Abstract

The Foldy-Lax hierarchy of equations for the scattering of electromagnetic waves by a monolayer of spheres was reduced to a set of algebraic equations and solved. The Percus-Yevick equation was used to describe the spatial correlation between the particles. Calculations of the optical densities of monolayers of Se spheres of diameters 0.2–0.3 μ were in good agreement with experimental results on particle-migration imaging films. The calculation also predicts an optimal particle size for such films.

© 1980 Optical Society of America

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References

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  1. See, e.g., P. Rouard and A. Meessen, “Optical properties of thin metal films,” in Progress in Optics, Vol. XV, edited by E. Wolf (North-Holland, Amsterdam, 1977), pp. 76–137, and references therein.
  2. W. L. Goffe, “Photographic migration imaging—a new concept in photography,” Photogr. Sci. Eng. 15, 304–308 (1971).
  3. A. L. Pundsack, “Phenomenological theory of the D log E curve for migration imaging,” Photogr. Sci. Eng. 18, 642–647 (1974).
  4. A. L. Pundsack, Y. C. Cheng, G. C. Hartmann, and L. M. Marks, “Optical properties of particle migration imaging film,” Appl. Opt. 17, 2650–2654 (1978).
    [PubMed]
  5. J. I. Treu, “Mie scattering, Maxwell Garnett theory, and the Giaever immunology slide,” Appl. Opt. 15, 2746–2750 (1976).
    [Crossref] [PubMed]
  6. S. Norrman, T. Andersson, C. G. Granqvist, and O. Hunderi, “Optical absorption in discontinuous gold films,” Solid State Commun. 23, 261–265 (1977).
    [Crossref]
  7. L. L. Foldy, “The multiple scattering of waves,” Phys. Rev. 67, 107–119 (1945).
    [Crossref]
  8. M. Lax, “Multiple scattering of waves,” Rev. Mod. Phys. 23, 287–310 (1951).
    [Crossref]
  9. M. Lax, “Multiple scattering of waves. II. The effective field in dense systems,” Phys. Rev. 85, 621–629 (1952).
    [Crossref]
  10. See, e.g., V. Twersky, “Coherent electromagnetic waves in pair-correlated random distributions of aligned scatterers,” J. Math. Phys. 19, 215–230 (1978), and references therein.
    [Crossref]
  11. See, e.g., J. P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic, New York, 1976).
  12. N. C. Mathur and K. C. Yeh, “Multiple scattering of electromagnetic waves by random scatterer of finite size,” J. Math. Phys. 5, 1619–1628 (1964).
    [Crossref]
  13. C. Liang and Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1418–1495 (1967).
  14. S. Levine and G. O. Olaofe, “Scattering of electromagnetic waves by two equal spherical particles,” J. Colloid Interface Sci. 27, 442–457 (1968).
    [Crossref]
  15. See, eg., P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1957), Chap. 13.
  16. S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).
  17. O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).
  18. See, eg., D. M. Brink and G. R. Satchler, Angular Momentum (Clarendon, Oxford, 1975).
  19. W. F. Koehler, F. K. Odencrantz, and W. C. White, “Optical constants of evaporated selenium films by successive approximations,” J. Opt. Soc. Am. 49, 109–115 (1959).
    [Crossref]
  20. A. L. Pundsack (private communication).
  21. R. J. A. Tough, “The transformation properties of vector multipole fields under a translation of coordinate origin,” J. Phys. A 10, 1079–1087 (1977).
    [Crossref]
  22. M. Danos and L. C. Maximon, “Multipole matrix elements of the translation operator,” J. Math. Phys. 6, 766–778 (1965).
    [Crossref]
  23. F. Lado, “Equation of state of the hard-disk fluid from approximate integral equations,” J. Chem. Phys. 49, 3092–3096 (1968).
    [Crossref]

1978 (2)

A. L. Pundsack, Y. C. Cheng, G. C. Hartmann, and L. M. Marks, “Optical properties of particle migration imaging film,” Appl. Opt. 17, 2650–2654 (1978).
[PubMed]

