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INT October 28, 2004Mihai Horoi - Central Michigan Univ1 New Approaches for Spin- and Parity-Dependent Shell Model Nuclear Level Density Mihai Horoi, Department.

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Presentation on theme: "INT October 28, 2004Mihai Horoi - Central Michigan Univ1 New Approaches for Spin- and Parity-Dependent Shell Model Nuclear Level Density Mihai Horoi, Department."— Presentation transcript:

1 INT October 28, 2004Mihai Horoi - Central Michigan Univ1 New Approaches for Spin- and Parity-Dependent Shell Model Nuclear Level Density Mihai Horoi, Department of Physics, Central Michigan University, Mount Pleasant, Michigan 48859, USA  Support from NSF grant PHY-02-44453 is acknowledged

2 INT October 28, 2004Mihai Horoi - Central Michigan Univ2 Plan of the Talk Part I: Methods for Shell Model NLD –Motivation –Sum on partitions vs moments of the whole density –Exponential Convergence Method –Fixed-J Configuration Centroids and Widths –Energy-Dependent Cutoff Description –PRC 67, 054309(2003), PRC 69, 041307(2004) Part II: Methods of Removal of the Center-of-Mass Spurious Contribution

3 INT October 28, 2004Mihai Horoi - Central Michigan Univ3 Hauser and Feshbach, Phys. Rev 87, 366 (1952)

4 INT October 28, 2004Mihai Horoi - Central Michigan Univ4 The Back-Shifted Fermi Gas Model for Nuclear Level Density

5 INT October 28, 2004Mihai Horoi - Central Michigan Univ5 A.Adams, G.Mitchell, J.F. Shriner Phys.Lett, B422, 13(1998) 26 Al sd-shell model, USD interaction

6 INT October 28, 2004Mihai Horoi - Central Michigan Univ6 Data: Table of Isotopes Theory: sd-shell model + USD interaction 28 Si: positive parity

7 INT October 28, 2004Mihai Horoi - Central Michigan Univ7 p - 10 2 -1960’s sd - 10 5 - 1980’s pf - 10 9 - 1990’s pf 5/2 -g 9/2 - 10 10 - 2006 Example: 76 Sr PRL 92, 232501 pf 5/2 -g 9/2 dimension 11,090,052,440 CMichSM code - m-scheme dimension 250,000,000 on one-processor machine - 150 Lanczos iterations/week

8 INT October 28, 2004Mihai Horoi - Central Michigan Univ8 12 particles in sd model space

9 INT October 28, 2004Mihai Horoi - Central Michigan Univ9 Nuclear Shell Model d = 2 (2 j + 1)

10 INT October 28, 2004Mihai Horoi - Central Michigan Univ10 Sum on Partitions vs Moments of the Whole Distribution 6 particles in pf 5/2 -g 9/2 New interaction A. Lisetskiy et al. PRC 2004

11 INT October 28, 2004Mihai Horoi - Central Michigan Univ11 12 particles in sd model space

12 INT October 28, 2004Mihai Horoi - Central Michigan Univ12 12 particles in sd model space

13 INT October 28, 2004Mihai Horoi - Central Michigan Univ13 6 particles in p-sd model space

14 INT October 28, 2004Mihai Horoi - Central Michigan Univ14 Exponential Convergence Method

15 INT October 28, 2004Mihai Horoi - Central Michigan Univ15 Exponential Convergence Method for fp-nuclei

16 INT October 28, 2004Mihai Horoi - Central Michigan Univ16 Exponential Convergence Method for fp-nuclei Central Michigan Shell Model (CMichSM) code Exact: -203.196 MeV

17 INT October 28, 2004Mihai Horoi - Central Michigan Univ17 r,s,.. – orbits, not states

18 INT October 28, 2004Mihai Horoi - Central Michigan Univ18 Fixed J Configura tion Centroids and Widths C. Jacquemin, Z. Phys. A 303, 135 (1981)

