1
Nuclear Physics Overview & IntroductionLanny Ray, University of Texas at Austin, Fall 2015
I. Nucleon+Nucleon System
II. Nuclear Phenomenology
III. Effective Interaction Theory
IV. Nuclear Structure
V. Nuclear Reactions
VI. Scattering Theory Applications
2
I. The Nucleon + Nucleon System
Topics to be covered include, but are not necessarily limited to:
Quantum numbers, symmetries, the deuteron
One-pion exchange potential
Phenomenological models
Meson exchange potentials
Effective chiral field theory models
Scattering – amplitudes, phase shifts, observables
Relativistic amplitudes
3
Quantum numbers and symmetries:
up
quark
down
quark
spin ½ ; isospin ½; parity = +
I3= ½ I3= -½
Isospin is an observed symmetry among most hadrons, e.g. the similarity in masses
of protons and neutrons, p+,-,0, S+,-,0, X+,-, kaons, etc. and derives from
the near equivalence of mass of the up and down quarks. The flavor
independence of QCD together with the approximate up/down mass equivalence
results in an isospin invariance in the nuclear interaction. Isospin symmetry is a
BIG DEAL in nuclear physics!
(particle physics sign convention)
The lowest energy configuration for the nucleon is zero orbital angular momentum,
spin ½, isospin ½, I3 = +½ for protons and – ½ for neutrons, with parity +.
Mproton = 938.28 MeV; Mneutron = 939.57 MeV where the small mass difference is
due to Coulomb repulsion and u-d quark mass difference.
4
Lowest mass nucleon resonances:
N*(1470): spin-parity ½+; isospin ½
D(1232): spin-parity 3/2+; isospin 3/2
which is a DS =1, DI=1 excitation of the nucleon
Pion:The lowest energy configuration is zero orbital angular
momentum, spin 0, parity = (+)(-)(-1)L = -, isospin 1
Symmetries: The wave function for identical Fermions must be anti-symmetric
and for hadrons (quarks) includes the spatial, spin, isospin (flavor), (and color)
components. For two nucleons or two quarks interchange of labels 1,2 must therefore
change the sign of the wave function. In this course we focus on the wave functions
of nucleons and mesons and will ignore their internal (color & flavor) QCD structures.
odd );1,2()1()2,1()2,1()2,1()2,1(
:issymmetry ion wavefunctNN combined theand
)1,2()1()2,1(
:ispart isospin The
)1,2()1()2,1(
:ispart spin The
)1,2()1()2,1(
:ispart spatial The
1
1
=++-==
+
-=
-=
-=
++
+
+
ISLISL
isospinspinspatial
isospin
I
isospin
spin
S
spin
spatial
L
spatial
5
The deuteron – the only nucleon+nucleon bound state
N
P
Orbital ang. mom = 0
Spin = 1
Parity = + (even)
Jp = 1+
Isospin = 0
(-1)L+S+I = -1
B.E. = 2.226 MeV
Perhaps the reason there is no di-proton bound state is that the Coulomb
replusion overcomes the nuclear attraction. If so then why isn’t there a
bound di-neutron?
Next we will derive the nuclear potential between two nucleons due to
the exchange of one pion. We will see that even this simple exchange
leads to spin & isospin dependent forces. When these interactions are
combined from many meson exchanges we will see what accounts for
the absence of a di-neutron in Nature.
6
Spin & Isospin review
( )
( )
( )
triplet1 S 1,
3)1(2
singlet 0S ,3
2/)1()1(2/)1(
element matrix theand
,,1,1
,,,,0,1
,,,,0,0
1(triplet)or 0(singlet)spin total tocouple and spins
|,
1
0,
0
1 rep.matrix in or ;,, :functions statespin
2 ; :spins 2 Combining
matrices Pauli theare ,, where
21
,,432
,,21
212
222
121
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
,
221
121
221
121
21
21
21
2
2
2
1
2
21
21
21
=
=-+=
=-
-+=+-+
-=
=
-+-=
---=
=
=
++=+=
=
SS
SSSS
SMsSMSSMssMS
mmSMmmMS
ms
ssssSssS
s
SSSS
ss
MMSSMMSS
SSSS
mm
ssSssS
s
zyx
7
Spin & Isospin review
( )
( )
( )
IIII
tt
MMIIMMII
IIII
mm
ttIttI
t
zyx
IIII
IMtIMIIMttMI
mmIMmmMI
mt
ttttIttTI
t
-+=+-+
-=
=
-+-=
---=
=
=
++=+=
=
,,43
,,21
21
222
121
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
,
221
121
221
121
21
21
21
2
2
2
1
2
21
21
2/)1()1(2/)1(
element matrix theand
,,1,1
,,,,0,1
,,,,0,0
triplet)-1(isoor singlet)-0(isoisospin total tocouple and isospins
|,
1
0,
0
1 rep.matrix in or ;,, :functions stateisospin
2 ;)(or :isospins 2 Combining
matrices Pauli theare ,, where
:is physicsnuclear
innotation usual the wherequantities analogous define isospin weFor
21
8
Nuclear interaction invariances
The nuclear force is invariant wrt to:
spatial rotation – total angular momentum conservation, but
orbital angular momentum conservation
spatial reflection – conservation of parity
time reversal
identical particle exchange – by including the isospin d.o.f. protons
and neutrons are treated as identical fermions (EM
effects are the exception) and hence the wave
function must be antisymmetric wrt interchange of
particle labels.
