Nuclear Physics (10 th lecture)

Transcription

1 ~Theta Nuclear Physics ( th lecture) Content Nuclear Collective Model: Rainwater approx. (reinder) Consequences of nuclear deforation o Rotational states High spin states and back bending o Vibrational states Monopole, dipole and quadrupole vibrations, Giant resonances, experiental observations The nuclear collective odel (contd.) Rainwater approxiation (reinder) Marriage of the liquid drop odel and the shell odel. Concept: nucleus = core + valence nucleon(s) deforable shell odel liquid drop Consequence #: Soe nuclei are defored also in ground state. Consequence #: Nuclear quadrupole oent: Here S is the rigidity of the core, and Q N is the quadrupole oent of the valence nucleon Consequence #: Nilsson-schee of shell-odel levels Better description of (defored) ground-state properties Q ZR U 4 5 r S Q N Further consequences of the core & potential deforation 4) No ore isotropy rotational excitation is possible 5) Deforability vibrational excitation is possible Rotation Classical physics also a sphere can rotate Quantu physics only defored objects can rotate Rotator: I oent of inertia Total angular oentu : J I j valence nucleon J I j Ij (not a rotation!) core rotation Ij, since I is perpendicular to the syetry axis, and j is precessing around it. This eans that: I J j JJ j j For the rotational excitations we have: J J j j In the ground state I =, which gives J = j t a rotational excitation I is increasing, so we get J=j+, j+, J j ~Q (kev) 5,, 5,, 5,, 5,,,5 4,5 J 6,5 8,5 5/ 9/ / 7/,8,7,6,5,4,,,5,5 4,5 5,5 6,5 7,5 8,5 5/ 9/ J / 7/ Sees that slightly J J j j 4 increases (?)

2 For the rotational excitations we have: J J j j For even-even nuclei because of parity conservation only J=j+, j+4, can occur, and of course j= (kev),, 8, 6, 4,,,,, 4, 6, 8,, J This way can be easured! It can also be calculated using the nuclear shape r(r) (deduced fro the quadrupole oent)! z theor r r d r MR y 5 Big surprise:??? eas theor Why does it change (increases) as J gets larger??? x Q? J 5 6 xplanation (age Bohr): Nuclei are superfluid because of the pairing (like Cooper-pairs in superconductors) Rotation: not like a rigid body, but ore like a surface wave Rigid body: v rω rot v ω Superfluid: no friction inside rot v the core stays still, only the surface rotates Fro rot v follows: v grad (velocity potential) Fluid is incopressible: div v Fro all these the velocity potential: Fro B B v grad the velocity field can be deterined. age Bohr 9-9 Nobel-prize 975 yz 7 M d r The rotational energy: M r v d r Fro these two I can be calculated Result: The angular oentu: I r r v wave rigid rigid age Bohr, Ben Mottelson: wave final observed rigid good description, taking into account even the pair-correlation! Nobel-prize 975:. Bohr, Mottelson, Rainwater for the discovery of the connection between collective otion and particle otion in atoic nuclei and the developent of the theory of the structure of the atoic nucleus based on this connection. 8

3 High spin states and back bending Producing high spin states: fusion of heavy ions I rp Since p is big, large angular oentu states can be fored x I ap bands xaple: MeV 6 O + Po 8 U 4 5 Nuclear excitation energy: rot II This is only the rotational energy! The nucleus ay store energy in other fors (e.g. deforation, internal excitation etc.)! 6 Lu 9 Yrast = ost rotating What is the reason? The picture is even ore coplicated Superfluid solid transition g s What is the cause for the transition? Coriolis-force breaks up a nucleon-pair Rotational alignent

4 Vibration It is convenient to give the instantaneous coordinate R(t) of a point on the nuclear surface at (, ) in ters of the spherical haronics λ R(t) R λμ ( t) Yλμ ( θ,φ) λ μ λ Due to reflection syetry λμ λ, = vibration: onopole (GMR observed for >4) (breathing ode of a copressible fluid) 8 MeV = vibration: dipole R(t) R t Y ( θ,φ) R t Y t Y t Y μ μ μ -μ, -, - = vibration: quadrupole R(t) R t Y, 5 t Y R t cos θ R 4 π because = for (using appropriate coord. sys.) (for ellipsoidal shape, R is a function of only ) The shape of the surface can be described by Y, = ±, ±,. In the case of an ellipsoid R = R(θ) hence =. / R t Y R t cos θ π because μ t for μ (, -t t and Y, - Y) J π = +, +, 4 +. Giant dipole resonance (observed for >6) 77 Giant quadrupole resonance (observed for >6) 6 MeV 4 μ μ μ R Y Y / Y Y Y, -, - Quantization of quadrupole vibration is called a quadrupole phonon, J π = +. This ode is doinant. For ost even-even nuclei, a low lying state with J π = + exists and near closed shells second haronic states can be seen MeV = vibration: quadrupole For a haronic oscillation: H v r B d dt / 5 C N N ω; ω B Orthogonal transforation to get a diagonal for: a a a a,,,,, a, μ C μ, Y,, a, Y Coonly new variables are introduced: a, cos, a, The three ain axis (k =,,) of the 5 ellipsoid are then: R k R R cos k 4 sin 5 These two variables can be considered as polar coordinates, thus every shape can be represented by a point in a D surface The case of the two-phonon state There are -vibrations, -vibrations one phonon, two phonon, etc For every single-phonon state J = J = + Because of J J J J we would expect J = +, +, +, +, 4 + J 6 4

5 But consider the quantu nubers: M J = J = M J = 4 J = J = Note: Only non-negative and M values are shown. The table is syetric for M < Only J = +, +, 4 + occurs! xaple: vibrational states in 4 Cd Nuber of phonons N = ω N = N = ω single-phonon state two-phonon states ω ground state 7 8 vibrations: octupole, hexadecupole etc. xperiental evidences for giant resonances Mainly (inelastic) scattering experients Giant dipole resonance (p,p ), (e,e ), (,p) etc. (, ), (d,d ) etc. Octupole (λ = ) odes with J π = can be observed in any nuclei. The collective Hailtonian (for = ) Rotation Vibrational (the ain axis rotate) potential energy kk H B C k the total kinetic energy The energy: coll vibr rot coll n I I 9 Giant Quadrupole (GQR) and Giant Monopole (GMR) resonances Many different vibrational odes have been established also experientally. GQR GMR Morsch, Sükösd et al. Phys Rev. C (98)