseven thethat detuning4, while the shift (I) an function odd one, of and it does not grow with 235
Volume 142, number 4,5
PHYSICS LETTERS A
detuning, as (1) does. Putting3.numbers in (19) we get ~Ø= 10 ‘° for N= l0~cm The authors are grateful to V.P. Gudkov for valuable discussions and to A.N. Moskalev, discussions with whom initiated this research and who took part in the first stage of the work.
11 December 1989
[3] V.E. Bunakov and V.P.physics, Gudkov, Tests of time-reversal invariance in neutron ed.in: N.R. Robertson (1987) p. 175. [4]J.B. French et al., in: Tests of time-reversal invariance in neutron physics, ed. N.R. Robertson (1987) p. 80. [5] C.A. Davis, L.G. Greeniaus, G.A. Moss, D.A. Hutcheon, C.A. Miller, R. Abe, A.W. Stetz, W.C. Olsen, G.C. Neilson, G. Roy and J. Uegaki, Phys. Rev. C 33 (1986) 1196. [6] L.M. Sarkov, M.S. Zolotorev and D.A. Melik-Pashaev, Pis’ma Zh. Eksp. Teor. Fiz 48 (1988) 134.
[7] S.E. Frish, Optical spectra of atoms (Moscow, 1963) [in
References [1] P.K. Kabir, in: NBS Special publication, No. 711. The investigation of fundamental interactions with cold neutrons, ed. G.L. Greene (National Bureau ofStandards, Washington, 1986) p. 81. [2] V.E. Bunakov, Phys. Rev. Lett. 60(1988)2250.
236
Russian]. [8] M.A. Bouchiat and C.J. Bouchiat, J. Phys. (Paris) 35 (1974) 899. [9] I.B. Khnplovich, Parity nonconservation in atomic physics (Moscow, 1988) [in Russian]. [10] A.N. Moskalev and S.G. Porsev, Yad. Fiz. 49 (1989) 1266. [11] N.B. Baranova and B.Ya. Zel’dovich, Mol. Phys. 38 (1979) 1085.
PHYSICS LETTERS A
II December 1989
THE POSSIBILITY TO STUDY THE BREAK OF TIME-REVERSAL INVARIANCE IN ATOMS M.G. KOZLOV and S.G. PORSEY Leningrad NuclearPhysics Institute, Gatchina, Leningrad district 188350, USSR Received 15 August 1989; revised manuscript received 20 September 1989; accepted for publication 2 October 1989 Communicated by V.M. Agranovich
The possibility to study T-odd electron—nucleon interaction by means of an optical experiment on the 6p1,2—6p3,2 transition in thallium is discussed.
During the last years many experiments were devoted to the investigation of P-odd and P,T-odd interactions in atoms and molecules. Here we want to draw attention to the possibility of studying T-odd electron—nucleus interactions in atomic experiments. Up to flow only the nucleon—nucleon sector was studied, where such interactions were discussed in refs. [1—3],and experiments [4,5] established the upper limits on the constants of the T-odd Hamiltonian. Our proposal is to study the k~Ecorrelation in the refractive index ofthe atomic gas, where k isthe wave vector of the light beam, and Eis the external electric field. On can study this correlation in the interference of two b$ms which cross the cell with the gas in opposite directions. A possible scheme of such an interferometer is shown in fig. 1. If the direction of the field E is changed to the opposite direction, the relative phase of the beams is also changed: L~Ø—_4L~nL 1 w/c,
5
J 3
-._________________
I -
,(
2
1
_.~__jJ~
_________________
U
Fig. 1. Scheme of interferometer for the study of k-E correlation in the refractive index. (1) Laser, (2) semitransparent plate, (3) cell, (4) ~./4plate, (5) detector. The arrow shows one ofthe possible directions ofthe electric field.
(1)
where ~n1 is the T-odd part of the refractive index, and L is the lehgth of the cell. If an additional phase shift ir/2 between two beams is introduced, the intensity on the detector linearly depends on the T-odd phase (1) (see fig. 2). The best sig*al-to-noise ratio is achieved if L = 2L0 (L0 is the absOrption length). The measured quan tity i~Øis governed by the external field. It gives an effectivetest on the systematics. Because of that, one can expect much higher sensitivity, in comparison with the measurements of the rotation of the plane
-_____________
-X/2
_____________
0
iiIZ
Fig. 2. Dependence of the light intensity on the detector from the T-odd phase ~Ø.
