15thNational Nuclear Physics
Summer School
June 15-27, 2003
Nuclear Structure
Erich Ormand
N-Division, Physics and Adv.
Technologies Directorate
Lawrence Livermore National
Laboratory
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Nuclear Structure
•
•
Of course, I can’t cover everything in just five hours
At least we speak a common language. It could be worse:
I could have to explain the rules to baseball!
2
Nuclear Structure
•
•
•
Nuclear physics is something of a
mature field, but there are still many
unanswered questions about nuclei.
The nuclear many-body problem is one
of the hardest problems in all of physics!
Do we really know how they are put
together?
— This is a fundamental question in nuclear
physics and we are now getting some
interesting answers – for example, threenucleon forces are important for structure
•
Atomic nuclei make up the vast majority
of matter that we can see (and touch).
How did they (and we) get there?
— One of the key questions in science
— Nucleosynthesis
— Structure plays an important role
3
Nuclear Structure
•
•
•
•
4
Exact methods exist up to
A=4
Computationally exact
methods for A up to 16
Approximate many-body
methods for A up to 60
Mostly mean-field pictures
for A greater than 60 or so
Nuclear Structure
•
•
This talk will basically be a theory talk
So, there will be formulae
5
Before we get started:
Some useful reference materials I
General references for nuclear-structure physics:
•
•
•
•
•
•
Angular Momentum in Quantum Mechanics, A.R. Edmonds,
(Princetion Univ. Press, Princeton, 1968)
Structure of the Nucleus, M.A. Preston and R.K. Bhaduri, (AddisonWesley,Reading, MA, 1975)
Nuclear Models, W. Greiner and J.A. Maruhn, (Springer Verlag, Berlin,
1996)
Basic Ideas and Concepts in Nuclear Physics, K. Heyde (IoP
Publishing, Bristol, 1999)
Nuclear Structure vols. I & II, A. Bohr and B. Mottelson, (W.A.
Benjamin, New York, 1969) &
Nuclear Theory, vols. I-III, J.M. Eisenberg and W. Greiner, (North
Holland, Amsterdam, 1987)
6
Before we get started:
Some useful reference materials II
References for many-body problem:
•
•
•
•
Shell Model Applications in Nuclear Spectroscopy, P.J. Brussaard and
P.W.M. Glaudemans, (North Holland, Amsterdam)
The Nuclear Many-Body problem, P. Ring and P. Schuck, (Springer
Verlag, Berlin, 1980)
Theory of the Nuclear Shell Model, R.D. Lawson, (Clarendon Press,
Oxford, 1980)
A Shell Model Description of Light Nuclei, I.S. Towner, (Clarendon
Press, Oxford, 1977)
Review for applying the shell model near the drip lines:
•
The Nuclear Shell Model Towards the Drip Line, B.A. Brown, Progress
in Particle and Nuclear Physics 47, 517 (2001)
7
Nuclear masses, what nuclei exist?
M ( Z , N , A)  Zmh  Nmn  BE ( Z , N , A)
•
Let’s start with the semi-empirical mass
formula, Bethe-Weizsäcker formual, or
also the liquid-drop model.
Z  N 
2
BE (Z,N, A)  aV A  aSurf A 2 3  asym
A

aCoul Z(Z 1)A1 3   Pair  Shell
One goal in theory is 
to
accurately describe the
binding energy
— There are global Volume, Surface,
Symmetry, and Coulomb terms
— And specific corrections for each nucleus
due to pairing and shell structure
aV  15.85 MeV
aSurf  18.34 MeV
aSymm  23.21 MeV
aCoul  0.71 MeV
8
Values for the parameters,
A.H. Wapstra and N.B. Gove, Nulc. Data
Tables 9, 267 (1971)
Nuclear masses, what nuclei exist?
Z  N

23
BE (Z,N, A)  aV A  aSurf A  asym
 aCoul Z(Z 1)A1 3
A
2
Liquid-drop isn’t too bad!
There are notable
problems though.

Can we do better and
what about the
microscopic structure?
From A. Bohr and B.R. Mottleson,
Nuclear Structure, vol. 1, p. 168
Benjamin, 1969, New York
Bethe-Weizsäcker formula
9

Nuclear masses, what nuclei exist?
•
•
How about deformation?
For each energy term, there are also shape factors dependent on
the quadrupole deformation parameters b and g
BSurf
2  5  2  5 
 1 
b   
b  cos 3g 
5  4   21  4  
BCoul
2
3




1
5
1
5
 1 
b  
b  cos 3g 

5  4   105  4  
2
3


R,   R0 1 a20Y20 ,   a22 Y22 ,   Y22 , 
a20  b cos g

a22  b

2 sin g
Note that the liquid drop always has a minimum for a spherical
shape!

