Technical University Munich - Max Planck SocietyTechnical University Munich May 29th, 2013 Nicolas K¨ohler Measuring resonant thermonuclear reaction rates Revision Theory Measurement

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Measuring resonant thermonuclear reaction ratesAdvisor-Seminar ”Nuclei in the Cosmos”, 2013

Nicolas Kohler

Technical University Munich

May 29th, 2013

Nicolas Kohler Measuring resonant thermonuclear reaction rates

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MeasurementSummary

Table of contents

1 Nonresonant reactionsReaction rate, cross section, S-factor, Gamow-window

2 Theory: Resonant reactionsNuclear statesResonant Reaction Rates

3 MeasurementDirect measurements26gAl(p, γ)27Si21Na(p, γ)22Mg

4 Summary


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Reaction rate, cross section, S-factor, Gamow-window

Revision: Nonresonant reactionsInside stars, particles almost behave like an ideal gas→ Maxwell-Boltzmann distribution of velocityNonresonant reaction rate (of particles of type 1 and 2):

r12 =

( 8µπτ3

) 12 N1N2

1 + δ12

∞∫0

E · σ(E) · exp(−Eτ

)dE

where τ = kBT , µ effective masscross section σ at low relative energies E (tunneling):

σ(E) =S(E)

E · exp(−2π

~

√µ

2E Z1Z2e2)

S(E) is the Astrophysical S-Factor


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Reaction rate, cross section, S-factor, Gamow-window

Revision: Nonresonant reactionsFor low relative energies: S(E) ≈ S0, thus:

r12 =

( 8µπτ3

) 12 N1N2

1 + δ12S0

∞∫0

exp(−Eτ−√

2µπZ1Z2e2

~1√E

)︸︷︷︸

Gamow−window

dE

The Gamow-window provides the relevant energy range for anuclear reaction by knowing the star’s temperature

Gamow-window for a given tem-perature T (from [Clayton, page302])


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Nuclear statesResonant Reaction Rates

Resonant reactions: Nuclear states

Nucleons cluster into bound states, comparable with boundenergy eigenstates of atoms (nucleus↔electrons) or hadrons(quarks↔(anti-)quarks)Due to interaction terms in the Hamiltonian, each energystate has a certain width Γ

Probability of the state having an energy between E andE + dE (Γ = ~/τ with τ lifetime of a state):

P(E) dE =Γ

2πdE

(E − Eres)2 +(

Γ2

)2

Eres is the energy of the according quasistationary state


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Nuclear states: The Shell modelPartly analogous to the atomic shell modelShells for protons and for neutrons are independent of eachotherIntermediate in shape between oscillator well and square wellCan be described in terms of two quantum numbers: n and l

energy levels of a harmonic oscillator(left), of a square well (right) and ofbound nucleons (middle) (from [Clayton,page 313])


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The Shell model: benefitsEvery closed shell of nucleons has J = 0All nuclei in groundstate having even numbers of protons andneutrons have J = 0Angular momentum J depends only on number of nucleons inexcess of closed shells

Nuclear level scheme of the 22Mg nucleus with excitationenergies and spin assignments (from [D’Auria])


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Resonant Reaction Rates in Astrophysics

Triple-Alpha-Reaction8Be formed by reaction of two 4He isunstable (T1/2 = 6.7(17) · 10−17 s)Abundance ratio NBe

Nα

= ρNAXα4

(h2

2πµαατ

) 32 exp(Qαα/τ)

≈ 10−9

(Xα = 1, ρ = 105gcm−3, T = 108K)Hoyle predicted existence of 7.654MeV 0+-resonance in 12C to explaincarbon abundance

Resonant reactions within the triple-alpha-process (from [Bishop])


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Resonant Reaction RatesRequirements: After tunneling Eres must be hit, conservationof angular momentum J and parity πResonant reaction rate (between particle 1 and 2):

r12 =(

8µπτ3

) 12 N1N2

1+δ12

∞∫0

E · σ(E) · exp(−Eτ

)dE

Consider reaction: X + a (→W ∗)→ b + Y , shortly:X(a, b)Y where X=1 and a=2, then:σ(E) = π~2

2µEΓaΓb

(E−Eres)2+ Γ24

if Γ = Γa + Γb is narrow, the widths Γa and Γb are almostindependant of E :

r12 =(

8π~4

µ3τ3

) 12 N1N2

1+δ12exp

(−Eres

τ

)ΓaΓb

Γ

∞∫0

Γ/2(E − Eres)2 + Γ2

4dE

︸︷︷︸=π


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Resonant Reaction Rates

All in all (for single resonances):

r12 =(

2πµτ

) 32 ~2 N1N2

1+δ122Jres+1

(2Ja+1)(2Jb+1) exp(−Eres

τ

)ΓaΓb

Γ

where 2Jres+1(2Ja+1)(2Jb+1) is the spin-statistical factor

for several resonances (inside the Gamow-window):

r12 =

(2πµτ

) 32~2 N1N2

1 + δ12

∑res

exp(−Eres

τ

) 2Jres + 1(2Ja + 1)(2Jb + 1)

