4th Week: Decays, Tunneling and Cross Sections: Difference between revisions

From PhyWiki
Jump to navigation Jump to search
 
(17 intermediate revisions by 4 users not shown)
Line 20: Line 20:


:<math>_{Z}^{A}\textrm{X}\to \; _{Z+1}^{A}\textrm{Y} + \;e^- +\bar{\nu_e} \ .</math>  
:<math>_{Z}^{A}\textrm{X}\to \; _{Z+1}^{A}\textrm{Y} + \;e^- +\bar{\nu_e} \ .</math>  
With,
:<math>\;M_n(Z,A)> M_n(Z+1,A)+ m_e</math>


From electric-charge conservation, the proton number in such decays is increased by one unit.
From electric-charge conservation, the proton number in such decays is increased by one unit.


Likewise, a <math>\beta^+</math>-decay occurs when a proton-rich nucleus emits a positron, therefore reducing the nuclear charge by one.  
Likewise, a <math>\beta^+</math>-decay occurs when a proton-rich nucleus emits a positron, therefore reducing the nuclear charge by one.  
Line 27: Line 32:
:<math>_{Z}^{A}\textrm{X}\to \; _{Z-1}^{A}\textrm{Y} + \;e^+ + \nu_e \ .</math>
:<math>_{Z}^{A}\textrm{X}\to \; _{Z-1}^{A}\textrm{Y} + \;e^+ + \nu_e \ .</math>


The final case, electron capture, is similar to <math>\beta^+</math>-decay because it is another way for a proton -rich nucleus to reduce its nuclear charge by the capture of an electron
with,
 
:<math>\;M_n(Z,A)> M_n(Z-1,A)+ m_e</math>
 
 
The final case, electron capture, is similar to <math>\beta^+</math>-decay because it is another way for a proton-rich nucleus to reduce its nuclear charge by the capture of an electron


:<math>_{Z}^{A}\textrm{X} + e^-\to \; _{Z-1}^{A}\textrm{Y} + \nu_e \ .</math>
:<math>_{Z}^{A}\textrm{X} + e^-\to \; _{Z-1}^{A}\textrm{Y} + \nu_e \ .</math>
With,
:<math>\;M_n(Z,A)+ m_e > M_n(Z-1,A)</math>


==Gamma Decay==
==Gamma Decay==
Line 35: Line 49:
When a nucleus changes its state from the higher energetic state to the lower energetic state an electromagnetic radiation will be emitted in terms of high energy photons. This process is known as a ''gamma decay'', and the decayed photon is called gamma-particle. In this case both the parent and daughter nuclei are the same, as the number of protons and neutrons do not change.  
When a nucleus changes its state from the higher energetic state to the lower energetic state an electromagnetic radiation will be emitted in terms of high energy photons. This process is known as a ''gamma decay'', and the decayed photon is called gamma-particle. In this case both the parent and daughter nuclei are the same, as the number of protons and neutrons do not change.  


=Transition probabilities through the potential barriers=
=Transmission probabilities through the potential barriers=
'''Bold text'''
 
[[Image:potential.jpg|right|300px|thumbnail|. Box potential plus square well]]
[[Image:potential.jpg|right|300px|thumbnail| Box potential plus square well]]


The transition probability in general is given by
The transmission probability in general is given by


:<math>T = \frac{j_{fin}}{j_{in}} \ ,</math>
:<math>T = \frac{j_{fin}}{j_{in}} \ ,</math>
Line 51: Line 65:
:<math>T \approx e^{-\frac{2}{\hbar} \int_{x_1}^{x_2} \sqrt{2m (V(x)-E)}} \ , </math>
:<math>T \approx e^{-\frac{2}{\hbar} \int_{x_1}^{x_2} \sqrt{2m (V(x)-E)}} \ , </math>


where <math>x_1</math> and <math>x_2</math> are turning points, <math>V(x)</math> is the potential. This is an important result that strictly applies to s-waves. This shows that a particle approaching the potential barrier from the right can reach the left side even if its energy is less than the barrier height. This is referred to as '''Tunnel effect''' and is importan for charged particle reactions in stars.
where <math>x_1</math> and <math>x_2</math> are turning points, <math>V(x)</math> is the potential.This shows that a particle approaching the potential barrier from the right can reach the left side even if its energy is less than the barrier height. This is referred to as '''tunnel effect''' and is important for charged particle reactions in stars.




