1. EXPERIMENTS WITH LARGE GAMMA DETECTOR ARRAYS Lecture IV Ranjan Bhowmik Inter University Accelerator Centre New Delhi -110067

2. Lecture IV SERC-6 School March 13-April 2,2006 2 ASSIGNMENT OF SPIN & PARITY

3. Lecture IV SERC-6 School March 13-April 2,2006 3 General Properties of Electromagnetic Radiation Individual nuclear states have unique spin and parity. For decay from (E i J i M i  i ) to (E f J f M f  f ), the electromagnetic radiation must satisfy the following relations: EnergyE  = E i - E f Multipolarity|J i - J f |  L  (J i + J f ) M-stateM = M i - M f Parity  =  i  f For time varying field, the vector potential A should satisfy the vector Helmholtz equation : The scalar Helmholtz equation has the following solution with states of good angular momentum L and parity (-1) L

4. Lecture IV SERC-6 School March 13-April 2,2006 4 ELECTRIC & MAGNETIC TRANSITIONS The corresponding Vector solutions are : Parity (-1) L+1 Parity (-1) L At large distances (kr » 1), Electric and magnetic fields complimentary : E(r ; E) = H(r ; M) H(r ; E) = -E(r ; M) At short distances (kr « 1) |E(r ; E)| >> |H(r ; E)| |H(r ; M)| >> |E(r ; M)| This justifies the names 'Electric' and 'Magnetic' for the two types of fields. Electric field interacts with charges  Electric multipole excitation Magnetic field interacts with currents (magnets)  Magnetic multipole excitation

5. Lecture IV SERC-6 School March 13-April 2,2006 5 ELECTRIC DIPOLE RADIATION The classical radiation field from an oscillating dipole is given by P ~ E  H ~ sin 2  r 2 which is maximum in a plane  to dipole direction [ zero at 0  ] The electric field is in the plane containing the dipole. Quantum mechanically, this correspond to a dipole field with L=1 M=0 with linear polarization along  P For an axially symmetric oscillating quadrupole field (Q 20 ) the radiation pattern P ~ E  H ~ sin 2  cos 2  r 2 [ zero at 0  & 90  ] Quadrupole field with L=2 M=0 with linear polarization along 

6. Lecture IV SERC-6 School March 13-April 2,2006 6 ANGULAR DISTRIBUTION OF MULTIPOLE RADIATION Angular distribution Z(  ) =| A(r, ,  ) | 2 is a function of  only For magnetic radiation, role of E & H are interchanged Similar angular distribution for electric and magnetic multipoles would differ in plane of polarization Adding all the M components incoherently would result in isotropic unpolarized radiation Electric dipole radiation at 90  Polarization M = 0 || to axis M = 1  to axis Electric Quadrupole radiation at 90  Polarization M = 1 || to axis M =2  to axis

7. Lecture IV SERC-6 School March 13-April 2,2006 7 ELECTROMAGNETIC TRANSITION PROBABILITY Since we are not interested in the orientation of either the initial or the final nucleus, we sum over all M f and average over all M i. Angular distribution of the photon would involve contributions from different allowed values of L & M. Since kR « 1, the transition probability  T fi decrease rapidly with L and the lowest allowed L is important. The transition probability for the nucleus decaying from a state |J i M i > to state |J f M f > by an interaction R is given by

8. Lecture IV SERC-6 School March 13-April 2,2006 8 MULTIPOLARITY OF TRANSITION For a change in angular momentum  L = |J i - J f | the dominant multipolarities are : JJ Same Parity  i =  f Opposite parity  i   f 0M1,E2 mixed radiation E1 1M1,E2 mixed radiation E1 2E2(M2,E3) M1 & E2 often have comparable strength