So, where does deformation come from?
10
Shell structure - evidence in atoms
•
Atomic ionization potentials show sharp discontinuities at shell
boundaries
From A. Bohr and B.R. Mottleson,
Nuclear Structure, vol. 1, p. 191
Benjamin, 1969, New York
11
Shell structure - neutron separation energies
•
So do neutron separation energies
From A. Bohr and B.R. Mottleson,
Nuclear Structure, vol. 1, p. 193
Benjamin, 1969, New York
12
More evidence of shell structure
Binding energies show preferred magic numbers
— 2, 8, 20, 28, 50, 82, and 126
Deviation from average binding energy
Experimental Single-particle Effect (MeV)
•
From W.D. Myers and W.J. Swiatecki, Nucl. Phys. 81, 1 (1966)
13
Origin of the shell model
•
Goeppert-Mayer and Haxel,
Jensen, and Suess proposed
the independent-particle shell
model to explain the magic
numbers
168
112
Harmonic
oscillator with
spin-orbit is a
reasonable
approximation
to the nuclear
mean field
70
40
20
8
2
M.G. Mayer and J.H.D. Jensen, Elementary Theory of Nuclear
Shell Structure,p. 58, Wiley, New York, 1955
14
Nilsson Hamiltonian - Poor man’s Hartree-Fock
•
Anisotropic harmonic oscillator
p2 m 2 2
  x x   y2 y 2   z2 z 2  2  0  s   0
2m 2
 0  41A1 3 MeV
 x   0e
 y   0e
 z   0e
5
b cosg 2  3
4
5
b cosg 2  3
4
5
b cosg
4

From J.M. Eisenberg and W. Greiner, Nuclear Models,
p 542, North Holland, Amsterdam, 1987
Oblate, g=60˚
Prolate, g=0˚
15
2
Nuclear masses
Shell corrections to the liquid drop
•
Shell correction
— In general, the liquid drop does a good job on the bulk properties
– The oscillator doesn’t!
— But we need to put in corrections due to shell structure
– Strutinsky averaging; difference between the energy of the discrete spectrum
and the averaged, smoothed spectrum
Discrete spectrum
Mean-field single-particle spectrum
Smoothed
Discrete
i
15
F
 gd
g(E)
i
N
g~    
20
g     i 
N
10
F
5

0
 gd
   i 
    i 
f 
 g 
~F
~
 g  d


E Discrete 
Smoothed spectrum
~F
f x   L x e  i 

20
40
60
80
E (MeV)
Shell  E Discrete  E Strutinsky
16

g2
is a modified Gaussian
g 
EStrutinsky   g~  d
0
2
2 g
Nilsson-Strutinsky and deformation
•
Energy surfaces as a function of deformation
From C.G. Andersson,et al., Nucl. Phys. A268, 205 (1976)
Nilsson-Strutinsky is a mean-field type approach that
allows for a comprehensive study of nuclear deformation
under rotation and at high temperature
17
Nuclear masses
Pairing corrections to the liquid drop
•
1 2
Pairing:  Pair Z,N, A  12A ;
= 0;
even - even(odd - odd)
odd - even(even - odd)

