Isospin mixing E.Farnea INFN Sezione di Padova 1. Isospin - PowerPoint PPT Presentation

Isospin mixing E.Farnea INFN Sezione di Padova 1. Isospin selection rules 2. Isospin mixing and forbidden E1 transitions: the 64 Ge case 3. Spin assignments 4. Measurements of linear polarization 5. Lifetime measurements 6. E1 transitions in mirror nuclei: the 67 As- 67 Se case

Selection rules Not all electromagnetic transitions are allowed! There are selection rules originated by the structure of the operators inducing the transition, or by the angular momentum coupling between the initial and final state and the emitted photon, for instance: Electromagnetic transitions connecting 0  0 states are forbidden An electromagnetic transition with multipolarity L can only connect a state with spin J i to a state with spin |J i -L| ≤J f ≤J i +L

Isospin selection rules Taking a very general expression describing the microscopic current density: 𝑞 𝑞 𝑞 𝑞 𝑓ℏ 𝑘 𝑂 𝑠 = 𝑓 𝑞 𝜀 𝑠 − 𝑠 𝑞 + 𝜀 𝑠 − 𝑠 + 𝑑 𝜈 𝑞 2𝑁 𝑞 𝑑 𝛼 × 𝜏 𝑞 𝜀 𝑠 − 𝑠 𝑞 𝑞 2𝑁 𝑞 2𝑁 𝑞 𝑞 𝑞 𝑓ℏ 𝑜 + 𝑑 𝜈 𝑜 2𝑁 𝑞 𝑑 𝛼 × 𝜏 𝑜 𝜀 𝑠 − 𝑠 𝑜 Since the charge can be written as e(1- τ 3 )/2, the electromagnetic interaction hamiltonian 𝐼(𝑀, 𝑁) = 1 𝑒𝜐 𝑑 𝑘 𝑂 𝑠 ∘ 𝐵 𝑀𝑁 𝑙𝑠 Naturally divides into two isoscalar and isovector components: 𝐼 𝑀, 𝑁 = 𝐼 0 𝑀, 𝑁 + 𝐼 1 (𝑀, 𝑁)

Isospin selection rules Calculating the transition amplitude: 2 Γ 𝛿 (𝑀) = 8𝜌𝑙 𝐾 𝑐 𝑁 𝑐 𝑈 𝑐 𝑈 3𝑐 𝐼(𝑀, 𝑁) 𝐾 𝑏 𝑁 𝑏 𝑈 𝑏 𝑈 3𝑏 𝑁,𝑁 𝑐 And extracting the T 3 dependency through the Wigner-Eckhart theorem, it can be «easily» shown that: Only electromagnetic transitions with Δ T=0, Δ T= ± 1 are allowed Electromagnetic transitions with Δ T= ± 1 in conjugate nuclei are identical

Isospin selection rules In the particular case of the E1 operator, expanding 𝑘 𝑂 𝑙𝑠 in power series of 𝑙𝑠 and using the long wave approximation, it can be shown that the isoscalar component vanishes. Therefore: Electromagnetic E1 transitions in conjugate nuclei have the same intensity In first approximation, in N=Z nuclei, Δ T=0, in other words only the isoscalar term should be considered; this means that: Electromagnetic E1 transitions in N=Z nuclei are forbidden

Isospin mixing States with different isospin (but same spin and parity) may mix through the Coulomb interaction, making E1 transitions possible also in N=Z nuclei: 1        | | N Z T T N Z   2 The amount of the admixture is of the order of a few percent, it is maximum in N=Z nuclei and increases with the atomic number Z up to the heaviest bound N=Z nucleus, 100 Sn J. Dobaczewski and I. Hamamoto, PLB 345 (1995) 181

64 Ge: forbidden transition? (13 – ) 8006.3 1442.4 (11 – ) 6563.9 1191.5 6065.0 (9 – ) (8+) (8+) 5372.4 (7+) 5175.8 5179.7 1819.8 1127.2 5025.8 1308.6 (7 – ) 1714.4 1768.5 4245.2 (5 – ) (5+) 528.4 (6+) 3717.2 (6+) 3716.8 3465.3 1047.3 3407.3 747.5 1664.8 (3 – ) 1252.1 1413.0 (3+) 2969.3 2669.9 (4+) 4+ 1090.9 2155.2 2052.3 576.2 2067.8 1579.0 1150.8 (2+) 677.0 2+ 1579. 0 901.5 901.5 0+ 0 64 32 Ge 32 P.J. Ennis et al., Nucl. Phys. A 535 (1991) 392

64 Ge: forbidden transition? (13 – ) 8006.3 1442.4 (11 – ) 6563.9 1191.5 6065.0 (9 – ) (8+) (8+) 5372.4 (7+) 5175.8 5179.7 1819.8 1127.2 5025.8 1308.6 (7 – ) 1714.4 1768.5 4245.2 (5 – ) (5+) 528.4 (6+) 3717.2 (6+) 3716.8 3465.3 1047.3 3407.3 747.5 1664.8 (3 – ) 1252.1 1413.0 (3+) 2969.3 2669.9 (4+) 4+ 1090.9 2155.2 2052.3 576.2 2067.8 1579.0 1150.8 (2+) 677.0 2+ 1579. 0 901.5 901.5 0+ 0 Intense E1 transition? Forbidden by 64 32 Ge 32 the isospin selection rules ... P.J. Ennis et al., Nucl. Phys. A 535 (1991) 392

