Abstract

A fluid-mechanical model is developed for the float-polishing process. In this model laminar flow between the sample and the lap results in pressure gradients at the grooves that support the sample on a fluid layer. The laminar fluid motion also produces supersmooth, damage-free surfaces. Quartz substrates for applications in high-stress environments were float polished, and their surfaces were analyzed by optical scatterometry, photoacoustic spectroscopy, and atomic force microscopy. The removal of 100 μm of material by a lapping–polishing process, with final float polishing, left low levels of subsurface damage, with a surface roughness of approximately 0.2-nm rms.

© 1994 Optical Society of America

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References

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  1. Y. Namba, H. Tsuwa, “Mechanism and some applications of ultra-fine finishing,” Ann. CIRP 27, 511–516 (1978).
  2. Y. Namba, Department of Mechanical Engineering, Chubu University, Kasugai, Aichi 487, Japan (personal communication, 1989).
  3. See Science of Optical Finishing, Vol. 9 of OSA 1990 Technical Digest Series (Optical Society of America, Washington, D.C., 1990).
  4. H. Schlichting, Boundary-Layer Theory, 7th ed. (McGraw-Hill, New York, 1979), Chap. 2.
  5. B. S. Bassey, Mechanics of Fluids, 4th ed. (Van Nostrand-Reinhold, New York, 1979), Chaps. 5 and 6.
  6. S. F. Soares, K. C. Jungling, W. K. Stowell, “Float polished quartz substrates,” in Science of Optical Finishing, Vol. 9 of OSA 1990 Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper JTuA3.
  7. M. S. Longuet-Higgins, “The statistical analysis of a random, moving surface,” Philos. Trans. R. Soc. London Ser. A 249, 321–387 (1957).
    [CrossRef]
  8. E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).
  9. A. C. Tam, “Applications of photoacoustic sensing techniques,” Rev. Mod. Phys. 58, 381–430 (1986).
    [CrossRef]
  10. M. Y. A. Raja, D. W. Reicher, S. R. J. Brueck, J. R. McNeil, D. E. Oates, “High-sensitivity surface-photoacoustic spectroscopy,” Opt. Lett. 15, 66–68 (1990).
    [CrossRef] [PubMed]
  11. G. Binnig, C. F. Quate, Ch. Gerber, “Atomic force microscope,” Phys. Rev. Lett. 56, 930–933 (1986).
    [CrossRef] [PubMed]
  12. S. A. C. Gould, B. Drake, C. B. Prater, A. L. Weisenhorn, S. Manne, H. G. Hansma, P. K. Hansma, J. Massie, M. Longmire, V. Elings, B. D. Northern, B. Mukherjee, C. M. Peterson, W. Stoockonius, T. R. Albrecht, C. F. Quate, “From atoms to integrated circuit chips, blood cells, and bacteria with the atomic force microscope,” J. Vac. Sci. Technol. A 8, 360–373 (1990).
    [CrossRef]
  13. D. R. Baselt, J. D. Baldeschwieler, “Lateral forces during atomic force microscopy of graphite in air,” J. Vac. Sci. Technol. B 10, 2316–2322 (1991).
    [CrossRef]
  14. P. Grutter, W. Zimmermann-Edling, D. Brodbeck, “Tip artifacts of microfabricated force sensors for atomic force microscopy,” Appl. Phys. Lett. 60, 2741–2743 (1992).
    [CrossRef]

1992 (1)

P. Grutter, W. Zimmermann-Edling, D. Brodbeck, “Tip artifacts of microfabricated force sensors for atomic force microscopy,” Appl. Phys. Lett. 60, 2741–2743 (1992).
[CrossRef]

1991 (1)

D. R. Baselt, J. D. Baldeschwieler, “Lateral forces during atomic force microscopy of graphite in air,” J. Vac. Sci. Technol. B 10, 2316–2322 (1991).
[CrossRef]

1990 (2)

