523-14

Wave functions• Shells • Orthogonality

523-14

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:

• lifts H-like degeneracy:

• “Far away” orbits: still hydrogen-like!

0 1 2 3 4 5

r [a.u.]

!20

!15

!10

!5

0

V(r

) [

a.u

.]

Sodium

�2s < �2p

�3s < �3p < �3d

523-14

Example: Na• Fill the lowest shells

• Use schematic potential

• Ground state: fill lowest orbits according to Pauli

• Excited states?

H0 |n⇤m�ms� = �n� |n⇤m�ms�

|300ms, 211 12, 211� 1

2, ..., 100 1

2, 100� 1

2i ⌘

|�0(Na)i

523-14

Closed-shell atoms• Neon

• Ground state

• Excited states

• operators: see later

• Note the H-like states

• Splitting?

• Basic shell structure of atoms understood ⇒ IPM

|�0(Ne)i = |211 12, 211� 1

2, ..., 100 1

2, 100� 1

2i

|n` (2p)�1i = a†n`a2p |�0(Ne)i

523-14

Periodic table

HF

Σ(2)F-RPA

523-14

Level sequence (approximately)

523-14

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 before

• Hamiltonian

• It is also possible to write before

p =�p · p r =

�r · r

rp =1

2

�1

pp · r + r · p1

p

⇥pr =

1

2

�1

rr · p+ p · r 1

r

⇥

H =p2

2m� �2

ma0

1

r

�2 = p2�r2 � r2p

⇥�2 = r2

�p2 � p2r

⇥

523-14

Detour (artificial)• Define “funny” operator

• When acting on eigenstate of Hamiltonian same effect as applying the operator

• Proof requires to show that

• Then it follows immediately that • Goal is now to factorize the “funny” operator

� = r2�p2 � 2mE

⇥2 � 2i�p · r�p2 � 2mE

⇥+ 4�2

�p2 � 2mE

⇥

H |E�m� = E |E�m�

r2�p2 � 2mH

⇥2

r2�H,p2

⇥=

2i�3ma0

(p · r + 2i�) 1r

� |E�m� = 4�4a20

|E�m�

523-14

Development• Works by defining

• Use to replace in and use • Inserting and replacing the square of the orbital angular

momentum by its eigenvalue, one finds

• Check that

• Note

• As before implies that the energy doesn’t change

P±� = rp

�p2 � 2mE

⇥± i�

�+ 12 ± 1

2

p

�p2 + 2mE

⇥

�� = P��±1P

±� � 4�2

��+

1

2± 1

2

⇥2

2mE

�� |E�� = 4�4a20

|E��

��±1

�P±� |E��

⇥=

4�4a20

�P±� |E��

⇥

P±� |E�� = p±E� |E�± 1�

�2 = p2�r2 � r2p

⇥r2 p · r = rpp� 2i�

�� = r2p�p2 � 2mE

⇥2+

�2�(�+ 1)

p2�p2 � 2mE

⇥2 � 2i�rpp�p2 � 2mE

⇥

�

523-14

More development• Normalization

• For bound states factor must break off for

• With the usual solutions with

• Go to momentum representation with • apply to

• insert and note phase choice!

��p±E�

��2 =4�4a20

⌅1 +

⇥�+

1

2± 1

2

⇤2 2ma20�2 E

⇧

1 + (�max + 1)22ma20�2 E = 0

En = � �22ma20

1

n2n = �max + 1

rp = i��

�

�p+

1

p

⇥

P±� |E�� = p±E� |E�± 1�

E

⇥p| rp�p2 � 2mE

⇥± i�

�+ 12 ± 1

2

p

�p2 + 2mE

⇥|n�⇤ = 2�2

a0i

⇤1�

(�+ 12 ± 1

2 )2

n2

⌅1/2⇥p|n�± 1⇤

523-14

Differential equation in momentum space• Final result

• For use upper result (rhs --> 0)

• Solution

⇤p2 +

�2a20n

2

⌅d

dp⇤p|n�⌅+

⌥⇧±⇤�+

1

2± 1

2

⌅+ 3

⌃p+

�2a20n

2

1

p

⇧1⇥

⇤�+

1

2± 1

2

⌅⌃�⇤p|n�⌅ =

2�a0

1�

��+ 1

2 ± 12

⇥2

n2

⌦1/2

⇤p|n�± 1⌅

� = �max

⇤�p2 +

�2a20n

2

⇥d

dp+ (n+ 3) p+

�2a20n

2

1

p(1� n)

⌅⇥p|n� = n� 1⇤ = 0

⇥p|n⇥ = n� 1⇤ = �n�=n�1(p) = Npn�1

�p2 + �2

a20n

2

⇥n+1

523-14

Ground state• Normalization

• Other wave functions: use lowering operator • Ground state wave function

|N |2 =24n+2(n!)2

�(2n)!