Experiment performed at LNL ISIS EUROBALL III n-Wall 32 S (125MeV) + 40 Ca (1mg/cm 2 ) + 12 mg/cm 2 197 Au backing EUROBALL III + n-Wall + ISIS Laboratori Nazionali di Legnaro

Level scheme (9813) (16+) 9628 (15 – ) The level scheme constructed (899) using the data from the 8914 (14 + ) 1621 EUROBALL III experiment 848 8066 12(+) 8007 (13 – ) does not extend significantly 873 the previously known level 1442 7193 10(+) scheme. On the other hand, it 6565 (11 – ) is confirmed that the 1665keV 1820 line is one of the strongest 1192 5373 9 (–) 5180 (8+) 1127 4246 7 (–) 1714 528 5 (–) Need spin and parity 3466 (6 + ) 3718 747 (3 – ) assignments to confirm the 2970 1413 1665 (tentative) E1 character!!! 2053 4 + 2068 1579 1579 1151 677 902 2 + 902 0 + 0 64 Ge

Assigning spins/parities The spin/parity of the «top» level (unknown) Level with can be determined, starting from the unknown «bottom» (known) level, by measuring the spin/parity character of the transition connecting the two levels, or, in other words, the angular momentum carried by the emitted photon How can we do it in practice? Level with known spin/parity Angular distributions Linear polarization

Angular distributions In order to measure angular distributions, we need to find a way of populating our nuclei of interest with a well defined distribution of spins This is the case with fusion-evaporation reactions, where the initial angular momentum is orthogonal to the beam direction: = 𝑠 𝑚 ∧ 𝑞 • Before particle evaporation and photon emission, the distribution of nuclei has well defined angular momentum and projection of the angular momentum on the beam axis (m=0), namely, they are in an aligned state • Since the evaporated particles carry at most a few units of angular momentum, the residual nuclei after evaporation are populated with a distribution of states symmetrically peaked around m=0, with width ( alignment ) σ

Angular distributions The angular distribution of photons emitted from an aligned state, having spin I 1 and populating a state with spin I 2 , has quite a simple expression: 4 cosϑ 𝑋 𝜘 = 𝐵 0 + 𝐵 2 (𝜏, 𝜀, 𝐽 1 , 𝐽 2 ) ∙ 𝑄 2 cosϑ + 𝐵 4 (𝜏, 𝜀, 𝐽 1 , 𝐽 2 ) ∙ 𝑄 Where A 2 , A 4 are the angular distribution coefficients depending on the spins of the levels and on the multipole mixing ratio δ and P 2 , P 4 are the Legendre polynomials. In principle the angular distribution coefficients should be determined through analysis of singles data. It can be shown that an isotropic gate will not perturb the angular distribution (thus it is possible to make this kind of analysis for weak channels identified through multiple γ coincidences) The finite size of the detectors translates into an «attenuation» of the angular distribution (each detector «sees» a range of angles)

Electromagnetic emission from Oriented Nuclei: Angular distributions s/ J Beam Angular distributions and correlations are only sensitive to the multipolarity L and θ

Some examples quadrupole transition dipole transition 1217 keV 66Ge (4 to 2), gate 521 keV 1510 keV 66Ge (5 to 4), gate 521 keV 2.8E+06 1.1E+06 1.0E+06 2.6E+06 1.0E+06 9.5E+05 2.4E+06 Intensity Intensity 9.0E+05 2.2E+06 8.5E+05 8.0E+05 2.0E+06 7.5E+05 7.0E+05 1.8E+06 6.5E+05 1.6E+06 6.0E+05 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 Angle (degrees) Angle (degrees) Symmetric with respect to θ =90 ° (only even values of the Legendre polynomials allowed)

Angular distribution for the 1665 keV line 1665 keV 10000 d  0.089(34) 5  4 d  3.93(70) 9000 8000 Intensity 7000 6000 5000 4000 0 20 40 60 80 100 120 140 160 180 Angle (degrees) Angular distributions strongly suggest that the observed transition is indeed 5  4, they are not conclusive, however, concerning the multipole mixing ratio δ

Angular distribution for the 1665 keV line 500 Χ 2 analysis Fixed s /J = 0.386, A 0 = 7077 Secondary 400 minimum (small δ ) 300 c Primary minimum 200 (large δ ) 100 0 -90 -70 -50 -30 -10 10 30 50 70 90 arctg( d ) Angular distributions strongly suggest that the observed transition is indeed 5  4, they are not conclusive, however, concerning the multipole mixing ratio δ

Linear polarization • γ -rays emitted by oriented nuclei are partially polarized. The polarization vector is different for E and M transitions • Compton scattering can be used to measure the degree of polarization through the dependency with the polarization vector. The Klein-Nishina cross section for linearly polarized γ -rays is the following: 2     s 2 ' ' d r E E E E         2 2 ' KN 0  2 sin cos  E    ' E d 4  E   E E     1 ( 1 cos ) 2 c m 0  :angle between the scattering plane and the initial polarization plane Legendre angular correlation coefficients   2 2 a (cos ) P      P ( ) Degree of Linear Polarizati on   a (cos ) P    For stretched transitions with L=1,2 this is maximum at θ =90 o H.D.Hamilton, The electromagnetic interaction in nuclear spectroscopy, North-Holland H.Morinaga and T.Yamazaki, In Beam Gamma-Ray Spectroscopy, North Holland

Degree of polarization for a dipole transition as a function of Atan δ . The quadrupole content is: d 2  Q  d 2 ( 1 )