Abstract

We find the first, simple, analytical expressions for the bound states of a short-ranged, spherically-symmetric deep center potential in terms of the eigenstates of total (spin plus Bloch plus envelope) angular momentum and the Kane 8×8 k ·p Hamiltonian. We find that the spatial extent of the deep center bound state is proportional to the Kane dipole. This physical size of the deep center bound state is in excellent agreement with both scanning-tunneling-microscopy and measured optical dipoles.

© 2001 Optical Society of America

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  1. G. M. Martin, “Optical assessment of the main electron trap in bulk semi-insulating GaAs,” Appl. Phys. Lett. 39, 747–749, (1981).
    [Crossref]
  2. P. Silverberg, P. Omling, and L. Samuelson, “Hole photoionization cross sections of EL2 in GaAs,” Appl. Phys. Lett. 52, 1689–1691, (1988).
    [Crossref]
  3. A. Chantre, G. Vincent, and D. Bois, “Deep-level optical spectroscopy in GaAs,” Phys. Rev. B 23, 5335–5359, (1981).
    [Crossref]
  4. G. A. Baraff and M. A. Schluter, “Electronic aspects of the optical-absorption spectrum of the EL2 defect in GaAs,” Phys. Rev. B 45, 8300–8309, (1992).
    [Crossref]
  5. G. A. Baraff, “Stress splitting of the EL2 zero-phonon line: Need for reinterpretation of the main optical transitions,” Phys. Rev. B 41, 9850–9859, (1990).
    [Crossref]
  6. M.R. Melloch, J. M. Woodall, E. S. Harmon, N. Otsu ka, F. H. Pollak, D.D. Nolte, R.M. Feenstra, and M. A. Lutz, “Low-temperature grown III–V materials,” Ann. Rev. Mater. Sci. 25, 547–600, (1995).
    [Crossref]
  7. M. R. Melloch, D. D. Nolte, J. M. Woodall, J. C. P. Chang, D. B. Janes, and E. S. Harmon, “Molecular beam epitaxy of nonstoichiometric semiconductors and multiphase material systems,” Crit. Rev. in Sol. St. and Mater. Sci. 21, 189–263, (1996).
    [Crossref]
  8. G. Lucovsky, “On the photoionization of deep impurity centers in semiconductors,” Sold. St. Comm. 3, 299–302, (1965).
    [Crossref]
  9. S. T. Pantelides, “The electronic structure of impurities and other point defects in semiconductors,” Rev. Mod. Phys. 50, 798–858, (1978).
    [Crossref]
  10. M. Jaros, “Wave functions and optical cross sections associated with deep centers in semiconductors,” Phys. Rev. B 16, 3694–3706, (1977).
    [Crossref]
  11. E. O. Kane, “Band structure of indium antimonide,” J. Phys. Chem. Sol. 1, 249–261, (1957).
    [Crossref]
  12. A. Baldereschi and N. O. Lipari, “Spherical model of shallow acceptor states in semiconductors,” Phys. Rev. B 8, 2697–2709, (1973).
    [Crossref]
  13. P. C. Sercel and K. J. Vahala, “Analytical formalism for determining quantum-wire and quantum-dot band structure in the multiband envelope function approximation,” Phys. Rev. B 42, 3690–3710, (1990).
    [Crossref]
  14. R. M. Feenstra, “Cross-sectional scanning tunneling microscopy of III-V semiconductor structures,” Semicond. Sci. Tech. 9, 2157–2168, (1994).
    [Crossref]
  15. R. M. Feenstra, J. M. Woodall, and G. D. Pettit, “Observation of bulk defects by scanning tunneling microscopy and spectroscopy: Arsenic antisites defects in GaAs,” Phys. Rev. Lett. 71, 1176–1179, (1993).
    [Crossref] [PubMed]
  16. M. E. Rose, Elementary Theory of Angular Momentum. (Wiley, New York, 1957).
  17. We have also assumed the deep center bound states to have a Gaussian distribution with a standard deviation of 0.078eV, as is consistent with the scanning-tunneling-spectroscopy measurements of Feenstra [15].
  18. R. E. Viturro, M. R. Melloch, and J. M. Woodall, “Optical emission properties of semi-insulating GaAs grown at low temperatures by molecular beam epitaxy,” Appl. Phys. Lett. 60, 3007–3009, (1992).
    [Crossref]

1996 (1)