 n  14 {BE N  2,Z   3BE N 1,Z 
+ 3BE N,Z   BE N  1,Z }
 p  14 {BE N,Z  2  3BE N,Z 1
+ 3BE N,Z   BE N,Z  1}
From A. Bohr and B.R. Mottleson,
Nuclear Structure, vol. 1, p. 170
Benjamin, 1969, New York
18
How well do mass formulae work?
•
Most formulae reproduce the known masses at the level of ~ 600 keV
heavier nuclei, and ~1 MeV for light nuclei
Mass predictions for Sn isotopes
and experiment
Predictions towards
the neutron-drip line
tend to diverge!
Why?
19
How well do mass formulae work?
•
Where is the neutron drip line?
What is the limit of stability?
What do we mean by a drip-line?
The neutron separation energy:
The mass
models don’t
agree as we go
away from
known masses!
S n  BE ( Z , N )  BE ( Z , N  1)
If Sn > 0, unbound to neutron
emission
The drip line is the point where all
nuclei with more neutrons have
Sn > 0
The location of
the neutron dripline in rather
uncertain!
20
Mass Formulae
Homework:
•
Use the Bethe-Weizsäcker fromula to calculate masses, and determine
the line of stability (ignore pairing and shell corrections)
— Show that the most stable Z0 value is
Z0 
A 2  M n  M p A 8aSymm  aCoul A 2 3 8aSymm
1
1
4
a
Coul
aSymm A 2 3
More advanced homework:

•
Assume symmetric fission and calculate the energy released.
Approximately at what A value is the energy released > 0?
21
What are the forces in nuclei?
•
Srart with the simplest case: Two nucleons
— NN-scattering
— The deuteron
•
From these we infer the form of the nucleon-nucleon interaction
— The starting point is, of course, the Yukawa hypothesis of meson exchange
– Pion, rho, sigma, two pion, etc.
•
However, it is also largely phenomenological
— Deuteron binding energy: 2.224 MeV
— Deuteron quadrupole moment: 0.282 fm2
— Scattering lengths and ranges for pp, nn, and analog pn channels
a pp  17.3  0.4 fm
ann  18.8  0.3 fm
a pn  23.75  0.01 fm
rpp  2.85  0.04 fm
rnn  2.75  0.11 fm
rpn  2.75  0.05 fm
– Unbound!
— Note that Vpp  Vnn  Vpn
•
Some of the most salient features are the Tensor force and a strong
repulsive core at short distances
22
NN-interactions
•
Argonne potentials
— R.B. Wiringa, V.G.J. Stoks, R. Schiavilla, PRC51, 38 (1995)
— Coulomb + One pion exchange + intermediate- and short-range
•
Bonn potential
— R. Machleidt, PRC63, 024001 (2001)
— Based on meson-exchange
— Non-local
•
,r,,s1,s2
Effective field theory
— C. Ordóñez, L. Ray, U. van Kolck, PRC53, 2086 (1996); E. Epelbaoum, W.
Glöckle, Ulf-G. Meißner, NPA637, 107 (1998)
— Based on Chiral Lagrangians
— Expansion in momentum relative to a cutoff parameter (~ 1 GeV)
— Generally has a soft core
•
All are designed to reproduce the deuteron and NN-scattering
23
NN-interactions
•
Pion exchange is an integral part of NN-interactions
— Elastic scattering in momentum space
g2 s1  qs1  q
Vlocal q  k k  
4 M 2 q2  m2
NN

— Or, through a Fourier transform, coordinate space ( 

 m c )
 e  r

g2 1 
3
3 
3s i  rˆs i  rˆ  s i  s j 
V 
m s i  s j  1 

2 
2
4M 3 
 r r  
 r
Tensor operator
•
Off-shell component present in the Bonn potentials
V

NN
g2 E  M E  M  s1  k s 2  k   s 2  k s1  k 



 

k,k   2
4 M k k2  m2 E  M E  M  E  M E  M 
— Non-local (depends on the energies of the initial and final states)
— Plays a role in many-body applications and provides more binding
24
Three-Nucleon interactions
•
First evidence for three-nucleon forces comes from exact calculations
for t and 3He
— Two-nucleon interactions under bind
– Note CD-Bonn has a little more binding due to non-local terms
•
Further evidence is provided by ab initio calculations for 10B
— NN-interactions give the wrong ground-state spin!
– More on this later
•
Tucson-Melbourne
— S.A. Coon and M.T. Peña,
PRC48, 2559 (1993)
— Based on two-pion exchange
and intermediate ’s
— The exact form of NNN is not
known

There is mounting evidence that threebody forces are very important
25
Isospin
•
Isospin is a spectroscopic tool that is based on the similarity between
the proton and neutron
— Nearly the same mass, qp=1, qn=0
— Heisenberg introduced a spin-like quantity with the z-component defining
the electric charge
— Protons and neutrons from an isospin doublet
tz  q 
1
2
p 
n 
— Add isospin using angular momentum algebra, e.g., two particles:
– T=1
T  1, Tz  1 
– T=0
T  1, Tz  1 
T  0,Tz  0 
T  1, Tz  0 
1

