Isospin • Check [Tˆ , Tˆ±]= Tˆ± 3 ± ˆ 1 ˆ+ ˆ • Then operators T1 = 2 T + T
1 + ⇥ Tˆ = Tˆ Tˆ 2 2i ⇥ Tˆ3
obey the same algebra as Jx,Jy,Jz 2 so spectrum identical and H ˆ S , T ˆ , Tˆ 3 simultaneously diagonal ! rm = rm m = 1 proton | s p | s t 2 1 neutron rms n = rmsmt = 2 | ⇥ | 2 ⇥ For this doublet T rm m = 1 ( 1 + 1) rm m | s t 2 2 | s t and T3 rmsmt = mt rmsmt | | States with total isospin constructed as for angular momentum
QMPT 540 Examples • Z=11, N=12 vs Z=12, N=11 • Total isospin of low-lying states --> ½
• Triplet of nuclei • Total isospin of ground state of 22Na --> 0 • But 22Ne and 22Mg ground states have total isospin --> 1
QMPT 540 Nuclei • Two particles outside closed shells • Different shells only Clebsch-Gordan constraint
• Uncoupled states in the same shell jm,jm = a† a† 0 | jm jm | • Note restriction due to Pauli principle: J=2j forbidden!
• Coupling jj,JM = (jmjm⇥ JM) jm,jm = (jm⇥ jm JM) jm ,jm | ⇥ | | ⇥ | | ⇥ mm mm 2j J = ( 1) (jmjm⇥ JM)( 1) | | jm,jm ⇥ mm J =(1) (jmjm⇥ JM) | | jm,jm ⇥ mm =(1)J | jj,JM ⇥ • Only even total angular momentum
jj,JM,T M = (jmjm JM)(1 mt 1 m TMT ) jmm ,jm m • With isospin | T ⇥ | 2 2 t | | t t ⇥ mm m m t t =(1)J+T +1 | jj,JM,T MT ⇥
• J+T odd! QMPT 540 40Ca + two nucleons • Spectrum • T=1 0+ in 42Sc below T=0 states due to “pairing” effect • Most nuclei: ground state lowest possible total isospin
QMPT 540 Orbits around 208Pb
• First empty proton level h9/2 for neutrons --> g9/2
QMPT 540 Other examples • 2 protons outside 208Pb
2 • (h9/2) --> J = 0,2,4,6,8 • parity +
QMPT 540 More • 2 neutrons or p and n
2 • (g9/2) (πh9/2 νg9/2) • J=0,2,4,6,8 J = 0,1,2,…,9 • parity + parity -
(πh9/2 νg9/2) 10-?
QMPT 540 More still • 16O plus 2 neutrons
(νd5/2 νs1/2) 2 (νd5/2)
QMPT 540 and • But in 18F • T=0 ground state • with J = 1 • Note J = 3 and 5
QMPT 540 New case • Excited states in 208Pb
QMPT 540 How about 16O? • Isospin coupling --> T = 0 or 1
QMPT 540 Improving excitation spectra beyond RPA
QMPT 540 Yet another example • 19O • 3 neutrons
QMPT 540 Nucleon-nucleon interaction • Shell structure in nuclei and lots more to be explained on the basis of how nucleons interact with each other in free space
• QCD • Lattice calculations • Effective field theory • Exchange of lowest bosonic states • Phenomenology
• Realistic NN interactions: describe NN scattering data up to pion production threshold plus deuteron properties • Note: extra energy scale from confinement of nucleons QMPT 540 (Effective) central force N.Ishii, S.Aoki, T.Hatsuda, Phys.Rev.Lett.99,022001(’07). The following diagram is contained.
This leads to the one pion exchange at the large spatial separation.
Repulsive core: 500 - 600 MeV Ishii talk at 2009 Oak Ridge workshop Attractive pocket: about 30 MeV
Both of these are smaller than we expect. This is answered by the quark mass dependence. Quark mass dependence of the central force: (1) mπ=380MeV: Nconf=2034 [28 exceptional configurations have been removed] 1 (2) mπ=529MeV: Nconf=2000 S0 (3) mπ=731MeV: Nconf=1000
Strong quark mass dependence is found.
In the light quark mass region, ! the repulsive core grows rapidly.