M. R. Melloch, D. D. Nolte, J. M. Woodall, J. C. P. Chang, D. B. Janes, and E. S. Harmon, “Molecular beam epitaxy of nonstoichiometric semiconductors and multiphase material systems,” Crit. Rev. in Sol. St. and Mater. Sci. 21, 189–263, (1996).
[Crossref]

1995 (1)

M.R. Melloch, J. M. Woodall, E. S. Harmon, N. Otsu ka, F. H. Pollak, D.D. Nolte, R.M. Feenstra, and M. A. Lutz, “Low-temperature grown III–V materials,” Ann. Rev. Mater. Sci. 25, 547–600, (1995).
[Crossref]

1994 (1)

R. M. Feenstra, “Cross-sectional scanning tunneling microscopy of III-V semiconductor structures,” Semicond. Sci. Tech. 9, 2157–2168, (1994).
[Crossref]

1993 (1)

R. M. Feenstra, J. M. Woodall, and G. D. Pettit, “Observation of bulk defects by scanning tunneling microscopy and spectroscopy: Arsenic antisites defects in GaAs,” Phys. Rev. Lett. 71, 1176–1179, (1993).
[Crossref] [PubMed]

1992 (2)

R. E. Viturro, M. R. Melloch, and J. M. Woodall, “Optical emission properties of semi-insulating GaAs grown at low temperatures by molecular beam epitaxy,” Appl. Phys. Lett. 60, 3007–3009, (1992).
[Crossref]

G. A. Baraff and M. A. Schluter, “Electronic aspects of the optical-absorption spectrum of the EL2 defect in GaAs,” Phys. Rev. B 45, 8300–8309, (1992).
[Crossref]

1990 (2)

G. A. Baraff, “Stress splitting of the EL2 zero-phonon line: Need for reinterpretation of the main optical transitions,” Phys. Rev. B 41, 9850–9859, (1990).
[Crossref]

P. C. Sercel and K. J. Vahala, “Analytical formalism for determining quantum-wire and quantum-dot band structure in the multiband envelope function approximation,” Phys. Rev. B 42, 3690–3710, (1990).
[Crossref]

1988 (1)

P. Silverberg, P. Omling, and L. Samuelson, “Hole photoionization cross sections of EL2 in GaAs,” Appl. Phys. Lett. 52, 1689–1691, (1988).
[Crossref]

1981 (2)

A. Chantre, G. Vincent, and D. Bois, “Deep-level optical spectroscopy in GaAs,” Phys. Rev. B 23, 5335–5359, (1981).
[Crossref]

G. M. Martin, “Optical assessment of the main electron trap in bulk semi-insulating GaAs,” Appl. Phys. Lett. 39, 747–749, (1981).
[Crossref]

1978 (1)

S. T. Pantelides, “The electronic structure of impurities and other point defects in semiconductors,” Rev. Mod. Phys. 50, 798–858, (1978).
[Crossref]

1977 (1)

M. Jaros, “Wave functions and optical cross sections associated with deep centers in semiconductors,” Phys. Rev. B 16, 3694–3706, (1977).
[Crossref]

1973 (1)

A. Baldereschi and N. O. Lipari, “Spherical model of shallow acceptor states in semiconductors,” Phys. Rev. B 8, 2697–2709, (1973).
[Crossref]

1965 (1)

G. Lucovsky, “On the photoionization of deep impurity centers in semiconductors,” Sold. St. Comm. 3, 299–302, (1965).
[Crossref]

1957 (1)

E. O. Kane, “Band structure of indium antimonide,” J. Phys. Chem. Sol. 1, 249–261, (1957).
[Crossref]

Baldereschi, A.

A. Baldereschi and N. O. Lipari, “Spherical model of shallow acceptor states in semiconductors,” Phys. Rev. B 8, 2697–2709, (1973).
[Crossref]

Baraff, G. A.

G. A. Baraff and M. A. Schluter, “Electronic aspects of the optical-absorption spectrum of the EL2 defect in GaAs,” Phys. Rev. B 45, 8300–8309, (1992).
[Crossref]

G. A. Baraff, “Stress splitting of the EL2 zero-phonon line: Need for reinterpretation of the main optical transitions,” Phys. Rev. B 41, 9850–9859, (1990).
[Crossref]

Bois, D.

A. Chantre, G. Vincent, and D. Bois, “Deep-level optical spectroscopy in GaAs,” Phys. Rev. B 23, 5335–5359, (1981).
[Crossref]

Chang, J. C. P.