JJssLLppr
JSLpprr
----
--
; ;; unchanged; is reversal-Under time
unchanged are ; ;operation parity Under ,,
9
One-Pion Exchange Potential (OPEP):
( )
( )
(1996)]. 2086 53, C Rev. Phys Kolck, van Ray, Ordonez, [see
theories.field chiral effective from derived that of version simplified a is Lagrangian This
spinor.nucleon component -2 theis and below), (see componentsr upper/lowe theoftion representa Pauli
theusing 2/ whereand 14,/relation Treiman -Goldberger thefrom
obtained is strength coupling where : termcouplingisovector scalar,-pseudo thefrom
follows n terminteractio The s.coordinate NN relativeon acts operator,spin nucleon theis
const.); coupling (axial 25.1 constant);decay (pion MeV 1322
operator,isospin nucleon theis fields,nucleon theare fields,pion 1I thedenote or where
22
1
:Lagrangianpion &nucleon order -lowest a with Starting
5
5
222
1
p
p
pp
pppp
pp
pp
pp
p
p
N
mkiNiNfmgg
gNiNg
N
gff
N
NNf
gNmiNm
NNANN
NN
aa
NN
A
a
aaAN
=
-+
===
=
+
--+-=L
N1
N2
p q
( ) ( )
.limit static In the .1set weand where
)(
)(2
)(2
222
0
2
2112
112222
qqqqiq
NNqVNNi
NqiNf
g
imq
iNqiN
f
g
rel
OPEP
bbAaaA
--==-=
-=
-
+-
-= p
p
ppp
M
10
( )( )( )
( )
( )( )
( )( )
( )( )( )( )
( )( )( )
( )( )
( ) ( )( )
( )
( )( )
( )
ignored. is range)-(zeropart function -delta thensapplicatioIn
)ˆ)(ˆ(3 :operator tensor thedefineswhich
)(3
4331
212
)(3
4331)ˆ)(ˆ(3
212)(
Finally,
)(3
4331ˆˆ3
3
give out to worked
becan sderivative theˆ and ),(4)/1( that Recalling
42
22
22
)(2
)(
space coordinateIn
2)(
Therefore, .element matrix fieldpion Also,
212112
3
212121221
2 2
3
21212212121
2 2
3
2
2
32
2121
2
223
3
2121
2
22
21
3
3
21
2
3
3
22
2121
2
p
p
p
p
p
p
p
p
p
p
pp
p
p
pp
p
ppp
p
ppp
p
pp
p
p
pp
pp
pp
-
-
+
++
=
-
+
++-
=
-
+
++-=
=
-=
=
+
=
+
-=
=
+
-=
=
-
-
--
-
rrS
rr
e
rmrmS
f
gm
rr
e
rmrmrr
f
gmrV
rrmrm
rrr
em
r
e
xx
rr
x
x
rrr
r
e
f
g
mq
eqd
f
g
mq
qqe
qd
f
g
qVeqd
rV
mq
f
gqV
rm
A
rm
AOPEP
ijijijji
rmrm
ji
j
j
rm
A
rqi
A
rqiA
OPEP
rqi
OPEP
AOPEP
ab
ba
Work out
this integral
Show
11
N+N phenomenological potentials
The earliest idea for the nuclear force originated with Hideki Yukawa’s paper in
Proc. Phys. Math. Soc. Japan 17, 48 (1935) which showed that the exchange of a
massive, spin 0 particle (meson) would generate an exponential potential of the
form exp(-mr)/r and that a mass of about 100 MeV would do the job. The discovery
of the muon in 1937 caused many to believe that the muon was the carrier of the
nuclear interaction which turned out to be wrong. The pi-meson or pion was not
discovered until 1947.
Np N
mass large range-short ,/
// 2
2
D
DD
D
mctcR
mcEt
mcE
12
Generally, the early phenomenological NN potential models included a minimum
number of spin-dependent terms which could account for the existence of a deuteron
But no di-neutron, and the limited scattering data, e.g. central, spin-orbit and tensor,
and they may or may not have included the theoretical OPEP. They only described the
deuteron properties (B.E., magnetic dipole moment, electric quadrupole moment, d-state
fraction) and N+N scattering data (scattering lengths and phase shifts) up to about 350 MeV
lab collision energy where single pion production begins, the inelastic scattering threshold.
An early, accurate model was introduced by R. Reid, Ann. Phys. (N.Y.) 50, 411 (1968).
It is often still used as a bench mark test for codes because it is relatively simple and
is local, ie. V = V(r) with no explicit momentum dependence.
Starting the 1970s meson exchange based theoretical models appeared and I will
summarize three – the Paris, Bonn and Nijmegen models.
Then in the 80s-90s effective chiral symmetry based models started appearing; I will
summarize the one I had the privilege of working on with Weinberg’s student and post-doc.
N+N phenomenological potentials
Show
13
N+N phenomenological potentials
Empirical knowledge:
the nuclear force depends on everything it can as allowed by the
underlying symmetries of QCD.
it is short range, ~few fm (10-15m)
N+N cross sections are ~4 fm2 = 40 mb
nuclear forces are very strongly repulsive at short distances
less than the proton radius, ~ 0.7 fm
only p+n forms a bound state and it is I=0, Jp = 1+
Recent review article:
R. Machleidt and D. R. Entem, Phys. Rep. 503, 1-75 (2011)
14
N+N phenomenological potentials
3S1 phase shift1S0 phase shift
Potential appears attractive at
lower energies, but becomes
repulsive at higher energies;
the increasing p.s. at low energy
also indicates a bound or nearly
bound state.