S. A. C. Gould, B. Drake, C. B. Prater, A. L. Weisenhorn, S. Manne, H. G. Hansma, P. K. Hansma, J. Massie, M. Longmire, V. Elings, B. D. Northern, B. Mukherjee, C. M. Peterson, W. Stoockonius, T. R. Albrecht, C. F. Quate, “From atoms to integrated circuit chips, blood cells, and bacteria with the atomic force microscope,” J. Vac. Sci. Technol. A 8, 360–373 (1990).
[CrossRef]

M. Y. A. Raja, D. W. Reicher, S. R. J. Brueck, J. R. McNeil, D. E. Oates, “High-sensitivity surface-photoacoustic spectroscopy,” Opt. Lett. 15, 66–68 (1990).
[CrossRef] [PubMed]

1986 (2)

G. Binnig, C. F. Quate, Ch. Gerber, “Atomic force microscope,” Phys. Rev. Lett. 56, 930–933 (1986).
[CrossRef] [PubMed]

A. C. Tam, “Applications of photoacoustic sensing techniques,” Rev. Mod. Phys. 58, 381–430 (1986).
[CrossRef]

1979 (1)

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).

1978 (1)

Y. Namba, H. Tsuwa, “Mechanism and some applications of ultra-fine finishing,” Ann. CIRP 27, 511–516 (1978).

1957 (1)

M. S. Longuet-Higgins, “The statistical analysis of a random, moving surface,” Philos. Trans. R. Soc. London Ser. A 249, 321–387 (1957).
[CrossRef]

Albrecht, T. R.

S. A. C. Gould, B. Drake, C. B. Prater, A. L. Weisenhorn, S. Manne, H. G. Hansma, P. K. Hansma, J. Massie, M. Longmire, V. Elings, B. D. Northern, B. Mukherjee, C. M. Peterson, W. Stoockonius, T. R. Albrecht, C. F. Quate, “From atoms to integrated circuit chips, blood cells, and bacteria with the atomic force microscope,” J. Vac. Sci. Technol. A 8, 360–373 (1990).
[CrossRef]

Baldeschwieler, J. D.

D. R. Baselt, J. D. Baldeschwieler, “Lateral forces during atomic force microscopy of graphite in air,” J. Vac. Sci. Technol. B 10, 2316–2322 (1991).
[CrossRef]

Baselt, D. R.

D. R. Baselt, J. D. Baldeschwieler, “Lateral forces during atomic force microscopy of graphite in air,” J. Vac. Sci. Technol. B 10, 2316–2322 (1991).
[CrossRef]

Bassey, B. S.

B. S. Bassey, Mechanics of Fluids, 4th ed. (Van Nostrand-Reinhold, New York, 1979), Chaps. 5 and 6.

Binnig, G.

G. Binnig, C. F. Quate, Ch. Gerber, “Atomic force microscope,” Phys. Rev. Lett. 56, 930–933 (1986).
[CrossRef] [PubMed]

Brodbeck, D.

P. Grutter, W. Zimmermann-Edling, D. Brodbeck, “Tip artifacts of microfabricated force sensors for atomic force microscopy,” Appl. Phys. Lett. 60, 2741–2743 (1992).
[CrossRef]

Brueck, S. R. J.

Church, E. L.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).

Drake, B.

S. A. C. Gould, B. Drake, C. B. Prater, A. L. Weisenhorn, S. Manne, H. G. Hansma, P. K. Hansma, J. Massie, M. Longmire, V. Elings, B. D. Northern, B. Mukherjee, C. M. Peterson, W. Stoockonius, T. R. Albrecht, C. F. Quate, “From atoms to integrated circuit chips, blood cells, and bacteria with the atomic force microscope,” J. Vac. Sci. Technol. A 8, 360–373 (1990).
[CrossRef]

Elings, V.

S. A. C. Gould, B. Drake, C. B. Prater, A. L. Weisenhorn, S. Manne, H. G. Hansma, P. K. Hansma, J. Massie, M. Longmire, V. Elings, B. D. Northern, B. Mukherjee, C. M. Peterson, W. Stoockonius, T. R. Albrecht, C. F. Quate, “From atoms to integrated circuit chips, blood cells, and bacteria with the atomic force microscope,” J. Vac. Sci. Technol. A 8, 360–373 (1990).
[CrossRef]

Gerber, Ch.