Line 94: Line 108:
where <math>j_l(kr)</math> are spherical Bessel functions, and <math>Y_{l,0}</math> are spherical harmonics.
where <math>j_l(kr)</math> are spherical Bessel functions, and <math>Y_{l,0}</math> are spherical harmonics.


=Cross section approaches=


We have two different approaches for the cross sections
We introduce the '''astrophysical factor''' S(E) which varies less with energy compared to the cross section and is given by
:<math>\;S(E)=\sigma(E) E e^{2 \pi \eta} </math>
Because S(E) is not as dependent as the cross section in energy, it is a very useful concept for Nuclear astrophysicist.
 
=Resonance reaction in a compound nucleus=
 
In the complexity of the compound nucleus we have two different cross section approaches for the reaction <math>i+j \rightarrow o+m </math>


Statistical Model (Hauser-Feshbach)
'''Statistical Model (Hauser-Feshbach)'''. In the case where the resonances are broader and close together, the cross section varies smoothly with energy and can be approached as
:<math>\sigma(j,o)={\pi^2 \over  k^2_{j}}{(1+\delta_{ij}) \over (2I_{i}+1)(2I_{j}+1)} \sum_{j,\pi} (2J_{n}+1){T_j(E,j,\pi)T_o(E,j,\pi) \over T_{tot}(E,j,\pi)}</math>
:<math>\sigma(j,o)={\pi^2 \over  k^2_{j}}{(1+\delta_{ij}) \over (2I_{i}+1)(2I_{j}+1)} \sum_{j,\pi} (2J_{n}+1){T_j(E,j,\pi)T_o(E,j,\pi) \over T_{tot}(E,j,\pi)}</math>


Individual resonance dominates (Breit Wigner)
'''Individual resonance dominates (Breit-Wigner)'''. When the reaction presents an isolated narrow resonances we can approximate the cross section by
:<math>\sigma(j,k)={\pi^2 \over  k^2_{j}}{(1+\delta_{ij}) \over (2I_{i}+1)(I_j+1)} \sum_n (2J_{n}+1){\Gamma_{j,n}\Gamma_{o,n} \over (E-E_{n})^{2}+(\Gamma_{n}/2)^{2}}</math>
:<math>\sigma(j,k)={\pi^2 \over  k^2_{j}}{(1+\delta_{ij}) \over (2I_{i}+1)(I_j+1)} \sum_n (2J_{n}+1){\Gamma_{j,n}\Gamma_{o,n} \over (E-E_{n})^{2}+(\Gamma_{n}/2)^{2}}</math>


Line 108: Line 127:
:<math>E_{\alpha } = E_{center of mass} \frac{M_{product}+M_{reaction}}{M_{reaction}},
:<math>E_{\alpha } = E_{center of mass} \frac{M_{product}+M_{reaction}}{M_{reaction}},
\; \; \; \; E_{center of mass} = E_{\alpha} \frac{M_{reaction}}{M_{reaction}+M_{product}}</math>
\; \; \; \; E_{center of mass} = E_{\alpha} \frac{M_{reaction}}{M_{reaction}+M_{product}}</math>
Example:
<math>_{}^{56}\textrm{Fe}(\alpha,2n)_{}^{58}\textrm{Ni}
\;\;\;</math>
nuclear reaction with alpha particle of 20 MeV
<math>\alpha +_{}^{56}\textrm{Fe}\Rightarrow  _{}^{60}\textrm{Ni}\Rightarrow  _{}^{58}\textrm{Ni}+n+n</math>
<math> E_{center of mass} = 18.6 MeV</math>

Latest revision as of 13:50, 25 February 2009

Nuclear Radiation

Figure 1. A table of known stable and radioactive nuclides

When looking at the table of nuclides it is obvious that the valley of stability is only a small percentage of all known nuclei. So most nuclei are considered unstable. To become stable these nuclides emit ionizing particles and radiation, resulting in a loss of energy from the original nuclei. This process is called radioactive decay.