M. R. Melloch, D. D. Nolte, J. M. Woodall, J. C. P. Chang, D. B. Janes, and E. S. Harmon, “Molecular beam epitaxy of nonstoichiometric semiconductors and multiphase material systems,” Crit. Rev. in Sol. St. and Mater. Sci. 21, 189–263, (1996).
[Crossref]

Chantre, A.

A. Chantre, G. Vincent, and D. Bois, “Deep-level optical spectroscopy in GaAs,” Phys. Rev. B 23, 5335–5359, (1981).
[Crossref]

Feenstra, R. M.

R. M. Feenstra, “Cross-sectional scanning tunneling microscopy of III-V semiconductor structures,” Semicond. Sci. Tech. 9, 2157–2168, (1994).
[Crossref]

R. M. Feenstra, J. M. Woodall, and G. D. Pettit, “Observation of bulk defects by scanning tunneling microscopy and spectroscopy: Arsenic antisites defects in GaAs,” Phys. Rev. Lett. 71, 1176–1179, (1993).
[Crossref] [PubMed]

Feenstra, R.M.

M.R. Melloch, J. M. Woodall, E. S. Harmon, N. Otsu ka, F. H. Pollak, D.D. Nolte, R.M. Feenstra, and M. A. Lutz, “Low-temperature grown III–V materials,” Ann. Rev. Mater. Sci. 25, 547–600, (1995).
[Crossref]

Harmon, E. S.

M. R. Melloch, D. D. Nolte, J. M. Woodall, J. C. P. Chang, D. B. Janes, and E. S. Harmon, “Molecular beam epitaxy of nonstoichiometric semiconductors and multiphase material systems,” Crit. Rev. in Sol. St. and Mater. Sci. 21, 189–263, (1996).
[Crossref]

M.R. Melloch, J. M. Woodall, E. S. Harmon, N. Otsu ka, F. H. Pollak, D.D. Nolte, R.M. Feenstra, and M. A. Lutz, “Low-temperature grown III–V materials,” Ann. Rev. Mater. Sci. 25, 547–600, (1995).
[Crossref]

Janes, D. B.

M. R. Melloch, D. D. Nolte, J. M. Woodall, J. C. P. Chang, D. B. Janes, and E. S. Harmon, “Molecular beam epitaxy of nonstoichiometric semiconductors and multiphase material systems,” Crit. Rev. in Sol. St. and Mater. Sci. 21, 189–263, (1996).
[Crossref]

Jaros, M.

M. Jaros, “Wave functions and optical cross sections associated with deep centers in semiconductors,” Phys. Rev. B 16, 3694–3706, (1977).
[Crossref]

Kane, E. O.

E. O. Kane, “Band structure of indium antimonide,” J. Phys. Chem. Sol. 1, 249–261, (1957).
[Crossref]

Lipari, N. O.

A. Baldereschi and N. O. Lipari, “Spherical model of shallow acceptor states in semiconductors,” Phys. Rev. B 8, 2697–2709, (1973).
[Crossref]

Lucovsky, G.

G. Lucovsky, “On the photoionization of deep impurity centers in semiconductors,” Sold. St. Comm. 3, 299–302, (1965).
[Crossref]

Lutz, M. A.

M.R. Melloch, J. M. Woodall, E. S. Harmon, N. Otsu ka, F. H. Pollak, D.D. Nolte, R.M. Feenstra, and M. A. Lutz, “Low-temperature grown III–V materials,” Ann. Rev. Mater. Sci. 25, 547–600, (1995).
[Crossref]

Martin, G. M.

G. M. Martin, “Optical assessment of the main electron trap in bulk semi-insulating GaAs,” Appl. Phys. Lett. 39, 747–749, (1981).
[Crossref]

Melloch, M. R.

M. R. Melloch, D. D. Nolte, J. M. Woodall, J. C. P. Chang, D. B. Janes, and E. S. Harmon, “Molecular beam epitaxy of nonstoichiometric semiconductors and multiphase material systems,” Crit. Rev. in Sol. St. and Mater. Sci. 21, 189–263, (1996).
[Crossref]

R. E. Viturro, M. R. Melloch, and J. M. Woodall, “Optical emission properties of semi-insulating GaAs grown at low temperatures by molecular beam epitaxy,” Appl. Phys. Lett. 60, 3007–3009, (1992).
[Crossref]

Melloch, M.R.

