15thNational Nuclear Physics Summer School June 15-27, 2003 Nuclear Structure Erich Ormand N-Division, Physics and Adv. Technologies Directorate Lawrence Livermore National Laboratory Lecture #2 This document was prepared as an account of work sponsored by an agency of the United States Government. Neither the United States Government nor the University of California nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or the University of California, and shall not be used for advertising or product endorsement purposes. This work was carried out under the auspices of the U.S. Department of Energy by the University of California, Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48. Low-lying structure The interacting Shell Model The interacting shell model is one of the most powerful tools available too us to describe the low-lying structure of light nuclei We start at the usual place: pr i2 r r H =

+ U ( ri ) + VNN ( ri rj ) U ( ri ) 0f1p N=4 2m i i< j i 0d1s N=2 Construct many-body states |i so that 1p N=1 0s N=0 i = Cin n n i (r1) i (r2 ) K 1 j (r1) j (r2 ) = M O A! l (r1) l (r2 ) K Calculate Hamiltonian matrix Hij=j|H|i Diagonalize to obtain eigenvalues H11 H12 L H 21 H 22 M O H K N1

H1N H NN = al+ K a +j ai+ 0 We want an accurate description of low-lying states 2 i (rA ) j (rA ) M l (rA ) Nuclear structure with NN-interaction NN-interaction determined from scattering and the deuteron Argonne, Bonn, Paris, Reid, etc. Phase shift and potential in 1S0 channel Strong repulsion at ~ 0.5 fm j|H| |H|i large Problem: Repulsion in strong interaction Infinite space! 3 Can we get around this problem? Effective interactions Choose subspace of n for a calculation (P-space) Include most of the relevant physics

Q -space (excluded - infinite) Effective interaction: Q P H eff Pi =Ei Pi Two approaches: Bloch-Horowitz H eff =PH + PH 1 QHP Ei QH Lee-Suzuki: Heff=PXHX-1P ij G kl QXHX-1P=0 4 + + + Effective interactions permit first-principles shell-model applications Impossible problem Difficult problem Two, three, four, A-body operators

+ + +K Q P Compromise between size of P-space and number of clusters Three-body clusters 5 The general idea behind effective interactions Q P-space defined by N max h H Heff has one-, two, three-, A-body terms +2 + P 22 33 Heff Exact reproduction -1 QXHX P=0

of N eigenvalues Q P 6 Interactions in real-world applications Ideally, we would like to use these fundamental interactions in our theory calculations In most cases this is not really practical as the the NN-interaction has a very strong repulsive core at short distances This means that in many-body applications an infinite number of states are needed as states can be scattered to high-energy intermediate states We need to use effective interactions Derived from some formal theory This is in principle possible and is difficult. But it is becoming practical now for light nuclei Assume they exist as the formal theory stipulates and determine it empirically to reproduce data This has permitted many studies in nuclear structure to go forward 7 Shell model applications The practical Shell Model 1. Choose a model space to be used for a range of nuclei E.g., the 0d and 1s orbits (sd-shell) for 16O to 40Ca or the 0f and 1p orbits for 40Ca to 120Nd 2. We start from the premise that the effective interaction exists 3. We use effective interaction theory to make a first approximation (Gmatrix) 4. Then tune specific matrix elements to reproduce known experimental levels 5. With this empirical interaction, then extrapolate to all nuclei within the

chosen model space 6. Note that radial wave functions are explicitly not included, so we add them in later The empirical shell model works well! But be careful to know the limitations! More on exact treatments later. 8 The Shell Model We write the Hamiltonian as H = a a a + a a a abcd j a j b ;JT V j c j d ;JT A (1+ ab )(1+ cd ) Start with closed inner core, e.g., for 24Mg, close the p-shell Active valence particles in a computationally viable model space, e.g., the 0d5/2, 0d3/2, 1s1/2 orbits for 24Mg + a + JT b

[[ a a ] JT 00 [ a c a d ] ] 00 No excitations up here 0f1p N=4 0d1s N=2 Allow all configurations in the valence space 1p N=1 0s N=0 Single-particle energy i Closed core. No excitations of the core allowed! Two-body residual interactions Energy is relative to 16O core 9 Building the shell-model basis states Need to construct the many-body basis states to calculate matrix elements of H Choose states with definite parity, Jz and Tz and let the Hamiltonian do the rest A very useful approach is a bitrepresentation known as the M-scheme 0f1p N=4 0d1s N=2 1p N=1 0s N=0

a 5+, 1 a 5+, 3 a 3+, 1 a 1+, 1 0 = 2 2jz 2 2 2 2 2 2 2 0 0 1 0 1 0 0 1 0 0 1 -5 -3

