1. Relativistic Kinematics for the Binding Energy of Nuclear Reactions
Yongkyu Ko
Yonsei University, Korea
Particle physics lecture 10 October 2016

2. Contents The Doppler effect for the deuteron binding energy
The binding energy measurement and lifetime of the 4943 keV state in 29Si.
Lifetime of the 6792 keV state in 15O and the astrophysical S factor

3. The Doppler effect for the deuteron binding energy
Contents
1. Abstract
2. Motivation
3. Non-relativistic determination of the binding energy
4. Relativistic determination without initial energy
5. Relativistic determination with initial energy
6. Lorentz transformation
7. Non-relativistic reduction
8. Experimental set-ups
9. Origin of the error
10. Other reactions
11. Summary and conclusion

4. Abstract
The deuteron binding energy is extracted from the neutron capture
Reaction with relativistic kinematics.
Negligible initial kinetic energy of the neutron causes a significant
uncertainty due to the Doppler effect.

5. Motivation

6. Non-relativistic determination of the binding energy

7. Relativistic determination without initial energy
Energy-momentum relation
Energy conservation
Momentum conservation
Lorentz invariant

8. Relativistic determination with initial energy
Energy-momentum relation
Energy conservation
Momentum conservation
Lorentz invariant

9. Lorentz transformation
Lab system
c. m system

10. Non-relativistic reduction

11. Experimental set-ups

12. Origin of the error
Theoretical error
Number of measurements : 52

13. Other reactions

14. Summery and conclusions
Selecting a certain angle of the incident neutron to the emitted
photon will lead to a reduced linewidth.
A possible geometry could make use of a target sample near
the core of a reactor with appropriate shielding from thermal
neutrons.
Using the difference of the penetration lengths for neutrons
and photons, shielding the target should not cause much
loss in flux of the incident neutron beam.
Careful geometric considerations can also explain the
discrepancies among the values of the deuteron binding
energies.

15. The binding energy measurement and lifetime of the 4943 keV state in 29Si.
Contents
1. Abstract
2. Motivation
3. Neutron capture reaction
4. Deuteron binding energy
5. Binding energy of 29Si
6. Lifetime measurement
7. Summary and conclusion

16. Abstract
Using a flat crystal spectrometer, the binding energy of the neutron capture reaction can be precisely determined by measuring the -ray energy with the Bragg law.
The most probable decay channel of the 29Si is a cascade decay via
4943 keV level in the neutron capture reaction 28Si(n, , )29Si.
The nucleus of the intermediate state has a considerable velocity, because it is recoiling by emitting the primary -ray.
Using angular correlation of the two successive -rays, coincidence measurement of the two -rays can give the information on the Doppler shift of the secondary -ray and determine the binding energy precisely.
The line shape of the secondary -ray may be free from Doppler broadening of the recoiling nucleus and give the information on the lifetime of the 29Si 4943 keV level.

17. Motivation Principle of the two axis flat crystal spectrometer
Measurement of the Bragg angle with the flat crystal spectrometer
Results in the scale of interferometer angle

18. Principle of Penning trap measurement
Cyclotron frequency

19. Comparison of the two experiments
Penning Trap
Flat crystal spectrometer

20. Neutron Capture Reaction
Energy levels of 29Si

21. Decay Scheme of 29Si
Feynman diagram for neutron capture reaction
Binding energy of neutron
Separation of binding energy into two parts

22. Deuteron Binding Energy
Energy-momentum conservation :
Velocity of the center of mass system
Nonrelativistic calculations
Relativistic calculations

23. Binding energy for 29Si Velocity of the center of mass system
Binding energy for the excited state of 29Si
Energy-momentum conservation for the capture reaction
Binding energy of the excited state

24. Energy-momentum conservation
for the decay of the intermediate state
Velocity of the intermediate nucleus
Binding energy of the intermediate state
Total binding energy

25. Angular correlation
Angular correlation function
For 010 dipole-dipole
For 1/23/21/2 dipole-dipole

26. GAMS4 Facility and the Reactor at Institut Laue Langevin
The through tube makes coincidence measurements possible!
Mean value gives more precise binding energy free from recoil effect.
Energy difference gives the information of the attenuation of the nucleus recoil in the environment of the nucleus.

