Abstract

We propose a simple atomic-beam collimator consisting of a blue-detuned cylindrical-hollow (doughnut) laser beam. Cold atoms from a magneto-optical trap are loaded into the doughnut beam and guided in the propagation direction of the doughnut beam. The cold atoms experience efficient Sisyphus cooling induced by the doughnut beam and a weak repumping beam and are collimated in the transverse direction. In the longitudinal direction the cold atoms experience small accelerations that are due to the absorption-induced heating of the doughnut beam and the repumping beam. The cooling, heating, and loss mechanisms in this atomic collimator are analyzed. Our study shows that an intense cold atomic beam with a minimum transverse velocity of 2 cm s-1 (a minimum transverse temperature of ∼1.4 μK) and a brightness of approximately 2×1013 atoms (sr-1 cm-2 s-1) can be obtained from the collimator.

© 1998 Optical Society of America

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  1. D. W. Keith, C. R. Ekstrom, Q. A. Turchette, and D. E. Pritchard, Phys. Rev. Lett. 66, 2693 (1991).
    [CrossRef] [PubMed]
  2. J. Fujita, M. Morinaga, T. Kishimoto, M. Yasuda, S. Matsui, and F. Shimizu, Nature (London) 380, 691 (1996).
    [CrossRef]
  3. G. Timp, R. E. Behringer, D. M. Tennant, J. E. Cunningham, M. Prentiss, and K. K. Berggren, Phys. Rev. Lett. 69, 1636 (1992).
    [CrossRef] [PubMed]
  4. P. Westphal, S. Koch, A. Horn, J. Schmand, and H. J. Andra, Phys. Rev. Lett. 56, 2784 (1997).
  5. W. D. Phillips and H. Metcalf, Phys. Rev. Lett. 48, 596 (1982); T. E. Barrett, S. W. D. Schwartz, M. D. Ray, and G. P. Lafyatis, Phys. Rev. Lett. 67, 3483 (1991).
    [CrossRef] [PubMed]
  6. W. Ertmer, R. Blatt, J. L. Hall, and M. Zhu, Phys. Rev. Lett. 54, 996 (1985).
    [CrossRef] [PubMed]
  7. R. E. Scholten, R. Gupta, J. J. McClelland, and R. J. Celotta, Phys. Rev. A 55, 1331 (1997).
    [CrossRef]
  8. A. Scholz, M. Christ, D. Doll, J. Ludwig, and W. Ertmer, Opt. Commun. 111, 155 (1994).
    [CrossRef]
  9. E. Riis, D. S. Weiss, K. A. Moler, and S. Chu, Phys. Rev. Lett. 64, 1658 (1990).
    [CrossRef] [PubMed]
  10. Z. T. Lu, K. L. Corwin, M. J. Renn, M. H. Anderson, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 77, 3331 (1996).
    [CrossRef] [PubMed]
  11. J. Yin, Y. Zhu, W. Wang, Y. Wang, and W. Jhe, J. Opt. Soc. Am. B 15, 25 (1998).
    [CrossRef]
  12. J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, Opt. Commun. 138, 287 (1997).
    [CrossRef]
  13. C. Paterson and R. Smith, Opt. Commun. 124, 121 (1996).
    [CrossRef]
  14. J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 2, 1707 (1985).
    [CrossRef]
  15. J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 6, 2023 (1989).
    [CrossRef]
  16. Yu. B. Ovchinnikov, J. Soding, and R. Grimm, JETP Lett. 61, 20 (1995).
  17. J. Soding, R. Grimm, and Yu. B. Ovchinnikov, Opt. Commun. 119, 652 (1995).
    [CrossRef]
  18. K. I. Lee, J. A. Kim, H. R. Noh, and W. Jhe, Proc. SPIE 2995, 279 (1997).
    [CrossRef]
  19. Yu. B. Ovchinnikov, I. Manek, and R. Grimm, Phys. Rev. Lett. 79, 2225 (1997).
    [CrossRef]
  20. V. I. Balykin and V. S. Letokhov, Opt. Commun. 64, 151 (1987).
    [CrossRef]

1998 (1)

1997 (5)

