Abstract

A method is described for obtaining the distribution of energy in an x-ray beam as a function of wavelength. The method consists of obtaining filtration data with suitable filter material, and analyzing this data in terms of Laplace transforms. The desired distribution of energy is obtained in terms of a sum of inverse Laplace transforms.

© 1952 Optical Society of America

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References

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  1. Glasser, Quimby, Taylor, and Weatherwax, Physical Foundations of Radiology (Paul B. Hoeber, Inc., New York, 1944), pp. 163–164.
  2. G. E. Bell, Brit. J. Radiology 9, N.S. 680 (1936).
    [Crossref]
  3. L. Silberstein, Phil. Mag. 15, 375 (1933).
  4. J. A. Victoreen, J. Appl. Phys. 14, 95 (1943).
    [Crossref]
  5. J. R. Greening, Proc. Phys. Soc. (London) 63, 11-A, 1227 (1950).
  6. D. E. A. Jones, Brit. J. Radiology 13, 95 (1940).
    [Crossref]
  7. S. M. Aly, Proc. Math. Phys. Soc. (Egypt) 4, 41 (1949).
  8. H. A. Kramers, see M. et L. de Broglie, Physique des Rayons X et Gamma (Paris, 1928), pp. 78–79.
  9. R. H. De Waard, Proc. Koninkl. Nederland. Akad. Wetenschap. 49, 944–954 (1946).
  10. J. A. Victoreen, J. Appl. Phys. 20, 1141 (1949).
    [Crossref]
  11. G. Doetsch, Theorie and Anwendung der Laplace-transformation (Verlag. Julius Springer, Berlin, 1937), p. 188.

1950 (1)

J. R. Greening, Proc. Phys. Soc. (London) 63, 11-A, 1227 (1950).

1949 (2)

S. M. Aly, Proc. Math. Phys. Soc. (Egypt) 4, 41 (1949).

J. A. Victoreen, J. Appl. Phys. 20, 1141 (1949).
[Crossref]

1946 (1)

R. H. De Waard, Proc. Koninkl. Nederland. Akad. Wetenschap. 49, 944–954 (1946).

1943 (1)

J. A. Victoreen, J. Appl. Phys. 14, 95 (1943).
[Crossref]

1940 (1)

D. E. A. Jones, Brit. J. Radiology 13, 95 (1940).
[Crossref]

1936 (1)

G. E. Bell, Brit. J. Radiology 9, N.S. 680 (1936).
[Crossref]

1933 (1)

L. Silberstein, Phil. Mag. 15, 375 (1933).

Aly, S. M.

S. M. Aly, Proc. Math. Phys. Soc. (Egypt) 4, 41 (1949).

Bell, G. E.

G. E. Bell, Brit. J. Radiology 9, N.S. 680 (1936).
[Crossref]

de Broglie, M. et L.

H. A. Kramers, see M. et L. de Broglie, Physique des Rayons X et Gamma (Paris, 1928), pp. 78–79.

De Waard, R. H.

R. H. De Waard, Proc. Koninkl. Nederland. Akad. Wetenschap. 49, 944–954 (1946).

Doetsch, G.

G. Doetsch, Theorie and Anwendung der Laplace-transformation (Verlag. Julius Springer, Berlin, 1937), p. 188.

Glasser,

Glasser, Quimby, Taylor, and Weatherwax, Physical Foundations of Radiology (Paul B. Hoeber, Inc., New York, 1944), pp. 163–164.

Greening, J. R.

J. R. Greening, Proc. Phys. Soc. (London) 63, 11-A, 1227 (1950).

Jones, D. E. A.

D. E. A. Jones, Brit. J. Radiology 13, 95 (1940).
[Crossref]

Kramers, H. A.

H. A. Kramers, see M. et L. de Broglie, Physique des Rayons X et Gamma (Paris, 1928), pp. 78–79.

Quimby,

Glasser, Quimby, Taylor, and Weatherwax, Physical Foundations of Radiology (Paul B. Hoeber, Inc., New York, 1944), pp. 163–164.

Silberstein, L.

L. Silberstein, Phil. Mag. 15, 375 (1933).

Taylor,

Glasser, Quimby, Taylor, and Weatherwax, Physical Foundations of Radiology (Paul B. Hoeber, Inc., New York, 1944), pp. 163–164.

Victoreen, J. A.

J. A. Victoreen, J. Appl. Phys. 20, 1141 (1949).
[Crossref]

J. A. Victoreen, J. Appl. Phys. 14, 95 (1943).
[Crossref]

Weatherwax,

Glasser, Quimby, Taylor, and Weatherwax, Physical Foundations of Radiology (Paul B. Hoeber, Inc., New York, 1944), pp. 163–164.