G. Binnig, C. F. Quate, Ch. Gerber, “Atomic force microscope,” Phys. Rev. Lett. 56, 930–933 (1986).
[CrossRef] [PubMed]

Gould, S. A. C.

S. A. C. Gould, B. Drake, C. B. Prater, A. L. Weisenhorn, S. Manne, H. G. Hansma, P. K. Hansma, J. Massie, M. Longmire, V. Elings, B. D. Northern, B. Mukherjee, C. M. Peterson, W. Stoockonius, T. R. Albrecht, C. F. Quate, “From atoms to integrated circuit chips, blood cells, and bacteria with the atomic force microscope,” J. Vac. Sci. Technol. A 8, 360–373 (1990).
[CrossRef]

Grutter, P.

P. Grutter, W. Zimmermann-Edling, D. Brodbeck, “Tip artifacts of microfabricated force sensors for atomic force microscopy,” Appl. Phys. Lett. 60, 2741–2743 (1992).
[CrossRef]

Hansma, H. G.

S. A. C. Gould, B. Drake, C. B. Prater, A. L. Weisenhorn, S. Manne, H. G. Hansma, P. K. Hansma, J. Massie, M. Longmire, V. Elings, B. D. Northern, B. Mukherjee, C. M. Peterson, W. Stoockonius, T. R. Albrecht, C. F. Quate, “From atoms to integrated circuit chips, blood cells, and bacteria with the atomic force microscope,” J. Vac. Sci. Technol. A 8, 360–373 (1990).
[CrossRef]

Hansma, P. K.

S. A. C. Gould, B. Drake, C. B. Prater, A. L. Weisenhorn, S. Manne, H. G. Hansma, P. K. Hansma, J. Massie, M. Longmire, V. Elings, B. D. Northern, B. Mukherjee, C. M. Peterson, W. Stoockonius, T. R. Albrecht, C. F. Quate, “From atoms to integrated circuit chips, blood cells, and bacteria with the atomic force microscope,” J. Vac. Sci. Technol. A 8, 360–373 (1990).
[CrossRef]

Jenkinson, H. A.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).

Jungling, K. C.

S. F. Soares, K. C. Jungling, W. K. Stowell, “Float polished quartz substrates,” in Science of Optical Finishing, Vol. 9 of OSA 1990 Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper JTuA3.

Longmire, M.

S. A. C. Gould, B. Drake, C. B. Prater, A. L. Weisenhorn, S. Manne, H. G. Hansma, P. K. Hansma, J. Massie, M. Longmire, V. Elings, B. D. Northern, B. Mukherjee, C. M. Peterson, W. Stoockonius, T. R. Albrecht, C. F. Quate, “From atoms to integrated circuit chips, blood cells, and bacteria with the atomic force microscope,” J. Vac. Sci. Technol. A 8, 360–373 (1990).
[CrossRef]

Longuet-Higgins, M. S.

M. S. Longuet-Higgins, “The statistical analysis of a random, moving surface,” Philos. Trans. R. Soc. London Ser. A 249, 321–387 (1957).
[CrossRef]

Manne, S.

S. A. C. Gould, B. Drake, C. B. Prater, A. L. Weisenhorn, S. Manne, H. G. Hansma, P. K. Hansma, J. Massie, M. Longmire, V. Elings, B. D. Northern, B. Mukherjee, C. M. Peterson, W. Stoockonius, T. R. Albrecht, C. F. Quate, “From atoms to integrated circuit chips, blood cells, and bacteria with the atomic force microscope,” J. Vac. Sci. Technol. A 8, 360–373 (1990).
[CrossRef]

Massie, J.

S. A. C. Gould, B. Drake, C. B. Prater, A. L. Weisenhorn, S. Manne, H. G. Hansma, P. K. Hansma, J. Massie, M. Longmire, V. Elings, B. D. Northern, B. Mukherjee, C. M. Peterson, W. Stoockonius, T. R. Albrecht, C. F. Quate, “From atoms to integrated circuit chips, blood cells, and bacteria with the atomic force microscope,” J. Vac. Sci. Technol. A 8, 360–373 (1990).
[CrossRef]

McNeil, J. R.

