Nuclear Stretched States
excited by
Electron Scattering

Richard A. Lindgren, University of Virginia
Benjamin L. Clausen, Loma Linda University

• Equations: |F_T(q)|², q_eff
• Data Sources: references

Tables

• Data — ¹⁰B, ¹²C, ¹³C, ¹⁴C, ¹⁴N, ¹⁶O, ¹⁷O, ¹⁸O, ²⁴Mg, ²⁶Mg, ²⁸Si, ³⁰Si, ³²S, ⁴⁸Ca, ⁵²Cr, ⁵⁴Fe, ⁵⁶Fe, ⁵⁸Ni, ⁶⁰Ni, ⁹⁰Zr, ²⁰⁸Pb
• Fit results — using harmonic oscillator (HO) and Woods-Saxon (WS) wave functions

Plots [all 89 graphs = 774 KB]

- Individually --- nucleus / stretched state (j_bj_a^-1) J_max / excitation energies (MeV)

• ¹⁰B (p_3/2p_3/2^-1) 3⁺ — 1.74
• ¹²C (d_5/2p_3/2^-1) 4^- — 19.5
• ¹³C (d_5/2p_3/2^-1) 4^- — 9.48, 16.08, 21.47
• ¹⁴C (d_5/2p_3/2^-1) 4^- — 11.67, 17.26, 24.4
• ¹⁴N (d_5/2p_3/2^-1) 4^- — 16.91, 18.5
• ¹⁶O (d_5/2p_3/2^-1) 4^- — 17.788, 17.881, 18.63, 18.975, 19.802, 20.51
• ¹⁷O (d_5/2p_3/2^-1) 4^- — 15.78, 17.06, 20.14, 20.7
• ¹⁸O (d_5/2p_3/2^-1) 4^- — 18.7, 20.36, 22.39
• ²⁴Mg (f_7/2d_5/2^-1) 6^- — 15.13
• ²⁶Mg (f_7/2d_5/2^-1) 6^- — 7.54, 9.17, 12.5, 12.88, 13.0, 13.97, 14.5, 15.36, 15.46, 16.50, 18.049
• ²⁸Si (f_7/2d_5/2^-1) 6^- — 14.356
• ³⁰Si (f_7/2d_5/2^-1) 6^- — 9.78, 10.57, 12.87, 14.34, 15.36, 17.03
• ³²S (f_7/2d_5/2^-1) 6^- — 10.98, 11.17, 11.94, 12.74, 13.26, 13.54, 14.29, 16.43, 17.16
• ⁴⁸Ca (g_9/2f_7/2^-1) 8^- — 9.138, 9.276, 9.953
• ⁵²Cr (g_9/2f_7/2^-1) 8^- — 8.1, 9.66, 15.47
• ⁵⁴Fe (g_9/2f_7/2^-1) 8^- — 8.34, 8.961, 9.98, 10.691, 13.263
• ⁵⁶Fe (g_9/2f_7/2^-1) 8^- — 8.69, 9.0, 10.27, 10.42, 10.74, 12.97, 13.74, 14.56, 15.65, 16.29
• ⁵⁸Ni (g_9/2f_7/2^-1) 8^- — 7.937, 8.808, 10.19, 11.239, 12.50
• ⁶⁰Ni (g_9/2f_7/2^-1) 8^- — 7.522, 8.433, 8.959, 9.172, 12.333, 12.505, 13.908, 14.84, 15.499, 16.08
• ⁹⁰Zr (h_11/2g_9/2^-1) 10^- — 7.36
• ²⁰⁸Pb (j_15/2i_13/2^-1) 14^- — 6.74

Equations — taken from: Clausen et al., Phys. Rev. C 38, 589 (1988)

- experimental form factor

| F_T(q) | ² = ds/dW • { [1 + 2 (E₀/M_T) sin²(q/2)] / (Z²s_M) } ¸ { ½ (q_m²/q²) + tan²(q/2) }

• F_T = transverse or magnetic form factor
• E₀ = incident electron energy
• M_T = target mass
• q = scattering angle
• Z = nuclear charge
• s_M = Mott cross section
• q_m = four momentum transferred from the electron to the nucleus
• q² = q_m² + w², where w is the difference between the incoming and outgoing electron energy