! attractive pocket is enhanced mildly. Ishii talk at 2009
Oak Ridge workshop The lattice QCD calculation at light quark mass region is quite important. Big energy perspective • High-energy data pp and pn
reactions and structure Energy scales
reactions and structure Two-body interactions and matrix elements 1 • To determine Vˆ = ( ⇥ V ⇤⌅)a† a† a a 2 | | ⇥ ⌅ ⇤ ⇥⇤⌅ • we need a basis and calculate ( ⇥ V ⇤⌅ ) for given interaction | |
• Simplest type: spin-independent & local (also for spinless bosons)
(r r V r r )=(Rr V R r ) 1 2| | 3 4 | | = (R R ) r V r = (R R ) (r r )V (r) ⇥ | | ⇤ • with 1 R = 2 (r1 + r2) r = r r 1 2
QMPT 540 Nucleon-nucleon interaction • Yukawa 1935 • short-range interaction requires exchange of massive particle e µr V (r)=V Y 0 µr 2 • mass of particle µ c = mc • mesons are the bosonic excitations of the QCD vacuum • many quantum numbers; most important: pion T=1, 0- lowest mass! • So one encounters also spin and isospin dependence
V = V (r) spin 1 · 2 V = V (r) isospin 1 · 2 Vs i = V ⇥ (r) 1 2⇥1 ⇥2 · · QMPT 540 Spin and isospin matrix elements
• Pauli spin matrices 1 2 · • represent 4 s1 s2 2 ·
• Use S = s1 + s2 1 • Then s s = S2 s2 s2 1 · 2 2 1 2 ⇥ • So coupled states are required
S M 1 2 SMS = (2S (S + 1) 3) S,S M ,M ⇤ S| · | ⌅ S S • Same for isospin
T M 1 2 TMT = (2T (T + 1) 3) T,T M ,M ⇤ T | · | ⌅ T T
QMPT 540 Realistic NN interaction • Required for NN scattering
1 ⇥ ⇥ ⇥ ⇥ 1 · 2 1 · 2 1 · 2 1 · 2 S12 S12 ⇥1 ⇥2 L SLS ⇥1 ⇥2 L2 L2 ⇥ · ⇥ L2 · L2 · ⇥ · ⇥ 1 · 2 1 · 2 1 · 2 1 · 2 (L S)2 (L S)2 ⇥ ⇥ · · 1 · 2 • plus radial dependence • Tensor force S (rˆ)=3( rˆ)( rˆ) 12 1 · 2 · 1 · 2 • Short-range interaction suggests use of angular momentum basis • Angular momentum algebra • Spherical tensor algebra • Often calculations are done in momentum space
QMPT 540 Operator content pion exchange • Decomposition of static pion exchange 2 2 2 1 f NN 3 1 Qc 2 Qc 1 2Q c V (Q, 0) = 2 · 2 2· 2 2 2· ⇥1 ⇥2 3 µ c µ + Q c · 1 2 2 2 1 2 + f NN c 2 2 2 · 2 2 ⇥1 ⇥2 3 c µ + Q c · 1 f 2 NN ⇥ ⇥ 3 µ2 1 2 1 2
(p p V p p )=(PpV P p )= p V p 1 2| | 3 4 | | P ,P | | ⇥ • or wave vectors 1 3 k V k = d r exp i(k k) r V (r) ⇧ | | ⌃ V { · } • Use exp iq r =4 i Y (rˆ)Y (qˆ)j (qr) { · } m m m • to find 4 2 k V k = dr r j ( qr ) V ( r ) with q = k k | | ⇥ V 0 | | • Yukawa 4 ⇥ V0 1 k VY k = ⇥ | | ⇤ V µ µ2 +(k k)2 4 q q e2 • Helps for Coulomb 1 2 when k = k k VC k = 2 ⇥ | | ⇤ V (k k) QMPT 540 Partial wave basis • Requires matrix elements of the form
kLM V k L M = dkˆ LM kˆ dkˆ kˆ L M k V (r) k L| | L⇥ L| ⇥ | L⇥ | | ⇥ • For Yukawa write 4⇤ V0 1 1 k VY (r) k = 2 2 2 µ +k +k ⇥ | | ⇤ V µ 2kk cos kk 2kk • and use 2 2 2 1 ⌅ µ + k + k⇤ 2 2 2 = (2⌥ + 1) Q⇥ P⇥(cos kk ) µ +k +k cos kk 2kk⇤ 2kk ⇤⇥=0 ⇥ ⇥ 2 2 2 ⌅ µ + k + k⇤ = 4⇤ Q Y ⇥ (kˆ)Y (kˆ ) ⇥ 2kk ⇥m ⇥m ⇤ ⇥=0 m= ⇥ ⇤ ⇥ ⇤ ⇤ 1 z +1 Q (z)= ln 0 2 z 1 • with Legendre functions ⇥ z z +1 Q (z)= ln 1 1 2 z 1 ⇥ 3z2 1 z +1 3 Q (z)= ln z 2 4 z 1 2 ⇥ 2 2 2 2 • yields (4⇤) V0 µ + k + k kLM V k L M = Q L L L,L ML,ML L | | ⇥ Vµ2kk 2kk ⇥QMPT 540 Example • Reid soft-core interaction (1968)
100
1 • solid S0 • no bound state ) MeV (
3 ) • dashed S1 r 0 ( V • deuteron • ??