See, e.g., V. Twersky, “Coherent electromagnetic waves in pair-correlated random distributions of aligned scatterers,” J. Math. Phys. 19, 215–230 (1978), and references therein.
[Crossref]

1977 (2)

R. J. A. Tough, “The transformation properties of vector multipole fields under a translation of coordinate origin,” J. Phys. A 10, 1079–1087 (1977).
[Crossref]

S. Norrman, T. Andersson, C. G. Granqvist, and O. Hunderi, “Optical absorption in discontinuous gold films,” Solid State Commun. 23, 261–265 (1977).
[Crossref]

1976 (1)

1974 (1)

A. L. Pundsack, “Phenomenological theory of the D log E curve for migration imaging,” Photogr. Sci. Eng. 18, 642–647 (1974).

1971 (1)

W. L. Goffe, “Photographic migration imaging—a new concept in photography,” Photogr. Sci. Eng. 15, 304–308 (1971).

1968 (2)

S. Levine and G. O. Olaofe, “Scattering of electromagnetic waves by two equal spherical particles,” J. Colloid Interface Sci. 27, 442–457 (1968).
[Crossref]

F. Lado, “Equation of state of the hard-disk fluid from approximate integral equations,” J. Chem. Phys. 49, 3092–3096 (1968).
[Crossref]

1967 (1)

C. Liang and Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1418–1495 (1967).

1965 (1)

M. Danos and L. C. Maximon, “Multipole matrix elements of the translation operator,” J. Math. Phys. 6, 766–778 (1965).
[Crossref]

1964 (1)

N. C. Mathur and K. C. Yeh, “Multiple scattering of electromagnetic waves by random scatterer of finite size,” J. Math. Phys. 5, 1619–1628 (1964).
[Crossref]

1962 (1)

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

1961 (1)

S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).

1959 (1)

1952 (1)

M. Lax, “Multiple scattering of waves. II. The effective field in dense systems,” Phys. Rev. 85, 621–629 (1952).
[Crossref]

1951 (1)

M. Lax, “Multiple scattering of waves,” Rev. Mod. Phys. 23, 287–310 (1951).
[Crossref]

1945 (1)

L. L. Foldy, “The multiple scattering of waves,” Phys. Rev. 67, 107–119 (1945).
[Crossref]

Andersson, T.

S. Norrman, T. Andersson, C. G. Granqvist, and O. Hunderi, “Optical absorption in discontinuous gold films,” Solid State Commun. 23, 261–265 (1977).
[Crossref]

Brink, D. M.

See, eg., D. M. Brink and G. R. Satchler, Angular Momentum (Clarendon, Oxford, 1975).

Cheng, Y. C.

Cruzan, O. R.

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

Danos, M.

M. Danos and L. C. Maximon, “Multipole matrix elements of the translation operator,” J. Math. Phys. 6, 766–778 (1965).
[Crossref]

Feshbach, H.

See, eg., P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1957), Chap. 13.

Foldy, L. L.

L. L. Foldy, “The multiple scattering of waves,” Phys. Rev. 67, 107–119 (1945).
[Crossref]

Goffe, W. L.

W. L. Goffe, “Photographic migration imaging—a new concept in photography,” Photogr. Sci. Eng. 15, 304–308 (1971).

Granqvist, C. G.

S. Norrman, T. Andersson, C. G. Granqvist, and O. Hunderi, “Optical absorption in discontinuous gold films,” Solid State Commun. 23, 261–265 (1977).
[Crossref]

Hansen, J. P.

See, e.g., J. P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic, New York, 1976).

Hartmann, G. C.

Hunderi, O.

S. Norrman, T. Andersson, C. G. Granqvist, and O. Hunderi, “Optical absorption in discontinuous gold films,” Solid State Commun. 23, 261–265 (1977).
[Crossref]

Koehler, W. F.

Lado, F.