P. Westphal, S. Koch, A. Horn, J. Schmand, and H. J. Andra, Phys. Rev. Lett. 56, 2784 (1997).

R. E. Scholten, R. Gupta, J. J. McClelland, and R. J. Celotta, Phys. Rev. A 55, 1331 (1997).
[CrossRef]

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, Opt. Commun. 138, 287 (1997).
[CrossRef]

K. I. Lee, J. A. Kim, H. R. Noh, and W. Jhe, Proc. SPIE 2995, 279 (1997).
[CrossRef]

Yu. B. Ovchinnikov, I. Manek, and R. Grimm, Phys. Rev. Lett. 79, 2225 (1997).
[CrossRef]

1996 (3)

C. Paterson and R. Smith, Opt. Commun. 124, 121 (1996).
[CrossRef]

Z. T. Lu, K. L. Corwin, M. J. Renn, M. H. Anderson, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 77, 3331 (1996).
[CrossRef] [PubMed]

J. Fujita, M. Morinaga, T. Kishimoto, M. Yasuda, S. Matsui, and F. Shimizu, Nature (London) 380, 691 (1996).
[CrossRef]

1995 (2)

Yu. B. Ovchinnikov, J. Soding, and R. Grimm, JETP Lett. 61, 20 (1995).

J. Soding, R. Grimm, and Yu. B. Ovchinnikov, Opt. Commun. 119, 652 (1995).
[CrossRef]

1994 (1)

A. Scholz, M. Christ, D. Doll, J. Ludwig, and W. Ertmer, Opt. Commun. 111, 155 (1994).
[CrossRef]

1992 (1)

G. Timp, R. E. Behringer, D. M. Tennant, J. E. Cunningham, M. Prentiss, and K. K. Berggren, Phys. Rev. Lett. 69, 1636 (1992).
[CrossRef] [PubMed]

1991 (1)

D. W. Keith, C. R. Ekstrom, Q. A. Turchette, and D. E. Pritchard, Phys. Rev. Lett. 66, 2693 (1991).
[CrossRef] [PubMed]

1990 (1)

E. Riis, D. S. Weiss, K. A. Moler, and S. Chu, Phys. Rev. Lett. 64, 1658 (1990).
[CrossRef] [PubMed]

1989 (1)

1987 (1)

V. I. Balykin and V. S. Letokhov, Opt. Commun. 64, 151 (1987).
[CrossRef]

1985 (2)

J. Dalibard and C. Cohen-Tannoudji, J. Opt. Soc. Am. B 2, 1707 (1985).
[CrossRef]

W. Ertmer, R. Blatt, J. L. Hall, and M. Zhu, Phys. Rev. Lett. 54, 996 (1985).
[CrossRef] [PubMed]

Anderson, M. H.

Z. T. Lu, K. L. Corwin, M. J. Renn, M. H. Anderson, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 77, 3331 (1996).
[CrossRef] [PubMed]

Andra, H. J.

P. Westphal, S. Koch, A. Horn, J. Schmand, and H. J. Andra, Phys. Rev. Lett. 56, 2784 (1997).

Balykin, V. I.

V. I. Balykin and V. S. Letokhov, Opt. Commun. 64, 151 (1987).
[CrossRef]

Behringer, R. E.

G. Timp, R. E. Behringer, D. M. Tennant, J. E. Cunningham, M. Prentiss, and K. K. Berggren, Phys. Rev. Lett. 69, 1636 (1992).
[CrossRef] [PubMed]

Berggren, K. K.

G. Timp, R. E. Behringer, D. M. Tennant, J. E. Cunningham, M. Prentiss, and K. K. Berggren, Phys. Rev. Lett. 69, 1636 (1992).
[CrossRef] [PubMed]

Blatt, R.

W. Ertmer, R. Blatt, J. L. Hall, and M. Zhu, Phys. Rev. Lett. 54, 996 (1985).
[CrossRef] [PubMed]

Celotta, R. J.

R. E. Scholten, R. Gupta, J. J. McClelland, and R. J. Celotta, Phys. Rev. A 55, 1331 (1997).
[CrossRef]

Christ, M.