Mukherjee, B.

S. A. C. Gould, B. Drake, C. B. Prater, A. L. Weisenhorn, S. Manne, H. G. Hansma, P. K. Hansma, J. Massie, M. Longmire, V. Elings, B. D. Northern, B. Mukherjee, C. M. Peterson, W. Stoockonius, T. R. Albrecht, C. F. Quate, “From atoms to integrated circuit chips, blood cells, and bacteria with the atomic force microscope,” J. Vac. Sci. Technol. A 8, 360–373 (1990).
[CrossRef]

Namba, Y.

Y. Namba, H. Tsuwa, “Mechanism and some applications of ultra-fine finishing,” Ann. CIRP 27, 511–516 (1978).

Y. Namba, Department of Mechanical Engineering, Chubu University, Kasugai, Aichi 487, Japan (personal communication, 1989).

Northern, B. D.

S. A. C. Gould, B. Drake, C. B. Prater, A. L. Weisenhorn, S. Manne, H. G. Hansma, P. K. Hansma, J. Massie, M. Longmire, V. Elings, B. D. Northern, B. Mukherjee, C. M. Peterson, W. Stoockonius, T. R. Albrecht, C. F. Quate, “From atoms to integrated circuit chips, blood cells, and bacteria with the atomic force microscope,” J. Vac. Sci. Technol. A 8, 360–373 (1990).
[CrossRef]

Oates, D. E.

Peterson, C. M.

S. A. C. Gould, B. Drake, C. B. Prater, A. L. Weisenhorn, S. Manne, H. G. Hansma, P. K. Hansma, J. Massie, M. Longmire, V. Elings, B. D. Northern, B. Mukherjee, C. M. Peterson, W. Stoockonius, T. R. Albrecht, C. F. Quate, “From atoms to integrated circuit chips, blood cells, and bacteria with the atomic force microscope,” J. Vac. Sci. Technol. A 8, 360–373 (1990).
[CrossRef]

Prater, C. B.

S. A. C. Gould, B. Drake, C. B. Prater, A. L. Weisenhorn, S. Manne, H. G. Hansma, P. K. Hansma, J. Massie, M. Longmire, V. Elings, B. D. Northern, B. Mukherjee, C. M. Peterson, W. Stoockonius, T. R. Albrecht, C. F. Quate, “From atoms to integrated circuit chips, blood cells, and bacteria with the atomic force microscope,” J. Vac. Sci. Technol. A 8, 360–373 (1990).
[CrossRef]

Quate, C. F.

S. A. C. Gould, B. Drake, C. B. Prater, A. L. Weisenhorn, S. Manne, H. G. Hansma, P. K. Hansma, J. Massie, M. Longmire, V. Elings, B. D. Northern, B. Mukherjee, C. M. Peterson, W. Stoockonius, T. R. Albrecht, C. F. Quate, “From atoms to integrated circuit chips, blood cells, and bacteria with the atomic force microscope,” J. Vac. Sci. Technol. A 8, 360–373 (1990).
[CrossRef]

G. Binnig, C. F. Quate, Ch. Gerber, “Atomic force microscope,” Phys. Rev. Lett. 56, 930–933 (1986).
[CrossRef] [PubMed]

Raja, M. Y. A.

Reicher, D. W.

Schlichting, H.

H. Schlichting, Boundary-Layer Theory, 7th ed. (McGraw-Hill, New York, 1979), Chap. 2.

Soares, S. F.

S. F. Soares, K. C. Jungling, W. K. Stowell, “Float polished quartz substrates,” in Science of Optical Finishing, Vol. 9 of OSA 1990 Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper JTuA3.

Stoockonius, W.

S. A. C. Gould, B. Drake, C. B. Prater, A. L. Weisenhorn, S. Manne, H. G. Hansma, P. K. Hansma, J. Massie, M. Longmire, V. Elings, B. D. Northern, B. Mukherjee, C. M. Peterson, W. Stoockonius, T. R. Albrecht, C. F. Quate, “From atoms to integrated circuit chips, blood cells, and bacteria with the atomic force microscope,” J. Vac. Sci. Technol. A 8, 360–373 (1990).
[CrossRef]

Stowell, W. K.

