1. Sizes

2. Positronium and e + -e - Annihilation Gamma-Gamma Correlations 2 E  /2E 0 1 dN/dE  E  /2E 0 1 dN/dE  Decay at rest: 2E  ≈E 0 ≈ 1.022 MeV 2-body decay   12 ≈180 0 3-body decay  continuum in E  and   2-body decay  line spectrum in E  and     Para Positronium Ortho Positronium

3. Gamma-Gamma Correlations 3 Radiation Detectors for Medical Imaging Positron e + (anti-matter) annihilates with electron e - (its matter equivalent of the same mass) to produce pure energy (photons, -rays). Energy and momentum balance require back-to-back (180 0 ) emission of 2 -rays of equal energy  Detectors (NaI(Tl)) Positron emission tomographic (PET) virtual slice through patient’s brain Administer to patient radioactive water: H 2 17 O radioactive acetate: 11 CH 3 COOX Observe 17 O or 11 C  + decay by

4. ANSEL Angular Correlation Experiment Gamma-Gamma Correlations 4 Top left: PET imaging experiment setup with two 1.5”x1.5” NaI(Tl) detectors (BICRON) on a slotted correlation table. A “point-like” 22 Na  source can be hidden from view. Top right: 22 Na  spectrum measured with NaI 1. Bottom right: angular correlation measurement. Second correlation setup: NaI(Tl) vs. HPGe

5. Gamma-Gamma Correlations 5 E 1 -E 2 -t Multi-Parameter Measurement  cascade ( 60 Ni) or PET experiment, full 3D energy-time distribution. Delay var. TFA/C FTD Gate Gener ator Pre Amp Amp Energy 1 Relative Time Pre Amp TFA/C FTD Gate Gener ator Amp TAC t Energy 2 Start Stop Counter DAQ Pulse Gener ator Counter Pulser Dead time measurement

6. Gamma-Gamma Correlations 6 E 1 -E 2 Dual-Parameter Measurement  cascade ( 60 Ni) or PET experiment, full 2D spectrum Delay var. TFA/ CFTD Gate Gener ator Pre Amp Amp Energy 1 Strobe Pre Amp TFA/ CFTD Gate Gener ator Amp Energy 2 Pulse Gener ator Counter Pulser Dead time measurement Coincid. OR DAQ

7. 2D Parameter Measurement Gamma-Gamma Correlations 7 2D Scatter Plot: Each point represents one event {E 1, E 2 } E2E2 E1E1 Projection on E 2 axis Projection on E 1 axis

8. Gamma-Gamma Correlations 8 Gated E 1 -E 2 Coincidence Measurement  cascade ( 60 Ni) or PET experiment, gates on   and   lines Delay var. SCA Gate Gener ator Amp Energy 1 Strobe Amp SCA Gate Gener ator Energy 2 Pulse Gener ator Counter Pulser Dead time measurement Coincid. OR DAQ

9. Gamma-Gamma Correlations 9 Absolute Activity Measurement N1N1 N2N2 N 12 TFA/ CFTD Gate Gene rator Pre Amp TFA/ CFTD Gate Gene rator Coincid ence Counter Coinc. Counter 1 Counter 2 1 2 Activity A Activity A [disintegrations/time], independent radiation types i =1,2 detection probabilities P i

10. Sizes

11. Symmetries of the Nuclear Mean Field Gamma-Gamma Correlations 11 Potential Bound States r  (r) 0 Nucleon-nucleon forces are dominantly radial r r n=0 n=1 n=2

12. Gamma-Gamma Correlations 12 Stationary Nuclear States Consequences of space-inversion and rotational invariance: Stationary states eigen states are also eigen states of Radial (r) and angular () degrees of freedom are independent  Separate d.o.f in Schrödinger Equ.  Product wave functions r Spherical Harmonics or (axial symmetry) Legendre Polynomials r n=0 n=1 n=2 I2+I2+ 1-1- 0+0+ EnEn

13. Gamma-Gamma Correlations 13 Electromagnetic Nuclear Transitions Conserved: Total energy (E), total angular momentum (I) and total parity (): I2+I2+ 1-1- 0+0+ EiEi I2+I2+ 1-1- 0+0+ EfEf Consider here only electric multipole transitions. Neglect weaker magnetic transitions due to changes in current distributions.

14. t-Dependent Electromagnetic Radiation Fields Gamma-Gamma Correlations 14 Point Charges Nuclear Charge Distribution Monopole ℓ = 0 Dipole ℓ = 1 Quadrupole ℓ =2 Protons in nuclei = moving charges  emits elm. radiation Heinrich Hertz Propagating Electric Dipole Field E. Segré: Nuclei and Particles, Benjamin&Cummins, 2 nd ed. 1977

15. Gamma-Gamma Correlations 15 Selection Rules for Electromagnetic Transitions Conserved: Total energy (E), total angular momentum (I) and total parity (): z mimi mfmf mm Coupling of Angular Momenta Quantization axis : z direction. Physical alignment of I possible (B field, angular correlation)

16. Gamma-Gamma Correlations 16 Spherical Harmonics P 1 |m| cos (m z x y   Plot Re Y l m  P 1 |m| cos (m Emission in z direction Emission perpend. to z

17.  Angular Correlations Gamma-Gamma Correlations 17 Simple example:  cascade 0  1  0 Mostly m=±1 emitted in z-direction Det 1 Det 2  1 =0 22 11 22 Det 1: Define z, select transition with maximum  intensity for   =0. Det 2: Measure emission patterns with respect to this z direction  determine m. General Expression for angular correlation Typically: n ≤ 2. Legendre Polynomials P n (cos)

18. E2  Angular Correlations Gamma-Gamma Correlations 18 Anisotropy Example: Rotational E2- cascade, m=±2 maximally emitted in z-direction After: G. Musiol, J. Ranft, R. Reif, D. Seeliger, Kern-u. Elementarteilchenphysik, VCH 1988

19. Gamma-Gamma Correlations 19

20. Gamma-Gamma Correlations 20