新学術科研費「素核宇融合による計算基礎科学物理学の進展研究会」
2011.12.3-5 賢島

A New Computational Approach
forRadiative Capture Reaction Rate
K. Yabana
University of Tsukuba
In collaboration withY. Funaki, T. AkahoriI will talk on our work towards…

3α → Cα12αα

In astrophysical environment,- reaction starts in total angular momentum L=0- proceeds either resonant (Hoyle state) or nonresonant- emission of E2 gamma-ray to yield bound 2+ state 12C at 4.44MeV
Computational approach is important- since experiment is not possible- but serious calculation was achieved only recently.

Recent 3α -> 12C reaction rate calculation by CDCC method
The first serious 3-body direct capture calculation below barrier energyααLarge direct capture cross section
at low energy
Nonresonant

Resonant (Hoyle)α

Dominant and large contribution ofDirect capture process at low temperature,107-108K
CDCC, nonresonant+resonant

NACRE

Need to solve up to R=2500-5000fm, 3-body boundary condition?, Jacobi coord. L=l=0 only.

Large reaction rate below 108K by Ogata et.al. hassignificant astrophysical implications- Shortening or disappearance of the red giant phaseof low mass star.
Independent examination of 3α reaction rate is quite important,but…
- Analytic asymptotic form of the wave function for three-charged
particles is not known.
- One must solve three-body tunneling problem in huge space(several thousands fm?).
- ‘Ogata’ and ‘Kamimura’ are well-known experts of 3-body reaction

Dominant and large contribution ofDirect capture process at low temperature,107-108K

Large direct capture cross section
at low energy
NonresonantResonant (Hoyle)

CDCC, nonresonant+resonant

NACRE

Need to solve up to R=2500-5000fm, 3-body boundary condition?, Jacobi coord. L=l=0 only.

50keV

αα2 ⋅ 2 ⋅ e2
= 50 keV ⇒ R = 120 fm
Rc.f. α-decay 50fm

Our idea to overcome the first difficulty.
- Analytic asymptotic form of the wave function for three-charged
particles is not known.

Erase the 3-body scattering state employing spectral representationof the 3-body Hamiltonian,
we then obtain ‘imaginary-time Schroedinger equation
starting from final (bound) wave function’.

2-body case:Thermal reaction rate at temperature 𝛽=1/𝑘𝑇2 2vσ fi ∝ (Ek − E f ) − β 2 kµvσ ∝ ∫ dk evσ fi

2 λ +1

散乱状態
 * 
 2
 (r )()
φφdrr
Mλµ k
∫ f

Combining two expressions, we may erase initial scattering states
散乱状態
2k 2
 −β 2 µ(Ek − E f )2λ +1 φ f M λµ φk φk M λµ + φ fvσ ∝ ∫ dk e()

2 λ +1
ˆ+= φ f M λµ e − βH Hˆ − E fPˆ M λµ φ fe− βH= ∑e− β Enφn (r )φn* (r ' ) +n
 2k 2    * φk (r )φk (r ' )dk exp − β2µ(2π )3 ∫
1Pˆ = 1 − ∑ φn φnn

Projector to remove bound states in the intial channel

M λµ = r λYλµ (rˆ )Imaginary-time method for reaction rate()

2 λ +1
ˆ+vσ fi ∝ φ f M λµ e − βH Hˆ − E f
Pˆ M λµ φ fψ (r , β )M λµ = ∑ riλYλµ (rˆi )Pˆ = 1 − ∑ φn φni∈ pnAlgorithm:
1. Prepare β=0 wave function from ‘final’ state(ψ (r , β = 0) = Hˆ − E f)

2 λ +1

+Pˆ M λµ φ f (r )2. Solve imaginary time equation−∂ψ (r , β ) = Hψ (r , β )∂βTaylor expansion methodψ (r , β + ∆β ) = Pˆ e − ∆βHψ (r , β )3. Take overlap with β=0 wave function
to obtain reaction ratek()−β≈ Pˆ ∑H kψ (r , β )kk!

 r (β ) ∝ ∫ drψ * (r ,0)M λµψ (r , β )Three-body scattering states are removed.
Next problem, how large space we need to solve the equation,
several hundred (thousand?) fm?

Test calculation: 16O(α,γ)20Ne( 2-body reaction, Simple potential model)u (r )
ψ (r , β ) = l Ylm (θ , φ )r  2 d 2  2l (l + 1)∂()−++ul (r, β ) = −Vrul (r, β )22∂β2µdr2µr


２通りの計算の比較。①散乱問題を解く ②虚時間法
Spatial box sizefor imaginary-time calculation
Scattering calc. & Up to 400fm
Calculation by ordinary method
and imaginary-time method
coincide each otherIf we calculate up to 400fm.

Up to 100fm
Up to 200fm

108K107K

For a given temperature β, how far we need to solve the equation?
u (r )
ψ (r , β ) = l Ylm (θ , φ )r  2 d 2  2l (l + 1)∂()−++ul (r, β ) = −Vrul (r, β )22∂βµµ2dr2rβ→∞では、最低エネルギーの固有状態に
β=100MeV-1− βE
ψ (r , β ) → e ϕ E (r )β=300MeV-1
β=1000MeV-1

Position of Gamow window
Rmax=400fm

Algorithm:
1. Prepare initial wave function
2 λ +1

Imaginary-time evolutionof the wave function Hˆ + E f ψ (r , β = 0) =  c - decrease in β-evolution by 1080
- 1020 difference in r at T=107K

+Pˆ M λµ φ f (r )2. Solve imaginary time equation−∂ψ (r , β ) = Hψ (r , β )∂β3. Take overlap to obtain reaction rate r (β ) ∝ ∫ drψ * (r ,0)M λµψ (r , β )r u f (r )2β=10MeV-1

β=1000MeV-1

Overlap will be taken with initial function

Large reaction rate below 108K by Ogata et.al. hassignificant astrophysical implications- Shortening or disappearance of the red giant phaseof low mass star.
Independent examination of 3α reaction rate is quite important,but…
- Analytic asymptotic form of the wave function for three-charged
particles is not known.
Imaginary time method- One must solve three-body tunneling problem in huge space(several thousands fm?).
Probably 200-300fm- ‘Ogata’ and ‘Kamimura’ are well-known experts of 3-body reaction

Final problem,How to calculate 3-body imaginary-time problem?
vσ fi ∝ φ f M λµ e− βHˆ()2 λ +1+
ˆH − EfPˆ M λµ φ fM λµ = ∑ riλYλµ (rˆi )i∈ pPˆ = 1 − ∑ φn φnnφf3-body wave function for 2+ 4.44 MeV excited state of 12C
confined within 30fm.

2122
H = − (∆1 + ∆ 2 + ∆ 3 ) +  + + 2
 r12 r23 r31 3-body Hamiltonian. Probably, nuclear force is not important.
What is the efficient representation for the wave function
and algorism to evolve it ?
- decrease in β-evolution by 1080
- 1020 difference in coordinate space r

SummaryTriple-alpha reaction rate.
Toward examining Ogata, Kan, Kamimura calculation:
1020 increase in reaction rate at low temperature.
- Imaginary-time method: rate calculation without scattering solution- For 2-body problem, it works.
- large space, 200-400fm(Coulomb potential is below the Gamow window)- How to solve 3-body imaginary-time evolution?