−100 note similarity to 0 0.5 1 1.5 2 2.5 atom-atom interaction r (fm)
QMPT 540 Two-particle states and interactions • Pauli principle has important effect on possible states • Free particles ⇒ plane waves p2 T = • Eigenstates of 2 m notation (isospin) • Use box normalization p s = 1 m t = 1 m pm m • Nucleons | 2 s 2 t⇥ | s t⇥
• Use successive basis transformations for two-nucleon states to survey angular momentum restrictions • Total spin & isospin; CM and relative momentum; orbital angular momentum relative motion; total angular momentum
QMPT 540 Antisymmetric two-nucleon states • Start with 1 p1ms mt ; p2ms mt = p1ms mt p2ms mt p2ms mt p1ms mt | 1 1 2 2 ⌅ ⌃2 {| 1 1 ⌅ | 2 2 ⌅ | 2 2 ⌅ | 1 1 ⌅} 1 1 = ( 1 ms ms SMS)(1 mt 1 mt TMT ) p1 p2 SMS TMT ) ⌃2 { 2 1 2 2 | 2 1 2 2 | | SM S TM T ( 1 m 1 m SM )(1 m 1 m TM ) p p SM TM ) 2 s2 2 s1 | S 2 t2 2 t1 | T | 2 1 S T } P = p + p • then 1 2 p = 1 (p p ) 2 1 2
p = pLM LM pˆ = pLM Y (pˆ) • and use | ⇥ | L⇥ L| ⇥ | L⇥ LML LM L LM L L p = pLM LM p = pLM ( 1) Y (pˆ) | ⇤ | L⇤⇥ L| ⇤ | L⇤ LML LML LML L Y ( p)=Y (⇥ , ⇤ + ⇥)=( 1) Y (pˆ) ⇥ LML LML p p LML 1 + 1 S ( 1 m 1 m SM )=( 1) 2 2 ( 1 m 1 m SM ) • as well as 2 s2 2 s1 | S 2 s1 2 s2 | S 1 + 1 T ( 1 m 1 m TM )=( 1) 2 2 ( 1 m 1 m TM ) 2 t2 2 t1 | T 2 t1 2 t2 | T QMPT 540 Antisymmetry constraints for two nucleons • Summarize p m m ; p m m = | 1 s1 t1 2 s2 t2 ⇤ 1 ( 1 ms 1 ms SMS)(1 mt 1 mt TMT ) Y (pˆ) ⇧2 2 1 2 2 | 2 1 2 2 | LML SMS TM⇤T LML 1 ( 1)L+S+T P p LM SM TM ) ⇥ | L S T 1 = ( 1 ms 1 ms SMS)(1 mt ⇥ 1 mt TMT ) Y (pˆ) ⇧2 2 1 2 2 | 2 1 2 2 | LML SMS TM⇤T LMLJMJ (LM SM JM) 1 ( 1)L+S+T P p (LS)JM TM ) ⇥ L S | J | J T • L + S + T must be odd! ⇥ – Notation T=0 T=1 3S 3 D 1S 1 1 0 1 3 P1 P0 3 3 D2 P1 ... 3P 3 F 2 2 1 D2 QMPT 540 Phase shifts 1968...
Dynamic Static
Nucleon correlations Phase shifts 1968...
Nucleon correlations