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Lec 20

The document discusses resonant reactions in nuclear astrophysics, focusing on the mathematical formulation of reaction rates and the significance of resonance energy. It highlights the importance of narrow resonances and their impact on stellar reaction rates, particularly at low temperatures. Additionally, it addresses the production and decay rates of nuclei in resonance states and the implications for nuclear reactions in stars.

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17 views7 pages

Lec 20

The document discusses resonant reactions in nuclear astrophysics, focusing on the mathematical formulation of reaction rates and the significance of resonance energy. It highlights the importance of narrow resonances and their impact on stellar reaction rates, particularly at low temperatures. Additionally, it addresses the production and decay rates of nuclei in resonance states and the implications for nuclear reactions in stars.

Uploaded by

issacalbert22
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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NUCLEAR ASTROPHYSICS

Lecture-20
Resonant reactions…
ANIL KUMAR GOURISHETTY
PHYSICS

1
Recap

Resonant reactions

2
2𝐽 + 1 𝑎 𝑏
𝜎(𝐸) = 𝜋ƛ2 1 + 𝛿12
2𝐽1 + 1 2𝐽2 + 1 𝐸 − 𝐸𝑅 2 + /2 2

Valid for isolated resonances → total width  (= a + b) is > separation of nuclear levels

Narrow resonance → total width  ≪ Eresonance i.e.  is constant over total resonance width

∞
8 1 −𝐸/𝑘𝑇
The stellar reaction rate is 𝜎𝑣 = න 𝐸 𝜎𝐵𝑊 𝐸 𝑒 𝑑𝐸
𝜋𝜇 𝑘𝑇 3/2 0
For narrow resonance, the change in E 𝑒 −𝐸/𝑘𝑇 is very small
∞
8 1
𝜎𝑣 = 3/2
𝐸𝑅 𝑒 −𝐸𝑅 /𝑘𝑇 න 𝜎𝐵𝑊 𝐸 𝑑𝐸
𝜋𝜇 𝑘𝑇 0

3
∞
𝑎 𝑏 𝑎 𝑏
න 𝜎𝐵𝑊 𝐸 𝑑𝐸 = 2𝜋 2 ƛ2 𝜔 where 𝜔𝛾 = Resonance Strength
 
0
∞
𝑎 𝑏 𝜋
At E = ER, 𝜎(𝐸)= 𝜎 𝐸 = 𝐸𝑅 = 4𝜋ƛ2𝑅 𝜔  2
 න 𝜎𝐵𝑊 𝐸 𝑑𝐸 = 𝜎
2 𝑅
0

Area under the resonance curve = Total width   height 𝜎𝑅


3/2
2𝜋 Stellar reaction rate per particle pair for a narrow
𝜎𝑣 = ℏ2 𝜔𝛾 𝑅 𝑒 −𝐸𝑅 /𝑘𝑇 𝑓
𝜇𝑘𝑇 resonance
Strength, width and resonance energy → Measurable parameters
Gamow peak = Resonance energy ER → nuclear burning at a narrow resonance
3/2
2𝜋
For many narrow resonances, 𝜎𝑣 = ෍ 𝜔𝛾 𝑖 𝑒 −𝐸𝑅/𝑘𝑇 𝑓
𝜇𝑘𝑇
𝑖

4
Rection A(x,)B   1 eV;
𝑥𝛾
At ER near Vc , 𝑥 is ~ MeV  𝜔𝛾 =  𝛾

 
At ER far below Vc , 𝑥 𝑥 ≪ 𝛾  𝜔𝛾 = 𝑥 𝛾  𝑥

Strength decreases very rapidly due to barrier penetration term in 𝑥
2𝜋 3/2
From the reaction rate 𝜎𝑣 = σ𝑖 𝜔𝛾 𝑖 𝑒 −𝐸𝑅/𝑘𝑇 𝑓
𝜇𝑘𝑇
 resonances with ER near kT dominates reaction rate

For low T, it is very important to know locations and strengths of low energy
resonances

5
Ex: 14N(p,)15O, T6 = 30, ratio of reaction rates at 10 keV and 2 MeV is 10312
Mean lifetime of nuclei and stars’ properties change dramatically for a small
difference in resonance energies ER.
For neutron induced resonant reactions (radiative capture),     holds for
nearly all resonance energies

Nuclei having states with long life-times can participate in the reaction. How to
find such number N12?

6
Production rate:  no. of nuclei in resonance state.
At high resonance energies, a >> b    a     b
3/2
𝑁1 𝑁2 𝑁1 𝑁2 2𝜋
𝑟12 = 𝜎𝑣 = ℏ2 𝜔 𝑏 𝑒 −𝐸𝑅/𝑘𝑇
1 + 𝛿12 1 + 𝛿12 𝜇𝑘𝑇

Decay rate: b determines life-time of b of nuclei N12


𝑁12 𝑏 𝑁12
𝑟12 = =
𝜏𝑏 ℏ
𝑁1 𝑁2 2𝜋 3/2 3
Combining, 𝑁12 = ℏ 𝜔 𝑒 −𝐸𝑅/𝑘𝑇 Saha equation
1+𝛿12 𝜇𝑘𝑇
Valid for all reactions when equilibrium is established
Ex: Triple  reaction → N(8Be)/N(4He) = 10−10 sufficient to bridge mass gaps at 5 & 8

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