F. Lado, “Equation of state of the hard-disk fluid from approximate integral equations,” J. Chem. Phys. 49, 3092–3096 (1968).
[Crossref]

Lax, M.

M. Lax, “Multiple scattering of waves. II. The effective field in dense systems,” Phys. Rev. 85, 621–629 (1952).
[Crossref]

M. Lax, “Multiple scattering of waves,” Rev. Mod. Phys. 23, 287–310 (1951).
[Crossref]

Levine, S.

S. Levine and G. O. Olaofe, “Scattering of electromagnetic waves by two equal spherical particles,” J. Colloid Interface Sci. 27, 442–457 (1968).
[Crossref]

Liang, C.

C. Liang and Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1418–1495 (1967).

Lo, Y. T.

C. Liang and Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1418–1495 (1967).

Marks, L. M.

Mathur, N. C.

N. C. Mathur and K. C. Yeh, “Multiple scattering of electromagnetic waves by random scatterer of finite size,” J. Math. Phys. 5, 1619–1628 (1964).
[Crossref]

Maximon, L. C.

M. Danos and L. C. Maximon, “Multipole matrix elements of the translation operator,” J. Math. Phys. 6, 766–778 (1965).
[Crossref]

McDonald, I. R.

See, e.g., J. P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic, New York, 1976).

Meessen, A.

See, e.g., P. Rouard and A. Meessen, “Optical properties of thin metal films,” in Progress in Optics, Vol. XV, edited by E. Wolf (North-Holland, Amsterdam, 1977), pp. 76–137, and references therein.

Morse, P. M.

See, eg., P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1957), Chap. 13.

Norrman, S.

S. Norrman, T. Andersson, C. G. Granqvist, and O. Hunderi, “Optical absorption in discontinuous gold films,” Solid State Commun. 23, 261–265 (1977).
[Crossref]

Odencrantz, F. K.

Olaofe, G. O.

S. Levine and G. O. Olaofe, “Scattering of electromagnetic waves by two equal spherical particles,” J. Colloid Interface Sci. 27, 442–457 (1968).
[Crossref]

Pundsack, A. L.

A. L. Pundsack, Y. C. Cheng, G. C. Hartmann, and L. M. Marks, “Optical properties of particle migration imaging film,” Appl. Opt. 17, 2650–2654 (1978).
[PubMed]

A. L. Pundsack, “Phenomenological theory of the D log E curve for migration imaging,” Photogr. Sci. Eng. 18, 642–647 (1974).

A. L. Pundsack (private communication).

Rouard, P.

See, e.g., P. Rouard and A. Meessen, “Optical properties of thin metal films,” in Progress in Optics, Vol. XV, edited by E. Wolf (North-Holland, Amsterdam, 1977), pp. 76–137, and references therein.

Satchler, G. R.

See, eg., D. M. Brink and G. R. Satchler, Angular Momentum (Clarendon, Oxford, 1975).

Stein, S.

S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).

Tough, R. J. A.

R. J. A. Tough, “The transformation properties of vector multipole fields under a translation of coordinate origin,” J. Phys. A 10, 1079–1087 (1977).
[Crossref]

Treu, J. I.

Twersky, V.

See, e.g., V. Twersky, “Coherent electromagnetic waves in pair-correlated random distributions of aligned scatterers,” J. Math. Phys. 19, 215–230 (1978), and references therein.
[Crossref]

White, W. C.

Yeh, K. C.

N. C. Mathur and K. C. Yeh, “Multiple scattering of electromagnetic waves by random scatterer of finite size,” J. Math. Phys. 5, 1619–1628 (1964).
[Crossref]

Appl. Opt. (2)

J. Chem. Phys. (1)

F. Lado, “Equation of state of the hard-disk fluid from approximate integral equations,” J. Chem. Phys. 49, 3092–3096 (1968).
[Crossref]

J. Colloid Interface Sci. (1)

S. Levine and G. O. Olaofe, “Scattering of electromagnetic waves by two equal spherical particles,” J. Colloid Interface Sci. 27, 442–457 (1968).
[Crossref]