Alpha Decay and Fission

Alpha decay is the transformation of a parent nucleus to a daughter nucleus through the emission of a helium nucleus. This decay is usually seen in heavier elements with too many protons. This causes excessive repulsion due to the coulomb force. In an attempt to get rid of this excessive repulsion the nucleus emits an alpha particle.

Alpha decay and nuclear fission are both considered to be a two body disintegration. In the case of an alpha decay, the daughter nucleus is at rest and most of the released energy is carried away by the alpha particle. For fission the energy is distributed between the two daughter nuclei.

Beta Decay

Figure 2. This is two examples of the beta decay process

There are three types of beta decays, -decay, -decay, and electron capture. All three of these decays can be characterized by and .

The first one, decay, occurs when a nucleus with an over abundance of neutrons transforms to a more stable nucleus by emitting an electron. This process can be denoted as:

With,

From electric-charge conservation, the proton number in such decays is increased by one unit.


Likewise, a -decay occurs when a proton-rich nucleus emits a positron, therefore reducing the nuclear charge by one.

with,


The final case, electron capture, is similar to -decay because it is another way for a proton-rich nucleus to reduce its nuclear charge by the capture of an electron

With,

Gamma Decay

When a nucleus changes its state from the higher energetic state to the lower energetic state an electromagnetic radiation will be emitted in terms of high energy photons. This process is known as a gamma decay, and the decayed photon is called gamma-particle. In this case both the parent and daughter nuclei are the same, as the number of protons and neutrons do not change.

Transmission probabilities through the potential barriers

Box potential plus square well

The transmission probability in general is given by

where is quantum-mechanical flux

Using the so called WKB approximation it can be calculated through the following formula

where and are turning points, is the potential.This shows that a particle approaching the potential barrier from the right can reach the left side even if its energy is less than the barrier height. This is referred to as tunnel effect and is important for charged particle reactions in stars.


Coulomb barrier

This is the energy barrier due to electrostatic interaction that two nuclei need to overcome so they can get close enough to undergo nuclear fusion. This energy barrier is given by the electrostatic potential energy:

where k is the Coulomb's constant, and equal to k = 8.9876×109 N m² C−2, and q1, q2 are the charges of the interacting nuclei.

For the Coulomb potential the transition probability is calculated using the WKB approximation

This is called Gamow factor and is important for the thermonuclear reaction rates for charged particles, where is the Sommerfeld parameter given by

Box potential

Figure 3. The final potential well

For a finite box potential shown in Fig. 3 the transmission coefficient is easily calculated using the Schrodinger equation.

where

Nuclear cross section

The nuclear cross section is used to characterize the probability that the nuclear reactions occur. Generally it can be written as

It is related to the tunneling probability through the expression

To calculate the cross section one needs to know the incoming flux which is described by plane wave

where are spherical Bessel functions, and are spherical harmonics.


We introduce the astrophysical factor S(E) which varies less with energy compared to the cross section and is given by

Because S(E) is not as dependent as the cross section in energy, it is a very useful concept for Nuclear astrophysicist.

Resonance reaction in a compound nucleus

In the complexity of the compound nucleus we have two different cross section approaches for the reaction

Statistical Model (Hauser-Feshbach). In the case where the resonances are broader and close together, the cross section varies smoothly with energy and can be approached as

Individual resonance dominates (Breit-Wigner). When the reaction presents an isolated narrow resonances we can approximate the cross section by

Unknown chapter at this moment (please, complete)


Example: nuclear reaction with alpha particle of 20 MeV