0375-9601 /89/$ 03.50 C Elsevier Science Publishers B.V. (North-Holland)
233
Volume 142, number 4,5
PHYSICS LETI’ERS A
11 December 1989
of polarization, where an accuracy 108 is achieved. If we assume ~ 10~’~ [6], then
a4 = /14H_4 + iOr/1’4H~q
L~n1 l0~°c/8L0w.
In this expression d4,
(2)
Below we shall study the limits that can be obtamed from expression (2) for the constants of the T-odd interaction. an interaction electron with the protonSuch includes two termsof (h the = c= I):
H~= h
2~
eä ,~(~y~ a,~p).
(3)
Here e and p are the wave functions of the electron and the proton; y,~,y~,a,~are the Dirac matrices; me, m~are the masses of the electron and the proton; h1 and h2 are the dimensionless constants. Assuming that the nucleus has only one valence proton and neglecting nucleus motion, one can reduce expression (3) to the form ~
m~
0E
EaE~,
For the sake ofsimplicity we have dropped the sums over the intermediate states in eq. (5). The z axis is assumed to be parallel to the wave vector k (k~/ 1k I v= ± 1). Taking into account that for the plane-polarized light H_4 = iqvE_4 = ivE/.,,,h and extracting the common phase of the amplitude a,~,we get a4=(a+~) exp(i~), where a=~IE/%J~,
(6) (7)
\ me
(4) where <~>is the proton Pauli matrix, averaged over the nucleus state, r is the electron coordinate, [ ] denotes the commutator. The Hamiltonian (4) has much in common with that of the P-odd weak interaction. So its matrix elements (ME) 3) must rapidly withfor the andgrow must turn to zero allnuclear but the charge Z (cc Z smallest orbital and total electron angular momenta / and j. As far as H~conserves parity and the total angular momentum of the atom F, we conclude that HTcan mix states (n, P1/2, F, mF) with (n’, P3/2, F, mF) (n, s112, mF) with (n’,case d312, F, mF). Theand suitable andF,rather simple is the (6p 112, 6p3,2, F’, m’F) Mi-transition in thallium, F, mF)—’( where calculations can be carried out explicitly for F=OandF’=l.
_O~O’7-d~)vE/J~i.
-
-
(8)
‘~= (OEOrdI~/~uI
In these expressions we assume that OEdl <
[g(z) —if(z)] (9) TD is the Doppler where N0 is the concentration, width, and functions g and ~(—iz)}, fdescribe the line profile: f+ig=.,/~exp(—z2)[l— =
1
—
N0
4it
,
(10)
~
External electric fields admix to this transition El transitions 6p—7s and 6p—6d, while T-odd interaction results in corrections to both Ml and El -transitions. As the ME ofHT are purely imaginary, these corrections to the amplitude have an additional phase 3t/2. Finally the amplitude can be written in the form 234
(5) the ME of the electric
and magnetic dipole moments, q= ±1 denotes spherical components of vectors, H4, E4 are magnetic 0E define and electric fields of the by light ô~-and the admixture of states thewave, operator H~and the external electric field,
1ö~(ë~y5a~~e)jYy,4y5p +
ji~are
h
mpme
+ OEd4E_4 + iO’~O~d’4E_4.
0
The complex detuning parameter z is (11) where us the collision width. The absorption length
Volume 142, number 4,5
PHYSICS LETTERS A
ii December 1989
is determined by the imaginary part of expression (9),
So, if our treatment of the experimental conditions is realistic, thallium experiments provide a rather sensitive test on the time-reversal invariance
L ~‘ = 2wn2,
in electron—nucleon interaction. Another source of the break of time-reversal invariance in atoms are the T-odd nuclear toroidal moments (first of them is the quadrupole toroidal moment) [10]. Because of the small spin of the thallium nucleus, these moments must turn to zero, and so thallium experiments cannot give any information about nuclear Todd forces. Of course, there must be some other possibilities of measuring constants of the T-odd interaction. It is clear from the above that T-odd effects must be larger of levels in the atoms same with parity largeand polarizabilities total angularand momenclose
(12) and we can rewrite formula (1) for L = 2L0 in the form = —8 (n/a)
(g/J).
