Sizes

Positronium and e+-e- Annihilation

Gam

ma-G

am

ma C

orr

ela

tion

s 2

10 32 ( ) 1.25 10 sec

#

n n

n principal quantum

7 33 ( ) 1.4 10 secn n 2

2 0 2 3

0

372

2.818 , . .

e erc

r fm class el radius

Eg/2E0 1

dN/dEg

Eg/2E0 1

dN/dEg

Decay at rest: 2Eg≈E0 ≈ 1.022 MeV2-body decay q12≈1800

3-body decaycontinuumin Eg and q12

2-body decayline spectrumin Eg and q12

q12

Para Positronium

Ortho Positronium

0I

1I

0E eV

48 10E eV

1

s

2

s

2

s

1

s

Gam

ma-G

am

ma

Corr

ela

tion

s3

Radiation Detectors for Medical Imaging

e e 2 (511 keV )

Positron e+ (anti-matter) annihilates with electron e- (its matter equivalent of the same mass) to produce pure energy (photons, g-rays). Energy and momentum balance require back-to-back (1800) emission of 2 g-rays of equal energy

g Detectors (NaI(Tl))

Positron emission tomographic (PET) virtual slice through patient’s brain

Administer to patient radioactive water: H2

17O radioactive acetate: 11CH3COOX

Observe 17O or 11C b+ decay by

ANSEL Angular Correlation Experiment

Gam

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tion

s 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” 22Na g source can be hidden from view.Top right: 22Na g spectrum measured with NaI1.Bottom right: - g g angular correlation measurement.

Second correlation setup: NaI(Tl) vs. HPGe

Gam

ma-G

am

ma

Corr

ela

tion

s 5

E1-E2-Dt Multi-Parameter Measurement

-g g cascade (60Ni) or PET experiment, full 3D energy-time distribution.

Dela

y v

ar.

TFA/CFTD

Gate Generator

PreAmp

Amp Energy 1

Relative Time

PreAmp

TFA/CFTD

Gate Generator

Amp

TACDt

Energy 2

Start

Stop

Counter

Counter

DAQ

Pulse Generator Counter

Pulser

Dead time measurement

Gam

ma-G

am

ma

Corr

ela

tion

s 6

E1-E2 Dual-Parameter Measurement

-g g cascade (60Ni) or PET experiment, full 2D spectrum

Dela

y v

ar.

TFA/CFTD

Gate Generator

PreAmp

Amp Energy 1

Strobe

PreAmp

TFA/CFTD

Gate Generator

Amp Energy 2

Pulse Generator Counter

Pulser

Dead time measurement

Coin

cid.

OR

DAQ

2D Parameter MeasurementG

am

ma-G

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ma

Corr

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tion

s 7

2D Scatter Plot:Each point represents one event {E1, E2}

E2

E1

Projection on E2 axis

Projection

on E1 axis

Gam

ma-G

am

ma

Corr

ela

tion

s 8

Gated E1-E2 Coincidence Measurement

-g g cascade (60Ni) or PET experiment, gates on g1 and g2 lines

Dela

y v

ar.

SCAGate

Generator

Amp Energy 1

Strobe

Amp

SCA

Gate Generator

Energy 2

Pulse Generator Counter

Pulser

Dead time measurement

Coin

cid.

OR

DAQ

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ma-G

am

ma

Corr

ela

tion

s 9

Absolute Activity Measurement

1 1 2 2

12 1 2 12 12

1 2 1 2

12 12

/

N A P N A P individual rates

P P P N A P coincidence rate

N N A P A PA singles coinc

N A P

N1

N2

N12

TFA/CFTD

Gate Generator

Pre Amp

Pre Amp

TFA/CFTD

Gate Generator

Coin

cid

en

ce Counter

Coinc.

Counter 1

Counter 2

1

2

Activity A

Activity A [disintegrations/time], independent radiation types i =1,2 detection probabilities Pi

Sizes

- g g Correlations in Nuclear Decay

SURVEY

Symmetries of the Nuclear Mean Field

Gam

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ma C

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ela

tion

s 1

1

1,...,

,

1: ( )

,

2

,( ) ( )

( )

ˆ

A

i ji j

central potential no angle depe

Invariance against space inversion rotati

Nuclearmean fieldU U r r

per nucleon A

U r U r

Conservation of parity and angular momentum

quantum dualit

nd

y

ons

n

H

e ce

2

1

22 2 2

2

2 2

ˆ ˆ ˆ ˆˆ( ) , 0 ,2

ˆ ˆ ˆ: ( ) ( )( )

ˆ ˆ ˆ ˆ ˆ( ) ( ) ( );

Ai

ii i

ix ix i ii

z i z i i z i i

pU r H H R

m

because p p and U r U rx

and R U r U R r U r and R p p

Potential

Bound States

r

Y(r)

