Atomic Effects on Nuclear Transitions

The following processes will be discussed:

Ante Ljubičić, Rudjer Bošković Institute, Zagreb, Croatia

Why these three processes?- large discrepancies between the theory and experiment,- interaction pictures for these processes have similar structure, they show interaction between two oscillators in the same atom, and- our simple theoretical model could remove these discrepancies

Introduction

Nuclear excitation in positron-electron annihilation

Nuclear excitation in electron transition

NEPEANEETTh: Osaka U. 1973 Exp: Osaka U. 1978

Th: U. Tenesee 1952. Exp: Kyoto U. 1972.

We can consider the NEET process as the two-step process , i.e. first the X-ray is emitted by the electron, and then subsequently absorbed by the nucleus.

Typical experimental set-up for the NEET investigations:

NEET process

2

2

22

NTNe

NTNR

eT

eRNEET

Ng

P

Therefore it could be expressed as

vacancieselectroncreatedofnumber

statesnuclearexcitedofnumberPNEET

Transition probability is defined as

However using this expression we obtain results which are too small compared to experiments.

In order to overcome this problem we introduced a simple model of Indistiguishable Quantum Oscillators ( IQO ). Using this model we were able to obtain reasonable agreement with experiments.

- Let us first assume that the two oscillators, with equal multipolarities and transition energies, are far away from each other, so that D >> λ . In that case they exchange real photons. It means that if electron oscillator with radiative width Γel >> ΓN emits photons, then number of photons absorbed by the nuclear oscillator will be proportional to Nabs ~ Γel ( ΓN / Γel ) ~ ΓN

- However if these two oscillators are so close that D < λ, then the two oscillators exchange virtual photons, and we can not distinguish between them.

eN

N e

N

N

e

e

- And this is exactly the basis of our model of two Indistinguishable Quantum Oscillators, the IQO model.

- Quite generally, the IQO model says that if we can not distinguishbetween the two oscillators then the two oscillators with the two individual line-widths behave as one oscillator with one line-width equal to the sum of the two individual line-widths.

- Two oscillators are indistinguishable if they have equal transition energy ω, equal multipolarity, and if the separation D between the two oscillators is less than the wavelength λ of the exchanged resonant photon.

In this case we would expect that they behave as one oscillator with the line-width equal to the sum of individual line-widths, i. e.

ΓTot ≈ Γel + ΓN

and number of counts absorbed by the second oscillator will be

Nabs ~ ΓTot ~ Γel + ΓN ~ Γel

ΓN → Γe + ΓN ≈ Γe

we only have to replace

)(

22 2

2

e

eTNe

eReRNEET f

gP N

2

2

22

NTNe

NTNR

eT

eRNEET

Ng

P

Now we can apply our IQO model to the NEET processes.

In our previous expression for PNEET

and we obtain

Nucleus Experiment References IQO model

237Np (2.1±0.6)×10-4 Saito et al. [1980] 1.3×10-4

197Au (5.0±0.6)×10-8 Kishimoto et al. [2006] 9.4×10-9

189Os ≤ 3×10-10 Ahmad et al. [2002] 3.6×10-10

Using this expression we have calculated several PNEET and compared them with the experimental results:

As we can see the agreement between the experiment and our calculations based on the IQO model is reasonable.

Nucleus Experiment References Theory References237Np 2.1 x 10-4 Saito et al. 1.5 x 10-7 Pisk et al ,1989

1980 8.5 x 10-9 Ho et al.,1991

3.1 x 10-12 Tkalya, 1992

1.3 x 10-4 Ljubicic et al.,1998

197Au 5 x 10-8 Kishimoto et al 3.5 x 10-5 Pisk et al.,1989

2006 4.2 x 10-7 Ho et al.,1991

1.4 x 10-7 Tkalya,1992

9.2 x1 0-9 Ljubicic et al.,1998

189Os 5.7 x 10-9 Shinohara et al. 2.5 x 10-7 Pisk et al.,1989

1987 1.2 x 10-9 Ho et al.,1991

< 3 x 10-10 Ahmad et al. 1.1 x 10-10 Tkalya, 1992

2002 1.3 x 10-10 Ahmad et al,2000

3.6 x 10-10 Ljubicic et al.,1998

NEPEA

E+ = E1078 – 2mc2 + |BK| ≈ 83 keV

annihilate with K-shell electrons, the 1078-keV gamma-ray is emitted and nuclear level of the same energy is excited.

The best case is 115In, because its nuclear level scheme is well known. First experiment by Kyoto group in 1972. Indium sample was irradiated by positrons from 22Na. Positrons slow down in the sample

1290.613/2+

1132.611/2+

1078.2

597.1

336.2

5/2+

9/2+

3/2-

1/2-

0.42 ps

. 0.25 ns

4.5 h

0.99 ps

0.07 ps

115In

Γ

e+

e+

and at resonant positron kinetic energy

Transition from the 336-keV metastable state was observed in the experiment.

Γ

e+

e+ - In their analysis they assumed that number of effective 115In atoms in the sample is ~ Γ1078 . Therefore from

Ngamma ~ σexp Φ+ NIn Γ1078

they obtained σexp ≈ 10-24 , but theory predicts σth ≈ 10-26

.

We could estimate this process using the IQO model.

The NEPEA process could also be treated as a system of two oscillators, and if the two oscillators are close enough we can replace

Γ1078 → ΓK >> Γ1078

Then for larger Γ we expect smaller cross section and better agreement with theory.

- We must check how close the two oscillators are, i.e. if we can apply the IQO model.

- To a first approximation we can define the indistinguishability factor βK for K-shell electron as the probability of finding it within the distance from the nucleus D < λ . In that case

0

22

0

22

drr

drr

K

K

K

- However it cannot be assumed that there is a sharp break between distinguishability and indistinguishability at D = λ, and it is necessary to introduce a simple model to allow for this.- It is assumed that each particle can be represented by a Gaussian

)2/exp( 22 rG

- Several other experiments were performed and all of them obtained cross sections which are several orders of magnitude larger than theoretical predictions.

- We can also calculate similar factor β+ for positrons and then re-analyze experimental result previously reported by Kyoto group.

18.0

0

22

0

22

drr

drrG

K

K

K

where Δ=λ/2. In that case we obtain

for 115In

Nucleus

Old approach, before 1982

New approach, after 1991

Experiment Theory(Present &

Chen)

Experiment(using IQO model)

Theory(Kaliman et

al.)

115In 10-24 10-26 1.2×10-26 2.0×10-26

111Cd 8.6×10-25 2.4×10-26 2.4×10-25 3.9×10-25

176Lu 9×10-22 1.2×10-24 2.7×10-26 2.2×10-26

103Rh σexp ≈ 1.3x10-24 cm2

107,109Ag σexp≈ 4.0x10-23 cm2

113In σexp ≈ 1.9x10-24 cm2

We re-analyzed 3 experiments using our IQO model and obtained good agreement with the most recent theoretical predictions of Kaliman et al.

Theories:- Grechukhin &

Soldatov- Pisk et al.- Horvat et al.- Kolomietz

Other experiments:

Conclusion:We have analyzed six experiments in which atomic effects could play important role in exciting nuclear levels.We have employed the model of IQO and quite generally obtained good agreement between the theory and experiment. Therefore I believe that the IQO model is a realistic one and we will use it in order to explain other processes in which nuclei interact with atomic electrons.