9. Lecture IV SERC-6 School March 13-April 2,2006 9 RADIATION FROM ORIENTED NUCLEI  Random orientation of nuclei : radiation is isotropic as all M i substates are to be added incoherently: radioactive decay  Nuclei oriented perpendicular to z-axis: fusion Populates large spins with M i ~ 0 by heavy ion fusion M i  0 nuclei decaying predominantly to M f  0 For L=1 M = 0  Emitted radiation maximum at  ~ 90   Polarization || to z-axis for Electric transition For L=2 M = 0,  1  Emitted radiation minimum at  ~ 90   Polarization || to z-axis for Electric transition L=  J for stretched transition  Nuclei oriented along z-axis : polarized nuclei M = L Angular distribution opposite; polarization reversed in sign

10. Lecture IV SERC-6 School March 13-April 2,2006 10 ALIGNMENT IN NUCLEAR REACTION In fusion reaction between even-even nuclei, compound nucleus is populated with high spin at M=0 state. Successive particle emission would broaden the M-distribution. Since the  -decay along the cascade is mostly stretched in nature (  J =L) the M- distribution of the decaying state J i would be centered around M=0 If the spin distribution is symmetric i.e. P(-M) = P(M) NUCLEAR ALIGNMENT Asymmetric spin distribution P(M) > P(-M) leads to NUCLEAR POLARIZATION Gaussian parameterization for oriented nuclei: P(M i ) ~ exp(-M i 2 /  2 ) /  i exp(-M i 2 /  2 )with  J i ~ 0.3

11. Lecture IV SERC-6 School March 13-April 2,2006 11 ANGULAR DISTRIBUTION IN FUSION Angular distribution of  -transitions can be measured by moving the detector to a different  and normalising the counting rate w.r.t. a fixed detector Shows pronounced anisotropy : W(  ) = 1 +a 2 P 2 (cos  ) +a 4 P 4 (cos  ) Symmetric about 90  W(  ) = W(  ) Only even orders allowed with N max  2L 'Beam in' & 'Beam out' directions equivalent Nucl. Phys. A95(1967)357

12. Lecture IV SERC-6 School March 13-April 2,2006 12 Theoretical angular Distribution The theoretical angular distribution from a state J i to a state J f by multipole radiation of order L, L' can be written as : where  K Statistical Tensor describing initial state population. Only even K allowed for symmetric M distribution Depends on the population width  Normalize to transitions with known multipolarity A K Geometrical factor depending on 3j, 6j, 9j symbols Sensitive to L-change in the high spin limit A K (J i LL'J f ) ~ A K (  J,L)

13. Lecture IV SERC-6 School March 13-April 2,2006 13 ANUGULAR DISTRIBUTION FOR PURE MULTIPOLES Angular distribution coeffs for pure multipoles in high spin limit for ideal initial M-distribution P(M) =1 for M=0 or  ½ JJ La2a2 a4a4 010.5000 02-0.357-.542 11-0.2500 12-0.1790.429 220.357-0.107

14. Lecture IV SERC-6 School March 13-April 2,2006 14 SYSTEMATICS OF L=2 TRANSITIONS Angular distributions for  J =2 very similar with a minimum at 90  For most transitions a 2 = +0.30  0.09 a 4 = -0.09  0.05 2  0 transitions show large deviation due to external perturbation Large anisotropy consistent with a narrow M-distribution  ~ 0.3 J PRL16(1966)1205

15. Lecture IV SERC-6 School March 13-April 2,2006 15 SYSTEMATICS OF DIPOLE TRANSITIONS Dipole transitions have a maximum at 90  a 2 -ve -a 2 ~ 0.4 - 0.6 If there is no change in parity, M1 can be mixed with E2 transitions Angular distribution sensitive to the mixing ratio  As the transitions are weak L=1 mostly seen in coincidence measurements E2 M1,E2 PRL16(1966)1205

