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

EXPERIMENTS WITH LARGE EXPERIMENTS WITH LARGE GAMMA DETECTOR ARRAYSGAMMA DETECTOR ARRAYS

Lecture IVLecture IV

Ranjan Bhowmik

Inter University Accelerator Centre

New Delhi -110067

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

ASSIGNMENT OF SPIN & PARITYASSIGNMENT OF SPIN & PARITY


General Properties of General Properties of Electromagnetic RadiationElectromagnetic Radiation

Individual nuclear states have unique spin and parity. For decay from (Ei Ji Mi i ) to (Ef Jf Mf f), the electromagnetic radiation must satisfy the following relations:

Energy E = Ei - Ef

Multipolarity |Ji - Jf| L (Ji + Jf)

M-state M = Mi - Mf

Parity = if

For time varying field, the vector potential A should satisfy the vector Helmholtz equation : 022 Ak

),()(),,( LMl Ykrjr The scalar Helmholtz equation has the following solution with states of good angular momentum L and parity (-1)L


ELECTRIC & MAGNETIC TRANSITIONSELECTRIC & 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


ELECTRIC DIPOLE RADIATIONELECTRIC DIPOLE RADIATION

The classical radiation field from an oscillating dipole is given by

P ~ E H ~ sin2 r2

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 (Q20) the radiation

pattern P ~ E H ~ sin2cos2 r2 [ zero at 0 & 90] Quadrupole field with L=2 M=0 with linear polarization along


ANGULAR DISTRIBUTION OF ANGULAR DISTRIBUTION OF MULTIPOLE RADIATIONMULTIPOLE 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


ELECTROMAGNETIC TRANSITION ELECTROMAGNETIC TRANSITION PROBABILITYPROBABILITY

Since we are not interested in the orientation of either the initial or the final nucleus, we sum over all Mf and average over all Mi . Angular distribution of the photon would involve contributions from different allowed values of L & M. Since kR « 1, the transition probabilityTfi decrease rapidly with L and the lowest allowed L is important.

The transition probability for the nucleus decaying from a state |JiMi > to state |JfMf > by an interaction R is given by

2

12

2!)!12(

)1(8)(

iilff

l

fi MJRMJk

ll

lRlT


MULTIPOLARITY OF TRANSITIONMULTIPOLARITY OF TRANSITION

For a change in angular momentum L = |Ji - Jf| the dominant multipolarities are :

J Same Parityi = f

Opposite parityi f

0 M1,E2mixed radiation

E1

1 M1,E2mixed radiation

E1

2 E2 (M2,E3)M1 & E2 often have comparable strength


RADIATION FROM ORIENTED NUCLEIRADIATION FROM ORIENTED NUCLEI

Random orientation of nuclei : radiation is isotropic as all Mi substates

are to be added incoherently: radioactive decay Nuclei oriented perpendicular to z-axis: fusion

Populates large spins with Mi ~ 0 by heavy ion fusionMi 0 nuclei decaying predominantly to Mf 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


ALIGNMENT IN NUCLEAR REACTIONALIGNMENT 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 Ji 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(Mi) ~ exp(-Mi2/2)/i exp(-Mi

2/2) with Ji ~ 0.3


ANGULAR DISTRIBUTION IN FUSIONANGULAR 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 +a2P2(cos) +a4P4(cos) Symmetric about 90

W() = W() Only even orders allowed with

Nmax 2L 'Beam in' & 'Beam out'

directions equivalentNucl. Phys. A95(1967)357


Theoretical angular DistributionTheoretical angular Distribution

The theoretical angular distribution from a state Ji to a state Jf by multipole radiation of order L, L' can be written as :

)(cos)()(1

)(cos1)(

KfiKievenK

K

KKevenK

PJLLJAJ

PaW

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

AK Geometrical factor depending on 3j, 6j, 9j symbols

Sensitive to L-change in the high spin limit

AK(JiLL'Jf) ~ AK(J,L)


ANUGULAR DISTRIBUTION FOR PURE ANUGULAR DISTRIBUTION FOR PURE MULTIPOLESMULTIPOLES

Angular distribution coeffs for pure multipoles in high spin limit for ideal initial M-distribution P(M) =1 for M=0 or ½