A. Scholz, M. Christ, D. Doll, J. Ludwig, and W. Ertmer, Opt. Commun. 111, 155 (1994).
[CrossRef]

Chu, S.

E. Riis, D. S. Weiss, K. A. Moler, and S. Chu, Phys. Rev. Lett. 64, 1658 (1990).
[CrossRef] [PubMed]

Cohen-Tannoudji, C.

Cornell, E. A.

Z. T. Lu, K. L. Corwin, M. J. Renn, M. H. Anderson, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 77, 3331 (1996).
[CrossRef] [PubMed]

Corwin, K. L.

Z. T. Lu, K. L. Corwin, M. J. Renn, M. H. Anderson, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 77, 3331 (1996).
[CrossRef] [PubMed]

Cunningham, J. E.

G. Timp, R. E. Behringer, D. M. Tennant, J. E. Cunningham, M. Prentiss, and K. K. Berggren, Phys. Rev. Lett. 69, 1636 (1992).
[CrossRef] [PubMed]

Dalibard, J.

Doll, D.

A. Scholz, M. Christ, D. Doll, J. Ludwig, and W. Ertmer, Opt. Commun. 111, 155 (1994).
[CrossRef]

Ekstrom, C. R.

D. W. Keith, C. R. Ekstrom, Q. A. Turchette, and D. E. Pritchard, Phys. Rev. Lett. 66, 2693 (1991).
[CrossRef] [PubMed]

Ertmer, W.

A. Scholz, M. Christ, D. Doll, J. Ludwig, and W. Ertmer, Opt. Commun. 111, 155 (1994).
[CrossRef]

W. Ertmer, R. Blatt, J. L. Hall, and M. Zhu, Phys. Rev. Lett. 54, 996 (1985).
[CrossRef] [PubMed]

Fujita, J.

J. Fujita, M. Morinaga, T. Kishimoto, M. Yasuda, S. Matsui, and F. Shimizu, Nature (London) 380, 691 (1996).
[CrossRef]

Grimm, R.

Yu. B. Ovchinnikov, I. Manek, and R. Grimm, Phys. Rev. Lett. 79, 2225 (1997).
[CrossRef]

Yu. B. Ovchinnikov, J. Soding, and R. Grimm, JETP Lett. 61, 20 (1995).

J. Soding, R. Grimm, and Yu. B. Ovchinnikov, Opt. Commun. 119, 652 (1995).
[CrossRef]

Gupta, R.

R. E. Scholten, R. Gupta, J. J. McClelland, and R. J. Celotta, Phys. Rev. A 55, 1331 (1997).
[CrossRef]

Hall, J. L.

W. Ertmer, R. Blatt, J. L. Hall, and M. Zhu, Phys. Rev. Lett. 54, 996 (1985).
[CrossRef] [PubMed]

Horn, A.

P. Westphal, S. Koch, A. Horn, J. Schmand, and H. J. Andra, Phys. Rev. Lett. 56, 2784 (1997).

Jhe, W.

J. Yin, Y. Zhu, W. Wang, Y. Wang, and W. Jhe, J. Opt. Soc. Am. B 15, 25 (1998).
[CrossRef]

K. I. Lee, J. A. Kim, H. R. Noh, and W. Jhe, Proc. SPIE 2995, 279 (1997).
[CrossRef]

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, Opt. Commun. 138, 287 (1997).
[CrossRef]

Keith, D. W.

D. W. Keith, C. R. Ekstrom, Q. A. Turchette, and D. E. Pritchard, Phys. Rev. Lett. 66, 2693 (1991).
[CrossRef] [PubMed]

Kim, J. A.

K. I. Lee, J. A. Kim, H. R. Noh, and W. Jhe, Proc. SPIE 2995, 279 (1997).
[CrossRef]

Kim, K.

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, Opt. Commun. 138, 287 (1997).
[CrossRef]

Kishimoto, T.

J. Fujita, M. Morinaga, T. Kishimoto, M. Yasuda, S. Matsui, and F. Shimizu, Nature (London) 380, 691 (1996).
[CrossRef]

Koch, S.

P. Westphal, S. Koch, A. Horn, J. Schmand, and H. J. Andra, Phys. Rev. Lett. 56, 2784 (1997).

Lee, K.