J. Math. Phys. (3)

See, e.g., V. Twersky, “Coherent electromagnetic waves in pair-correlated random distributions of aligned scatterers,” J. Math. Phys. 19, 215–230 (1978), and references therein.
[Crossref]

N. C. Mathur and K. C. Yeh, “Multiple scattering of electromagnetic waves by random scatterer of finite size,” J. Math. Phys. 5, 1619–1628 (1964).
[Crossref]

M. Danos and L. C. Maximon, “Multipole matrix elements of the translation operator,” J. Math. Phys. 6, 766–778 (1965).
[Crossref]

J. Opt. Soc. Am. (1)

J. Phys. A (1)

R. J. A. Tough, “The transformation properties of vector multipole fields under a translation of coordinate origin,” J. Phys. A 10, 1079–1087 (1977).
[Crossref]

Photogr. Sci. Eng. (2)

W. L. Goffe, “Photographic migration imaging—a new concept in photography,” Photogr. Sci. Eng. 15, 304–308 (1971).

A. L. Pundsack, “Phenomenological theory of the D log E curve for migration imaging,” Photogr. Sci. Eng. 18, 642–647 (1974).

Phys. Rev. (2)

L. L. Foldy, “The multiple scattering of waves,” Phys. Rev. 67, 107–119 (1945).
[Crossref]

M. Lax, “Multiple scattering of waves. II. The effective field in dense systems,” Phys. Rev. 85, 621–629 (1952).
[Crossref]

Q. Appl. Math. (2)

S. Stein, “Addition theorems for spherical wave functions,” Q. Appl. Math. 19, 15–24 (1961).

O. R. Cruzan, “Translational addition theorems for spherical vector wave functions,” Q. Appl. Math. 20, 33–40 (1962).

Radio Sci. (1)

C. Liang and Y. T. Lo, “Scattering by two spheres,” Radio Sci. 2, 1418–1495 (1967).

Rev. Mod. Phys. (1)

M. Lax, “Multiple scattering of waves,” Rev. Mod. Phys. 23, 287–310 (1951).
[Crossref]

Solid State Commun. (1)

S. Norrman, T. Andersson, C. G. Granqvist, and O. Hunderi, “Optical absorption in discontinuous gold films,” Solid State Commun. 23, 261–265 (1977).
[Crossref]

Other (5)

See, e.g., P. Rouard and A. Meessen, “Optical properties of thin metal films,” in Progress in Optics, Vol. XV, edited by E. Wolf (North-Holland, Amsterdam, 1977), pp. 76–137, and references therein.

See, e.g., J. P. Hansen and I. R. McDonald, Theory of Simple Liquids (Academic, New York, 1976).

See, eg., D. M. Brink and G. R. Satchler, Angular Momentum (Clarendon, Oxford, 1975).

See, eg., P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1957), Chap. 13.

A. L. Pundsack (private communication).

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Figures (9)

FIG. 1
FIG. 1

Electron micrograph of the cross section of a particle migration imaging film.

FIG. 2
FIG. 2

Electron micrograph of a migration imaging film taken normal to the surface.

FIG. 3
FIG. 3

Solid line: particle size distribution of a typical migration imaging film. Dashed line: distribution used in the calculation.

FIG. 4
FIG. 4

Three of the nine (or six) radial distribution functions for the case where mean coverage fraction f ¯ = 0.66.

FIG. 5
FIG. 5

Comparison of calculated optical density with experimental results. The horizontal marks at 0.24 indicate the background density assumed for the layer of aluminum in the film.

FIG. 9
FIG. 9

Calculations of the optical densities of films with the same mean coverage fraction f ¯ = 0.66, showing the variation of the peak height with changing particle size.

