Methods to directly measure nonresonant stellar reaction rates
Tanja Geib
Outline
1.
Theoretical background:
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–
–
–
2.
Reaction rates
Maxwell-Boltzmann-distribution of velocity
Cross-section
Gamow-Window
Experimental application using the example of the
pp2-chain reaction in the Sun
–
–
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–
Motivation and some more theory
Historical motivation
3He(α,γ)7Be as important onset reaction
Prompt and activation method
Reaction Rates
Nuclear Reaction Rate:
particle density of type X
reaction cross section
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
Sun
velocity-distribution
Inside a star, the particles clearly do
not move with a mono-energetic
velocity distribution. Instead, they
have their own velocity distributions.
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 MaxwellBoltzmann 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
Generalization to a Maxwell-Boltzmann
velocity-distribution
After some calculation, including the change into CMS, one obtains:
12 is entered to avoid double-counting of particle
pairs if it should happen that 1 and 2 are the
same species
In terms of the relative energy (E=1/2 μv2 ) this means
 8
r12  
 
1/ 2



N1 N 2  3 / 2 
 E
  E ( E ) exp   dE
0
1  12
  
Cross-Section
The only quantity in the reaction rate that we have not treated yet is the crosssection, 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
Radial Schrödinger equation for s-waves
d 2u 2 
 2 E  V (r ) u  0
2
dr

is solved by the ansatz
u I  A exp(iKx)  B exp(iKx)
u II  C exp(x)  D exp(x)
u III  F exp(ikx)  G exp(ikx)
This leads to transmission coefficient
Tˆ 
KB
kG
2
2
 2

 exp
2 (V1  E ) ( R1  Ro )
 

for low-energy s-wave
transmission at a squarebarrier potential
Cross-Section
We generalize this to a Coulomb-potential by
dividing the shape of the Coulomb-tail into
thin slices of width dr.
Total transmission coefficient for s-wave:
 2

Tˆ  Tˆ1  Tˆ2  ..... Tˆn  exp  2 (Vi  E ) ( Ri 1  Ri )
  i

 2 Rc



n


exp

2

V
(
r
)

E
dr




R
0
 

Reminder: If angular momentum not equal zero, then V(r)  V(r) + centrifugal
barrier
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
ˆ
T  exp
 

Z1 Z 2 e 
2E


2
Cross-Section
•Quantum-mechanical interaction between two particles is always
proportional to a geometrical factor:
1
2
   
E
deBroglie wavelength
• 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.
Finally, our considerations lead to defining the cross-section at low energies
as:
 2
S (E)
 (E) 
exp
E
 

Z1Z 2 e 
2E


2
Cross-Section
The figure on the left shows the
measured cross section as a
function of the laboratory energy
12
of protons striking a C target.
The observed peak corresponds
to a resonance.
12C(p,g)13N
 (E) 
 2
S (E)
exp
E
 

Z1Z 2 e 2 
2E


Gamow-Window
Entering the cross section into the reaction rate, we obtain:
 8
r12  
 
1/ 2




N1 N 2 3 / 2
E
 b
  S ( E ) exp
 dE
1  12
E 

0
Z1Z 2e 2   
b  2
 
 2
1/ 2
with
Using mean value theorem for integration, we bring the equation to the form
 8
r12  
 
1/ 2




N1 N 2 3 / 2
E
 b
 S0  exp
 dE
1  12
E 

0
to pull out the essential physics/ evolve the Gamow-window.
Log scale plot
Gamow-Window
We know that
r12 
area under the curve
This is where the action
happens in thermonuclear
burning!
This overlap function is approximated by
a Gaussian curve: the Gamow-Window.
The Gamow-Window provides the relevant
energy range for the nuclear reaction.
Linear scale plot
Gamow-Window

A Gaussian curve is characterized by its
expectation value E0 and its width  :
1/ 3
 b 
E0     1.22 Z12 Z 22 T6 2 keV
 2 
4
1/ 2
2 2
5 1/ 6
E0   0.75 Z1 Z 2 T6

3
2/3




E0 tells us where we find the Gamow-window.  provides us with the relevant
energy range.
Knowing the temperature of a star, we are able to determine where we
have to measure in the laboratory.
Astro-Physical S-Factor (12C(p,g)13N)
How does
S0 look like?
A given temperature defines
the Gamow-window. For stars,
inside the Gamow-window,
S(E) is slowly varying.
Approximate the astrophysical factor by its value
at E0 :
S0  S ( E0 )
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
Proton-Proton-Chain
4p  4He + 2e+ + 2n + Qeff
Netto:
p + p  d + e+ + n
p + d  3He + g
86%
3He
14%
+ 3He  4He + 2p
3He
99.7%
PP-I
Qeff= 26.20 MeV
+ 4He  7Be + g
+ e-  7Li + n
7Li + p  2 4He
7Be
0.3%
+ p  8B + g
8B  8Be + e+ + n
7Be
PP-II
2 4He
Qeff= 25.66 MeV
PP-III
Qeff= 19.17 MeV
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
99.7%
Necessary to measure reaction rates
at high accuracy. Here: with the help of
3He(α,γ)7Be as the onset of neutrino- 7Be + e-  7Li + n
7Li + p  2 4He
producing reactions
0.3%
+ p  8B + g
8B  8Be + e+ + n
7Be
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
Experimentally we get the cross section over:
 exp
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
Yield

N Beam  
Background reduction
• underground, at the
energy range we are
interested in: about 10 h to
see one background
event
surface
• 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
Laboratory for Underground Nuclear Astrophysics at
Laborati Nazionali del Gran Sasso (LNGS)
Luna
target
accelerator
detector
Credits to Matthias Junker at LNGS-INFN for
making the LNGS picture available
Prompt-γ-Method
Experimental Set-Up
Schematic view of the target chamber
Prompt-γ-Method
background
1st
GS
1st
GS
signal
1st
Measured γ-ray spectrum at
Gran Sasso LUNA
accelerator facility
GS
Prompt-γ-Method
Overview on available S-factor values and extrapolation
Activation Method
Experimental 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
Summary





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 𝑆 0 = 0.560 ± 0.017 𝑘𝑒𝑉
References







Donald D. Clayton, Principles of Stellar Evolution
and Nucleosynthesis (University of Chicago Press,
Chicago, 1983)
Christian Iliadis, Nuclear Phyics of Stars (WILEYVCH 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
ECM 
mt
Ebeam
mt  mbeam
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)