-1 1 3 5 -3 -1 1 3 -1 0d5/2 Closed core. No excitations of the core allowed! 0d3/2 0 =2 2 + 2 4 + 27 + 210 =1172 1 1s1/2 A single integer represents a complicated Slater Determinant 0 Basis states - How many are there? To do a calculation we need states with fixed Jz All J-values are contained in Jz=0 or Jz=1/2

Counting the number of basis states Order-of-magnitude estimate n particles, and Nsps single-particle states Nsps in the sd-shell = 12 (0d5/2=6, 0d3/2=4, 1s1/2=2) Nsps in the fp-shell = 20 (0f7/2=8, 0f5/2=6, 1p3/2=4, 1p1/2=2) p n N sps N sps Dim p n n n 2 12 12 12! 24 Mg(sd - shell) = = 245025; 4 4 4!8! 60 20 20 Z(fp - shell) = 3.4 1010 10 10 Includes states of all J and Jz Number of Jz=0 divide by a factor of ten Number of states with a given J

N ( J ) = N ( J z = J ) N ( J z = J 1) Tools of the trade - Lanczos Setup Hamiltonian matrix j|H|i and diagonalize Lanczos algorithm Bring matrix to tri-diagonal form H v1 =1 v1 + 1 v 2 H v 2 =1 v1 + 2 v 2 + 2 v 3 H v 3 = H v 4 = Note that after each iteration, we need to re-orthogonalize! 2 v 2 + 3v3 + 3v 4 3v3 + 4 v 4 + 4 v5 nth iteration computes 2nth moment H 0 = i H i = Tr [ H ] Prove that Lanczos computes 1 moments vi = i N i i H n n

= Tr [( H H 0 ) ] But you cant find eigenvalues with calculated moments Eigenvalues converge to extreme (largest and smallest) values ~ 100-200 iterations needed for 10 eigenvalues (even for 108 states) 2 Shell-model codes Oak Ridge (1969) Coefficients of Fractional parentage (CFP) j J = [ j N N 1 N J jJ } j J ] [ j N 1 J j ] DUSM (1989) Permutation groups J ANTOINE (1991 & 1999) M-scheme Apply matrix on-the-fly

Large dimensions J Glasgow (1977) Good Jz (M-scheme) J restored in diagonalization NATHAN (1998) J-projected similar to ANTOINE Hybrid M-scheme-CFP code OXBASH (1985) J-projected M-scheme Smaller matrices REDSTICK (now) RITSSCHIL (1985) CFP 3 Similar to ANTOINE M-scheme Three-body interactions Parallel architecture REDSTICK basis state ordering

Ordering of basis states speeds up the calculation Need a FAST lookup scheme Construct proton and neutron many-body Slater determinants Order by Parity, Jz and Protons Neutrons 1 5 0+1=1 1 6 0+2=2 2 5 2+1=3 2 6 2+2=4 3 3 4+1=5

3 4 4+2=6 4 3 6+1=7 4 4 6+2=8 5 3 8+1=9 4 8+2=10 1 1 2 2 3 3 4 4

5 5 5 6 1 10+1=11 6 2 10+2=12 7 1 12+1=13 7 2 12+2=14 Jz=-1 Jz=-1 Jz=0 Jz=0 h Lanczos vector Jz=1 6

Jz=1 7 6 4 For each proton SD store start in Lanczos vector: pos(i) For each neutron SD store relative position in Jz, parity list: pos(j) Position in Lanczos vector determined by summing two integers k= pos(i)+ pos(j) Same algorithm can be used to sort and find two-body matrix elements Limits truncations Partition truncations not possible Much faster! Lanczos vector points to p & n SDs (less memory) REDSTICK Applying the Hamiltonian pn part On-the-fly H v =H ci i =ci i H i j =cj j i j i

j With large dimensions, it may be better not to pre-calculate and store the Hamiltonian matrix Find states connected by one-body proton or neutron operator pn-part is the hardest Product of two one-body operators pn + H pn =vijkl i j k+ l ijkl Pre-sort connections by one-body operators; store operator and phase Loop over all initial proton-neutron SDs i=pk nl For each pk loop over all pm connected by one-body operator For each nl loop over all nn connected by onebody operator but limited in Jz, parity, and h by final proton state pm Update Lanczos vector: Position=pos(m n) pos(m)+pos( )+pos(n 5 Initial PN-SD pn mp + kp nn + ln v Jump List #1 Jump List #2 REDSTICK Applying the Hamiltonian Parallel execution Note loop over Ndim states Divide loop over NCPU independent processors For load balance use do i=1,Ndim,NCPU Creates updated Lanczos vector on each processor Total Lanczos vector with Global sum

