II. Spontaneous symmetry breaking

Transcription

1 . Spontaneous symmetry breaking

2 .1 Weinberg s chair Hamiltonian rotational invariant eigenstates of good angular momentum: M > have a density distribution that is an average over all orientations with the weight D MK ( ψθφ) Why do we see the chair shape? Spontaneously broken symmetry States of different M are so dense that the tiniest interaction With the surroundings generates a wave packet that is well oriented. a >= c M Ma > M

3 h energy scale of rotational levels ~ 10 h J energy distance of rotational levels h J angular momentum -49 ev ~ ev J h ~ evj[kg m s -1 ] Tiniest external fields generate a superposition of the JM> that is oriented in space, which is stable. Spontaneous symmetry breaking Macroscopic ( infinite ) system

4 The molecular rotor NH = Axial rotor + = J J J H + + = 3 1 1) ( 1 K K E 0 ], [ 0 ], [ 0 ], [ 3 = = = J H J H J H z + + = J J J H

5 eigenstates :, M, K > a > probability amplitude for orientation of rotor : < ψ, θ, φ, M, K 1/ + 1 >= ( ψ, θ, φ) 8π D MK Wigner D function imψ D MK ( ψ, θ, φ) = e dmk ( θ ) e ikφ a > descibes the "intrinsic" structure that spontanously breaks rotational symmetry.

6 Microscopic ( finite system ) Rotational levels become observable. Spontaneous symmetry breaking = Appearance of rotational bands. Energy scale of rotational levels in molecules : h ~ 10-6 ev << intrinsic scale :10-1 ev

7 E( ) = B( + 1) = J hω = E( + 1) E( ) = B( + 1) HCl Rotational bands are the manifestation of spontaneous symmetry breaking. Microwave absorption spectrum

8 Born-Oppenheimer Approximation. Electronic motion Vibrations hωel ~ 1eV hω vib ~ 10 1 ev Rotations hω rot ~ 10 4 ev. CO

9 . The collective model Most nuclei have a deformed axial shape. The nucleus rotates as a whole. (collective degrees of freedom) The nucleons move independently inside the deformed potential (intrinsic degrees of freedom) The nucleonic motion is much faster than the rotation (adiabatic approximation)

10 E E in Ψ Ψ in + E ( x ) Ψ ( ψ, θ, φ) Φ Ψ ( ψ, θ, φ) ν rot rot K rot Nucleons are indistinguishable The nucleus does not have an orientation degree of freedom with respect to the symmetry axis. K = 0 E Axial symmetry E in + ( + Ψ = 8π + 1) K R 3 (φ) Ψ 1/ 1 (,, ) D ψ θ φ MK Φ ikφ in = e Ψin K 3

11 Rotational bands in Limitations: rotational energy scale : h ~ 10 MeV ~ intrinsic scale Single particle and collective degrees of freedom become entangled at high spin and low deformation Er Adiabatic regime Collective model

12 .3 Microscopic approach: Mean field theory + concept of spontaneous symmetry breaking for interpretation. Retains the simple picture of an anisotropic object going round.

13 Rotating mean field (Cranking model): Start from the Hamiltonian in a rotating frame H ' = t + v1 ω j z t v j z 1 kinetic energy effective two - body interaction angular momentum Reaction of the nucleons to the inertial forces must be taken into account Mean field approximation: find state > of (quasi) nucleons moving independently in mean field V mf generated by all nucleons. h' >= e ' >, h' = t + Vmf -ωj z, selfconsistency : { >, v1} h' mean field hamiltonian in the rotating frame (routhian) V mf Selfconsistency : effective interactions, density functionals (Skyrme, Gogny, ), Relativistic mean field, Micro-Macro (Strutinsky method).

14 Rotational response Low spin: simple droplet. High spin: clockwork of gyroscopes. Quantization of single particle motion determines relation J(ω). Uniform rotation about an axis that is tilted with respect to the principal axes is quite common. New discrete symmetries Mean field theory: Tilted Axis Cranking TAC S. Frauendorf Nuclear Physics A557, 59c (1993)

15 Spontaneous symmetry breaking Full two-body Hamiltonian H Mean field approximation Mean field Hamiltonian h and m.f. state h >=e >. Symmetry operation S and + SH ' S = H ' Spontaneous symmetry breaking + Sh' S h', and < S > 1 All states S > are mean field solutions with the same energy E' =< S + H ' S >=< H ' >. Symmetry restoration c i S i >

16 Which symmetries can be broken? H ' = H ω J z is invariant under (ψ ) R z - rotation about z - axis Broken by m.f. rotational bands Combinations of discrete operations R ( π ) - rotation about z - axis by angleπ z P - space inversion TR ( π ) - time reversal with rotation y Obeyed by m.f. broken by m.f. spin parity sequence doubling of states

17 Deformed charge distribution h' = t + V mf ωj z Rotation about the z - axis RH ' z < R + z = H ' but Rh' R + z R > 1 is sharply peaked. z z R( ψ ) z h'. = i e ψ J z nucleons on high-j orbits specify orientation Rotational degree of freedom and rotational bands. All State of orientations ψ >= R z ( ψ ) > good angular momentum have the same energy. >= 1 ψ e i ψ > π dψ.

18 163 Er deformed sotropy broken 00 Pb spherical sotropy conserved

19 Current in rotating 16 Yb J. Fleckner et al. Nucl. Phys. A339, 7 (1980) Lab frame Body fixed frame Moments of inertia reflect the complex flow. No simple formula.

20 .3 Discrete symmetries Combinations of discrete operations R ( π ) z P TR ( π ) y - rotation about z - axis by angleπ - space inversion - time reversal with rotation

21 Common bands P = 1 TR ( π ) R ( π ) z y = 1 - space inversion - time reversal with rotation - rotation about z - axis by π PAC solutions (Principal Axis Cranking) Rz ( π ) >= = α + n e i πα > TAC solutions (planar) (Tilted Axis Cranking) Many cases of strongly broken symmetry, i.e. no signature splitting signatureα

22 Rotational bands in 163 Er

23 Chiral bands

24 Examples for chiral sister bands Pr πh νh / 11/ Rh πg 1 νh 59 9 / 11/ Nd πh νh / 11/

25 Chirality t is impossible to transform one configuration into the other by rotation. mirror

26 Only left-handed neutrinos: Parity violation in weak interaction mirror mass-less particles

27 Reflection asymmetric shapes, two reflection planes Simplex quantum number S S = parity R( π ) P z >= e iσπ π = > ( ) σ Parity doubling

28 6 Th

29 .4 Spontaneous breaking of isospin symmetry S = 0 T = 1 S = 1 T = 0 Form a condensate isovector pair field

30 r = zˆ r = yˆ np = = = = 0 The relative strengths of pp, nn, and pn pairing are determined by the isospin symmetry nn np nn pp pp = = 0

31 Symmetry restoration sorotations (strong symmetry breaking collective model) intrinsic state : isorotational state : > D isorotational energy : E(T,T Experimental : E ( T T ) T ( T + 1), θ T T z 0 ( ϑ, φ,0) > z 1 θ ) =< exp = z = H intrinsic 75MeV A > + T ( T + 1) θ