S. F. Soares, K. C. Jungling, W. K. Stowell, “Float polished quartz substrates,” in Science of Optical Finishing, Vol. 9 of OSA 1990 Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper JTuA3.

Tam, A. C.

A. C. Tam, “Applications of photoacoustic sensing techniques,” Rev. Mod. Phys. 58, 381–430 (1986).
[CrossRef]

Tsuwa, H.

Y. Namba, H. Tsuwa, “Mechanism and some applications of ultra-fine finishing,” Ann. CIRP 27, 511–516 (1978).

Weisenhorn, A. L.

S. A. C. Gould, B. Drake, C. B. Prater, A. L. Weisenhorn, S. Manne, H. G. Hansma, P. K. Hansma, J. Massie, M. Longmire, V. Elings, B. D. Northern, B. Mukherjee, C. M. Peterson, W. Stoockonius, T. R. Albrecht, C. F. Quate, “From atoms to integrated circuit chips, blood cells, and bacteria with the atomic force microscope,” J. Vac. Sci. Technol. A 8, 360–373 (1990).
[CrossRef]

Zavada, J. M.

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).

Zimmermann-Edling, W.

P. Grutter, W. Zimmermann-Edling, D. Brodbeck, “Tip artifacts of microfabricated force sensors for atomic force microscopy,” Appl. Phys. Lett. 60, 2741–2743 (1992).
[CrossRef]

Ann. CIRP (1)

Y. Namba, H. Tsuwa, “Mechanism and some applications of ultra-fine finishing,” Ann. CIRP 27, 511–516 (1978).

Appl. Phys. Lett. (1)

P. Grutter, W. Zimmermann-Edling, D. Brodbeck, “Tip artifacts of microfabricated force sensors for atomic force microscopy,” Appl. Phys. Lett. 60, 2741–2743 (1992).
[CrossRef]

J. Vac. Sci. Technol. A (1)

S. A. C. Gould, B. Drake, C. B. Prater, A. L. Weisenhorn, S. Manne, H. G. Hansma, P. K. Hansma, J. Massie, M. Longmire, V. Elings, B. D. Northern, B. Mukherjee, C. M. Peterson, W. Stoockonius, T. R. Albrecht, C. F. Quate, “From atoms to integrated circuit chips, blood cells, and bacteria with the atomic force microscope,” J. Vac. Sci. Technol. A 8, 360–373 (1990).
[CrossRef]

J. Vac. Sci. Technol. B (1)

D. R. Baselt, J. D. Baldeschwieler, “Lateral forces during atomic force microscopy of graphite in air,” J. Vac. Sci. Technol. B 10, 2316–2322 (1991).
[CrossRef]

Opt. Eng. (1)

E. L. Church, H. A. Jenkinson, J. M. Zavada, “Relationship between surface scattering and microtopographic features,” Opt. Eng. 18, 125–136 (1979).

Opt. Lett. (1)

Philos. Trans. R. Soc. London Ser. A (1)

M. S. Longuet-Higgins, “The statistical analysis of a random, moving surface,” Philos. Trans. R. Soc. London Ser. A 249, 321–387 (1957).
[CrossRef]

Phys. Rev. Lett. (1)

G. Binnig, C. F. Quate, Ch. Gerber, “Atomic force microscope,” Phys. Rev. Lett. 56, 930–933 (1986).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

A. C. Tam, “Applications of photoacoustic sensing techniques,” Rev. Mod. Phys. 58, 381–430 (1986).
[CrossRef]

Other (5)

Y. Namba, Department of Mechanical Engineering, Chubu University, Kasugai, Aichi 487, Japan (personal communication, 1989).

See Science of Optical Finishing, Vol. 9 of OSA 1990 Technical Digest Series (Optical Society of America, Washington, D.C., 1990).

H. Schlichting, Boundary-Layer Theory, 7th ed. (McGraw-Hill, New York, 1979), Chap. 2.

B. S. Bassey, Mechanics of Fluids, 4th ed. (Van Nostrand-Reinhold, New York, 1979), Chaps. 5 and 6.

S. F. Soares, K. C. Jungling, W. K. Stowell, “Float polished quartz substrates,” in Science of Optical Finishing, Vol. 9 of OSA 1990 Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper JTuA3.