2


With T=0, symmetry under p  n

26
1

2


Isospin
•
For Z protons and N neutrons
Tz  Z  N
Z N T  Z  N
•
•
•
Even-even N=Z: T=0
N Z: T=Tz
Odd-odd N=Z: T=0 or T=1 (Above A=22, essentially degenerate)
T=1 multiplet
Analog states reached via t =2t
22Mg
22Ne
22Na
T=0 singlet
•
If Vpp=Vnn=Vpn, isospin-multiplets have the same energy and isospin is
a good quantum number
27
Isospin
•
Of course, Vpp  Vnn  Vpn
T=1 multiplet
22Mg
22Na
22Ne
•
T=0 singlet
NN-interaction has scalar, vector, and tensor components in isospin
space
1 pp
v  v nn  vTpn1 

3
v 1  v pp  v nn 
v 0 
v 2  vTpn1  v pp  v nn  2
isoscalar
isovector
isotensor
— Note that the Coulomb interaction contributes to each component and is
the largest!!!
e
2
q
 e 2 1 t z i1 t z  j  4  1 t z i  t z  j  t z it z  j 
2
4
28
Coulomb-displacement energies
T=1 multiplet
22Mg
22Ne
•
22Na
T=0 singlet
We apply the Wigner-Eckart theorem
 T 0 T 
0
BE i,T,Tz   1 
 i,T  H i,T  
Tz 0 Tz 
T Tz  T 1 T 
1

i,T
H
i,T  
1 



Tz 0 Tz 
T Tz  T 2 T 
2
1 
 i,T  H i,T 
Tz 0 Tz 
T Tz
29
Isobaric-Mass-Multiplet Equation
 T 1 T 
T Tz

1

  
T
T
0
 z z 
 T 2 T 
T Tz

1




Tz 0 Tz 
•
Tz
T 2T  1T  1
3Tz2  T T  1
2T 1T 2T  1T  12T  3
The Isobaric-Mass-Multiplet Equation (IMME)

T=1 multiplet
22Mg
22Ne
22Na
Isovector
~ 3-15 MeV (~ A2/3)
BE i, T , Tz   ai  biTz  ciTz2
T=0 singlet
Isoscalar
Dominated by the strong interaction
~ 100’s MeV
30
Isotensor
~200-300 keV
Coulomb-displacement energies
•
Can we use the IMME to predict the proton drip-line?
— Binding energy difference between mirror nuclei
BE (i,Tz  T)  BE (i,Tz  T)  2biT
— In a T-multiplet, often the binding energy of the neutron-rich mirror is
measured
Calculated with theory

BE (i,Tz  T)  BE exp (i,Tz  T)  2biT
Simple estimate from a charged
sphere with radius r=1.2A1/2

3 Ze
EC 
5 R
3 e2 2 3
E C 
A 2T 
5 r0

2
31
30 keV uncertainty
Coulomb-displacement energies
•
Map the proton drip-line up to A=71 using Coulomb displacement with
an error of ~ 100-200 keV on the absolute value
— B.A. Brown, PRC42, 1513 (1991); W.E. Ormand, PRC53, 214 (1996); B.J.
Cole, PRC54, 1240 (1996); W.E. Ormand, PRC55, 2407 (1997); B.A. Brown et
al., PRC65, 045802 (2002)
Use nature nature to
give us the strong
interaction part, i.e.,
the a-coefficient by
adding the Coulombdisplacement to the
experimental binding
energy of the neutronrich mirror
Yes!
Coulomb displacement
energies provide an accurate
method to map the proton drip
line up to A=71
32
Now, lets start to get at the structure of nuclei
•
•
For two particles we use Schrodinger equation
For three and four, we turn to Faddeev and Faddeev-Yakubovsky
formulations
  1   2   3
Jacobi coordinates
1
R  13 ri  rj  rk 
i 
Ti  j   k 
E  H0
2
2
yi 
ri  12 rj  rk 
p
i
3
H0  
1
ri  2 rj  rk 
i 2m i
1
Ti  V jk  V jk
Ti
E