M.R. Melloch, J. M. Woodall, E. S. Harmon, N. Otsu ka, F. H. Pollak, D.D. Nolte, R.M. Feenstra, and M. A. Lutz, “Low-temperature grown III–V materials,” Ann. Rev. Mater. Sci. 25, 547–600, (1995).
[Crossref]

Nolte, D. D.

M. R. Melloch, D. D. Nolte, J. M. Woodall, J. C. P. Chang, D. B. Janes, and E. S. Harmon, “Molecular beam epitaxy of nonstoichiometric semiconductors and multiphase material systems,” Crit. Rev. in Sol. St. and Mater. Sci. 21, 189–263, (1996).
[Crossref]

Nolte, D.D.

M.R. Melloch, J. M. Woodall, E. S. Harmon, N. Otsu ka, F. H. Pollak, D.D. Nolte, R.M. Feenstra, and M. A. Lutz, “Low-temperature grown III–V materials,” Ann. Rev. Mater. Sci. 25, 547–600, (1995).
[Crossref]

Omling, P.

P. Silverberg, P. Omling, and L. Samuelson, “Hole photoionization cross sections of EL2 in GaAs,” Appl. Phys. Lett. 52, 1689–1691, (1988).
[Crossref]

Otsu ka, N.

M.R. Melloch, J. M. Woodall, E. S. Harmon, N. Otsu ka, F. H. Pollak, D.D. Nolte, R.M. Feenstra, and M. A. Lutz, “Low-temperature grown III–V materials,” Ann. Rev. Mater. Sci. 25, 547–600, (1995).
[Crossref]

Pantelides, S. T.

S. T. Pantelides, “The electronic structure of impurities and other point defects in semiconductors,” Rev. Mod. Phys. 50, 798–858, (1978).
[Crossref]

Pettit, G. D.

R. M. Feenstra, J. M. Woodall, and G. D. Pettit, “Observation of bulk defects by scanning tunneling microscopy and spectroscopy: Arsenic antisites defects in GaAs,” Phys. Rev. Lett. 71, 1176–1179, (1993).
[Crossref] [PubMed]

Pollak, F. H.

M.R. Melloch, J. M. Woodall, E. S. Harmon, N. Otsu ka, F. H. Pollak, D.D. Nolte, R.M. Feenstra, and M. A. Lutz, “Low-temperature grown III–V materials,” Ann. Rev. Mater. Sci. 25, 547–600, (1995).
[Crossref]

Rose, M. E.

M. E. Rose, Elementary Theory of Angular Momentum. (Wiley, New York, 1957).

Samuelson, L.

P. Silverberg, P. Omling, and L. Samuelson, “Hole photoionization cross sections of EL2 in GaAs,” Appl. Phys. Lett. 52, 1689–1691, (1988).
[Crossref]

Schluter, M. A.

G. A. Baraff and M. A. Schluter, “Electronic aspects of the optical-absorption spectrum of the EL2 defect in GaAs,” Phys. Rev. B 45, 8300–8309, (1992).
[Crossref]

Sercel, P. C.

P. C. Sercel and K. J. Vahala, “Analytical formalism for determining quantum-wire and quantum-dot band structure in the multiband envelope function approximation,” Phys. Rev. B 42, 3690–3710, (1990).
[Crossref]

Silverberg, P.

P. Silverberg, P. Omling, and L. Samuelson, “Hole photoionization cross sections of EL2 in GaAs,” Appl. Phys. Lett. 52, 1689–1691, (1988).
[Crossref]

Vahala, K. J.

P. C. Sercel and K. J. Vahala, “Analytical formalism for determining quantum-wire and quantum-dot band structure in the multiband envelope function approximation,” Phys. Rev. B 42, 3690–3710, (1990).
[Crossref]

Vincent, G.

A. Chantre, G. Vincent, and D. Bois, “Deep-level optical spectroscopy in GaAs,” Phys. Rev. B 23, 5335–5359, (1981).
[Crossref]

Viturro, R. E.

R. E. Viturro, M. R. Melloch, and J. M. Woodall, “Optical emission properties of semi-insulating GaAs grown at low temperatures by molecular beam epitaxy,” Appl. Phys. Lett. 60, 3007–3009, (1992).
[Crossref]

Woodall, J. M.