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Figures (11)

Fig. 1
Fig. 1

Schematic of the apparatus for float polishing. The groove pattern consists of two superimposed diamond-turned spirals. The groove profile is shown in the inset.

Fig. 2
Fig. 2

Fluid motion in float polishing. The fluid motion at consecutive laminae is shown. ν1 and νv are the laminar and vortex components of the lamina with velocity ν′ relative to the lower lamina with velocity ν. ν0 is the velocity of the upper plane relative to the lower plane.

Fig. 3
Fig. 3

(a) Inclined-slipper bearing; (b) float-polishing bearing. Pressure gradients near the peaks of grooves are sufficient to support a load.

Fig. 4
Fig. 4

In this simplified one-dimensional model the pressure gradients near a quadratic boss are calculated to investigate the feasibility of float polishing. g(x) is the groove function, α is the groove height, and ν0 is the velocity of the moving plane. The frictional forces and pressure on a differential element are as shown.

Fig. 5
Fig. 5

Nomarski photomicrograph taken at 10× and a Talystep profile with a vertical resolution of 2 nm/div and horizontal resolution of 3 μm/div of the surface of a float-polished quartz sample.

Fig. 6
Fig. 6

Optical scatter measurements with p-polarized incident light and both p and s polarizations measured. In both cases the scatter from surface A with only 10 μm of material removed (a) and (b) is approximately an order of magnitude higher than that for surface B with 100 μm of material removed (c), (d) by float polishing.

Fig. 7
Fig. 7

SAW signals for the two quartz surfaces. The signal for surface B is near the noise floor of the instrument, indicating that the density of subsurface damage is low.

Fig. 8
Fig. 8

Optical-lever AFM.

Fig. 9
Fig. 9

A 1.6 μm × 1.6 μm AFM image of (a) float-polished quartz surface A and (b) float-polished quartz surface B.

Fig. 10
Fig. 10

AFM topographic images of float-polished quartz. Surface A (10-μm material removed) had a roughness of less than 0.5-nm rms, while the roughness for surface B (100-μm material removed) was approximately twice as better at 0.2-nm rms. The noise floor of the instrument is approximately 0.1-nm rms and is also shown.

Fig. 11
Fig. 11

Three-dimensional atomic force micrograph of surface B.

Equations (14)

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

p d y d z - ( p + d p ) d y d z + ( τ + d τ ) d x d z - τ d x d z = 0 ,
d p d x = d d y ( η d ν d y ) ,
y 2 2 d p d x = η ν + c 1 y + c 2 .
c 1 = [ g ( x ) 2 d p d x + η ν 0 g ( x ) ]
y 2 2 d p d x = η ( ν - ν 0 ) + y [ g ( x ) 2 d p d x + η ν 0 g ( x ) ] ,
q 0 = 0 g ( x ) d y 0 b d z ν = b [ - d p d x g 3 ( x ) 12 η + ν 0 g ( x ) 2 ] .
d p d x = 12 η [ ν 0 2 g 2 ( x ) - q 0 b g 3 ( x ) ]
d p d x = 12 η [ ν 0 2 ( β x 2 + α ) 2 - q 0 b ( β x 2 + α ) 3 ] .
p ( x ) 12 η { ν 0 x 4 α [ 1 ( β x 2 + α ) + 1 α ] - q 0 x 2 b α × [ 1 2 ( β x 2 + α ) 2 + 3 4 α ( β x 2 + α ) + 3 4 α 2 ] } + C ,
tan - 1 ( b x β α ) ( b x β α )
q 0 = 2 ν 0 a b γ ( γ + 1 ) ( 2 γ 2 + 3 γ + 3 ) ,
p p 0 - 1.2 η ν 0 x α 2 ,
g ( x ) = n = - β ( x - n c ) 2 + α ,
g ( x ) = n = - β ( x - n c ) 2 + α + m = - β * ( x - m c * ) 2 + α * .

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