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, Opt. Commun. 138, 287 (1997).
[CrossRef]

Lee, K. I.

K. I. Lee, J. A. Kim, H. R. Noh, and W. Jhe, Proc. SPIE 2995, 279 (1997).
[CrossRef]

Letokhov, V. S.

V. I. Balykin and V. S. Letokhov, Opt. Commun. 64, 151 (1987).
[CrossRef]

Lu, Z. T.

Z. T. Lu, K. L. Corwin, M. J. Renn, M. H. Anderson, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 77, 3331 (1996).
[CrossRef] [PubMed]

Ludwig, J.

A. Scholz, M. Christ, D. Doll, J. Ludwig, and W. Ertmer, Opt. Commun. 111, 155 (1994).
[CrossRef]

Manek, I.

Yu. B. Ovchinnikov, I. Manek, and R. Grimm, Phys. Rev. Lett. 79, 2225 (1997).
[CrossRef]

Matsui, S.

J. Fujita, M. Morinaga, T. Kishimoto, M. Yasuda, S. Matsui, and F. Shimizu, Nature (London) 380, 691 (1996).
[CrossRef]

McClelland, J. J.

R. E. Scholten, R. Gupta, J. J. McClelland, and R. J. Celotta, Phys. Rev. A 55, 1331 (1997).
[CrossRef]

Moler, K. A.

E. Riis, D. S. Weiss, K. A. Moler, and S. Chu, Phys. Rev. Lett. 64, 1658 (1990).
[CrossRef] [PubMed]

Morinaga, M.

J. Fujita, M. Morinaga, T. Kishimoto, M. Yasuda, S. Matsui, and F. Shimizu, Nature (London) 380, 691 (1996).
[CrossRef]

Noh, H.

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, Opt. Commun. 138, 287 (1997).
[CrossRef]

Noh, H. R.

K. I. Lee, J. A. Kim, H. R. Noh, and W. Jhe, Proc. SPIE 2995, 279 (1997).
[CrossRef]

Ovchinnikov, Yu. B.

Yu. B. Ovchinnikov, I. Manek, and R. Grimm, Phys. Rev. Lett. 79, 2225 (1997).
[CrossRef]

J. Soding, R. Grimm, and Yu. B. Ovchinnikov, Opt. Commun. 119, 652 (1995).
[CrossRef]

Yu. B. Ovchinnikov, J. Soding, and R. Grimm, JETP Lett. 61, 20 (1995).

Paterson, C.

C. Paterson and R. Smith, Opt. Commun. 124, 121 (1996).
[CrossRef]

Prentiss, M.

G. Timp, R. E. Behringer, D. M. Tennant, J. E. Cunningham, M. Prentiss, and K. K. Berggren, Phys. Rev. Lett. 69, 1636 (1992).
[CrossRef] [PubMed]

Pritchard, D. E.

D. W. Keith, C. R. Ekstrom, Q. A. Turchette, and D. E. Pritchard, Phys. Rev. Lett. 66, 2693 (1991).
[CrossRef] [PubMed]

Renn, M. J.

Z. T. Lu, K. L. Corwin, M. J. Renn, M. H. Anderson, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 77, 3331 (1996).
[CrossRef] [PubMed]

Riis, E.

E. Riis, D. S. Weiss, K. A. Moler, and S. Chu, Phys. Rev. Lett. 64, 1658 (1990).
[CrossRef] [PubMed]

Schmand, J.

P. Westphal, S. Koch, A. Horn, J. Schmand, and H. J. Andra, Phys. Rev. Lett. 56, 2784 (1997).

Scholten, R. E.

R. E. Scholten, R. Gupta, J. J. McClelland, and R. J. Celotta, Phys. Rev. A 55, 1331 (1997).
[CrossRef]

Scholz, A.

A. Scholz, M. Christ, D. Doll, J. Ludwig, and W. Ertmer, Opt. Commun. 111, 155 (1994).
[CrossRef]

Shimizu, F.

J. Fujita, M. Morinaga, T. Kishimoto, M. Yasuda, S. Matsui, and F. Shimizu, Nature (London) 380, 691 (1996).
[CrossRef]

Smith, R.