(13) The Stark ME which enter the expression (8) can be extracted from the oscillator strengths of the conesponding transitions [7]. The largest terms are connected with the 6p—7s and 6p—6d transitions. The result of the calculation is a =
\ 6P3/2 112iH~ RyI 6p1,2 1> (6.1 X l0~ <
—4.4 x ~o—3 <7s 1,2 1121H~ E RyI 6d3,2 i > ~ 1 0~V/cm~ (14)
)
The ME of the operator H~can be calculated in the quasi-classical approximation [8,9], 6P3/2 11 H~I 6p < 1/2 1> 8[h = —6.2x10 1 +(me/mp)h2](2i Ry) (15) ,
<7s112 11 HT I 6d3,2 1> 8[h =0.5x 10 1 (me/mp)h2] (2i Ry) —
.
(16)
The second term in (14) can be neglected, and we obtain -~
E = —3.8 x 10— ‘°[h1 + (m~/m~)h2] l0~V/cm~ (17)
In the discussion of the experimental conditions we shall follow ref. [9]. If the temperature in the cell 3, r=o.l8x isl09c’, 1200°C,then N0=0.66x 1018 cm r’D= l.7x l09c~,and if we choose the detuning parameter x= (w w 0) /ID =5, then the absorption length L0 1 m and g/f~500. The electric intensity E in expression (15) is limited by the gas discharge. It can be increased by adding buffer gas. Taking a moderate value(17) E= that l0~the V/cm, we obtain from eqs. (2), (13) and following limit on the constants h 1 and h2 can be set:
turn F. Theseconditions are fulfilled for the rare-earth atoms. But it seems impossible to make calculations for them. In fact, the amplitude (5) is determined by the wave functions of the second order ofthe perturbation theory. In our case there are few leading terms in the sums over intermediate states, but in general it is not so. Moreover, even for other hyperfine components of the same electron transitions in thallium, calculations are more complicated, and within the accuracy of our approach the result is undertermined because of the large cancellations of the leading terms. Summing up, we think, that the transition in thalhum discussed above is the most clear case to study the break of time-reversal invariance in atoms by means of optical experiment. Now we shall dwell on the spurious effects. As was pointed out in ref. [11], macroscopic dissipation processes break time-reversal invariance and lead to the same correlation k-E in the refractive index. In fact the electric field willthis cause ionmust flowbeincompenthe cell. In Stationary conditions flow sated by the flow of neutral atoms. That will result
—
Ih1 +(m~/m~)h2l
(18)
in a Doppler shift of different sign for two opposite beams. The corresponding phase shift will be N~ (19) 1’D dg/dx f where CNN0 and v are the concentration and speed of
~
ions. It follows from expression (19) that this shift is an iseven thethat detuning4, while the shift (I) an function odd one, of and it does not grow with 235
Volume 142, number 4,5
PHYSICS LETTERS A
detuning, as (1) does. Putting3.numbers in (19) we get ~Ø= 10 ‘° for N= l0~cm The authors are grateful to V.P. Gudkov for valuable discussions and to A.N. Moskalev, discussions with whom initiated this research and who took part in the first stage of the work.
11 December 1989
[3] V.E. Bunakov and V.P.physics, Gudkov, Tests of time-reversal invariance in neutron ed.in: N.R. Robertson (1987) p. 175. [4]J.B. French et al., in: Tests of time-reversal invariance in neutron physics, ed. N.R. Robertson (1987) p. 80. [5] C.A. Davis, L.G. Greeniaus, G.A. Moss, D.A. Hutcheon, C.A. Miller, R. Abe, A.W. Stetz, W.C. Olsen, G.C. Neilson, G. Roy and J. Uegaki, Phys. Rev. C 33 (1986) 1196. [6] L.M. Sarkov, M.S. Zolotorev and D.A. Melik-Pashaev, Pis’ma Zh. Eksp. Teor. Fiz 48 (1988) 134.
[7] S.E. Frish, Optical spectra of atoms (Moscow, 1963) [in
References [1] P.K. Kabir, in: NBS Special publication, No. 711. The investigation of fundamental interactions with cold neutrons, ed. G.L. Greene (National Bureau ofStandards, Washington, 1986) p. 81. [2] V.E. Bunakov, Phys. Rev. Lett. 60(1988)2250.
236
Russian]. [8] M.A. Bouchiat and C.J. Bouchiat, J. Phys. (Paris) 35 (1974) 899. [9] I.B. Khnplovich, Parity nonconservation in atomic physics (Moscow, 1988) [in Russian]. [10] A.N. Moskalev and S.G. Porsev, Yad. Fiz. 49 (1989) 1266. [11] N.B. Baranova and B.Ya. Zel’dovich, Mol. Phys. 38 (1979) 1085.