Methods to directly measure non-resonant stellar reaction

rates

Tanja Geib

Outline

1. Theoretical background:– Reaction rates– Maxwell-Boltzmann-distribution of velocity– Cross-section– Gamow-Window

2. Experimental application using the example of the pp2-chain reaction in the Sun

– Motivation and some more theory– Historical motivation– 3He(α,γ)7Be as important onset reaction– Prompt and activation method

Reaction Rates

Nuclear Reaction Rate:

reaction cross section

particle density of type X

flux of particles of type a as seen by particles X

Important: this reaction rate formula only holds when the flux of particles has a mono-energetic (delta-function) velocity distribution of just

Generalization to a Maxwell-Boltzmann velocity-distribution

Inside a star, the particles clearly do not move with a mono-energetic velocity distribution. Instead, they have their own velocity distributions.

Sun

Looking at the figure, one can see, that particles inside the Sun (as well inside stars) behave like an ideal gas. Therefore their velocity follows a Maxwell-Boltzmann distribution.

Generalization to a Maxwell-Boltzmann velocity-distribution

The reaction rate of an ideal gas velocity distribution is the sum over all reaction rates for the fractions of particles with fixed velocity:

Here the Maxwell-Boltzmann distribution enters via

is entered to avoid double-counting of particle pairs if it should happen that 1 and 2 are the same species

Generalization to a Maxwell-Boltzmann velocity-distribution

12

After some calculation, including the change into CMS, one obtains:

0

2/3

12

21

2/1

12 exp)(1

8dE

EEE

NNr

In terms of the relative energy (E=1/2 μv2 ) this means

Cross-Section

The only quantity in the reaction rate that we have not treated yet is the cross-section, which is a measure for the probabitlity that the reaction takes place if particles collide. We will now motivate its contributions.

• Tunneling/ Transmission through the potential barrier

repulsive square-well potential

Cross-Section

)exp()exp(

)exp()exp(

)exp()exp(

ikxGikxFu

xDxCu

iKxBiKxAu

III

II

I

Radial Schrödinger equation for s-waves

is solved by the ansatz

This leads to transmission coefficient

)()(22

expˆ112

2

oRREVGk

BKT

0)(222

2

urVEdr

ud

for low-energy s-wave transmission at a square-barrier potential

We generalize this to a Coulomb-potential by dividing the shape of the Coulomb-tail into thin slices of width .

Cross-Section

dr

i

iiin RREVTTTT )()(22

expˆ.....ˆˆˆ121

Reminder: If angular momentum not equal zero, then V(r) V(r) + centrifugal barrier

cR

RndrErV

0

)(22

exp

Total transmission coefficient for s-wave:

Cross-Section

Inserting the Coulomb potential, one obtains:

Solving the integral, and again using that the incident s-wave has small energies compared to the Coulomb barrier height, we get:

2

212

2expˆ eZZ

ET

• We account for the corrections arising from higher angular momenta by inserting the “Astrophysical S-Factor” S(E), which “absorbs” all of the fine details that our approximations have omitted.

Cross-Section

E

12

• Quantum-mechanical interaction between two particles is always proportional to a geometrical factor:

deBroglie wavelength

Finally, our considerations lead to defining the cross-section at low energies as:

2

212

2exp)(

)( eZZEE

ESE

Cross-Section

12C(p,g)13NThe figure on the left shows the measured cross section as a function of the laboratory energy of protons striking a target. The observed peak corresponds to a resonance.

C12

2

212

2exp)(

)( eZZEE

ESE

Gamow-Window

0

2/3

12

21

2/1

12 exp)(1

8dE

E

E

bES

NNr

2/12

21

22

eZZ

b

Entering the cross section into the reaction rate, we obtain:

with

Using mean value theorem for integration, we bring the equation to the form

to pull out the essential physics/ evolve the Gamow-window.

0

02/3

12

21

2/1

12 exp1

8dE

E

E

bS

NNr

Gamow-Window

This is where the action happens in thermonuclear burning!

Log scale plot

Linear scale plot

We know that

12r area under the curve

This overlap function is approximated by a Gaussian curve: the Gamow-Window.

The Gamow-Window provides the relevant energy range for the nuclear reaction.

Gamow-Window

A Gaussian curve is characterized by its expectation value and its width :

keV22.12

3/1222

21

3/2

0 TZZb

E

6/1522

21

2/10 75.0

3

4TZZE

D0E

tells us where we find the Gamow-window. provides us with the relevant energy range.

0E

Knowing the temperature of a star, we are able to determine where we have to measure in the laboratory.

6

6

A given temperature defines the Gamow-window. For stars, inside the Gamow-window, S(E) is slowly varying.

Astro-Physical S-Factor (12C(p,g)13N)

)( 00 ESS

How does look like?

Approximate the astro-physical factor by its value at :

0S

0E

Nuclear Reactions in the Sun

• core temperature: 15 Mio K• main fusion reactions to convert hydrogen into helium:

• proton-proton-chain• CNO-cycle

• nuclear reactions in the Sun are non-resonant

PP-I

Qeff= 26.20 MeV

p + p d + e+ + n

p + d 3He + g

3He + 3He 4He + 2p

86% 14%

3He + 4He 7Be + g

2 4He

7Be + e- 7Li + n 7Li + p 2 4He

7Be + p 8B + g 8B 8Be + e+ + n

99.7% 0.3%

PP-II

Qeff= 25.66 MeV PP-III

Qeff= 19.17 MeV

Netto: 4p 4He + 2e+ + 2n + Qeff

Proton-Proton-Chain

Homestake-Experiment

Basic idea: if we know which reactions produce neutrinos in the Sun and are able to calculate their reaction rates precisely, we can predict the neutrino flux.