C. Paterson and R. Smith, Opt. Commun. 124, 121 (1996).
[CrossRef]

Soding, J.

Yu. B. Ovchinnikov, J. Soding, and R. Grimm, JETP Lett. 61, 20 (1995).

J. Soding, R. Grimm, and Yu. B. Ovchinnikov, Opt. Commun. 119, 652 (1995).
[CrossRef]

Tennant, D. M.

G. Timp, R. E. Behringer, D. M. Tennant, J. E. Cunningham, M. Prentiss, and K. K. Berggren, Phys. Rev. Lett. 69, 1636 (1992).
[CrossRef] [PubMed]

Timp, G.

G. Timp, R. E. Behringer, D. M. Tennant, J. E. Cunningham, M. Prentiss, and K. K. Berggren, Phys. Rev. Lett. 69, 1636 (1992).
[CrossRef] [PubMed]

Turchette, Q. A.

D. W. Keith, C. R. Ekstrom, Q. A. Turchette, and D. E. Pritchard, Phys. Rev. Lett. 66, 2693 (1991).
[CrossRef] [PubMed]

Wang, W.

Wang, Y.

J. Yin, Y. Zhu, W. Wang, Y. Wang, and W. Jhe, J. Opt. Soc. Am. B 15, 25 (1998).
[CrossRef]

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, Opt. Commun. 138, 287 (1997).
[CrossRef]

Weiss, D. S.

E. Riis, D. S. Weiss, K. A. Moler, and S. Chu, Phys. Rev. Lett. 64, 1658 (1990).
[CrossRef] [PubMed]

Westphal, P.

P. Westphal, S. Koch, A. Horn, J. Schmand, and H. J. Andra, Phys. Rev. Lett. 56, 2784 (1997).

Wieman, C. E.

Z. T. Lu, K. L. Corwin, M. J. Renn, M. H. Anderson, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 77, 3331 (1996).
[CrossRef] [PubMed]

Yasuda, M.

J. Fujita, M. Morinaga, T. Kishimoto, M. Yasuda, S. Matsui, and F. Shimizu, Nature (London) 380, 691 (1996).
[CrossRef]

Yin, J.

J. Yin, Y. Zhu, W. Wang, Y. Wang, and W. Jhe, J. Opt. Soc. Am. B 15, 25 (1998).
[CrossRef]

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, Opt. Commun. 138, 287 (1997).
[CrossRef]

Zhu, M.

W. Ertmer, R. Blatt, J. L. Hall, and M. Zhu, Phys. Rev. Lett. 54, 996 (1985).
[CrossRef] [PubMed]

Zhu, Y.

J. Opt. Soc. Am. B (3)

JETP Lett. (1)

Yu. B. Ovchinnikov, J. Soding, and R. Grimm, JETP Lett. 61, 20 (1995).

Nature (London) (1)

J. Fujita, M. Morinaga, T. Kishimoto, M. Yasuda, S. Matsui, and F. Shimizu, Nature (London) 380, 691 (1996).
[CrossRef]

Opt. Commun. (5)

A. Scholz, M. Christ, D. Doll, J. Ludwig, and W. Ertmer, Opt. Commun. 111, 155 (1994).
[CrossRef]

J. Soding, R. Grimm, and Yu. B. Ovchinnikov, Opt. Commun. 119, 652 (1995).
[CrossRef]

J. Yin, H. Noh, K. Lee, K. Kim, Y. Wang, and W. Jhe, Opt. Commun. 138, 287 (1997).
[CrossRef]

C. Paterson and R. Smith, Opt. Commun. 124, 121 (1996).
[CrossRef]

V. I. Balykin and V. S. Letokhov, Opt. Commun. 64, 151 (1987).
[CrossRef]

Phys. Rev. A (1)

R. E. Scholten, R. Gupta, J. J. McClelland, and R. J. Celotta, Phys. Rev. A 55, 1331 (1997).
[CrossRef]

Phys. Rev. Lett. (7)

Yu. B. Ovchinnikov, I. Manek, and R. Grimm, Phys. Rev. Lett. 79, 2225 (1997).
[CrossRef]