E
I3
-1
n+n
0
p+n
1
p+p
I=1
I=0(-2.2 MeV)
Few MeV unbound
Low-lying N+N states
15
N+N phenomenological potentials
Consider a spherical square well
with a weakly bound s-wave state:
-=-
=
=-===
-
-
-
)cot()sin(
)cos( :Rat Matching
/2;/)(2 ;)( );sin()( 22
0
kRke
e
kR
kRk
EVEkeRrukrRru
R
R
BB
r
Now consider s-wave scattering from this potential at low kinetic energy E0 where:
( )
1for cot
221
cot
cot
gives for solving and sin
cos
gives Rrat matching and /)(2
where)( and sin)(
are functions wave-s the where/2
0
02
0
2
00
2
00
+
+-
-
+=
-
+=
=-=
-==
=
-
-
-
-
RkRkk
ikRik
ikRkk
ikRkkeS
SeSe
eSeik
Rk
Rkk
VEk
eSeRrurkRru
Ek
oo
o
Rik
RikRik
RikRik
rikrik
o
oo
oo
oo
r
V(r)
0
V0R
EB
E0
. where01 approx. is state bound for the which cot
1
is slope theand integer,any is m where, cot
1
is state bound weakly for theshift phaseenergy low the
,by shift phase the torelatedmatrix -S theis
00
2
0
kkRRRkkk
RRkk
km
eSS
k
o
i
---
=
-
+
=
+
p
16
N+N phenomenological potentials
RkRki
RRkkk
eSeRru
VEkrkRru
kiikkrkkr
krk
ikEkeRru
VEk
kukr
krRruRr
reerere
iEE
eigenstateenergycomplex
B
B
rikrik
BB
BBB
rik
B
tEittiEitiE
B
B
B
/1 and 0~~
if 01
~1
cot
1
get weabove as steps same thefollowing and
)(
/)(2 ,sin)(
:are solutions scattering The
~cot
sin
cos
is Rat b.c. The
~/2 where)(
bygiven state, decaying a toingcorrespond wave
outgoingan bemust solution external The ./)(2 where
complex is However, .0)0( b.c. by the eliminated is )cos(
irregular the where)sin()( is for w.f.radial The
decay. lexponentia ),()()(
isfunction wave theof dependence-time
thesuch that ~
where(1968)] 265 109,A Phys. Nucl.
Berggren, [see state bound-antior a
complex, is eigenstateenergy then theunboundslightly is state theIf
0
2
00
2
2
0
/~
//)~
(/
00
-+
-
=
-=
-==
+===
-===
-=
=
=
===
-=
-
-----
r
V(r)
0
V0R
EB
E0
Re
Im
EB
kB
complex energy &
momentum plane
17
N+N phenomenological potentials
0=
E (lab K.E.)
0
Increases for
attractive V(r)
For weakly bound state
For unbound state close to zero
(depends on details of potential)
18
Reid Soft-Core NN Potential (1968)
OPEP + empirical sum of Yukawa potentials in the form:
Notation for N+N states:
)()()()( rVrVrVrV TLSC ++=
( ) S
MM
LJSLJ
J
S
SMLMJMSMLMMJ
SLJ
hgfdpsLSL
SL
=
+=
==+
,
12
|,
),,,,,(2,1,0 ;1,0 ;
Clebsch-Gordan coefficient
19
Reid soft-core NN potential in MeV: Isospin = 1
reversal-Time and
Parityunder invariant are
and that Show 12 SLS
20
Reid soft-core NN potential in MeV: Isospin = 0
21
Reid soft-core and hard-core deuteron radial
wave functions for L = 0 and 2
s state
d state
22
For low energy p+p, n+n
states in L = 0, I = 1, S = 0
This is the only nuclear interaction
for the 1S0, I=1 state.
It is insufficient to bind p+p or n+n.
Far too weak
to bind p+n
The tensor potential is
non-central (deformed)
and couples |L-L’|=2 states.
It’s strong attraction, acting
through the L=2 p+n state
(3D1) allows the p+n
to have a stable bound state – deuteron!
p, 2p, w exchange
plus empirical terms
Paris NN Potential
For low energy p+n states in
L=0, I = 0, S = 1 (3S1) the central
and tensor potentials are:
23
Paris NN Potential
24
Paris NN Potential
25
Bonn NN Potential
In Advances in Nuclear
Physics, Vol. 19, p. 189,
(1989).
( )
w
,)2/(
,, e.g. ector, v
,4
meson-sigma e.g. scalar, ,
pions e.g. scalar,-pseudo ,
nsinteractio Lagrangian
vvvv
v
v
int
s
s
s
int
ps5
ps
ps
int
i
M
fg
g
ig
N
=
---=
=
-=
L
L
LMeson exchanges included
26
Bonn NN Potential
Pseudo-scalar
interaction
Tensor
interaction
27
Bonn NN Potential
28
Bonn NN Potential
29
Relevant mesons for N+N interactions
30
Nijmegen NN potential
The model includes p,h,h‘,,w,,,,S* and the J=0 parts of the Pomeron,
f, f’ and A2 mesons.
Spin operators
31
Nijmegen NN potential
32
Nijmegen NN potential
33
Nijmegen NN potential
34
An example of modern N-N interaction models based on QCD
using effective field theories as pioneered by S. Weinberg in the 70’s
Effective Chiral LagrangiansOrdonez, Ray, van Kolck, Phys. Rev. C 53, 2086 (1996)
(low energy modes of QCD – pion, nucleon, D resonance fields)
vertex. theentering lines (nucleon)fermion ofnumber theis and sderivative ofnumber theis where
2 :index vertex counting-power in the n termsinteractio Expand 2
fd
d f -+=DL
35
Effective chiral field theory model
36
Effective chiral field theory model
37
The total list of spin-isospin operators in the model:
Effective chiral field theory model
Isospin and reversal-TimeParity, conserve operators all that Show
38
Effective chiral field theory model
the whole enchilada
39
Effective chiral field theory model
The 25 fitting parameters of the model
pND coupling
pN derivative
couplings
NN contact
interaction
effective
couplings
40
Effective chiral field theory model
41
Scattering: Schrodinger eq., boundary conditions, phase shifts, scattering
amplitudes, observables
First, consider the scattering of two neutral, spin 0 particles which interact by a
spherically symmetric, finite range potential V(r):
Solve the Sch.Eq.
numerically
in this region
Match to asymptotic
boundary conditions
at any large r where
V(r) vanishes
42
43
Beam (particles/area/time)
Target
(nuclei/area)
Beam area AB
solid angle
subtended
by detector:
DW=AD/R2
detector
acceptance
area AD
Scattering definitions for the
Differential Cross Section
( ) angle. area/solid of units hasch whi
beam areaper nuclei target ofNumber /particles/ beam ofNumber
/particles/ detected ofNumber
BB AtA
t
d
d
D
DDW=
W
In nuclear physics typical units are mb/sr (milli-barns/steradian) where
1 barn = 10-24 cm2; 1 mb = 10-27 cm2 = 0.1 fm2
44
Scattering observables from the asymptotic incoming and scattered wave functions
ikze :wave
plane incoming Z axis
r
ef
ikr
)( wavespherical
outgoing modulated
( )
2
frame CMin scattering elastic
2
Theory
222
)()(
/)( rate particle Detected ; intensity Beam
ffv
v
d
d
t
tvRref
tA
tvAe
in
out
outR
ikr
B
inB
ikz
=W
DDW
DDW=
D
D=
We must solve the Schrodinger eq., match to the above boundary conditions
to obtain f(), and then we can directly compare to the experimental d/dW.