27. Lifetime Measurement Doppler Shift Attenuation Method
Attenuation factor
Damping oscillation model
a: damping factor
Lifetime
Debye frequency

28. Doppler Shift Attenuation Method by the linewidth
Lorentzian shape with natural linewidth
Doppler shifted lineshape

29. Doppler shifted lineshape
Maxwell Boltzmann distribution

30. Summary and conclusion
The velocity of the recoiling nucleus due to the emission of a primary gamma ray is significantly so large (39km/s) that the Doppler shift of the secondary gamma ray reach cos eV.
Precision measurement of gamma ray with flat crystal
spectrometer should consider such a large Doppler shift and
Doppler broadening.
A coincidence measurement of gamma rays with angular
correlation could determine the binding energy precisely.
A coincidence measurement of gamma rays with angular
correlation could determine the lifetime of nuclear excited state.
The lineshape of the Doppler shifted gamma ray could give the
lifetime of the excited state with correction of the thermal motion
of the nucleus and compare with the lifetime by measuring the
attenuation factor.

31. Lifetime of the 6792 keV state in 15O and the astrophysical S factor
Contents
1. Abstract
2. Motivation
3. Lifetime measurement
4. Binding energy and energy level
5. Motion of the recoil nucleus
6. Results
7. Summary and conclusion

32. Abstract
The capture reaction 14N(p, , )15O is an important process in the hydrogen burning CNO cycle to determine astrophysical quantities.
The lifetime of the 6792 keV state in 15O plays an important role in determining the astrophysical S factor for the thermonuclear reaction.
The lifetime can be measured by the Doppler shift attenuation method and the recent measurements do not agree because the attenuation factor is near unity and thus the lifetime is too short to be measured by the method.
Precise relativistic calculation for the cascade decay would give a good explanation for the discrepancy of the experiments and especially the effect of the primary was investigated which was not considered in the experiments.

33. Motivation CNO cycle in stars * Stars with mass > 1.5 solar mass
* All stars at the end of main sequence lifetime
* Red giant branch
14N(p, )15O reaction is the slowest reaction

34. Astrophysical Importance
Detailed understanding of the neutrino spectrum of our sun
Age determination of globular cluster
Neutrino spectrum of our sun
M55 globular cluster

35. Energy levels of 15O
Measurement of astrophysical S factor

36. Comparison of recent experiments
Phys. Rev. C77 (2008)
F( )>0.98, Lifetime < 0.77 fs
Phys. Rev .Lett. 87 (2001)
F( )=0.93, Lifetime =1.6 fs

37. Lifetime Measurement Doppler Shift Attenuation Method
Usual formula in the literature.
Our calculation
Attenuation factor
Program SRIM2011
Our approach : The effect of the primary
Complete relativistic calculation

38. Binding Energy and energy level
Decay Scheme of 15O
Feynman diagram for proton capture reaction
Binding energy of proton
Separation of binding energy into two parts

39. The first part of the transition
Energy momentum conservation

40. The second part of the transition
Energy momentum conservation

41. Motion of the Recoil nucleus
Energy-momentum
Velocity distribution
Velocity

42. Results Velocity diagram for the intermediate nucleus
Comparison of velocities for Ein=300 keV and 318 keV

43. Doppler Shift Attenuation Factor
Ein=300
11.534
11.554
0.966
0.964
11.875
11.896
0.938
0.936
Ein=318
Unit : keV

44. Comparison of experimental data and calculations
Upper curve : Ein=318 keV
Lower curve : Ein=300 keV

45. Summary and conclusion
Since the attenuation factor is nearly unity, the effect of the primary
gamma ray contribute to the attenuation factor substantially in the
framework of relativistic kinematics.
The two experiments would be consistent, if the same kinematics is used, because small different incident kinetic energies give slightly different Doppler shift.

46. Thank you for your attention.