M. R. Melloch, D. D. Nolte, J. M. Woodall, J. C. P. Chang, D. B. Janes, and E. S. Harmon, “Molecular beam epitaxy of nonstoichiometric semiconductors and multiphase material systems,” Crit. Rev. in Sol. St. and Mater. Sci. 21, 189–263, (1996).
[Crossref]

M.R. Melloch, J. M. Woodall, E. S. Harmon, N. Otsu ka, F. H. Pollak, D.D. Nolte, R.M. Feenstra, and M. A. Lutz, “Low-temperature grown III–V materials,” Ann. Rev. Mater. Sci. 25, 547–600, (1995).
[Crossref]

R. M. Feenstra, J. M. Woodall, and G. D. Pettit, “Observation of bulk defects by scanning tunneling microscopy and spectroscopy: Arsenic antisites defects in GaAs,” Phys. Rev. Lett. 71, 1176–1179, (1993).
[Crossref] [PubMed]

R. E. Viturro, M. R. Melloch, and J. M. Woodall, “Optical emission properties of semi-insulating GaAs grown at low temperatures by molecular beam epitaxy,” Appl. Phys. Lett. 60, 3007–3009, (1992).
[Crossref]

Ann. Rev. Mater. Sci. (1)

M.R. Melloch, J. M. Woodall, E. S. Harmon, N. Otsu ka, F. H. Pollak, D.D. Nolte, R.M. Feenstra, and M. A. Lutz, “Low-temperature grown III–V materials,” Ann. Rev. Mater. Sci. 25, 547–600, (1995).
[Crossref]

Appl. Phys. Lett. (3)

G. M. Martin, “Optical assessment of the main electron trap in bulk semi-insulating GaAs,” Appl. Phys. Lett. 39, 747–749, (1981).
[Crossref]

P. Silverberg, P. Omling, and L. Samuelson, “Hole photoionization cross sections of EL2 in GaAs,” Appl. Phys. Lett. 52, 1689–1691, (1988).
[Crossref]

R. E. Viturro, M. R. Melloch, and J. M. Woodall, “Optical emission properties of semi-insulating GaAs grown at low temperatures by molecular beam epitaxy,” Appl. Phys. Lett. 60, 3007–3009, (1992).
[Crossref]

Crit. Rev. in Sol. St. and Mater. Sci. (1)

M. R. Melloch, D. D. Nolte, J. M. Woodall, J. C. P. Chang, D. B. Janes, and E. S. Harmon, “Molecular beam epitaxy of nonstoichiometric semiconductors and multiphase material systems,” Crit. Rev. in Sol. St. and Mater. Sci. 21, 189–263, (1996).
[Crossref]

J. Phys. Chem. Sol. (1)

E. O. Kane, “Band structure of indium antimonide,” J. Phys. Chem. Sol. 1, 249–261, (1957).
[Crossref]

Phys. Rev. B (6)

A. Baldereschi and N. O. Lipari, “Spherical model of shallow acceptor states in semiconductors,” Phys. Rev. B 8, 2697–2709, (1973).
[Crossref]

P. C. Sercel and K. J. Vahala, “Analytical formalism for determining quantum-wire and quantum-dot band structure in the multiband envelope function approximation,” Phys. Rev. B 42, 3690–3710, (1990).
[Crossref]

M. Jaros, “Wave functions and optical cross sections associated with deep centers in semiconductors,” Phys. Rev. B 16, 3694–3706, (1977).
[Crossref]

A. Chantre, G. Vincent, and D. Bois, “Deep-level optical spectroscopy in GaAs,” Phys. Rev. B 23, 5335–5359, (1981).
[Crossref]

G. A. Baraff and M. A. Schluter, “Electronic aspects of the optical-absorption spectrum of the EL2 defect in GaAs,” Phys. Rev. B 45, 8300–8309, (1992).
[Crossref]

G. A. Baraff, “Stress splitting of the EL2 zero-phonon line: Need for reinterpretation of the main optical transitions,” Phys. Rev. B 41, 9850–9859, (1990).
[Crossref]

Phys. Rev. Lett. (1)

R. M. Feenstra, J. M. Woodall, and G. D. Pettit, “Observation of bulk defects by scanning tunneling microscopy and spectroscopy: Arsenic antisites defects in GaAs,” Phys. Rev. Lett. 71, 1176–1179, (1993).
[Crossref] [PubMed]

Rev. Mod. Phys. (1)

S. T. Pantelides, “The electronic structure of impurities and other point defects in semiconductors,” Rev. Mod. Phys. 50, 798–858, (1978).
[Crossref]

Semicond. Sci. Tech. (1)

R. M. Feenstra, “Cross-sectional scanning tunneling microscopy of III-V semiconductor structures,” Semicond. Sci. Tech. 9, 2157–2168, (1994).
[Crossref]

Sold. St. Comm. (1)

G. Lucovsky, “On the photoionization of deep impurity centers in semiconductors,” Sold. St. Comm. 3, 299–302, (1965).
[Crossref]

Other (2)

M. E. Rose, Elementary Theory of Angular Momentum. (Wiley, New York, 1957).

We have also assumed the deep center bound states to have a Gaussian distribution with a standard deviation of 0.078eV, as is consistent with the scanning-tunneling-spectroscopy measurements of Feenstra [15].