Loop over all initial proton-neutron SDs i=pk nl steps of NCPU For each pk loop over all pm connected by one-body operator For each nl loop over all nn connected by one-body operator but limited in Jz, parity, and h by final proton state pm Update Lanczos vector: Position=pos( pos(m m)+pos( n) )+pos(n Loop over all initial proton-neutron SDs i=pk nl steps of NCPU For each pk loop over all pm connected by one-body operator For each nl loop over all nn connected by one-body operator but limited in Jz, parity, and h by final proton state pm Update Lanczos vector: Position=pos( pos(m m)+pos( n) )+pos(n

Loop over all initial proton-neutron SDs i=pk nl steps of NCPU For each pk loop over all pm connected by one-body operator For each nl loop over all nn connected by one-body operator but limited in Jz, parity, and h by final proton state pm Update Lanczos vector: Position=pos( pos(m m)+pos( n) )+pos(n Global Sum ~ NCPU speed up 6 Loop over all initial proton-neutron SDs i=pk nl steps of NCPU For each pk loop over all pm connected by one-body operator For each nl loop over all nn connected by one-body operator but limited in Jz, parity, and h by final proton state pm Update Lanczos vector: Position=pos( pos(m

m)+pos( n) )+pos(n Loop over all initial proton-neutron SDs i=pk nl steps of NCPU For each pk loop over all pm connected by one-body operator For each nl loop over all nn connected by one-body operator but limited in Jz, parity, and h by final proton state pm Update Lanczos vector: Position=pos( pos(m m)+pos( n) )+pos(n Loop over all initial proton-neutron SDs i=pk nl steps of NCPU For each pk loop over all pm connected by one-body operator For each nl loop over all nn connected by one-body operator but limited in Jz, parity, and h by final proton state pm Update Lanczos vector: Position=pos( pos(m

m)+pos( n) )+pos(n REDSTICK Reorthogonalization Parallel execution v j =v j v i v j i< j Not that trivial (but also not the main bottleneck in the calculation) Let each processor be responsible for a sub set of Lanczos vectors written to disk Each processor has a copy of the current vector Accumulate sum overlaps a n = v i v j on each node, n i< j Global sum a = a n then v =v a and normalize n Not quite the same as sequential reorthogonalization, but seems to work fine (also used in MFD). 7 Shell-model codes - Performance REDSTICK or ANTOINE

8 Simple application of the shell model A=18, two-particle problem with 16O core Two protons: 18Ne (T=1) One Proton and one neutron: 18F (T=0 and T=1) Two neutrons: 18O (T=1) Example: 18O Question # 1? How many states for each Jz? How many states of each J? There are 14 states with Jz=0 N(J=0)=3 N(J=1)=2 N(J=2)=5 N(J=3)=2 N(J=4)=2 9 Simple application of the shell model, cont. Example: Question #2 What are the energies of the three 0+ states in 18O? Use the Universal SD-shell interaction (Wildenthal) 0 5 / 2 =3.94780 1s1 / 2 =3.16354 0 3 / 2 =1.64658 Measured relative to 16O core Note 0d3/2 is unbound

0 5 / 2 0 5 / 2 ; J =0, T =1 V 0 5 / 2 0 5 / 2 ; J =0, T =1 =2.8197 0 5 / 2 0 5 / 2 ; J =0, T =1 V 0 3 / 2 0 3 / 2 ; J =0, T =1 =3.1856 0 5 / 2 0 5 / 2 ; J =0, T =1 V 1s1/ 21s1/ 2 ; J =0, T =1 =1.0835 1s1/ 21s1/ 2 ; J =0, T =1 V 1s1/ 21s1/ 2 ; J =0, T =1 =2.1246 1s1/ 21s1/ 2 ; J =0, T =1 V 0 3 / 2 0 3 / 2 ; J =0, T =1 =1.3247 0 3 / 2 0 3 / 2 ; J =0, T =1 V 0 3 / 2 0 3 / 2 ; J =0, T =1 =2.1845 20 Simple application of the shell model, cont. Example: Finding the eigenvalues 3.522 3.522 1.964 1.964 -0.830 -1.243 -1.348 -1.616 -2.706 -3.421 11.341 10.928 10.823 10.555 9.465 8.750 2+ -6.445 -7.732 -7.851 -8.389 -9.991