E. Riis, D. S. Weiss, K. A. Moler, and S. Chu, Phys. Rev. Lett. 64, 1658 (1990).
[CrossRef] [PubMed]

Z. T. Lu, K. L. Corwin, M. J. Renn, M. H. Anderson, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett. 77, 3331 (1996).
[CrossRef] [PubMed]

D. W. Keith, C. R. Ekstrom, Q. A. Turchette, and D. E. Pritchard, Phys. Rev. Lett. 66, 2693 (1991).
[CrossRef] [PubMed]

W. Ertmer, R. Blatt, J. L. Hall, and M. Zhu, Phys. Rev. Lett. 54, 996 (1985).
[CrossRef] [PubMed]

G. Timp, R. E. Behringer, D. M. Tennant, J. E. Cunningham, M. Prentiss, and K. K. Berggren, Phys. Rev. Lett. 69, 1636 (1992).
[CrossRef] [PubMed]

P. Westphal, S. Koch, A. Horn, J. Schmand, and H. J. Andra, Phys. Rev. Lett. 56, 2784 (1997).

Proc. SPIE (1)

K. I. Lee, J. A. Kim, H. R. Noh, and W. Jhe, Proc. SPIE 2995, 279 (1997).
[CrossRef]

Other (1)

W. D. Phillips and H. Metcalf, Phys. Rev. Lett. 48, 596 (1982); T. E. Barrett, S. W. D. Schwartz, M. D. Ray, and G. P. Lafyatis, Phys. Rev. Lett. 67, 3483 (1991).
[CrossRef] [PubMed]

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Figures (6)

Fig. 1
Fig. 1

Laser collimator of a cold atomic beam based on CHDB guiding and CHDB Sisyphus cooling. MOT, PBS, M, λ/4, and PA stand for magneto-optical trap, polarized beam splitter, mirror, quarter wave plate and pinhole aperture, respectively.

Fig. 2
Fig. 2

(a) Energy-level diagram of a three-level 85Rb atom interacting with the blue-detuned CHDB and the repumping beam. (b) Dressed-state levels of the three-level atom in the blue-detuned CHDB.

Fig. 3
Fig. 3

Optical potentials U1(r) (solid curves), U2(r) (dotted curves), and U3(r) (dashed curves) around the dark center of the CHDB versus the radial position r for several values of the CHDB power P0.

Fig. 4
Fig. 4

Mean spontaneous-emission rate γsp of the guided atoms in the CHDB versus the detuning δ for the 85Rb atoms.

Fig. 5
Fig. 5

The transverse Sisyphus cooling time τSisy and cooling rate γSisy versus the detuning δ for the 85Rb atoms.

Fig. 6
Fig. 6

A computer-controlled atom lithography scheme based on the DHB guiding and DHB-induced Sisyphus cooling. AB, DHB, HL, and SS stand for atomic beam, dark hollow beam, hollow lens, and sample stage, respectively.

Equations (15)

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

I(r)=2P0πw2 2r2w2 exp-2r2w2,
U1,2(r, δ)=2 23 I(r)Γ22Is+δF1,221/2-δF1,2.
U1,2(r, δ)=112 Γ2δF1,2 I(r)Is.
I(ξ)I0 exp-2ξ2wa2,(rr0),
½mdξdt2+U1(ξ)=Er.
|t|=2w¯a2vr ln4δ2Γ2 IsI0 S/δ3Er S-11/2,
w¯a=1Zguid 0Zguidwa(Z)dZ
Psp=1-exp-- 13 ΓSdt=1-exp-mw¯aΓδ vr.
γspn¯˙pspv¯r2r¯apd 1-exp-mw¯aΓδ vr.
-23 Δhfsδ+Δhfs prmsk2+11-qev+1qr=0,
a¯=a¯CHDB+a¯RPBvrecoil1+1-qevqrγsp,
vrecoil=km
v¯Z=v¯Z0+at¯guidZguid=v¯Z0tguid+½at¯guid2.
τSisy=1γSisy4.5 δ+ΔhfsΔhfs(1-qev)γsp.
LacZguidv¯z γac2.31×10-2σRbp1mkBT1/2 Zguidv¯z,

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