45
Solving the Schrodinger Eq. for spin 0, neutral particles scattering from a
spherically symmetric potential
( )
b.c. tomatchingout when cancels valueassumed its and
arbitrary is )( of valueThe .conditionsboundary tomatched becan w.f.radial thewhere
position any to0 from starting outward iterated becan method)(Stormer equation This
)()()()(2)2(
thatexpansionsTaylor usingshown becan it size step numerical a Adopting
.0)0( that require we allat )( finiteFor
)1(
2)(
2)(
where)()()( :form in the iswhich
0)()()1(
2
Sch.Eq. theinto substitute and )dependence azimuthal (no )()(
)( :Expand
mass reducedbody -2 theis and K.E. theis 2
where; 0)()(2
frame, CMbody -2 in the be to take which weEq.r Schrodinge the,)(
42
2
2
2
22
22
0
0
2222
hu
r
hhrurvhruhruhru
h
urr
rrVErv
rurvru
ruErVrdr
d
Yr
rur
kErErV
EVT
=
++++-++
=
+---=
=
=
+-
+-
=
==
-+
-
=+
=
O
We have in mind potentials that are too strong to allow perturbation expansions but
require the S.E. to be solved. In general, analytic solutions are not available. Solutions
for the phase shifts, scattering amplitude and diff. Xsec must be obtained numerically.
Show
Partial Wave
Expansion
46
Boundary conditions at large r, i.e. well outside the short-range nuclear potential:
( )
( )
( )
( )
b.c. theofpart radial thelikelook tostarting is which sin)2/sin(
ctadd/subtraan are termsmiddle the where,2
real. is scattering elasticfor but complex becan shift phase where, 2
)(
written,is form alconvention
The ).(by affected be could e)undetecabl is phase (overall wavesspherical two theof phase relative
and amplitudes The . largeat solutions w.f. thefrom follows which structure waveoutgoing & incoming the
support tocontinue but to altered be to w.f.radial asymptotic expect the should present we is )(When
2/cos)(
isally asymptotic which )()(function irregular an
define also We./)cos()( e.g. ,0at )(irregular diverge which )( functionsNeumann spherical
theare solutionsother the0 When solutions.t independen two,supports thereforeandorder -2nd is Sch.Eq. The
phase. relativedependent -an with wavesspherical outgoing and incoming of sum a as waveplane therepresents This
22
12/sin)(
then),()( define and ;2/sin1
)(
poly. Legendre are functions, Bessel spherical are where),()()12( :Expand
CMbody - twoin the /2 where; ),(
)2/(
)2/(2)2/()2/()2/(
22/
0
2/)2/()2/(
0
p
pppp
pp
pppp
p
p
p
p
-
-----
--
--+--
=
+-=
-+-=
-
-
-=
-==
=
-=-=-
=-
+=
=+
krii
kriikrikrikri
ikriiikri
kr
kr
ikriikrikrikri
kr
kr
ikz
ikrikz
r
eekrA
eeeeeiA
eeeeeiA
ru
rV
r
rV
krkrG
krkrnkrG
krkrkrnrkrn
V
eeeei
eei
krkrF
krkrjkrFkrkr
krj
PjPkrjie
Ekr
ekfer
47
( )
( ) ( ) ( )
( )( )
. wrt sderivativefirst denote primes where
get then and for Solve )()()(
)()()(
)(
)(
via)continuous and(smooth conditionsboundary these tomatched becan solution numerical The
)2/(1sin where, )()()(
)2/sin()2/cos(sin)2/sin(sin)2/sin()(
function waveradial theof form asymptoticearlier the toReturning
data. Xsec fit the toadjustedfreely are shifts phase the where),(*
1 where,)(sin)12(1
),(
as or ,modulation wavespherical outgoing heidentify t we
where, ),()(sin)12(1
)()()12(),(
yields which /)12(4 requirespart waveplane theparticularin , b.c. match the toat )( Requiring
)(sin)2/sin(4
12),(
).(for contributenot do in the zero-non where, )()(
4
12)(
)(),()(
givesexpansion function wave theinto above thengSubstituti
,
,
2
)2/(
2
2/2/2/
0
2/
0
)2/(
00
0
r
CkRiFkRGCkRF
kRFikRGCkRF
Ru
Ru
ieeCkriFkrGCkrFA
krikrekrAeekrAru
kfffd
d
eeeiPek
kf
amplitudescattering
r
ekfe
r
ePeei
kPkrjir
kiAr
Peekrr
Ar
rVemPr
ruY
r
rurr
mmm
mmm
mnum
mnum
ii
ikrii
r
iiii
ikrikz
ikrii
kr
krii
kr
im
pppp
p
pp
p
p
ppp
p
p
++
++=
-=++=
-+-+-=+-
==W
==+=
+=
+++
+=
+-+
+===
-
--
=
-
=
-
=
=
Boundary conditions at large r, i.e. well outside the short-range nuclear potential:
48
Consider scattering of two, spin 0 charged particles with combined
short-range nuclear interaction plus a Coulomb interaction. The Coulomb
interaction is infinite range, so the asymptotic waves become Coulomb
distorted plane waves and Coulomb distorted spherical waves:
Point-like Coulomb:
VC(r)=Z1Z2e2/r
Coulomb potential
for finite charge
distribution
Solve Sch.Eq. numerically
With potential VN+VC
Match numerical w.f. to
regular and irregular Coulomb
wave functions which are
solutions of the radial Sch.Eq.