Equations (45)

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E 0 ( r ) = ̂ e i k z ,
E ( r | { R } ) = E 0 ( r ) + α = 1 s l = 1 N s × V α d 3 r Γ ( r , r + R l α ) · E ( r + R l α | { R } ) ,
Γ ( r , r ) = n 2 1 4 π ( k 2 1 ) e i k | r r | | r r | ,
E ( r ) = E 0 ( r ) + α = 1 s ρ α d 2 R V α d 3 r × Γ ( r , r + R ) · E α ( r + R | R ) ,
E α ( r | R ) = E 0 ( r ) + V α d 3 r Γ ( r , r + R ) · E α ( r + R | R ) + β = 1 s ρ β d 2 R g α β ( | R R | ) V β d 3 r × Γ ( r , r + R ) · E α β ( r + R | R , R ) ,
E α β ( r + R | R , R ) E α ( r + R | R )
E ( r ) = ̂ e i k z + 2 π i k α = 1 s ρ α f α ( ± ) e i k | z | ,
f α ( ± ) = k 2 4 π ( n 2 1 ) × V α d 3 r e i k z ( 1 ) · E α ( r | o ) .
E α ( r | o ) = ̂ e i k z + V α d 3 r Γ ( r , r ) · E α ( r | o ) + β = 1 s ρ β d 2 R g α β ( R ) V β d 3 r × Γ ( r , r + R ) · E β ( r | o ) .
̂ e i k z = 1 2 l = 1 i l 1 ( 2 l + 1 ) l ( l + 1 ) [ M l 1 ( 1 ) ( r ) + N l 1 ( 1 ) ( r ) ] .
E α ( r | o ) = l = 1 [ a l M α M l 1 ( 1 ) ( r ) + a l E α Ñ l 1 ( 1 ) ( r ) ] .
f α ( ± ) = ̂ i 2 k l = 1 ( ± 1 ) l ( 2 l + 1 ) ( b l M α ± b l E α ) ,
n ( 2 l + 1 ) i l b l M α = 2 l ( l + 1 ) a l M α ( ψ l ψ l n ψ l ψ l ) α ,
n ( 2 l + 1 ) i l b l E α = 2 l ( l + 1 ) a l E α ( n ψ l ψ l ψ l ψ l ) α ,
Γ ( r , r ) = n 2 1 4 π ( i k 3 l m ( 2 l + 1 ) l ( l + 1 ) ( l m ) ! ( l + m ) ! × [ M l m ( 1 , 3 ) ( r ) M l m ( 4 , 1 ) * ( r ) + N l m ( 1 , 3 ) ( r ) N l m ( 4 , 1 ) * ( r ) ] 4 π r ̂ r ̂ δ ( r r ) ) ,
( a , b ) = { a if r > r b if r > r .
d 2 R g α β ( R ) Γ ( r , r + R ) = n 2 1 4 π i k 3 l l m ( 2 l + 1 ) l ( l + 1 ) ( l m ) ! ( l + m ) ! { M l m ( 1 ) ( r ) × [ A l l m α β * M l m ( 1 ) * ( r ) + B l l m α β * N l m ( 1 ) * ( r ) ] + N l m ( 1 ) ( r ) [ A l l m α β * N l m ( 1 ) * ( r ) + B l l m α β * M l m ( 1 ) * ( r ) ] }
M l m ( 4 ) ( r + R ) = l m [ A l m l m ( R ) M l m ( 1 ) ( r ) + B l m l m ( R ) N l m ( 1 ) ( r ) ] ,
d 2 R g α β ( R ) A l m l m ( R ) = δ m m A l l m α β .
V α d 3 r M l m ( 1 ) * ( r ) · M l m ( 1 ) ( r ) = C δ l l δ m m ( ψ l ψ l n ψ l ψ l ) α ,
V α d 3 r M l m ( 1 , 3 ) ( r ) M l m ( 4 , 1 ) * ( r ) · M l m ( 1 ) ( r ) = C δ l l δ m m [ ( ζ l ψ l n ζ l ψ l ) α M l m ( 1 ) ( r ) i n M l m ( 1 ) ( r ) ] ,
