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I. Nuclear symmetries and quantum numbers. I.1 Fermi statistics. Fermi statistics. Antisymmetric wave function. Fermi level. N. i. Second quantization:. Fermi level. Multi configuration shell model. Complete basis. Big matrix diagonalization. I.2 Interactions and symmetries. - PowerPoint PPT Presentation
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I. Nuclear symmetries and quantum numbers
I.1 Fermi statistics
Antisymmetric wave function Fermi statistics
neutron 1proton 1downspin 1 upspin 1
numbers quantum orbital },,{ state quantumeach on nucleon oneOnly
33
3
zz
z
nni
Fermi level
i
N
Second quantization:
excitation hole-particle ||
state ground 0|.....|
],[
nucleon a anihilates
nucleon a creates
nucleon no - vacuum 0|
1
''''
hp
N
iiiiiiii
i
i
ccph
cc
cccccc
c
c
Fermi level
Multi configuration shell model
Complete basis
...|
|||
''''
''
hphphhpp
hhpp
phhpph
cccc
cc
Big matrix diagonalization
I.2 Interactions and symmetriesInteraction strong electromag. weak
Exchanged boson mesons photon W,Z
Translation yes yes yes
Lorentz yes yes yes
Space inversion yes yes no
Rotation yes yes yes
Isorotation yes no no
Time reversal yes yes yes
I.3 Translational invarianceSpatial:
kkkkkk sspparr ,,
conserved. momentumlinear Total
0],[,1
Nk
kpPHP
Time: tt Total energy E conserved.
I.4 Lorentz invariance
Low energy – Galilei invariance
kkkkkkkk ssumppturr ,,
energy otal t2
mass ofcenter 1
mass total
],[
2
intrinsic
,1
,1
MPEE
rmM
R
mMMPHR
Nkkk
Nkk
High energy – Lorentz invariance
MMcE
PcEE
i massrest
energy otal t2
ntrinsic
222intrinsic
Mass spectrograph
The rest mass and rest energy 2McE
Creation of rest energy (mass) from kinetic energy. A high energycosmic sulfur nucleus (red) hits an silver nucleus generating a sprayof nuclei (blue, green) and pions (yellow).
I.5 Space inversion invariance
kkkkkk sspprr ,,:P
kk ll
:P
state theofparity |||
0],[
PP H
Quantum number
1 D
N
zyx NnnnE
)1(
)2/3()2/3(
3 D
E1 M1
Parity of electromagnetic dipole decay
I.6 Rotational invariance
kkkkkk sspprre
)1(,)1(,)1(:)(
R
conserved. momentumangular Total
spin
momentumangular orbital
0],[
,1
,1
Nkk
kkkNkk
sS
prllL
SLJHJ
But not spin or orbital separately!
3D rotations form a non-Abelian group
cyclic ],[
cyclic ],[
cyclic ],[
zyx
zyx
zyx
jijj
siss
lill
Lie algebra of group
2SU
1|)1)((|
|)1(|
||0],[0],[0],[
2
22
IMMIMIIMJ
IMIIIMJ
IMIIMMIMJJJHJHJ
z
zz
Spherical harmonics eigenfunctions of orbital angular momentum
),()1(),(),(),(
cyclic )(
22 lmlmlmlmz
x
YllYlmYYl
yz
zyil
lml
lm YY )(P
Spinors
10
1)(down spin
01
1)( upspin
1001
00
0110
matrices Pauli
2 particles 1/2spin
z
z
zyx ii
s
I
IMIM
I
P
notation picspectrosco
quotednot usually substates magneticenergy same theall have , projection m. a.
A odd .... 3/2, 1/2,or A even ... 2, 1, 0, momentumangular
)1( parity ginterestinnot momentumlinear
:statesnuclear of numbers quantum good
Spectroscopic notation
l
l
)(by parity changes
away carries0 has
Way to measure spinsand parities of groundand excites states
Alpha decay caused by strong and electromagnetic interaction
Angular momentum couplingBit complicated because of Quantization and non-commuting components
||||
rulesSelection
21321
213
IIIIIMMM
sljjIIMI
z
2133
numbers quantum
Clebsch-Gordan-Coefficients
||||
||||
||||
21321
21333322112211
321
22113322112133
3
21
IIIII
IIMIMIMIMIMIMI
MMM
MIMIMIMIMIIIMI
I
MM
Spin orbit coupling
Spin orbit coupling
l
j
l
l
(-)
.... 5, 4, 3, 2, 1, 0, .... h, g, f, d, p, s,
notation
Particle statesHole states
Pbch208| Pbcp
208|
Two particle states
Occ Jpp16
' |}{
Selection rules for electromagnetictransitions
Multipolarity of the photon – its angular momentum
|||| ifi III
The transition with the lowest multipole dominates.
ons transitimagnetic )(
ons transitielectric )(1
Pure M1
Pure M1
Pure E2
Pure E1
No transition
For alpha decay hold the general rules of angular momentum conservation too.
I.7 Isorotational invarianceStrong interaction same for n-n, p-p, n-p –charge independent.
Conservation of isospin (also for particle processes caused by strong interaction).
cyclic ],[,,0],[ 321321 TiTTTTTHT
1- 1 meson
0 1 meson
1 1 meson
1/2- 1/2 neutron 1/2 1/2 proton 0 0 hyperon
particle
0
3
tt
10
1)(neutron
01
1)(proton
1001
00
0110
matrices Pauli
2 particles 1/2isospin
3
3
321
ii
t
2/)(3 NZT Same orbital wave state
Total state must be antisymmetric.
HeNHO 414216
2/452/45 3 TT
2/432/45 3 TT
2/432/43 3 TT
Isobar analogue states
209
I.8 Time reversal invariance
kkkkkk sspprrtt ,,:T
nconjugatiocomplex : KKi yT
number. quantum aimply not does 0],[y.antiunitar is
HT T
angle in center of mass system
diff
eren
tial c
ross
sect
ion
Reaction A+B C+D has same probability as C+D A+B“detailed balance”
Random interaction
}4
exp{2
)(
ondistributiWigner
2
22
2 DDP
}
2exp{2()(
on distributi ThomasPorter :yprobabilit emission neutron
0
02/1000
)(n
)(n)(
n)(
n)(
n ΓΓΓΓΓP