H
0



E  H 0  V23 1  V23 P12P23  P13P23 1
Identical particles

W. Glöcle in Computational Nuclear Physics,
Springer-Verlag, Berlin, 1991
Exact methods exist for A  4
33
Three-particle harmonic oscillator
•
For fun, look at the three-particle harmonic oscillator
2
pi2 1
H 
  mi 2 ri  rj 
2 i j
i 2m i
Jacobi coordinates
R
1
3
r1  r2  r3 
y
2 1
 r1  r2   r3 
3 2

r
1
2
r1  r2 

Show that we can rewrite H as:
 P 2
 pr2 3
  py2 3
2 2
2 2
H    2 m r    2 m y  R
2m
 2m
 2M
M  3m
 n l m r  Rn l rYl m rˆ;
r r
r
r r
r
r
 n l m y   Rn l y Yl m yˆ 
y y
y
y y
L  lr  ly
Finally, we have to couple L to
the spin S and anti-symmetrize

Harmonic oscillators are always a convenient way to
start solving the problem
34
y
y
More than four particles
•
•
•
•
This is where life starts to get very hard!
Why?
Because there are so many degrees of freedom.
What do we do?
— Green’s Function Monte Carlo
exact Oˆ exact
 trial Oˆ e bH trial
 lim
 bHˆ
b  
e
 trial
trial
ˆ
— Coupled-cluster

C ij ai a j C ijkl ai a j ak al 
ijkl
  e ij
ref
— Shell model
– I’ll tell you about this approach

35
Many-body Hamiltonian
•
Start with the many-body Hamiltonian
pi2
H 
 VNN ri  rj 
i 2m
i j
•
Introduce a mean-field U to yield basis
 pi2

 H    U ri  VNN ri  rj  U ri 
2m
 i j
i 
i
0f1p N=4
0d1s N=2
1p N=1
0s N=0
Residual interaction
•
•
The mean field determines the shell structure
In effect, nuclear-structure calculations rely on perturbation
theory
The success of any nuclear structure calculation depends on
the choice of the mean-field basis and the residual interaction!
36
Single-particle wave functions
•
•
With the mean-field, we have the basis for building many-body states
This starts with the single-particle, radial wave functions, defined by
the radial quantum number n, orbital angular momentum l, and zprojection m
 nlm r   Rnl rYlm rˆ 
— Now include the spin wave function:
•
 Ss
1
2 z
Two choices, jj-coupling or ls-coupling

— Ls-coupling
nlms r   nlms r  Ss  Rnl rYlm r  Ss
z
1
2 z
z
1
2 z
— jj-coupling is very convenient when we have a spin-orbit (ls) force


 nljj r   Rnl r Yl rˆ   
z

Yl rˆ    1
S
2

jjz
S
1
2

jjz
  lm 12 sz | jjz Ylm rˆ  1Ssz
2
msz
37
Multiple-particle wave functions
•
•
Total angular momentum, and isospin;  12 s z
Anti-symmetrized, two particle, jj-coupled wave function
T

j1 j 2
JMTTz

 1   2
  n1 l1 j1 r1    n 2 l2 j 2 r2   1
T
T
1
2
1
2
TTz
JM
j1  j 2 J T

r1   n l j r2 
JM
n 2 l2 j 2
1 1 1

21 12 
— Note J+T=odd if the particles occupy the same orbits
•

Anti-symmetrized, two particle, LS-coupled wave function
LS
JMTTz


S JM
L
L 

l1 l 2 L S T
S
S
  n1 l1 r1    n 2 l2 r2   1
 n 2 l2 r1    n1 l1 r2    1 1   1 2 



2
2


 1   2
T
T
1
2
1
2
TTz
21 12 
38
Two-particle wave functions
•
Of course, the two pictures describe the same physics, so there is a
way to connect them
— Recoupling coefficients ( ˆj  2 j 1)
j j
 JMTT
1 2
•

z
l1

ˆ
ˆ
ˆ
ˆ

  j1 j1LSl2
L
LS

1
2
1
2
S
j1 
 LS
j 2  JMTTz
J 

Note that the wave functions have been defined in terms of r1 and r2 ,
but often we need them in terms of the relative coordinate r  r1  r2
— We can do this in two ways
– Transform the operator
1 Fl r1,r2  Y
l
ˆ  Ylarge
ˆ
V rQuadrupole,