With VC(r)=Z1Z2e2/r
Next, include the Coulomb interaction:
49
( )
( )
( )( )( )( )
( ) ( )
( )
( )
( )
=
+-
---
=
-
+-
+
++=
=++=
-=
+
+
--=
++-=
++- ++=
=
++=
++
++=
+-
+-
0
2
,
,
2))2/(ln(sin
2
))2ln((2
))ln((
0
0
)2/(
)2/(
22
21
1
)(sin121
)(),(
amplitude scattering complete theand
singet can which wefrom )()()()(
tomatched bemust )(function waveradial integratedy numericall thenscalculatio do To
)2/(sin2)(
amp. scattering Coulombwith
wavespherical outgoing distorted Coulomb waveplaneincident distorted Coulomb a of form thehaswhich
,multiplied is phase Coulomb when the,)(1
)(1)(
)(
gives wavespartial
over Summing wave.spherical outgoing and waveplaneincident an tomatchingsolution scattering a itself is
)(solution fact thein and phase Coulomb the todue waveplaneincident an of form in the quitenot is This
sin2/sin
2/sin)()()(|)(
is potentialnuclear theoutside w.f.radial theof form general The
.parameter) d(Sommerfel ,)/(e
andfunction Gammacomplex theis , 1
arctan1Re
1Imarctan
shift phase Coulomb theis ,2/cos)(
2/sin)(
:allyAsymptotic
02
h
h
p
p
h
h
h
h
p
p
h
hh
p
p
Peek
fkf
eCkriFkrGCekrFeRu
ru
ek
f
efrzrik
eYr
krFe
krF
eeekr
eCkrkriFkrGCkrFru
kZZ
i
i
krkrG
krkrF
Cii
C
CiCCCCiCi
mnum
num
ii
C
krkri
C
zrkziCi
C
kriCii
kriC
r
CCCC
Rr
r
C
r
C
C
C
C
nuc
50
Next, include spin, e.g. a spin ½ particle scattering from a potential,
or a proton scattering from a Jp = 0+ nucleus
( ) ( )
( ) frame. reference z)y,(x,
coordinate in the plane scattering ˆ,ˆ,ˆ theof rotations
underinvariant bemust and structurespin 22 thehaving
whilescoordinate momentum on these dependonly can
. matrices Pauli by the drepresente are ...spin theand
ˆ ;ˆ ;ˆ
:vectors
unit theare quantities kinematic region the asymptotic In the .symmetries from determined is of form possible The
. scattering theis where,
form thehassolution full theand state mixed ain bemay waveincoming thegeneralIn
)(
)(
where, incomingfor similarly and ,components , of mixture a becan wavespherical outgoing then the
statespin in the is waveplane incoming theIf . , where, states basis 1/2spin theDefine
operator.spin with theformed becan t scalar tha trivial-nononly theis which )()()(
2221
1211
i
22212122111112121i
2
1
21
2212
2111
21
21
spinwith
PNK
fod
kk
kkP
kk
kkN
kk
kkK
matrixff
ff
r
ee
r
efcfcfcfcecccc
c
ccc
r
effe
r
effe
SLrVrVrV
ikr
ii
ikz
ikrikz
i
ikrikz
ikrikz
zz
socentral
+
+=
=
-
-=
=+
++++++=
=+=
++
++
-==
+
M
M
MM
x
y
z
k
k
kk +kk
51
Spin ½ particle scattering from a spin-dependent potential
( ) ( )
-=
-=+-=
+
+=
-
-
-
-
-
-
-
fige
igef
e
eiN
Ngf
N
P
N
K
P
N
K
P
N
K
P
N
K
NPK
s
s
s
s
i
i
i
i
ysxs
s
revtimeparity
M
1M
1
1
and 0
0cossinˆ
thenaxis- x the torelative
angle azimuthalan makes plane scattering theand axis-z towards
directed is beam theIf .ˆ :ismatrix scattering theof form
generalmost theand symmetries required esatisfy th ˆ and Only
ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
and ˆ
ˆ
ˆ
ˆ
ˆ
ˆ
reversal.- timeandparity under invariant be alsomust matrix scattering The
ˆ ,ˆ ,ˆ ,matrix)(unit
:bygiven vectors,above theof nscombinatioscalar from dconstructe beonly can
matrix scattering therotations coordinateunder invariant be To
Show
52
Spin ½ particle scattering from a spin-dependent potential
++=
++=
++
+
+=
+-=+
+-
+=
=+
++
+=
-+=
+=
==
-
-
--
)()()()(
)()(
)()(
0
)(2
)(
10
)(
)(2
)(
10
)(
2
1,0
1
2
0
2
1
1,
2
0
0
)()(14
as expanded becan finally and
1)(4
as expressed becan waveplane incoming The
1212
1
inversely and
)1(2
where, 12
1
12
2 where,
1212
1
are state for the solutions The . and both of seigenstate are
nscombinatio thesuch that and of mixturesin expand toneed weSo
2)1(
2
similarly and )1(2
whereoperator theof eigenstate and
not is w.f. thisbecause troubleintoget we where)(
before, did weas function wave theexpand weIf . and amplitudes
scattering and shifts phase get the and matching expansion, wavepartial thedo Next,
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
21
bbaa
baikz
ba
bbb
aaa
ruruikr
krFikr
e
Y
SLYY
SLYY
SLL
YY
YYYSL
YYSL
SL
Yr
ru
gf
YY
YY
YY
YYY
YYY
p
p
Ref. Rodberg and Thaler,
“Introduction to The Quantum Theory of
Scattering,” (1967), Ch.11, pg. 278-285.
