Shells Orthogonality. Wave functions

Size: px

Start display at page:

Download "Shells Orthogonality. Wave functions"

Branden McCarthy
6 years ago
Views:

1 Shells Orthogonality Wave functions

2 Effect of other electrons in neutral atoms Consider effect of electrons in closed shells for neutral Na large distances: nuclear charge screened to 1 close to the nucleus: electron sees all 11 protons approximately: 0!5 V(r) [a.u.]!10!15 Sodium! r [a.u.] lifts H-like degeneracy: s < p 3s < 3p < 3d Far away orbits: still hydrogen-like!

3 Example: Na Fill the lowest shells Use schematic potential H 0 n m m s = n n m m s Ground state: fill lowest orbits according to Pauli 300m s, 11 1, 11 1,...,100 1, i 0(Na)i Excited states?

4 Neon Closed-shell atoms Ground state 0(Ne)i = 11 1, 11 1,...,100 1, i Excited states n` (p) 1 i = a n`ap 0(Ne)i operators: see later Note the H-like states Splitting? Basic shell structure of atoms understood IPM

5 Periodic table HF F-RPA Σ ()

6 Level sequence (approximately)

7 Hydrogen again Relevant references for factorization technique Am J Phys 55, 913 (1987) Am J Phys 46, 658 (1978) Factorization with the aim to go to momentum space! Consider r p = 1 Hamiltonian p = p p before r = r r 1 p p r + r p1 p r = 1 1 p r r p + p r 1 r H = p m 1 ma 0 r It is also possible to write = p r r p before = r p p r

8 Detour (artificial) Define funny operator = r p me i p r p me +4 p me When acting on eigenstate of Hamiltonian H E m = E E m same effect as applying the operator Proof requires to show that r p mh r H, p = i 3 (p r +i ) 1 ma 0 r Then it follows immediately that Goal is now to factorize the funny operator E m = 4 4 a 0 E m

9 Development Works by defining P ± = r p p me + 1 ± i ± 1 p +me p Use = p r rp to replace r in and use p r = r p p i Inserting and replacing the square of the orbital angular momentum by its eigenvalue, one finds = r p p me + ( + 1) p p me Check that = P ±1 P ± ± 1 me i r p p p me Note E = 4 4 a 0 E As before ±1 P ± E = 4 4 a 0 energy doesn t change P ± E = p ± E E ± 1 P ± E implies that the

10 Normalization More development For bound states factor must break off for With the usual solutions Go to momentum representation with apply to p r p p me ± i p ± E = 4 4 a ± 1 1+( max + 1) ma 0 E =0 E n = P ± E = p ± E E ± ± 1 p ma 0 p +me n = a 0 i ma 0 E 1 with n = n max +1 r p = i 1 p + 1 p ( + 1 ± 1 ) n 1/ p n ± 1 insert E and note phase choice!

11 Final result Differential equation in momentum space p + a 0 n d dp p n + ± + 1 ± 1 +3 p + a 0 n 1 p a ± 1 p n = + 1 ± 1 1/ p n ± 1 n = max For use upper result (rhs --> 0) p d + a 0 n dp +(n + 3) p + 1 a (1 n) 0 n p p n = n 1 =0 Solution p n = n 1 = n =n 1 (p) =N p n 1 p + a 0 n n+1

12 Normalization Ground state N = 4n+ (n!) (n)! a 0 n n+3 Other wave functions: use lowering operator Ground state wave function 10 =4 a 0 5/ 1 p + a 0

13 Atoms: (e,e) reaction Direct knockout reactions Nuclei: (e,e p) reaction [and others like (p,p), (d, 3 He), (p,d), etc.] Physics: transfer large amount of momentum and energy to a bound particle; detect ejected particle together with scattered projectile construct spectral function Impulse approximation: struck particle is ejected Other assumption: final state ~ plane wave on top of N-1 particle eigenstate (more serious in practical experiments) but good approximation if ejectile momentum large enough If relative momentum large enough, final state interaction can be neglected as well -> PWIA = plane wave impulse approximation Cross section proportional to spectral function