1  r2   4   l=2, component
l r1  is
l r2 
2l  1

l
and very important
2l  1
Fl r1,r2  
Pl cos  V r1  r2 sin 12d12

2 0
39
00

Two-particle wave functions in relative coordinate
•
Use Harmonic-oscillator wave functions and decompose in terms of
the relative and center-of-mass coordinates, i.e.,
 
r  r1  r2 ;
•
•
 
R  r1  r2 2
Harmonic oscillator wave functions are a very good approximation to
the single-particle wave functions
We have the useful transformation
 r    r 
L M 
n1 l1
1
n1 l1
1

 M nlNL;n l n l  r    R
L M 
11 2 2
nl
NL
nlNL

—
—
—
—
2n1+l2+2n2+l2=2n+l+2N+L
Where the M(nlNL;n1l1n2l2) is known as the Moshinksy bracket
Note this is where we use the jj to LS coupling transformation
For some detailed applications look in Theory of the Nuclear Shell Model,
R.D. Lawson, (Clarendon Press, Oxford, 1980)
40
Many-particle wave function
•
•
To add more particles, we just continue along the same lines
To build states with good angular momentum, we can bootstrap up
from the two-particle case, being careful to denote the distinct states
— This method uses Coefficients of Fractional Parentage (CFP)


j NJM   j N 1J jJ j NJ
J 
•

j N 1J   j

JM
Or we can make a many-body Slater determinant that has only a
specified Jz and Tz and project J and T

n l
1 n l
i i j i j zi
1,2, , A 
j j
j j jz j
A!
n l
l l j l j zl
r1 n l j j r2 
r1 n l j j r2 
n l
n l
r1 n l j j r2 
n l
i i i zi
j j
j zj
l l l zl
i i j i j zi
j j
j j jz j
l l j l j zl
rA 
rA 
rA 
In general Slater determinants are more convenient

41
Second Quantization
•
•
Second quantization is one of the most useful representations in manybody theory
Creation and annhilation operators
—
—
—
—
Denote |0 as the state with no particles (the vacuum)
a+i creates a particle in state i;
ai 0  i , ai i  0
ai 0  0
ai annhilates a particle in state i; ai i  0 ;
Anticommuntation relations:
ai ,aj  ai,a 
j



a
,a

a
,a
 i j   j i  ij
•
Many-body Slater determinant


 i r1  i r2 
1  j r1   j r2 
 i rA 
 j rA 
A!
 l r1  l r2 
 l rA 
42
 al
a j ai 0
l  j  i
Second Quantization
•
Operators in second-qunatization formalism
— Take any one-body operator O, say quadrupole E2 transition operator
er2Y2, the operator is represented as:
O   j O i aj ai
ij
where j|O|i is the single-particle matrix element of the operator O
— The same formalism exists for any n-body operator, e.g., for the NNinteraction

V
1
ij V kl A ai a j al ak   ij V kl A ai a j al ak

4 ijkl
i j,kl
ij V kl
A
 ij V kl  ij V lk
– Here, I’ve written the two-body matrix element with an implicit assumption
that it is anti-symmetrized, i.e.,
rr    i r ai ai

2
i

43
Second Quantization
•
Matrix elements for Slater determinants
acekl ac am aeklm  0 al ak ae ac aa ac am am al ak ae aa 0
 0 al ak ae ac aa ac al ak ae aa 0  acekl ac aekl
  0 al ak ae ac aa al ak ae ac aa 0   acekl acekl
 1
Second quantization makes the computation of expectation
values for the many-body system simpler
44
Second Quantization
•
Angular momentum tensors
— Creation operators rotate as tensors of rank j
— Not so for annihilation operators
a˜ jm  1
•
j  jz
a jm
Anti-symmetrized, two-body state


1
ja j b : JM,TTz  
a j a 1  a j b 1
2
2
1 ab

45

JM
TTz
0
The mean field
•
One place to start for the mean field is the harmonic oscillator
— Specifically, we add the center-of-mass potential
HCM
2
1
1
m2
2 2
2
 Am R   mri  
ri  rj 
2
2
2A
i
i j
— The Good:

– Provides a convenient basis to build the many-body Slater determinants
– Does not affect the intrinsic motion
– Exact separation between intrinsic and center-of-mass motion
— The Bad:
– Radial behavior is not right for large r
– Provides a confining potential, so all states are effectively bound
46