0

Nucleon-nucleon forces are dominantly radial

1 2 1 2( , ) ( )U r r U r r

r

U r

r n=0

n=1

n=2

Gam

ma-G

am

ma C

orr

ela

tion

s 1

2

Stationary Nuclear States

ˆ

2

ˆ: ( )

ˆ ˆ ˆ ˆ ˆ ˆ, ( ) 0

ˆ ( ) (

, 0 ,

)i

Lz z

n n n

z z

z z z

Hj r E j r sta

Operator for rotations R e

H R H L H

tio

L

nary states

2 2ˆ ( , ) ( 1) ( , )

ˆ ( , ) ( , )

ˆ ( ) ( )

m m

z m m

m m

L Y Y

L Y m

Parity

Y

Y Y

Consequences of space-inversion and rotational invariance:Stationary states eigen states are also eigen states of

Radial (r) and angular ( , q f) degrees of freedom are independent Separate d.o.f in Schrödinger Equ. Product wave functions

r

U r

ˆ ˆˆ , zand L LH

, , ( , , ) ( ) ( , )n mn m r j r Y

Spherical Harmonics or (axial symmetry)Legendre Polynomials

r n=0

n=1

n=2

Ip

2+

1-

0+

En

Gam

ma-G

am

ma C

orr

ela

tion

s 1

3

Electromagnetic Nuclear Transitions

Conserved: Total energy (E), total angular momentum (I) and total parity (p):

222

.

( ) ( , )mi

Initial Nucl State

j r Y

Ip

2+

1-

0+

Ei

Ip

2+

1-

0+

Ef

i fi fi fE E E I I I

111

.

( ) ( , )mf

Final Nucl State

j r Y ( ) ( , )I

mE

Photon WF

r Y

Consider here only electric multipole transitions.Neglect weaker magnetic transitions due to changes in current distributions.

t-Dependent Electromagnetic Radiation Fields

Gam

ma-G

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ma C

orr

ela

tion

s 1

4

Point Charges

NuclearCharge

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, 2nd ed. 1977

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tion

s 1

5

Selection Rules for Electromagnetic Transitions

Conserved: Total energy (E), total angular momentum (I) and total parity (p):

i fi fi fE E E I I I

iiI

f

fI

iI

z

mi

mf

fI

mg

,

.

, ,

[ 1( )]

, 1 , 1

:

fi i

i f

i fi f

i f

i

f

i

i

f

Angular momenta I I

direction undetermined

Projections conserved

m m m m m

Electric dipole radiation

I

I I I m m

I

m

I I

Coupling of Angular Momenta

Quantization axis : z direction. Physical alignment of I possible (B field, angular correlation)

Gam

ma-G

am

ma C

orr

ela

tion

s 1

6

Spherical Harmonics P1|m|cos (m )f

z

x

yq

f

r

Plot Re Ylm P1

|m|cos (m )f

( , , ) 1, ( ), cos( )

sin( ) cos( )

sin( ) sin( )

cos( )

mLr r P m

x r

y r

z r

11y

10y

2 21 1

0 1

1/3 2 1/3

( ) ( 0) ( ) ( 1) ( )W P m Y P m Y

21 21

21 20

1( ) 1 cos

2

( ) sin

Y

Y

Emission in z direction

Emission perpend. to z

-g g Angular Correlations

Gam

ma-G

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ma C

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tion

s 1

7

Simple example: cascade 0 1 0Mostly Dm=±1 emitted in z-direction

0

I

1

0

1

2

0, 1

0

m

0

Det 1

Det 2

Q1=0

Q2

g1

g2

1 2 21 2 1 1

21211

2

( ) (0) ( )

( ) ( )

11 cos

2

m m

m

W W W

W Y

Det 1: Define z, select transition with maximum g intensity for q1=0.

Det 2: Measure emission patterns with respect to this z direction determine Dm.

22 2 21 2 0 0

22

( ) cos cos

1 cos ....

nn n n

n nW A P A

A

General Expression for - g g angular correlation

Typically: n ≤ 2. Legendre Polynomials Pn(cosq)

E2 -g g Angular Correlations

Gam

ma-G

am

ma C

orr

ela

tion

s 1

8

2 2

4

2 4

4 2 0

2

1 0.1020 cos 0.0091 cos

E ETheoretical

highest order P

W P P

Anisotropy

max0 0

201

(90 ) (180 ):

(180 )

n

nn

W WA A

W

Example: Rotational E2-g cascade, Dm=±2 maximally emitted in z-direction

After: G. Musiol, J. Ranft, R. Reif, D. Seeliger, Kern-u. Elementarteilchenphysik, VCH 1988

Gam

ma-G

am

ma C

orr

ela

tion

s 1

9

No More Correlations

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ma-G

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ma

Corr

ela

tion

s 2

0