14 Start with Hydrogen Ground state wave function (e,e) removal amplitude (e,e) data for atoms 1s (p) = 3/ 1 (1 + p ) 0 a p n =1, =0 = p n =1, =0 = 1s (p) Hydrogen 1s wave function seen experimentally Phys. Lett. 86A, 139 (1981)

15 Helium IPM description is very successful Closed-shell configuration 1s Reaction more complicated than for Hydrogen DWIA (distorted wave impulse approximation) S = dp N 1 n a p N 0 agreement with IPM! 1 Phys. Rev. A8, 494 (1973)

16 Other closed-shell atoms Spectroscopic factor becomes less than 1 Neon p removal: S = 0.9 with two fragments each 0.04 IPM not the whole story: fragmentation of sp strength Summed strength: like IPM IPM wave functions still excellent Example: Argon 3p S = 0.95 Rest in 3 small fragments Differential cross section (10 3 a.u p (a.u.) Argon 3p

17 Fragmentation in atoms ~All the strength remains below (above) the Fermi energy in closed-shell atoms Fragmentation can be interpreted in terms of mixing between and a N 0 a a a N 0 with the same global quantum numbers Example: Argon ground state Ar + ground state excited state also and (3p) 1 = a 3p (3s) 1 = a 3s N 0 = (3s) (3p) 6 (s) (p) 6 (1s) N 0 = (3s) (3p) 5 (s) (p) 6 (1s) N 0 = (3s) 1 (3p) 6 (s) (p) 6 (1s) (3p) 4s = a 3p a 3p a 4s N 0 = (4s) 1 (3s) (3p) 4 (s) (p) 6 (1s) (3p) nd = a 3p a 3p a nd N 0 = (nd) 1 (3s) (3p) 4 (s) (p) 6 (1s)

18 Argon spectroscopic factors s strength also in the continuum: Ar ++ + e note vertical scale red bars: 3s fragments exhibit substantial fragmentation 8%

19 Requires DWIA (e,e p) data for nuclei Distorted waves required to describe elastic proton scattering at the energy of the ejected proton Consistent description requires that cross section at different energy for the outgoing proton is changed accordingly Requires substantial beam energy and momentum transfer Initiated at Saclay and perfected at NIKHEF, Amsterdam Also done at Mainz and currently at Jefferson Lab, VA Momentum dependence of cross section dominated by the corresponding sp wave function of the nucleon before it is removed

20 Inclusion of SO potential and magic numbers Sign of SO? s V s r 0 V s = 0.44V 1 r Consequence for d dr f(r) 0f 7 0g 9 0h 11 0i 13 Noticeably shifted Correct magic numbers!

21 Phenomenological! Calculated from Woods- Saxon plus spin-orbit Neutron levels as a function of A

22 Momentum profiles for nucleon removal Closed-shell nuclei NIKHEF data, L. Lapikás, Nucl. Phys. A553, 97c (1993)

23 But... Spectroscopic factors substantially smaller than simple IPM

24 08 Pb sp levels Remember 0f

25 Fragmentation patterns 08 Pb(e,e p) NIKHEF data: Quint thesis S(s 1/ )=0.65 other data: n(s 1/ )=0.75 very different from atoms

26 Fragmentation patterns 08 Pb(e,e p) NIKHEF data: Quint thesis start of strong fragmentation also very different from atoms

27 Fragmentation patterns 08 Pb(e,e p) NIKHEF data: Quint thesis deeply bound states: strong fragmentation again different from atoms

28 Simple interpretation! Mougey et al., Nucl. Phys. A335, 35 (1980) 16 O data from Saclay Momentum Energy

29 100 MeV missing energy 70 MeV/c missing momentum complete IPM domain Recent Pb experiment SRC also LRC

Other electrons. ε 2s < ε 2p ε 3s < ε 3p < ε 3d

Other electrons. ε 2s < ε 2p ε 3s < ε 3p < ε 3d Other electrons Consider effect of electrons in closed shells for neutral Na large distances: nuclear charge screened to 1 close to the nucleus: electron sees all 11 protons approximately:!!&! " # $ %