• Same idea by Raymond Davis jr and John Bahcall in the late 1960´s: Homestake Experiment

• purpose: to collect and count neutrinos emitted by the nuclear fusion reactions inside the Sun

• theoretical part by Bahcall: expected number of solar neutrinos had been computed based on the standard solar model which Bahcall had helped to establish and which gives a detailed account of the Sun's internal operation.

Homestake-Experiment

• experimental part by Davis: • in Homestake Gold Mine, 1 478 m underground (to protect from cosmic

rays)• 380 m3 of perchloroethylene (big target to account for small probabiltiy of

successful capture)• determination of captured neutrinos via counting of radioactive isotope of

argon, which is produce when neutrinos and chlorine collide

• result: only 1/3 of the predicted number of electron neutrinos were detected

Solar neutrino puzzle: discrepancies in the measurements of actual solar neutrino types and what the Sun's interior models predict.

Homestake-Experiment

Possible explanations:

• The experiment was wrong.

• The standard solar model was wrong.

• Reaction rates are not accurate enough.

• The standard picture of neutrinos was wrong. Electron neutrinos could

oscillate to become muon neutrinos, which don't interact with chlorine

(neutrino oscillations).3He + 4He 7Be + g

7Be + e- 7Li + n 7Li + p 2 4He

7Be + p 8B + g 8B 8Be + e+ + n

99.7% 0.3%Necessary to measure reaction rates at high accuracy. Here: with the help of 3He(α,γ)7Be as the onset of neutrino-producing reactions

Motivation

We will take a look at the 3He(α,γ)7Be reaction as:

The nuclear physics input from its cross section is a

major uncertainty in the fluxes of 7Be and 8B

neutrinos from the Sun predicted by Solar models

As well: major uncertainty in 7Li abundance obtained

in big-bang nucleosynthesis calculations

Critical link: important to know with high accuracy

Measuring the reaction rate of 3He(α,γ)7Be

Q= 1,586 MeV 429 keV

There are two ways to measure that the 3He(α,γ)7Be reaction occured:

• prompt γ method: measuring the γ´s emitted as the 7Be* γ-decays into the 1st excited or the ground state

• activation method: measuring the γ´s that are emitted when the radioactive 7Be decays

Basic Measuring Idea

BeamN

Yieldexp

Experimentally we get the cross section over:

where: • the yield is the number of γ events counted• NBeam is the number of beam particles counted• ρ is the number of target particles per unit area

Background reduction

• underground, at the energy range we are interested in: about 10 h to see one background event

• using the equation mentioned before, we can approximate that our 3He(α,γ)7Be reaction provides about 70 events an hour.

thanks to the shielding: the yield is significantly higher than the background and can therefore be clearly seperated from it

surface

Lunaaccelerator

detector

target

Credits to Matthias Junker at LNGS-INFN for making the LNGS picture available

Laboratory for Underground Nuclear Astrophysics at Laborati Nazionali del Gran Sasso (LNGS)

Prompt-γ-Method

Schematic view of the target chamber

Experimental Set-Up

Prompt-γ-Method

Measured γ-ray spectrum at Gran Sasso LUNA accelerator facility

GS

GS

GS

background

signal

1st

1st

1st

Prompt-γ-MethodOverview on available S-factor values and extrapolation

Activation MethodExperimental Set-Up at Gran Sasso LUNA2

Schematic view of the target chamber used for the irradiations

Activation Method

Offline γ-counting spectra from detector LNGS1

Activation Method

Astrophysical S-factor at lower panel, uncertainties at upper panel

Knowing the temperature of e.g. the Sun, we can specify the relevant energy range for a nuclear reaction

An important reaction to research the interior of the Sun as well as big-bang nucleosynthesis is 3He(α,γ)7Be

Energies related to Sun temperatures are technically not feasible: extrapolation demands high accuracy measurements

Necessary to reduce background The weighted average over results of both methods

(prompt and activation) provides an extrapolated S-factor of

Summary

References

Donald D. Clayton, Principles of Stellar Evolution and Nucleosynthesis (University of Chicago Press, Chicago, 1983)

Christian Iliadis, Nuclear Phyics of Stars (WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, 2007)

F. Confortola et al., arXiv: 0705.2151v1 (2007) F. Confortola et al., Phys. Rev. C 75, 065803 (2007) Gy. Gyürky et al., Phys. Rev. C 75, 035805 (2007) C. Arpesella, Appl. Radiat. Isot. Vol. 47, No. 9/10, pp.

991-996 (1996) D. Bemmerer et al., arXiv: 0609013v1 (2006)

Zusatz-Folie

beambeamt

tCM E

mm

mE

Example: using a α-Beam at an energy of 300 keV, which corresponds to an relative energy of 129 keV accords to a temperature of 207 MK (which is more than ten times higher than in the Sun: need for extrapolation)