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Chiral Symmetry Breaking in Nuclei. J.H. Hamilton 1 , S.J. Zhu 1,2,3 , Y.X. Luo 1,4, , A.V. Ramayya 1 , J.O. Rasmussen 4 , J.K. Hwang 1 , S. Frauendorf 5, V. Dimitrov 5 , G.M. Ter-Akopian 6 , and A.V. Daniel 6 1 Vanderbilt Univ. , 2 Tsinghua Univ., 3 JIHIR, ORNL, 4 LBNL, Berkeley

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chiral symmetry breaking in nuclei
Chiral Symmetry Breaking in Nuclei

J.H. Hamilton1, S.J. Zhu1,2,3, Y.X. Luo1,4, , A.V. Ramayya1, J.O. Rasmussen4,

J.K. Hwang1, S. Frauendorf5, V. Dimitrov5,

G.M. Ter-Akopian6, and A.V. Daniel6

1 Vanderbilt Univ. , 2 Tsinghua Univ.,

3 JIHIR, ORNL, 4 LBNL, Berkeley

5 Univ. Notre Dame, 6 JINR, Dubna

chirality in particle physics
Chirality in particle physics
  • Spin parallel or antiparallel to a particle’s momentum.
  • The two orientations corresponds to a right- handed and left-handed system.
  • Left-handed neutrinos introduced chiral asymmetry into the world.




Nuclei with only two types of particles should be achiral.

Frauendorf et al. noted chiral bands can be associated with angular momentum in triaxial nuclei when the total angular momentum is not along any axis.

Chirality is a geometrical concept that derives only from the orientation of the angular momentum with respect to the triaxial shape.

RZ : the rotation about the Z axis, T : Time reversal

chiral doubling
Chiral doubling
  • When the angular momentum has substantial components along all three axes of the tri-axial nucleus, then there are 2 energetically equivalent orientations of the angular momentum vector with the short, intermediate and long axes forming right or left handed systems with respect to the angular momentum.
  • This can give rise to two DI = 1 bands with same parity and very close energy.
examples of chiral doublets
Examples of chiral doublets
  • Odd-odd nuclei (Z~59, N~75) where the angular momentum is composed of odd h11/2 proton along short axis, h11/2 neutron hole along long axis and collective rotation along intermediate axis, 134Pr.
  • Recently reported “The best chiral properties observed to date were discovered in the 104Rh nucleus involving the nh11/2 x pg9/2-1 configuration where the valence proton and neutron play opposite roles to those in the A=130 region.” (PRL92, 2004)
In order to demonstrate the general nature of chirality it is important to find examples of chiral sister bands with a different qp composition.
  • Here we report the first observation of a pair of chiral vibrational bands in several even-even nuclei with A=100 - 112.

Experimental Details

  • Source : 252Cf with T1/2 = 2.6 y, SF : 3 %
  • Strength : 62 mCi
  • Sandwiched between two 10 mg/cm2 Fe foils inside a 3 in diameter plastic ball.
  • Detectors : Gammasphere with 102 Comp. supressed Ge detectors.
  • 6X1011triple and higher fold coincidence events.
  • Compressed and less compressed cubes and time-gated cubes for analysis with Radware program.
Evidence for Triaxiality in A=100-110 region
  • One and two phonon gamma bands in 104,106Mo Guessous et al., PRL 75 (1995).
  • Energies of gamma bands in Ru isotopes, Shannon et al., PL B336 (1994).
  • Hua et al., PR C69 (2004) reported evidence for triaxiality in odd A 103,105,107Mo but with 101,103Zr having more axially symmetric shapes.
  • Our studies have provided new insight into tri-axiality in this region.
  • Extended the one and two phonon gamma bands in 104,106Mo and gamma bands in 108,110,112Ru.
  • Studied 99,101Y, 103,105,107Nb, 105,107,109Tc and 109,111,113Rh.
  • Carried out Tri-axial-Particle-plus-Rotor calculations for Y, Nb, Tc and Rh nuclei.



Chiral doublet bands




4753.2 (14-)


4049.4 (12 -)



2 = 0.34


Red : band4

Green : band 5

2+ 0+ and 4-  3+g


TAC predicts Chiral Vibrational Bands.