19 INT October 28, 2004Mihai Horoi - Central Michigan Univ19 Shell Model vs Fixed-J Centroids and Widths Density of States 28 Si: 12 particles in sd, Tz=0

20 INT October 28, 2004Mihai Horoi - Central Michigan Univ20 Shell Model vs Fixed-J Centroids and Widths Density of States 28 Si: 12 particles in sd, Tz=0

21 INT October 28, 2004Mihai Horoi - Central Michigan Univ21 Spin Cutoff Factor Zeroth-Order: S.S.M. Wong, Nuclear Spectroscopy, Oxford 1986, p. 45,171 28 Si: 12 particles in sd, Tz=0

22 INT October 28, 2004Mihai Horoi - Central Michigan Univ22 Shell Model 28 Si: 12 particles in sd, Tz=0

23 INT October 28, 2004Mihai Horoi - Central Michigan Univ23 Shell Model 28 Si: 12 particles in sd, Tz=0

24 INT October 28, 2004Mihai Horoi - Central Michigan Univ24 Zeroth-Order 28 Si: 12 particles in sd, Tz=0

25 INT October 28, 2004Mihai Horoi - Central Michigan Univ25 Zeroth-Order 28 Si: 12 particles in sd, Tz=0

26 INT October 28, 2004Mihai Horoi - Central Michigan Univ26 Summary of Part I Shell Model NLD look very promising, at least up to the particle emission threshold. More comparison with experimental data necessary. J-dependent SM NLD are very accurately described by a sum of finite range Gaussians with fixed-J centroids and widths, if one knows with good precision the energy of g.s. and yrast states. We derived explicit expression to calculate fixed-J centroids and widths. Exponential Convergence Method (ECM) proves to be a very powerful tool for finding yrast and non-yrast energies, by doing shell model calculations in truncated model spaces. J-dependent SM NLD are reasonably well described by spin cutoff formula with exact cutoff factor, except for higher J’s, but not very well described by spin cutoff formula with zeroth-order cutoff factor. Improvement in estimating cutoff factor requires knowledge of higher order moments.