C = 4 π k 3 n ( n 2 1 ) l ( l + 1 ) 2 l + 1 ( l + m ) ! ( l m ) !
1 = b l M α C l M α β l ρ β [ P l l α β b l M β + Q l l α β b l E β ] ,
1 = b l E α C l E α β l ρ β [ P l l α β b l E β + Q l l α β b l M β ] ,
C l M α = ( ψ l ψ l n ψ l ψ l ) α ( ζ l ψ l n ζ l ψ l ) α ,
C l E α = ( n ψ l ψ l ψ l ψ l ) α ( n ζ l ψ l ζ l ψ l ) α ,
P l l α β = i l l l ( l + 1 ) l ( l + 1 ) A l l 1 α β * = 2 l + 1 2 [ l ( l + 1 ) l ( l + 1 ) ] 1 / 2 × p = even p ! [ ( p / 2 ) ! ] 2 2 p + 1 2 p [ l ( l + 1 ) + l ( l + 1 ) p ( p + 1 ) ] × ( l l p 0 0 0 ) ( l l p 1 1 0 ) H p α β ,
Q l l α β = i l l l ( l + 1 ) l ( l + 1 ) B l l 1 α β * = 2 l + 1 2 [ l ( l + 1 ) · l ( l + 1 ) ] 1 / 2 × p = even p ! [ ( p / 2 ) ! ] 2 2 p + 1 2 p [ ( p + l l ) ( p l + l ) × ( l + l + 1 + p ) ( l + l + 1 p ) ] 1 / 2 × ( l l p 1 0 0 0 ) ( l l p 1 1 0 ) H p α β .
( j 1 j 2 j 3 m 1 m 2 m 3 )
H p α β = 2 π 0 d R R g α β ( R ) h p ( 1 ) ( k R ) .
f ¯ = ( α = 1 s ρ α ) π d ¯ 2 4 .
I I 0 = | 1 π k 2 α l ρ α ( 2 l + 1 ) ( b l M α + b l E α ) | 2 ,
D T = log 10 ( I / I 0 ) .
X l m ( θ , ϕ ) = P l m ( cos θ ) e i m ϕ ,
P l m ( x ) = ( 1 x 2 ) m / 2 l ! 2 l d l + m d x l + m ( x 2 1 ) l .
M l m ( n ) ( r ) = curl [ r f l ( n ) ( k r ) X l m ( θ , ϕ ) ] ,
N l m ( n ) ( r ) = 1 k curl M l m ( n ) ( r ) ,
M l m ( 4 ) ( r + R ) = l m [ A l m l m ( R ) M l m ( 1 ) ( r ) + B l m l m ( R ) N l m ( 1 ) ( r ) ] ,
N l m ( 4 ) ( r + R ) = l m [ A l m l m ( R ) N l m ( 1 ) ( r ) + B l m l m ( R ) M l m ( 1 ) ( r ) ] ,
A l m l m ( R ) = p [ l ( l + 1 ) + l ( l + 1 ) p ( p + 1 ) ] × ( l l p 0 0 0 ) C l m l m ( p , R ) ,
B l m l m ( R ) = p [ ( p + l + l ) ( p l + l ) ( l + l + 1 + p ) × ( l + l + 1 p ) ] 1 / 2 ( l l p 1 0 0 0 ) C l m l m ( p , R ) ,
C l m l m ( p , R ) = ( 1 ) m 2 l + 1 l ( l + 1 ) i l + p l ( 2 p + 1 ) × ( l l p m m m m ) ( ( l m ) ! ( l + m ) ! ( p + m m ) ! ( l + m ) ! ( l m ) ! ( p m + m ) ! ) 1 / 2 × h p ( 2 ) ( k R ) X p m m ( R ̂ ) .
g α β ( r ) 1 = C α β ( r ) + γ d 3 r [ g α γ ( r ) 1 ] ρ γ C γ β ( | r r | ) ,
g α β ( r ) = 0 for r < ( d α + d β ) / 2 ,
C α β ( r ) = 0 for r > ( d α + d β ) / 2 ,