J L a2 a4

0 1 0.500 0

0 2 -0.357 -.542

1 1 -0.250 0

1 2 -0.179 0.429

2 2 0.357 -0.107


SYSTEMATICS OF L=2 TRANSITIONSSYSTEMATICS OF L=2 TRANSITIONS

Angular distributions for J =2 very similar with a minimum at 90

For most transitions

a2 = +0.30 0.09a4 = -0.09 0.05

20 transitions show large deviation due to external perturbation

Large anisotropy consistent with a narrow M-distribution ~ 0.3 J

PRL16(1966)1205


SYSTEMATICS OF DIPOLE SYSTEMATICS OF DIPOLE TRANSITIONSTRANSITIONS

Dipole transitions have a maximum at 90a2 -ve -a2 ~ 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


MIXING RATIO MIXING RATIO

If for transition between states Ji Jf 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

if

if

JLJ

JLJ

)(cos)()(1)( KfiKievenK

K PJLLJAJW )()(2)(

11

)( 22 ifKifKifKfiK JLLJFJLLJFLLJJFJLLJA

To extract from measured W(), K must be estimated from a model of

P(M) or extracted from pure E2 angular distribution


DETERMINATION OF MIXING RATIO 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 a2 & a4 important to identify the spin change J


ANGULAR CORRELATIONANGULAR 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, q2For decay from symmetric M-distribution all K are even


ANGULAR CORRELATIONANGULAR 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 :

evenK

KKK PAAW )(cos)2()1(1)(

where AK(1), AK(2) are the coefficients characterising the two transitions and is the angle between the detectors.


ANGLAR CORRELATION : ANGLAR CORRELATION : SYMMETRY PROPERTIESSYMMETRY 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 )()(),,( 2121 WWWd product of angular distributions

NPA563(1993)301

Integration over angle of one detector

)(),,( 1212 WWd Integration over all detectors gives the angular distribution

Angular distribution from angular correlations using large array


Similarity between angular distribution Similarity between angular distribution & angular correlation& angular correlation


Anisotropy in angular distributionAnisotropy in angular distribution 'Gated angular distribution'

extracted from the angular correlation W(1,2) by summing over all 2

Anisotropy defined as

)()(

)()(2

BA

BAA

whereA ~ 0 or 180B ~ 90

Sensitive to J & Gating with unknown L

possible

Mixing Angle

PRC53(1996)2682

E2

E1 M1/E

2

E2/M1

Three possible solutions !!need linear polarization data


Directional Correlation from Directional Correlation from Oriented NucleiOriented Nuclei

Useful information about J can be obtained by measuring coincidences between two detectors, one near 90 and the other near 0with respect to beam directionIf 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)


DCO RatioDCO 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


EXPERIMENTAL DCO RATIOEXPERIMENTAL DCO RATIO

Gate on E2 transition

607 keV transition E2

484, 506, 516, 568, 617 keV transitions dipole

PRC47(1993)87

E2 gate

93Tc


DCO Ratio : advantagesDCO 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 < 30 or 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


Sensitivity of DCO Ratio to mixing Sensitivity of DCO Ratio to mixing parameterparameter

EPJA17(2003)153

Two solutions, need polarization data !!


POLARIZATION MEASUREMENTSPOLARIZATION 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)711NIM163(1979)377NIMA362(1995)556NIMA378(1996)516NIMA430(1999)260


POLARIZATION FORMALISMPOLARIZATION FORMALISM

Polarization in a nuclear reaction :

where J0 , J90 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)


Measurement of PolarizationMeasurement 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


Detection of Compton-scattered Detection of Compton-scattered radiationradiation

Two Ge detectors : one as scatterer and other as detector of scattered radiation

Need large efficiency for coincident detection

Identified as E = E1 + E2

Experimental Asymmetry

||

||

)(

)()(

NNEa

NNEaEA

a(E) corrects for any instrumental effect between horizontal & vertical plane


Different Designs of PolarimeterDifferent Designs of PolarimeterG

AM

MA

SP

HER

E

CLO

VER


CLOVER as a PolarimeterCLOVER 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


Measurement of PolarizationMeasurement of Polarization

Electric

Magnetic


Polarization Measurement in Polarization Measurement in 163163LuLu

PRL86(2001)5866NPA703(2002)3


Polarization measurement in Polarization measurement in 163163LuLu

Confirmation of the wobbling mode in 163Lu through combined angular distribution and linear polarization measurement


Polarization-Direction Correlation PDCO Polarization-Direction Correlation PDCO Polarization-Polarization Correlation PPCOPolarization-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


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