May 1, Pierre NZABAHIMANA Survey of Nuclear Physics PHY802 Instructor: α αprof. Scattering Witek Nazarewicz

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1 α α Scattering Pierre NZABAHIMANA Survey of Nuclear Physics PHY802 Instructor: Prof. Witek Nazarewicz May 1, 2017

2 What people can learn from α α Scattering? α α Scattering is identical particles scattering problem. The even orbital angular momentum (i.e, l = 0, 2, 4,..) are possible to define partial wave in alpha-alpha scattering because this system has even party and zero spin. Alpha-alpha scattering can be studied as scattering of 8 Be which is very bound at ground state. Time delay between the projected particles and target (interaction time) at resonance energy.

3 Introduction (a) Scattering of two identical particles, [Shehadeh, Z.F., 2017]. (b) Incident and target alpha nucleus each has 4 particles[elhatisari, S, et al, 2015]

4 (a) Hulthen Potential [Bhoi J.et al, 2017] (b) Phase shift[bhoi J.et al, 2017] The repulsion is need to prevent the two alpha particles from coming closer each other.

5 Model of α α scattering in 3-D Figure: Incident and Scattered waves In three dimension, the scattering problem is given: ψ(r) e ik r + f (θ) e ik r at r (1) r

6 3-D Scattering Alpha-alpha scattering is symmetric, then the scattering amplitude will be; ( f in (θ) + f tar (π θ) = 1 (2l + 1)e iδ ( l sinδ l Pl (cosθ) + P l ( cosθ )) k l=0 at r Where δ l is phase shift, l is angular momentum quantum number. P l ( x) = ( 1) l P l (x),

7 Resonance α alpha scattering We considered the resonance scattering to occur in the region of low energy, this implies for; kr 0 1 The resonance scattering parameters are: Γ, k, E r, r 0 and δ l, which are energy width, wave vector resonance energy and resonance phase shift respectively. In resonance the phase shift is a function of incident energy E as follows: ( ) δ l = tan 1 Γ (2) 2(E r E) Where,Γ = 2V l 2 mr 0, v l = (kr 0) 2l [(2l 1)!!] 2, and k 2 = 2mEr 2.

8 Resonance scattering amplitude for low energy at l = 0 So, the resonance scattering amplitude becomes: F = (2l + 1) k Γ 2P l (cosθ) 2(E r E) + iγ (3) Figure: Scattering amplitude, used data from [Bhoi, J. and Laha, 2017] and [Heydenburg, N.P. and Temmer, G.M., 1956]

9 Time delay Time delay t d ; t d = dδ l de = 2 2 Γ 2 Γ 4(E Er) 2 + Γ 2 (4) Figure: Time delay at resonance energy, data used from [Bhoi, J. and Laha, 2017] and [Heydenburg, N.P. and Temmer, G.M., 1956]

10 Very recent result in Alpha-alpha Scattering Figure: Comparison btn calculated nonrelativistic and relativistic diff. cross section shown as solid and dashed lines respectively with solid circles measured value at E α = 280Mev [Shehadeh, Z.F., 2017]

11 Conclusion Alpha-alpha scattering exists only for l even. At resonant energy probability of particles to pass through cross section is higher. The projected particle interact with target longer at resonant energy. Alpha-alpha scattering can be used to construct the cluster model of light nuclei or to study interaction between two nucleus and nucleus systems.

12 Bibliography Bhoi, J. and Laha U., Hulthen potential models for α α and α He 3 elastic scattering. Pramana 88(3), p.42. Shehadeh, Z.F., Analyses of alpha-alpha elastic scattering data in the energy range MeV. Journal of the Korean Physical Society, 70(2), pp Bhoi, J. and Laha, U., Elastic scattering of light nuclei through a simple potential model. Physics of Atomic Nuclei, 79(3), pp Elhatisari, S., Lee, D., Rupak, G., Epelbaum, E., Krebs, H., Lähde, T.A., Luu, T. and Meißner, U.G., Ab initio alpha alpha scattering. Nature, 528(7580), pp Heydenburg, N.P. and Temmer, G.M., Alpha-Alpha scattering at low energies. Physical Review, 104(1), p.123.

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