��

a0n

⇥2n+3

⇥10 = 4

⇧2

�

⇤�a0

⌅5/2 1�p2 + �2

a20

⇥2

523-14

Direct knockout reactions• Atoms: (e,2e) reaction • Nuclei: (e,e’p) reaction [and others like (p,2p), (d,3He), (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

523-14

(e,2e) data for atoms• Start with Hydrogen • Ground state wave function • (e,2e) removal amplitude

⇥1s(p) =23/2

�

1(1 + p2)2

�0| ap |n = 1, ⇤ = 0⇥ = �p |n = 1, ⇤ = 0⇥ = �1s(p)

Hydrogen 1s wave function “seen” experimentally Phys. Lett. 86A, 139 (1981)

523-14

Helium• IPM description is very successful • Closed-shell configuration • Reaction more complicated than for Hydrogen • DWIA (distorted wave impulse approximation)

agreement with IPM! → 1

1s2

Phys. Rev. A8, 2494 (1973)

S =⇥

dp����N�1

n | ap |�N0 ⇥

��2

523-14

Other closed-shell atoms• Spectroscopic factor becomes less than 1 • Neon removal: S = 0.92 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 S = 0.95 • Rest in 3 small fragments

2p

0 1 2 3 p (a.u.)

0.0

0.4

0.8

1.2

1.6

2.0

Diff

eren

tial c

ross

sect

ion

(10−

3 a.u

Argon

3p

3p

523-14

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

• with the same “global” quantum numbers • Example: Argon ground state • Ar+ ground state • excited state • also • and

a� |�N0 �

a�a⇥a†⇤ |�N0 �

|�N0 � = |(3s)2(3p)6(2s)2(2p)6(1s)2�

|(3p)�1� = a3p |�N0 � = |(3s)2(3p)5(2s)2(2p)6(1s)2�

|(3s)�1� = a3s |�N0 � = |(3s)1(3p)6(2s)2(2p)6(1s)2�

|(3p)�24s� = a3pa3pa†4s |�N

0 � = |(4s)1(3s)2(3p)4(2s)2(2p)6(1s)2�|(3p)�2nd� = a3pa3pa

†nd |�N

0 � = |(nd)1(3s)2(3p)4(2s)2(2p)6(1s)2�

523-14

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

8%

523-14

(e,e’p) data for nuclei• Requires DWIA • 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

523-14

Inclusion of SO potential and magic numbers• Sign of SO?

• Consequence for

• Noticeably shifted • Correct magic numbers!

V�s

�� · s

�2

⇥r201r

d

drf(r)

0f 72

0g 92

0h 112

0i 132

V�s = �0.44V

523-14

Neutron levels as a function of A

• Phenomenological! • Calculated from Woods-

Saxon plus spin-orbit

523-14

Momentum profiles for nucleon removal • Closed-shell nuclei • NIKHEF data, L. Lapikás, Nucl. Phys. A553, 297c (1993)

523-14

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

523-14

Remember• 208Pb sp levels

0f

523-14

Fragmentation patterns• 208Pb(e,e’p) NIKHEF data: Quint thesis

• S(2s1/2)=0.65

• other data: • n(2s1/2)=0.75

• very different from atoms

523-14

Fragmentation patterns• 208Pb(e,e’p) NIKHEF data: Quint thesis

• start of strong fragmentation • also very different from atoms

523-14

Fragmentation patterns• 208Pb(e,e’p) NIKHEF data: Quint thesis

• deeply bound states: strong fragmentation • again different from atoms

523-14

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

Moment

um

Energy

523-14

Recent Pb experiment• 100 MeV missing energy • 270 MeV/c missing momentum • complete IPM domain

SRCalso LRC