ΓaΓbΓ︸︷︷︸

=: ωγ = resonance strength


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Direct measurements26gAl(p, γ)27Si21Na(p, γ)22Mg

Measurement of resonant reaction rates

Accreting ONe white dwarf forms a nova (from [Bishop])


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Astrophysical motivationDepending on nova type, different burning cycles are involvedPredominant in ONe novae: NeNa cycle on seed nuclei 20NeUnstable 22Na β-decays (T1/2 = 2.6 yr) to 22Ne where 1.275MeV γ-ray is emitted

Unstable 26Al β-decays(T1/2 = (7.1± 0.2) · 105

yr) to 26Mg where 1.809MeV γ-ray is emittedUnknown rate of21Na(p, γ) was mainsource of uncertainty

NeNa and MgAl cycle (from [Bishop])


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Astrophysical motivation

Decay scheme of 22Na to 22Ne with radiation of 1.275MeV γ-ray (from http://www.nndc.bnl.gov/)


http://www.nndc.bnl.gov/

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Astrophysical motivation

Decay scheme of 26Al to 26Mg withradiation of 1.809 MeV γ-ray (fromhttp://www.nndc.bnl.gov/)



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Direct measurements of resonant reactions

If either X or a of X(a, b)Y is unstable → require radioactivetarget or radioactive beamTRIUMF at Vancouver: Radioactive Beams LaboratoryExperimental setup of DRAGON at TRIUMFMeasuring the resonance strength of 26Al(p, γ)27Si atEcm = 184 keV using inverse kinematics21Na(p, γ)22Mg Reaction from Ecm = 200 to 1103 keV usinginverse kinematics


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Drawing of the DRAGON facility at TRIUMF (from [D’Auria, page 16])


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DRAGON Electromagnetic Separator

Magnetic dipoles seperateinto momentum-chargefractionsmv2

r1= qvB ⇔ mv

q = pq = Br1

→ select charge stateElectrostatic dipoles seperateinto different kinetic energies

q|~E | = mv2

r2

⇔ |~E|r22 = mv2

2q = p2

2qm = pq

p2m

→ select massInterior construction of the DRAGON electrostatic dipolebender (from [Bishop])


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Measuring resonance strength of 26gAl(p, γ)27Si @184 keVMeasurement:

26gAl has l = 5, p hasl = 1

2

Only 112

+, 102

+, 92

+

states are allowed92

+ is strongly suppressed

Nuclear level scheme for 27Si with Q-valuefor 26gAl(p, γ)27Si (from http://www.nndc.bnl.gov/)




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Measuring resonance strength of 26gAl(p, γ)27Si @184 keVResults:

≈ 49 hours of data @ average beam intensity of 2.5 · 109 s−1

Also ”off-resonance” background runCandidate events selected by a narrow window (200 ns) ontime-of-flight (TOF) set around coincidence peak

DSSD Energy (MeV)0 0.5 1 1.5 2 2.5 3 3.5 4

SeparatorTOF(arb.units)

6000

6500

7000

7500

8000

8500

9000

9500

10000

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time-of-flight for coincident γ-ray/heavy-ion events versus de-tected particle energy for the 5.122MeV run, 28 ± 6 candidate events(out of 4.4 · 1014 particles) (from[Ruiz])


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Measuring resonance strength of 26gAl(p, γ)27Si @184 keVResults:

Calculating resonance strength ωγ = 2εYλ2

MHMAl+MH

, λ is thede-Broglie wavelength in the C.M. system, ε is the beamenergy loss per target atom per unit area, Y is the reactionyield

Percentage contribution to error

E beam ∆ ∆ηD SSD ∆ηB GO ∆ηsep ∆ηS i4+ ∆N Tot. sys. ωγ(M eV ) error (µeV)

5.226 5% 1% 13% 2% 5% 3% 15% 35± 5sys± 4stat5.122 5% 1% 13% 2% 5% 8% 17% 36± 6sys± 8stat

Resulting resonance strengths and their associated errors (from [Ruiz])

Result reduces the reaction rate over the region of theGamow-window for this resonance by nearly a factor of 1.2→ For a given temperature, more 26gAl will survive a novaexplosion


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21Na(p, γ)22Mg Reaction from Ecm = 200 to 1103 keV

Gamow-window @T = 200 MK:Maximum Gamow-Peak E0 = 203.3 keVEffective width ∆E = 136.7 keVEnergy windowE0 ± ∆E