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Figures (4)

Fig. 1.
Fig. 1.

F = 1 2 , F z = 1 2 ; L = 0 , J = 1 2 ( L B = 0 ) is the part of |d〉 which has conduction-band character. The spatial extent, α -1, of the bound state is in excellent agreement with STM images by Feenstra [14, 15].

Fig. 2.
Fig. 2.

F = 1 2 , F z = 1 2 ;L=1, J = 3 2 ( L B = 1 ) is the part of |d〉 which has light-hole character. Yellow and green, respectively, denote Bloch and envelope wave functions.

Fig. 3.
Fig. 3.

The antisymmetric linear combination of orbitals |1〉+|2〉-|3〉-|4〉 has the same symmetry as the |L=1, Lz =0〉 envelope function in Fig.2.

Fig. 4.
Fig. 4.

The solid line is the calculated cross-section (from Eq. (6)) for emission from the AsGa antisite to the valence band. The circles denote measurements by Silverberg [2].

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

d = N [ E d E V E G f L = 0 ( r ) F = 1 2 , F z = 1 2 ; L = 0 , J = 1 2 ( L B = 0 ) +
+ 2 3 E C E d E G f L = 1 ( r ) F = 1 2 , F z = 1 2 ; L = 1 , J = 3 2 ( L B = 1 ) +
+ 1 3 E C E d E G f L = 1 ( r ) F = 1 2 , F z = 1 2 ; L = 1 , J = 1 2 ( L B = 1 ) ]
[ E + E + 2 3 ] E ( E E G ) = { P 2 k 2 for E E G , or E < 0 P 2 α 2 for 0 < E < E G
α 1 = < s z z E G E d ( E G E d )
2 < s z z for deep center ,
v = N [ E E V E G f L = 1 ( r ) F = 1 2 , F z = 1 2 ; L = 1 , J = 1 2 ( L B = 0 ) +
+ 2 3 E C E E G f L = 2 ( r ) F = 1 2 , F z = 1 2 ; L = 2 , J = 3 2 ( L B = 1 ) +
+ 1 3 E C E E G f L = 0 ( r ) F = 1 2 , F z = 1 2 ; L = 0 , J = 1 2 ( L B = 1 ) ]
v = N [ E E V E G f L = 1 ( r ) F = 3 2 , F z = 1 2 ; L = 1 , J = 1 2 ( L B = 0 ) +
+ 1 3 E C E E G f L = 0 ( r ) F = 3 2 , F z = 1 2 ; L = 0 , J = 3 2 ( L B = 1 ) +
+ 1 3 E C E E G f L = 2 ( r ) F = 3 2 , F z = 1 2 ; L = 2 , J = 3 2 ( L B = 1 ) +
+ 1 3 E C E E G f L = 2 ( r ) F = 3 2 , F z = 1 2 ; L = 2 , J = 1 2 ( L B = 1 ) ]
f = 2 3 f ( d , l h ) + 1 3 f ( d , h h )
f ( d , h h ) g h h ( E ) = 4 3 π [ 1 E d E ] E P E G { E d 3 / 2 ( E G E d ) 1 / 2 ( E ) 1 / 2 ( E G E ) 3 / 2 [ E d ( E G E d ) + ( E ) ( E G E ) ] 2 }
× [ E G ( α R C / 2 ) E d ( α R C / 2 ) + ( E G E d ) ] ,
f ( d , lh ) glh ( E ) = 4 3 π [ 1 E d E ] E P E G { E d 3 / 2 ( E G E d ) 1 / 2 ( E ) 1 / 2 ( m H H m 0 E P ) 3 / 2 [ E d ( E G E d ) + ( E ) ( m H H m 0 E P ) ] 2 }
× [ E G ( α R C / 2 ) E d ( α R C / 2 ) + ( E G E d ) ] .

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