53
Spin ½ particle scattering from a spin-dependent potential
( ) resp. , and tocorrespond and solutions where|
: toequal are and with ,,, of ionseigenfunct are and that noteFinally
.0at cos
)(cossin
1)1(
12
41
and, before as sin where)1(1
:are 0 angle plane scattering with , amplitudes scatteringdependent -spin andt independen-spin The
matrix scattering theof definition above theusing
sinsin)1(12
41
sinsin)1(12
41)(
gives for expansions above thengsubstituti and
sinsin14
)(
givesexpansion wavepartial theinto ngSubstituti
sin)2/sin()(
wheresin yield which sderivative-log the viaRrat b.c. thematch toBoth
0)()()1()(2
)1(
2
0)()()(2
)1(
2
SchEq. radial thegives seigenvalue theinserting andexpansion thengSubstituti
21
21)()(
21
21222)()(
0
)()(
0
1
)()(
12
),(),()()(
11
1211
1
)()(
0
)()(
),(
)()()()(
)2/(),(),(
),(),(
)(22
12
2
2
22
)(22
12
2
2
22
21
),(
21
21
21
21
)()(
21
)()(
21
21
)()(
21
21
),(
),(
-=+=
=
=--=-++
==-
=++==
=
++=
-++
+
+++
+
+++
+-
=
=
+-++
+-
=
+++
+-
-
-
-
JJYJMMLMY
MSLJJ
d
dPCC
kYCC
kfig
eCPCCk
ff
gf
r
effe
Yeekr
e
Yeekr
eer
eekr
eer
eekrru
eC
rurVrVrdr
d
rurVrVrdr
d
ba
MM
MLMSL
M
JL
z
ba
baba
baibaba
ikrikz
biaiikr
biaiikr
ikz
r
ba
bbiaaiikr
ikz
r
kribai
r
ba
baiba
b
socent
a
socent
SL
SL
ba
ba
ba
ba
ba
ba
YY
YY
Y
YY
p
p
p
p
p
p
M
Show
54
We are almost done, but we still need to see how to calculate observables from
the scattering amplitudes f, g. For this we need to introduce the spin density
matrix formalism which makes calculating any and all observables a breeze.
( )( )( ) ( )( )( )
( ) ( )
==
=
=
===
=====
===
===
=
===
2
2
*
12
*
21
2
1*
212
1
,
,,,
2
ii
i
ˆ 1/2spin for and
0
1
0
* 010
exampleFor . theis ˆ whereˆ and
states basis theusingelement matrix theis ˆ whereˆˆˆˆ
ˆˆˆˆˆˆ
get wematricesunit insertingby and 1 where statespin arbitrary an For
matrix)(unit , and 0), are others all 1, row (
0
1
0
example,for rep.matrix in wherestates basisspin
thebe Let . of rep.matrix transpose-complexor adjoint, theis where
ˆˆˆ where statespin and ˆoperator spin aConsider
ccc
cccccccc
c
matrixdensityspin
AAATrATrA
AAAAAA
cc
i
AAAAA
jiij
jijiij
jiij
ji
ijji
ji
jiij
ji
jjiijj
ji
ii
i
ii
i
i
i
iiijj
th
11
1
ith column jth row
Spin Density Matrix
55
Spin Density Matrix
The spin density matrix has the same dimensions as the spin operators, (2s+1).
It should be possible to express the former in terms of the latter,
( ) ( )
states.spin ingcorrespond with on polarizatiarbitrary produces operatorsrotation unitary using system
coordinate theRotating .0over averaging and 8.50 ,2.0 and 4.0,6.0 case In this
device.Gerlach -Stern a from in 40% and in 60%contain may sample prepared a example,For .
is which 0 randomover Averaging ed.undetermin is but ,4/ requires which ,0 and
Therefore . statein half and statein be willhalf particles, spin of sample unpreparedan For
).2cos( ),2sin()2sin( ),2cos()2sin( wherecos ,sin :eParametriz
polarized? 50%or d,unpolarize is particles of sampleor beam, asay that mean toit does what So
.1 and ,Im2 ,Re2 , statespin in particle oneFor
ˆ
0spin For
spin for quantity vector a is which theis where, ,
ˆ)12(ˆˆ
,)12(ˆˆˆˆ such that normal-ortho andHermitian are opsspin the,,,For
ˆˆˆˆ Trace, the takeand ˆleft with thefrom Operate
,, matrices Pauli unit thee.g. operator,spin theis ˆ andt coefficien a is whereˆˆ Expand
o2
2
2
1
21
2
2
2
121
21
2222
2
2
12
*
12
*
1
2
1
21
21
0
21
21
21
21
21
21
0
21
21
vectors
PPPcc
dunpolarize
PPP
cc
PPPecec
PPPccPccPccPc
c
PP
onpolarizatiPPAA
SAsSTr
ssSSTrSSTr
SSTrASTrS
iSASA
yxz
yxz
zyx
ii
zyxzyx
i
ii
iii
jjj
ijijijzyx
i
ijijj
zyx
th
ii
i
ii
======
====
==
-=-====
=++-===
=
+=
+=
====
=+=
=+==
=
+=
-
p
11
1
1
56
Spin Observables
Now let’s use this fancy machinery to calculate observables. For spin ½ + spin 0
systems recall that the differential cross section is (see slide 50):
( )( )
( )( )
( )
W+W
W-W=
W-W==
+++=W
+=
+++=
=+++===W
dddd
dddd
PP
ddddAA
poweranalyzingAPfggfPd
d
N
gf
gfgfTrP
iNgfgfPTrTrd
d
beam
N
beam
N
y
beam
N
beam
N
NN
beam
NNN
beam
iii
//
//1
2
//
wheredifferencedown vsupon polarizati beam aor asymmetry,right -left a measure wealy,Experiment
axis). beam about the by rotation under equivalent are (these directions oppositein flippedon polarizati beam
but with direction scattering fixed ain yields measureor ,scatteringright left vs measure weif ariseslatter The
asymmetry. scatteringor thedefines which 1**
,ˆdirection or plane, scattering the to that issection cross the toscontribute