27 INT October 28, 2004Mihai Horoi - Central Michigan Univ27 The Center-of-Mass Problem nucl-th/0111068

28 INT October 28, 2004Mihai Horoi - Central Michigan Univ28 Nuclear Shell Model N

29 INT October 28, 2004Mihai Horoi - Central Michigan Univ29 The Center-of-Mass Problem

30 INT October 28, 2004Mihai Horoi - Central Michigan Univ30 No E (MeV) Ex (MeV) J T 1 -40.0000 0.0000 0.0000 0.0000 2 -39.6000 0.4000 1.0000 0.0000 3 -39.0000 1.0000 1.0000 1.0000 4 -38.8000 1.2000 2.0000 0.0000 5 -38.8000 1.2000 2.0000 0.0000 6 -37.6000 2.4000 3.0000 0.0000 7 -37.0000 3.0000 3.0000 1.0000 8 -36.0000 4.0000 4.0000 0.0000 9 -29.6000 10.4000 1.0000 0.0000 10 -29.4000 10.6000 0.0000 1.0000 11 -29.0000 11.0000 1.0000 1.0000 12 -29.0000 11.0000 1.0000 1.0000 13 -28.8000 11.2000 2.0000 0.0000 14 -28.2000 11.8000 2.0000 1.0000 15 -28.2000 11.8000 2.0000 1.0000 16 -27.0000 13.0000 3.0000 1.0000 17 -10.0000 30.0000 0.0000 0.0000 18 -9.6000 30.4000 1.0000 0.0000 19 -9.6000 30.4000 1.0000 0.0000 20 -9.0000 31.0000 1.0000 1.0000 21 -8.8000 31.2000 2.0000 0.0000 22 -8.8000 31.2000 2.0000 0.0000 23 -8.2000 31.8000 2.0000 1.0000 24 -7.6000 32.4000 3.0000 0.0000 25 0.4000 40.4000 1.0000 0.0000 No E (MeV) Ex (MeV) J T 1 -60.0000 0.0000 0.0000 0.0000 2 -60.0000 0.0000 0.0000 0.0000 3 -60.0000 0.0000 0.0000 0.0000 4 -59.6000 0.4000 1.0000 0.0000 5 -59.6000 0.4000 1.0000 0.0000 6 -59.6000 0.4000 1.0000 0.0000 7 -59.6000 0.4000 1.0000 0.0000 8 -59.4000 0.6000 0.0000 1.0000 9 -59.0000 1.0000 1.0000 1.0000 10 -59.0000 1.0000 1.0000 1.0000 11 -59.0000 1.0000 1.0000 1.0000 12 -59.0000 1.0000 1.0000 1.0000 13 -59.0000 1.0000 1.0000 1.0000 14 -58.8000 1.2000 2.0000 0.0000 15 -58.8000 1.2000 2.0000 0.0000 16 -58.8000 1.2000 2.0000 0.0000 17 -58.8000 1.2000 2.0000 0.0000 18 -58.8000 1.2000 2.0000 0.0000 19 -58.2000 1.8000 2.0000 1.0000 20 -58.2000 1.8000 2.0000 1.0000 21 -58.2000 1.8000 2.0000 1.0000 22 -57.6000 2.4000 3.0000 0.0000 23 -57.6000 2.4000 3.0000 0.0000 24 -57.0000 3.0000 3.0000 1.0000 25 -57.0000 3.0000 3.0000 1.0000 26 -56.0000 4.0000 4.0000 0.0000 27 0.0000 60.0000 0.0000 0.0000 p-sd s-p-sd 2 particles 6 particles N = 1

31 INT October 28, 2004Mihai Horoi - Central Michigan Univ31 Dimensions of Nonspurious Spaces Example: s-p-sd, 6 particles J N=1(K=1) N=0 0 4 =4 2 1 2+4+3=9 4 2 4+3+1=8 3 3 3+1 =4 1 41 =1 0 Total 26

32 INT October 28, 2004Mihai Horoi - Central Michigan Univ32

33 INT October 28, 2004Mihai Horoi - Central Michigan Univ33 C. Jacquemin, Z. Phys. A 303, 135 (1981) Fixed J Restricted Configura tion Widths

34 INT October 28, 2004Mihai Horoi - Central Michigan Univ34

35 INT October 28, 2004Mihai Horoi - Central Michigan Univ35 N   Nonspurious Level Density

36 INT October 28, 2004Mihai Horoi - Central Michigan Univ36 20 Ne: 20 particles in s-p-sd-pf shell model space

37 INT October 28, 2004Mihai Horoi - Central Michigan Univ37 20 Ne: 20 particles in s-p-sd-pf shell model space

38 INT October 28, 2004Mihai Horoi - Central Michigan Univ38 20 Ne: 20 particles in s-p-sd-pf shell model space

39 INT October 28, 2004Mihai Horoi - Central Michigan Univ39 Nonspurious Level Density: (0+2)  

40 INT October 28, 2004Mihai Horoi - Central Michigan Univ40 10 B: 10 particles in s-p-sd-pf shell model space

41 INT October 28, 2004Mihai Horoi - Central Michigan Univ41 10 B: 10 particles in s-p-sd-pf shell model space

42 INT October 28, 2004Mihai Horoi - Central Michigan Univ42 Nonspurious Level Density: General

43 INT October 28, 2004Mihai Horoi - Central Michigan Univ43 Summary We derived explicit expressions to calculate fixed- J centroids and widths for restricted set of configurations, such N   configurations We found recursive formulae to calculate the dimensions of nospurious spaces We found recursive formulae for calculating exactly the nonspurious level density when one knows the level density for a restricted set of configurations, such N   configurations Using our method of calculating the level density for restricted set of configurations we can calculate very accurately the nonspurious level density

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