2 = 134.9− 271.6 keV

Nuclear level scheme of the 22Mg nucleus with excita-tion energies and spin assignments, on the right-handside the Gamow-windows are given for temperatures inGK (from [D’Auria])


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21Na(p, γ)22Mg Reaction from Ecm = 200 to 1103 keVResults:For each resonance best charge state was chosen while otherswhere filtered out using slits

Detected particle energy versustime-of-flight for coincident γ-ray/heavy-ion events for the 22Mgresonance level at Eres = 212 keV(from [Bishop])


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21Na(p, γ)22Mg Reaction from Ecm = 200 to 1103 keV

Results:

Thick target yield YY = λ2

2mNa+mp

mp

(dEρdx

)−1ωγ arctan

(∆EΓres

)

0

1

2

3

4

5

6

7

8

202.5 205 207.5 210 212.5 215 217.5 220 222.5

21Na(p,γ)22Mg

2.5 torr4.6 torr

Y=5.76 ± 0.88

c.m. Energy (keV)

22M

gY

ield

X10

-12

Thick target yield data forthe 21Na(p, γ)22Mg reac-tion, with the solid line show-ing the nominal target thick-ness for 4.6 Torr (Inten-sity of the 21Na beam .109 21Na per second) (from[Bishop])


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21Na(p, γ)22Mg Reaction from Ecm = 200 to 1103 keVResults:

Calculating resonance strength from thick target yield YY (∞) = λ2

2mNa+mp

mp

(dEρdx

)−1ωγ

with λ de-Broglie wavelength and dEρdx stopping power each

evaluated at the resonance energy

Ex MeV Ec.m. keV keV meV

5.714 205.7±0.5 1.03±0.21

5.837 329 0.29

5.962 454±5 0.86±0.29

6.046 538±13 11.5±1.36

6.246 738.4±1.0 219±25

6.329 821.3±0.9 16.1±2.8 556±77

6.609 1101.1±2.5 30.1±6.5 368±62

Resulting resonance strengths and their associated errors (from [D’Auria])


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Astrophysical results21Na(p, γ)22Mg(β+)22Na is favoredLower amount of 22Na was found22Na production takes place earlier in the outburst

Lower strength associatedwith the Ecm = 188 keVresonance in 26gAl(p, γ)favors the synthesis of26gAl in nova outburstsMore 26gAl will survive anova explosion

NeNa and MgAl cycle (from [Bishop])


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Summary

Resonant nuclear reaction ratesResonances of nuclei dominate cross sections by orders of magnitude

Resonant reactions in nuclear astrophysicsMany elements can only be built inside stars at relatively low energiesbecause of resonances

Measuring resonant nuclear reactionsDirect measurements via radioactive beams or targetsGamow-window gives relevant energy range for important resonancesResonant reaction rates need to be known for testing theoretical models


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LiteratureDonald D. ClaytonPrinciples of Stellar Evolution and NucleosynthesisThe University of Chicago Press, 1983.

J. M. D’Auria, R. E. Azuma, S. Bishop, L. Buchmann et al.The 21Na(p, γ)22Mg reaction from Ecm = 200 to 1103 keV in novae and x-ray burstsPhysical Review C 69, 2004.

C. Ruiz, A. Parikh, J. Jose, L. Buchmann et al.Measurement of the Ecm = 184 keV resonance strength in the 26gAl(p, γ)27Si reactionhttp://dragon.triumf.ca/docs/Al26finaldraft.pdf, 2006.

D.A. Hutcheon, S. Bishop, L. Buchmann, M.L. Chatterjee et al.The DRAGON facility for nuclear astrophysics at TRIUMF-ISAC: design, construction and operationNuclear Instruments and Methods in Physics Research A 498, 2003.

J. M. D’Auria, P. WaldenNuclear Astrophysics with DRAGON and TUDATRIUMF Financial Report 2000-2001.

Shawn BishopNuclear Astrophysics II Lecture at TUMNovae Explosions and Resonant Reaction Rate Measurements, 2012.

S. Bishop, R. E. Azuma, L. Buchmann, A. A. Chen et al.21Na(p, γ)22Mg Reaction and Oxygen-Neon NovaePhysical Review Letters 90, 62501, 2003.


http://dragon.triumf.ca/docs/Al26finaldraft.pdf

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DRAGON at ISAC1


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DRAGON at ISAC1


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DRAGON target chamber


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DRAGON target system

H /He gas cell2Collimatorinsert

Fill tube fromrecycling

Feedthruconnectors

Elastic monitordetectors

Schematic representation of the inner components of the DRAGONwindowless gas target system (from [Hutcheon, page 192])


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Technical University Munich - Max Planck SocietyTechnical University Munich May 29th, 2013 Nicolas K¨ohler Measuring resonant thermonuclear reaction rates Revision Theory Measurement