which direction on polarizati beamonly The section. cross aldifferenti dunpolarize the,
where**
state.spin incoming the
is and ,ˆ where, **ˆ
0
00N0
00N000
22
0
21
0
21
p
11
111MMMM
targetpolarization
yield
targetpolarization
yield
Show
57
Spin Observables
( ) ( )
*)Im(2
.quantity for , amplitudesfor n informatiot independen 3 theaccessesrotation spin The
.invariance reversal by time required as just or potentialorbit -spin thefromon polarizati Induced
)*Re(2**
rotation.-spin a called is on,polarizati induced theis where
as thiscan write or wedirection in the
bygiven is on,polarizati itsor particle, spin outgoing theofoperator spin theof n valueexpectatio The
21
K0P00
K0P0
rd
N000
00N0021
21
N0000
i0j0N000i0j0,N0000
21
21
fgTrD
QDgf
APP
AgfgfgfTrTrP
DPDPPPd
d
PTrPd
diP
KP
y
NNN
final
N
beam
jNi
final
i
final
i
beamfinal
i
ii
i
final
iifinal
i
==
=
==++=
+=W
+=W
=
MM
11MM
MM1MM
MM
targetpolarization
polarization
Rotation of P in
scattering plane
phase. relative a plus magnitudes two
and amplitudescomplex for angle
scatteringeach at quantitiest independen Three
)*Re(2
*)Im(2 ;
:tsmeasurement independen Three
00N00N0000
0
22
0
==
=+=
gf
gfAP
fgQgf
High Resolution Spectrometer
Lo
s A
lam
os
Mes
on
Ph
ysi
cs F
acil
ity (
LA
MP
F)
58
Spin Observables
George Igo
UCLAJerry Hoffmann
UT Austin
59
Finally, we can treat the N+N scattering problem which is spin ½ x ½
, , , , :quantitiesspin theand
products,sor their tenand ˆ ;ˆ ;ˆ
:quantities kinematic theusing symmetries from determined is of form possible The
. scattering 44 theis where,
form thehassolution full theand
state mixed ain bemay waveincoming The
nucleons. target and projectilerepresent
, subscripts whereetc., , )(
can write weso target,and projectile theof spins theinvolves now waveplane incoming The
)()()(
,,
i
41312111
12
NN
tjpitptptp
ikr
ii
ikz
ikr
tptptptptp
ikz
Tsocentral
kk
kkP
kk
kkN
kk
kkK
matrixr
ee
tpr
effffe
SVSLrVrVrV
+
+=
=
-
-=
=+
++++
++
1
M
MM
x
y
z
k
k
kk +kk
=++=
t
t
p
p
ttttppppi
c
c
c
c
cccc
2
1
2
1
2121 )()(
60
Spin ½ x ½ scattering
Kinematic
Tensors
1
K.P
P.N
N.K
K
N
P
KiKj
PiPj
NiNj
KiPj+KjPi
KiNj+KjNi
PiNj+PjNi
Tensors of the
Spin operators
1p1t
p.t
p+t
p-t
pxt
i,pj,t+j,pi,t
Tensor
Rank
0
1
2
The N+N scattering amplitude is
constructed from these spin and
kinematic tensors such that rotational,
parity and time-reversal symmetries
are satisfied. The most general form
for spin ½ x ½ is:
other.each relatedlinearly becan
thatliterature in theappear formsOther
Moravcsik. of that is form above The
.amplitudes scatteringt independen 5
are thereand ,0 amplitude NNFor
etc. , ˆ
nucleons,
targetand projectile refer to 1,2 labels where
)(
)(
)()(
)1(1
21
2121
2121
=+
=
-+
+++
++-+=
b
N
hg
hgm
cba
N
KK
PPNN
NNNN
M
Show
61
Spin ½ x ½ scattering
( )
( )
( )
. angle mixing and , shifts phasewith
)2cos()2sin(
)2sin()2cos(2
ismatrix - theandmatix 22 a becomes states. ,both of mixtures are that wavespartial spherical outgoing to
, statesin wavespartial incomingfor ,both match must we matching of instead
whereabove as ideas same thefollowbut dcomplicate morebit a are conditions matching andsolution numerical The
0)(11)(2
)()()1()(2)1(
:1
0)(11)(2
)()()(2)1(
:1
:solve toeqns. diff. radial coupled get two we2 and ;1For above.shown as proceeds
b.c the tomatching and solutions theand , wavepartialeach for equation er)(Schroding diff.
radial oneonly is therestates ,1for and 0For zero. are others all and
12
)1(2 0
12
)1(6 1
0 2 0
12
)1(6 0
12
)2(2 1
1 J 1
) of indep. are(they belowgiven are 11 , of elementsmatrix where)(
potential tensor theis with deal tohave we thingnew themodels scattering NN alconventionFor
11
2)(
)(2
:,:,
1,1
122
2
222
2
1,1
122
2
222
2
121212
111
111
JJLJL
i
J
i
J
i
J
i
J
JLLJLL
L
LLLLLLL
JJL
LJTJLsocent
JJL
JLTLJsocent
zzzT
JLJLJL
JLJLJL
eei
eieiCS
SCLL
LLuuiFGCFu
ruJLSJLrVrurVLrVEr
LL
dr
dJL
ruJLSJLrVrurLVrVEr
LL
dr
dJL
LLS
JL
JLLSS
J
J
J
JJJ
J
J
JJ
J
JJ
JJ
JJJLSJJLSSrV
+=-=
+
+
+-=
+-=
=+=
++
=-
++-+
+-+=
=-
--+
+--=
=-=
====
+
--
+
+-
+
+
+
+-+
-+
+
+=+=-=
+=-=-=
1
=L
=L
62
Spin ½ x ½ scattering
( )
( )
( )
( )
0)( ),()()( ,)(
)1(
1)( and
coupled et,spin triplfor ,1
coupled et,spin triplfor ,
coupled et,spin triplfor ,1
uncoupled et,spin triplfor ,1
singletspin for ,1
and
)1(2)1( where, )(
)1())(1( where, sin)(
)()()12(
)1(2)1( where, )()()12(
cos ,)()12(
bygiven are amplitudeshelicity the wavespartial of In terms
.amplitudeshelicity are theand momentum c.m. NN theis where,
sin2cos)(
sin)(cos2
