1. Chapter 3, Part3 Nuclear Chemistry CHEM 396 by Dr
Chapter 3, Part3 Nuclear Chemistry CHEM 396 by Dr. Ahmad Hamaed Fall 2014

2. Solar System Expansion
About 6% of hydrogen inside the Sun’s core is now burnt and then He accumulates in the core owing to its greater mass.
Most of the hydrogen burning moves to a layer around the helium.
As hydrogen fuel is consumed, the temperature of the core decreases. However, as the amounts of helium is increased and the hydrogen depleted, the core contracts.

3. Solar System Expansion, continued
Owing to the gravitational attractions of helium, the core contracts, and its temperature increases.
This heats up the outer layers of hydrogen, and results in an expansion of the outer mantle of the star, which in turn results in a cooler surface, so that the star irradiates more red color and is referred to as a “Red Giant”.
Around a five eons from now, the sun will pass through a Red Giant, at which time its diameter should expand sufficiently to engulf the inner planets of the solar system.

4. Types of eon and Periods
Phanerozoic:
540 million years ago (mya through today).
Proterozoic
2.5 million years ago (bya) to 540 mya.
Archeo-zoic
3.9 to 2.5 billion years ago (bya)
Hadean
4.6 to 3.9 billion years ago (bya)

5. CNO Cycle for Helium Production 10% of the solar energy for the stars with m≥1.5M
12C6 + 1H N γ Q = 1.94 MeV y
13N C6 + 0β+1 + e- + υ Q = 1.20 MeV min
13C6 + 1H N7 + γ Q = 7.55 MeV 3×106 min
14N6 + 1H O γ Q = 7.29 MeV 3×108 y
15O N7 + 0β+1 + e- + υ Q = 1.74 MeV min
15N7 + 1H C6 + 4He2 Q = 4.86 MeV y
The overall reaction is:
4 1H He2 + 20β e- + 2υ Q = 26.7MeV 3×108 y

6. Shell Structure of a 20M Star Just Before Supernova Explosion
The last steps of production of heavy elements (up to Fe/Ni) occurs rather rapidly in a few thousands years. When the nuclear fuel for fusion is exhausted the star collapses and results in a supernova.

7. Particle Accelerators
In order to induce a nuclear reaction it is necessary that the projectile particles have sufficient kinetic energy to overcome the coulombic barrier created by the repulsion between the positive charges of the projectile and the target nucleus.
Many types of accelerators are in use today, such as the single-stage accelerators, the Van De Craaff accelerators, the multiple-stage linear accelerators, and the cylotron.

8. Cyclotrons
First cyclotron constructed in 1931 by Lawrence and Livingston to overcome the difficulties inherent in the length of high energy linear accelerators.
The following figure illustrates the basic operating principles of the cyclotron.

9. Principle of a Cyclotron

10. Principle of a Cyclotron
The particles are accelerated in spiral paths inside two semicircular flat metallic cylinders called dees (D’s), placed in a flat vacuum chamber.
The two dees are connected to a high frequency alternating voltage. The volume within the dees corresponds to a equipotential condition just as does the volume within the drift tubes of linear accelerators.

11. The principle of a cyclotron
The dees and their vacuum chamber are placed between the two poles of a magnet so that the ion beam is constrained to flat circular paths inside the dees. At the gap between the dees the ions experience acceleration due to the potential difference between the dees. This causes an increase in radius when they enter the next dee. The beam particles originate at the ion source at the center of the cyclotron, as they spiral they acquire a constant increase in energy for each passage across the gap between the dees.
The target is located either inside the vacuum (internal beam) or the beam can be taken out into an evacuated chamber or through a thin window (external beam).

12. Cyclotrons, continued
Kinetic energy is directly depends on r2 the radius of the beam’s path and (B2) the magnetic beam field strength:
The frequency in Hz could be obtained from the following equation:
The maximum energies available in conventional cyclotrons of the constant frequency type are about 25MeV for 1H1 and 2H1 and 50MeV for 4He2.

13. The Development of the Concept of Relativistic Mass
Hendrik Lorentz introduced the ideas of longitudinal and transverse electromagnetic masses of an electron in his paper “Electromagnetic Phenomena in a System Moving with Any Velocity Less Than That of Light”, 1994.
According to Lorentz, mass is the ratio of force to acceleration rather than the ratio of momentum to velocity. He wrote equations for mass parallel to the direction of motion and mass perpendicular to the direction of motion.