- theoretical form factor

| F_T(q) | ² = { f_c.m. f_fs ¹/_Z • [(q ^h/_2p)/(2Mc)] • [(J+1)/(2J)]^½ • Ö(2j_a+1) Ö(2j_b+1) • < j_a j_b ½ -½ | J 0 > • 1/Ö2 å_a [C_a Z_a g_a^s I_a] }²

• for unique (1 ^h/_2p w) stretched transitions of the type (j_bj_a^-1) J_max
  • j_b = l_b + ½, the largest angular momenta found in the first open shell
  • j_a = l_a + ½, the largest angular momenta found in the last filled shell
  • J_max = j_a + j_b
  • l_b = l_a + 1

• f_c.m.(q) = exp[(bq/2)² / A]
• f_fs = finite-size correction of Simon et al.
• C_a = Ö2 for a=0,1 and C_a = 1 for a=n,p
• Z_n = (Z₀+Z₁)/Ö2 and Z_p = (Z₀-Z₁)/Ö2
• g₀^s = m_n+m_p and g₁^s = m_n-m_p
• g_n^s = 2m_n and g_p^s = 2m_p
• I_a(q) = ò₀^¥ y_aa(r) j_J-1(qr) y_ba(r) r² dr
  • for harmonic oscillator wave functions, I_a = { (bq)^J-1 exp[-(bq/2)²] } ¸ { 2^(J-1)/2 [ (J+1)!! (J-1)!! ]^½ }

- effective momentum transfer

q_eff = q • [1 + 3Z•(1.44 MeV-fm) / (2E₀R) ]

Data Sources

• ¹⁰B — Ansaldo et al., Nucl. Phys. A322, 237 (1979) AND Hicks et al., Phys. Rev. Lett. 60, 905 (1988)
• ¹²C — Hicks et al., Phys. Rev. C 30, 1 (1984)
• ¹³C — Hicks et al., Phys. Rev. C 34, 1161 (1986)
• ¹⁴C — Plum [Ph.D. Thesis, University of Massachusetts, Amherst (1985)] AND Plum et al., Phys. Rev. C 40, 1861 (1989)
• ¹⁴N — Bergstrom et al., Phys. Rev. C 29, 1168 (1984)
• ¹⁶O — Hyde-Wright [Ph.D. thesis, Massachusetts Institute of Technology (1984)]
• ¹⁷O — Manley et al., Phys. Rev. C 34, 1214 (1986)
• ¹⁸O — Manley et al., Phys. Rev. C 34, 1214 (1986)
• ²⁴Mg — personal communication AND Zarek et al., Phys. Rev. Lett. 38, 750 (1977) / Phys. Rev. C 29, 1664 (1984)
• ²⁶Mg — Plum [Ph.D. Thesis, University of Massachusetts, Amherst (1985)]
• ²⁸Si — personal communication AND Yen et al., Phys. Lett. 93B, 250 (1980) / Phys. Rev. C 27, 1939 (1983) / Phys. Lett. 289B, 22 (1992)
• ³⁰Si — personal files AND Knox et al., Bull. Am. Phys. Soc. 34, 1812 (1989)
• ³²S — personal files AND Clausen et al., Phys. Rev. Lett. 65, 547 (1990)
• ⁴⁸Ca — Wise et al., Phys. Rev. C 31, 1699 (1985) AND Wise [Ph.D. Thesis, University of Virginia, (1976)]
• ⁵²Cr — Sober et al., Phys. Rev. C 38, 654 (1988)
• ⁵⁴Fe — personal files AND Lindgren et al., Phys. Rev. Lett. 46, 706 (1981)
• ⁵⁶Fe — personal files AND Maruyama et al., Bull. Am. Phys. Soc. 26, 591 (1981)
• ⁵⁸Ni — personal files AND Lindgren et al., Phys. Rev. Lett. 40, 504 (1978)
• ⁶⁰Ni — personal files AND Lindgren et al., Phys. Rev. Lett. 47, 1266 (1981)
• ⁹⁰Zr — personal communication AND Hei,ND
• ²⁰⁸Pb — Lichtenstadt et al., Phys. Rev. C 20, 497 (1979)

Last update --- October 14, 1999 / Ben Clausen