16. Lecture IV SERC-6 School March 13-April 2,2006 16 MIXING RATIO  If for transition between states J i  J f two multipolarities L, L' are allowed,  is the ratio of the reduced nuclear matrix elements  a real number -      Sign of  depends on the relative phase of the nuclear matrix elements Angular distribution To extract  from measured W(  ),  K must be estimated from a model of P(M) or extracted from pure E2 angular distribution

17. Lecture IV SERC-6 School March 13-April 2,2006 17 DETERMINATION OF MIXING RATIO  Angular distribution of  -rays sensitive to  J and mixing ratio Solid curve : pure L=2 Dotted curve : pure L=1 Dashed & dot-dashed curve: mixed transition  = -1 & +1 Large interference effects for  J =1 Knowledge of both a 2 & a 4 important to identify the spin change  J

18. Lecture IV SERC-6 School March 13-April 2,2006 18 ANGULAR CORRELATION Weak transitions in a  -cascade can only be identified in  coincidence measurements Angular correlation W(  1,  2,  ) can be calculated theoretically if M-state population is known with sum over all variables K, K1, K2, q1, q2 For decay from symmetric M-distribution all K are even

19. Lecture IV SERC-6 School March 13-April 2,2006 19 ANGULAR CORRELATION As a special case, we consider radioactive decay of a cascade of  - transitions. Because of the random orientation of the 4 + state populated by  -decay, all  K zero. By summing over all other indices the angular correlation is obtained as : where A K (1), A K (2) are the coefficients characterising the two transitions and  is the angle between the detectors.

20. Lecture IV SERC-6 School March 13-April 2,2006 20 ANGLAR CORRELATION : SYMMETRY PROPERTIES Symmetric M distribution, 'beam in' & 'beam out' equivalent W(  1,  2,  ) = W(  -  1,  -  2,  ) Additional symmetries involving   -  and   +  NIMA313(1992)421 Integration over out-of-plane angle  product of angular distributions NPA563(1993)301  Integration over angle of one detector Integration over all detectors gives the angular distribution Angular distribution from angular correlations using large array

21. Lecture IV SERC-6 School March 13-April 2,2006 21 Similarity between angular distribution & angular correlation

22. Lecture IV SERC-6 School March 13-April 2,2006 22 Anisotropy in angular distribution 'Gated angular distribution' extracted from the angular correlation W(  1,  2) by summing over all  2 Anisotropy defined as where  A ~ 0  or 180   B ~ 90  Sensitive to  J &  Gating with unknown L possible Mixing Angle PRC53(1996)2682 E2 E1 M1/E2 E2/M1 Three possible solutions !! need linear polarization data

23. Lecture IV SERC-6 School March 13-April 2,2006 23 Directional Correlation from Oriented Nuclei Useful information about  J can be obtained by measuring coincidences between two detectors, one near 90  and the other near 0  with respect to beam direction If the detectors are sensitive to both radiations  1 &  2 we can distinguish between (i)  1 in detector 1  2 in detector 2 (ii)  2 in detector 1  1 in detector 2 DCO = W(  1,  1 ;  2,  2 )/W(  1,  2 ;  2,  1 )

24. Lecture IV SERC-6 School March 13-April 2,2006 24 DCO Ratio Ignoring  dependence we get DCO ratio ~ [W(  1 ;  1 )*W(  2 ;  2 )] / [W(  1 ;  2 )*W     )] = [W(  1 ;  1 )/ W(  1 ;  2 )] * [W(  2 ;  2 )/W     )] If both radiations  1 and  2 have the same multipolarity, they have similar angular distribution and DCO ratio =1 If they have different multipolarity i.e. L=1 for  1 and L=2 for  2 both terms greater than 1 and DCO ~ 2 Exchange of angles or exchange of gating multipolarity would invert the ratio Generalization valid only for Stretched transitions ! Some papers have inverted definition i.e. NIMA275(1989)333

25. Lecture IV SERC-6 School March 13-April 2,2006 25 EXPERIMENTAL DCO RATIO Gate on E2 transition 607 keV transition E2 484, 506, 516, 568, 617 keV transitions dipole PRC47(1993)87 E2 gate 93 Tc