Band-(4) : Zero Phonon

Band-(5): One phonon

Tilted axis cranking calculations yield e= 0.29 and g= 31o.
  • TAC calculations indicate the chirality is generated by nh11/2 particle coupled to the short axis and a mixed nd5/2,g7/2 hole coupled to the long axis with collective rotation along the intermediate axis.
  • The angular momentum vector moves with increasing rotational frequency from the plane spanned by the intermediate and long axis through an aplanar orientation into the plane spanned by the short and intermediate axis, in contrast to odd-odd nuclei where it moves from the long-short plane toward the intermediate axis.
Thus the 106Mo structure and motion of the equilibrium position of the angular momentum are quite different from the odd-odd nuclei.
  • The angular momentum moves rather rapidly from one into another principal plane with a very shallow minimum of the aplanar orientation. These indicate the chirality has a dynamical character. It appears as a low energy vibration which correspond to slow excursions of the angular momentum vector into left handed and right handed regions rather than substantial tunneling between the left and right handed configurations in odd-odd nuclei.
  • The two bands in 106Mo correspond to the zero and one phonon states of the chiral vibration.
  • The 106Mo case is unusual in that zero phonon chiral band decays only to the one phonon gamma band and the one phonon band decays only to the two phonon gamma band. This is an interesting unsolved question for theory.

New type chiral bands

  • Chiral vibrational bands in even-even nucleus
  • TAC calculations indicate that in 106Mo chirality has a dynamical character
  • Two bands are low energy zero and one phonon chiral vibrations
  • This different mechanism of generating chirality helps prove the general nature of chirality in nuclei
crossover to cascade transition strengths and transitions between the pairs of doublets
Crossover to cascade transition strengths and transitions between the pairs of doublets.
  • In some cases cascade transitions are stronger and then in others the DI=1 transitions are stronger. In some cases there are crossing transitions going up in spins(104Rh) and in other cases (106Mo) crossing transitions occur only at the bottom.
  • There is an unknown phase factor that connects the right -and left-hand terms. Differences in phase strongly influence both of the above properties. So E2/M1 ratios and crossing transitions between doublet pairs are not unique signatures of chirality.
100 zr
  • Ground band strongly axial symmetric prolate deformed

b2 =0.37

  • An excited band from 0+ to 12+ is nearly spherical

b2 =0.12 .

  • The two new doublet bands can be a) axial symmetric b) tri-axial.
  • The 5- band heads for the doublets decay to the 4+ and 6+ ground band states. We measured the half-lives of both 5- states to be < 8nsec by triple coincidence technique for time windows 4, 8, 16, 20, 28, 48, 72, 100, 300, 500 nsec.
  • These short life times are not consistent with DK=5 if these are axially symmetric states.
  • Thus the life times and branching ratios indicate these bands are tri-axial.






energy differences between states of same spin
Energy differences between states of same spin
  • 106Mo and 110Ru have very similar energy differences.
  • 100Zr has the largest most nearly zero (degenerate) levels observed in any nucleus. Thus it could have the best chiral doublet bands.
  • In 112Ru the energy differences decrease rapidly toward zero as in 104,105Rh and 134Pr and then cross zero as in 134Pr. This crossing could indicate that these nuclei are soft chiral vibrations away from the crossing point where they pass through a region of chiral rotation.
The energy difference to first order is Plank’s constant times tunneling frequency between the right and left handed terms.
  • However, Frauendorf has recently noted there is a phase factor connecting the two terms and it could change with spins. It is possible that a phase factor that slowly varies with spin could explain the crossing of zero.
  • This phase factor also strongly influences the intensity ratios of cross-over to cascade transitions and transitions between bands.
  • Observed first pairs of degenerate doublet bands in even-even neutron-rich nuclei, 106Mo, 110,112Ru and 100Zr.
  • In tri-axial 106Mo and 110Ru, these bands have properties of soft chiral vibrations where neutron particles and holes provide the conditions for tri-axial shape.
  • In tri-axial 112Ru they look more like the chiral rotors observed in other nuclei.
  • These data on chiral vibrations provide evidence that chiral behavior is a more general property of nuclei.
  • The 100Zr doublets are likely also chiral vibrational bands. This is unexpected since it’s ground state is axially symmetric and strongly deformed. Thus there is new physics here.
  • 100Zr is the first case of coexistence of strongly deformed and nearly spherical shapes coexisting with triaxial shape.
  • Problem of how to explain the crossing of zero of the energy differences is an important major problem. This may be related to the phase factor between the right- and left- symmetry states.