sin2cos)(
:Arndt) R. (from follows as angles mixing and shifts phase thefrom obtained are amplitudes 5 therecord For the
0
:1,1215
:1,1214
:1,1213
212
211
453
5
45300
4
2
3
4532
2
1
1
5,4,3,2,15321
1221
4532121
5342
453212
=-=+
=
-=
=
-=
-=
-=
++++==
++-+==
++=
+-++=++=
=+=
+-=
-=
+++--=
+-=
++++=
+=+=
+=-=
-=-=
==
=
++
-
--
xGxxGxPxFdx
xdP
JJxG
ST
ST
ST
ST
ST
TJJTJJTDxPDh
TJJTTJJDxGDh
xFDxGTJh
TJJJTTJDxGDxFTJh
xxPTJh
hkhhh
hhg
hhhhhm
hhhc
hhhhha
JJJJ
J
JJLJLiJ
JJLJLiJ
JJLJLiJ
JLLiJ
JLiJ
JJJJ
J
JJ
JJJJ
J
JJ
J
JJJJ
JJJJ
J
JJJJ
J
JJ
k
k
k
ki
ki
63
Spin ½ x ½ scattering
:arerest theall and
d
d
:is target and beam polarizedon with polarizati inducednucleon scattered theand
1d
d
: targetand beam polarizedth section wi cross The
2222
where
name) special (not measuremenspin triplea ,
rotationspin or r on transfepolarizati isthe just sometimesor
ferspin trans ,
ncorrelatiospin ,
power analyzing particle target the
power analyzing particle beam the
nucleon scattered in the inducedon Polarizati the
section cross aldifferenti dunpolarize theis
0,,00,00,0000,
00,,000,000,0
222222
0
000
00000
00000
00000
0000,000000
0,000000
0,000000
00000
ijkjtibikitikibNkscatt
ijjtibiitiib
jkijki
ijjiji
jiji
ijij
NNiNi
yNiNi
NiNi
MPPKPDPPP
APPAPAP
bhgmca
MX
DDX
KX
AX
AXX
AXX
PXX
X
+++=W
+++=W
+++++=
=
=
=
=
==
==
==
=
:include sobservablecommon some and :readsnotation the whereˆˆˆˆ
)39,1(1978) (Paris) Phys. de J al.et (Bystricky :is sobservablespin for notations usedcommonly theof One
ly.respective labels,nucleon target and beam are (2) (1), where
onspolarizati (t) target and (b) beamfor ˆ
:is nucleons colliding theof statespin initial for thematrix density spin The
targetbeam,recoil,scatt,41
,
)2()1(,,)2()1(41
XSSSSTrX
PPPP
jilkijkl
ji
jijtibtb
MM
1
=
+++=
64
Spin ½ x ½ scattering
(using Bystricky’s amplitude definition)
Moravcsik
a
b
c
m
g
h
Bystricky
(a+b)/2
f/2
e/2
(a-b)/2
c/2
-d/2
Bystricky
a
b
c
d
e
f
Moravcsik
a+m
a-m
2g
-2h
2c
2b
65
Dirac representation of the N+N scattering amplitudes
The above forms for the N+N scattering amplitudes are given in terms of the
Pauli spin operators and are referred to as the Pauli representation. We can also
represent this same physical information using Dirac matrices in the so-called
Dirac representation. This new form provides the essential input for relativistic
nuclear structure and scattering theories.
N+N c.m.kk
-
initial k -
final
k
( )
( ) ( )
21212121
/),()()(),()()(
bygiven states,spin final-to-intialarbitrary from tioningfor transi values
nexpectatio respective their using related becan tionsrepresenta twoThese
),(
:is matrices Dirac of in terms amplitude scattering NN shell)-(on The
/ˆ ,/ˆ ,/ˆ :where
)()()()()(),(
above)than different (slightly istion representamatrix Pauli The
212
5
21
5
121
5
2
5
121
21212121
ss
T
s
T
sssss
TAVPs
ppqqnnnn
ktsfkukutsFkuku
FFFFFtsF
kkkkpkkkkqkkkkn
qEqDqCqBqAtsf
=--
++++=
+
++=--==
+++++=
11
66
Dirac representation of the N+N scattering amplitudes
( )
( )
( )
( )( ) ( )( )( ) ( ) ( )
tables.following in thegiven are results The related. be totionsrepresenta two theallows
amplitudes Pauli theofeach for termscollecting and )( invariants Lorentz theallout Working
.,, amplitudes Pauli toscontributescalar Lorentz theofelement matrix that thissee We
);( whereˆ4/
in results using productsspin thereducing and
4
)()(
,,),(4
)()(),()()(
gives mscalar ter for theelement matrix out the Working. , state 1/2spin and
mass.nucleon is andenergy totalis where, ,2
)(
10
01,
2,
2)( and
2)(
bygiven spinors-4energy positive are )( where
21
1
22
111
222
1112
2
2
2
1
1
21
22
112
2
21
*
2
*
1
2
1
0
2121
2121
2121
S,P,V,A,T
CBA
kkqkkkniqkqkkkikkkk
BAiBABA
mE
k
mE
k
mE
k
mE
k
m
mE
mE
k
mE
kmE
k
mE
ktsF
m
mE
kukutsFkuku
ccc
c
mEmE
k
m
mEku
mE
k
m
mE
mE
k
m
mEku
mE
km
mEku
ku
aaa
ss
cmcmcmcm
T
s
T
scm
ss
cmcmcmcm
T
s
T
sScm
ssSss
T
ss
T
ss
T
s
T
ssss
s
-=+=--=+=
+=
+
+
-
+
+
-
+=
+
-
+
+
--
+
-
+
=--
=
+
-
+=
-
+
+=
+
+=
+
+
=
11
11
10
01
10
0111
11
1
11
1
Show
67
Dirac representation of the N+N scattering amplitudes
[from McNeil, Ray, Wallace, Phys. Rev. C 27, 2123 (1983)]
This matrix eqn. is derived for p + nucleus scattering. To use it for N+N scattering
in the c.m. omit row 4, use N+N c.m. k, Ecm and set A=1.
68
Dirac representation of the N+N scattering amplitudes
[from McNeil, Ray, Wallace, Phys. Rev. C 27, 2123 (1983)]
For N+N scattering use N+N c.m. k, Ecm and set A=1.
69
This concludes Chapter 1:
The Nucleon+Nucleon System