5.726 4.440 4.320 3.781 2.180 0+ -12.171 0.000 Set up the Hamiltonian matrix We can use all 14 Jz=0 states, and well recover all 14 J-states But for this example, well use the two-particle J=0 states |(0d5/2)2J=0 |(1s1/2)2J=0 1 + |(0d3/2)2J=0 10.7153 1.0835 3.1856 H = 1.0835 8.4517 1.3247 3.1856 1.3247 1.1087 2 3+ 4+ Effective interactions with the Lee-Suzuki method

Choose P-space for A-body calculation, with dimension dp P-space basis states: P and Q-space basis states: Q Need dP solutions, |k, in the infinite space, i.e., H k = E k k Write X=e- =QP Qe He P = 0; dP Q P = Q k k P ; k k P P k = kk P dP P H eff P = P (1+ ) + k 1 2 1 2 + P P k E k k P P (1+ ) P

P P dP P (1+ ) P = + Hermitian effective interaction P k k P k The matrix P|Heff|P exactly reproduces dp solutions of the full problem 22 For an n-body cluster in Heff, we must first solve the n-body problem! Find Hn-eff iteratively n-particles bound in oscillator potential Steps: A=2 H2-eff Use H2-eff for A=3 H3-eff Use H3-eff for A=4 H4-eff 23 Effective interactions really work He with the effective-field theory Idaho-A potential 4

Effective interactions improve convergence! Are EFT potentials useful for nuclear-structure studies? 24 But, we do need to use big computers Complex computational problem! Example: 10B, Nmax=4 H3-eff 39,523,066 3-particle matrix elements j|H|i matrix dimension: 581,740 581,740 =1.7 1011 Easy for H2-eff ~ 1-2 CPU-hr for lowest ten states H3-eff : j|H|i has 2.2 109 non-zero elements! 100 CPU-hr Three-body effective interaction takes 24-48 hours True three-body interaction ~ 1 week The future: H3-eff with Nmax=6 H4-eff with Nmax=4 25 Results with three-body effective interactions Binding energies with Av8 400 keV 1.8 MeV, Clustering? Three-body effective interactions represent a significant improvement and give results within

400 keV of the GFMC Model space Oscillator parameter 26 Excitation spectra with NN-interactions So far, things are looking pretty good! 27 Excitation spectrum of 8Be Experiments hint at a new excited state in 8Be Excitation energy ~10-30 MeV Previous theory studies were unable to predict the existence of such a state In large model spaces we find an intruder band 10h model space and 2108 basis states! No-core ANTIONE 28 Excitation spectrum of 8Be

Rotational Band What is the nature of this state? 9-10 MeV Is it real? 4+ 2+ 0+ Stable 0+, 2+, 4+ rotational band -1 I ~ 1 1.1 h MeV Ex appears to be stabilizing ~10-15 MeV or d E= J ( J + 1) 2I 30 Beta-vibration of the ground-state cluster? 25 Excitation energy seems too high 20 15(MeV)

Super-deformed prolate shape? + 0 x E Shell-Model Exponential 1/Nmax Gaussian 10 5 0 0 2 4 6 8 Nmax 29 10 12 14 16 The NN-interaction clearly has problems Note inversion of spins!

The NN-interaction by itself does not describe nuclear structure Also true for A=11 30 How about the three-body interaction? Tucson-Melbourne 3 Three-body interaction in a nucleus The three-nucleon interaction plays a critical role in determining the structure of nuclei 32 More on three-body interactions in a nucleus Gamow-Teller and M1 transitions Sensitive to spin-orbit force Because s is a generator of SU(4) and transitions in different representations are forbidden, i.e., B(GT)=0!! NN-interaction tends to preserve SU(4) Spin-orbit breaks SU(4) The three-nucleon interaction has a strong spinorbit component 33 Summary of ab initio studies Significant progress towards an exact understanding of nuclear structure is being made! These are exciting times!!

Extensions and improvements: Determine the form of the NNN-interaction Implementation of effective operators for transitions Four-body effective interactions Effective field-theory potentials; are they any good for structure? Integrate the structure into some reaction models (R-matrix) Questions and open problems to be addressed: Is it possible to improve the mean field? Can we improve the convergence of the higher h states? Unbound states. Can we use a continuum shell model? How high in A can we go? Can we use this method to derive effective interactions for conventional nuclear structure studies? 34