26. Lecture IV SERC-6 School March 13-April 2,2006 26 DCO Ratio : advantages Can be used for weak transitions More sensitive to angular distribution i.e. W(  ) 2 Ideal for small arrays with limited number of angle combinations Not overly sensitive to choice of angles 75  <   < 105   2 150  DCO similar for both M1 & E2 transitions if  J =1 Large interference effect for mixed transitions DCO ambiguity for  J=0, 1  1=90   =0  gate on L=2

27. Lecture IV SERC-6 School March 13-April 2,2006 27 Sensitivity of DCO Ratio to mixing parameter EPJA17(2003)153 Two solutions, need polarization data !!

28. Lecture IV SERC-6 School March 13-April 2,2006 28 POLARIZATION MEASUREMENTS Angular distribution for both E1 and M1 similar; maximum at 90  Can be distinguished by polarization measurement Stretched E1 transition has polarization vector in-plane stretched M1 transition has polarization vector perpendicular to plane Maximum polarization at  = 90  Can be studied in (i) singles (ii) in coincidence with another detector (PDCO) (iii) measuring polarization of both detectors (PPCO) RMP31(1959)711 NIM163(1979)377 NIMA362(1995)556 NIMA378(1996)516 NIMA430(1999)260

29. Lecture IV SERC-6 School March 13-April 2,2006 29 POLARIZATION FORMALISM P olarization in a nuclear reaction : where J 0, J 90 are the average intensities of the Electric vector in plane with the beam direction & perp. to the plane. Angular distribution : Polarization : Maximum at 90  with a value for pure E1, M1 or E2:  = +1 (E1,E2) ;-1 (M1)

30. Lecture IV SERC-6 School March 13-April 2,2006 30 Measurement of Polarization Compton Scattering is sensitive to the polarization direction Vertically polarized photons would be preferentially scattered in the horizontal plane Klein-Nishina formula Maximum sensitivity at  ~ 90 

31. Lecture IV SERC-6 School March 13-April 2,2006 31 Detection of Compton-scattered radiation Two Ge detectors : one as scatterer and other as detector of scattered radiation Need large efficiency for coincident detection Identified as E  = E 1 + E 2 Experimental Asymmetry a(E  ) corrects for any instrumental effect between horizontal & vertical plane

32. Lecture IV SERC-6 School March 13-April 2,2006 32 Different Designs of Polarimeter GAMMASPHERE CLOVER

33. Lecture IV SERC-6 School March 13-April 2,2006 33 CLOVER as a Polarimeter Polarization sensitivity Q = A/P where P is polarization of the incident radiation Large polarization sensitivity Q ~ 13% at 1 MeV Large Compton detection efficiency ~ 40% at 1 MeV Measurement in singles or in coincidence NIMA362(1995)556

34. Lecture IV SERC-6 School March 13-April 2,2006 34 Measurement of Polarization Electric Magnetic

35. Lecture IV SERC-6 School March 13-April 2,2006 35 Polarization Measurement in 163 Lu PRL86(2001)5866 NPA703(2002)3

36. Lecture IV SERC-6 School March 13-April 2,2006 36 Polarization measurement in 163 Lu Confirmation of the wobbling mode in 163 Lu through combined angular distribution and linear polarization measurement

37. Lecture IV SERC-6 School March 13-April 2,2006 37 Polarization-Direction Correlation PDCO Polarization-Polarization Correlation PPCO With the availability of a large array of Clover detectors, we can measure the polarization of one or both  -rays in coincidence. This results in additional information in the form of PDCO (where one polarization is measured) or PPCO where both polarizations are measured. Combined with DCO this provides a powerful tool for spin assignment. I  4 +  2 + NIMA430(1999)260

38. Lecture IV SERC-6 School March 13-April 2,2006 38