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

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

Lecture IV SERC-6 School March 13-April 2, 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

Lecture IV SERC-6 School March 13-April 2, 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

Lecture IV SERC-6 School March 13-April 2, 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 

Lecture IV SERC-6 School March 13-April 2, 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

Lecture IV SERC-6 School March 13-April 2, 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

Lecture IV SERC-6 School March 13-April 2, 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

Lecture IV SERC-6 School March 13-April 2, 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

Lecture IV SERC-6 School March 13-April 2, 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

Lecture IV SERC-6 School March 13-April 2, 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

Lecture IV SERC-6 School March 13-April 2, 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)

Lecture IV SERC-6 School March 13-April 2, 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 a4a

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

Lecture IV SERC-6 School March 13-April 2, SYSTEMATICS OF DIPOLE TRANSITIONS Dipole transitions have a maximum at 90  a 2 -ve -a 2 ~ 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

Lecture IV SERC-6 School March 13-April 2, 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

Lecture IV SERC-6 School March 13-April 2, 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

Lecture IV SERC-6 School March 13-April 2, 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

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

Lecture IV SERC-6 School March 13-April 2, 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

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

Lecture IV SERC-6 School March 13-April 2, 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

Lecture IV SERC-6 School March 13-April 2, 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 )

Lecture IV SERC-6 School March 13-April 2, 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

Lecture IV SERC-6 School March 13-April 2, 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

Lecture IV SERC-6 School March 13-April 2, 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    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

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

Lecture IV SERC-6 School March 13-April 2, 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

Lecture IV SERC-6 School March 13-April 2, 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)

Lecture IV SERC-6 School March 13-April 2, 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 

Lecture IV SERC-6 School March 13-April 2, 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

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

Lecture IV SERC-6 School March 13-April 2, 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

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

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

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

Lecture IV SERC-6 School March 13-April 2, 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

Lecture IV SERC-6 School March 13-April 2,