14. The Development of the Concept of Relativistic Mass
Energy, acceleration and velocity are correlated to masses. Therefore (m) in the previous equations are named relativistic mass.
Therefore, either the frequency or the magnetic field or both must be modified to compensate for the increasing mass in order to have the beam particles arrive at the gap between the D’s..

15. The Development of the Concept of Relativistic Mass
This is accomplished only in synchrocyclotrons while the magnetic field is varied in sector focused cyclotrons using super conducting magnets.
In synchrocyclotrons, the magnet has spiral ridges often which provide sufficient modification of the field to allow the acceleration of particles to much higher energies than obtained from conventional cyclotrons.

16. Sector-Focused Cyclotron
A 150MeV P+ sector focused cyclotron showing the magnet with its spiral-ridged pole pieces and ¼ -wave D-design.

17. Chapter 4 Mechanisms and Models of Nuclear Reactions

18. The Reaction Cross Section
The probability for a nuclear reaction to occur is expressed in terms of the reaction cross-section.
The geometric cross-section that a nucleus presents to a beam of particles is πr2. If we use 6×10-15 m as an average value for the nuclear radius, the value of πr2 becomes 3.14 (6×10-15)2 ≈ m2.

19. The Reaction Cross Section
Fig 4.1: Excitation functions for reactions between 4He ions and 54Fe target nuclei. The kinetic energy of the projectile is in the laboratory system.

20. Resonance and Tunneling
Experimentally, it is found that nuclear reactions sometimes occur at energies less than that required by the Coulomb barrier. This is called tunneling.
As a projectile approaches a target nucleus in a nuclear reaction, the probability that there will be overlap and hence interaction of their wave functions increases.

21. Resonance and Tunneling
An example is provided by the reaction of protons with lithium:
(a)
(b)
(c)
For the above reaction the value of Ecb(min) is 1.3MeV. However, due to tunneling the reactions begin to occur at lower proton energies. At an energy of 0.15 MeV about 0.1% of the protons penetrate the Coulomb barrier, at 0.3 MeV about 1%, and at 0.6 MeV about 20%.
The reaction cross-section is closely related to the excited energy states of the compound nucleus.

22. Resonance and Tunneling
Four excited states levels are shown for 8Be* in the following example:
Fig 4.2: Yield curves for the reaction between protons and 7Li, leading to different excited levels in 8Be, followed by decay to stable end products.

23. Resonance and Tunneling
In the previous curve, to the left of the figure, the (p,γ), (p, n), and (p,α) excitation functions are shown as function of the proton kinetic energy. The maximum cross-section for reaction (b) occurs at a proton kinetic energy of 0.44MeV, which, together with the Q-value, 17.2 MeV, of the reaction 7Li + 1H Be, leads to an excitation energy of 17.6 MeV, which exactly matches an excited level of the same energy in 8Be*.
At an excitation energy of MeV another energy level is reached in the compound nucleus leading to its decay into 7Be + n, reaction (c).
The excitation energy is achieved from the release of nuclear binding energy (17.25 MeV) and from the proton kinetic energy. The amount needed is = 1.93 MeV. In order to conserve momentum, the proton must have (8/7)×1.93 = 2.21 MeV in kinetic energy. The increase in cross-section when the total excitation energy matches an excited energy level of the compound nucleus is known as resonance.

24. Resonance and Tunneling
Reaction (b) is still used for the production of γ-radiation (17MeV).
While reaction (c) is used as a source of mono-energetic neutrons.
The energy of the neutrons from reaction (C) is a function of the proton energy and the angle between the neutron and the incident proton beam.
A necessary requirement, however, is that the threshold energy (1.64 × (8/7) = 1.88 MeV) must be exceeded, the Q-value for reaction (C) BEING MeV.

25. Neutron Capture and Scattering
Unlike charged particles, no Coulomb barrier hinders neutrons from reaching the target nucleus. This leads to generally higher reaction cross-sections for neutrons, particularly at very low energies.
Neutron-induced processes are among the most important and well studied nuclear reactions.
We have seen that the geometric cross-section of a target nucleus is in the order m2.
Experimentally, the cross-sections for capture of energetic “fast” neutrons (En≥ 1MeV) are often close to10-28 m2. However, for neutrons whose kinetic energy is in the range 1-100eV, some nuclei show very large cross sections- as high as m2. This is due to electron capture where the compound nucleus is excited to one of its discrete energy levels (resonance capture).