Brit. J. Radiology (2)

G. E. Bell, Brit. J. Radiology 9, N.S. 680 (1936).
[Crossref]

D. E. A. Jones, Brit. J. Radiology 13, 95 (1940).
[Crossref]

J. Appl. Phys. (2)

J. A. Victoreen, J. Appl. Phys. 14, 95 (1943).
[Crossref]

J. A. Victoreen, J. Appl. Phys. 20, 1141 (1949).
[Crossref]

Phil. Mag. (1)

L. Silberstein, Phil. Mag. 15, 375 (1933).

Proc. Koninkl. Nederland. Akad. Wetenschap. (1)

R. H. De Waard, Proc. Koninkl. Nederland. Akad. Wetenschap. 49, 944–954 (1946).

Proc. Math. Phys. Soc. (Egypt) (1)

S. M. Aly, Proc. Math. Phys. Soc. (Egypt) 4, 41 (1949).

Proc. Phys. Soc. (London) (1)

J. R. Greening, Proc. Phys. Soc. (London) 63, 11-A, 1227 (1950).

Other (3)

H. A. Kramers, see M. et L. de Broglie, Physique des Rayons X et Gamma (Paris, 1928), pp. 78–79.

Glasser, Quimby, Taylor, and Weatherwax, Physical Foundations of Radiology (Paul B. Hoeber, Inc., New York, 1944), pp. 163–164.

G. Doetsch, Theorie and Anwendung der Laplace-transformation (Verlag. Julius Springer, Berlin, 1937), p. 188.

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Figures (3)

Fig. 1
Fig. 1

Analytical fits for filtration data as a function of thickness of copper filter.

Fig. 2
Fig. 2

Distribution of x-ray energy as a function of wavelength for the different approximations to the filtration data.

Fig. 3
Fig. 3

Percentage of x-ray energy below a given wavelength or above a given kilovoltage.

Tables (1)

Tables Icon

Table I Constants used in obtaining the analytical fits to the experimental filtration data.

Equations (46)

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I ( x ) = λ 0 f ( λ ) e - μ ( λ ) x d λ ,
λ 0 f ( λ ) d λ = 1 .
ϕ ( μ ) = f ( λ ) ( d λ / d μ ) .
z = μ - μ 0 = μ - μ ( λ 0 )
Φ ( z ) = ϕ ( μ ) ,
J ( x ) = 0 Φ ( z ) e - z x d z ,
J ( x ) = I ( x ) e μ 0 x .
J ¯ ( x ) = a 1 exp ( - b 1 x 1 2 ) + a 2 exp ( - b 2 x 1 2 ) + .
Φ ( z ) = a 1 b 1 2 π 1 2 z - 3 2 exp ( - b 1 2 4 z ) + a 2 b 2 2 π 1 2 z - 3 2 exp ( - b 2 2 4 z ) + .
a 1 + a 2 + = 1.
ϕ ( μ ) = a 1 b 1 2 π 1 2 ( μ - μ 0 ) - 3 2 exp ( - b 1 2 4 ( μ - μ 0 ) ) + .
f ( λ ) = ϕ ( μ ) d μ / d λ .
J ¯ ( x ) = exp ( - b x 1 2 ) ,
J ¯ ( x ) = a 1 exp ( - b 1 x 1 2 ) + a 2 exp ( - b 2 x 1 2 ) ,
J ¯ ( x ) = a 1 exp ( - b 1 x 1 2 ) + a 2 exp ( - b 2 x 1 2 ) + a 3 exp ( - b 3 x 1 2 ) .
0.05 A λ 0.2 A
J ( x ) = [ d / ( x + d ) ] 1 2
J ( x ) = a exp ( - b x 1 2 ) .
J ( x ) = a exp ( - b [ ( x + d ) 1 2 - d 1 2 ] ) .
y = log 10 ( 100 I x / I 0 )
d I / d x < 0 for x > 0 I 0 as x ,
λ 0 f ( λ ) d λ = 1 ;
J ( x ) = e μ 0 x I ( x ) ,
J ( 0 ) = 1 ,             d J / d x < 0 ,             J 0             as             x .
d J / d x = - 0 e - z x z Φ ( z ) d z < 0 .
J ( x ) = [ a / ( x + a ) ] 1 2
f ( λ ) = d μ d λ [ a π ( μ - μ 0 ) ] 1 2 exp ( - a ( μ - μ 0 ) ) .
a exp ( - b x 1 2 ) ,             ( b > 0 ) .
J ¯ ( x ) = n = 1 N a n exp ( - b n x 1 2 ) ,             ( b n > 0 ) ,
ϕ ( μ ) = f ( λ ) d λ d μ = n = 1 N a n b n 2 π 1 2 ( μ - μ 0 ) 3 2 exp [ - b n 2 4 ( μ - μ 0 ) ] .
J ¯ = a 1 e - b 1 X + a 2 e - b 2 X .
( X 1 , J 1 ) ,             ( X 1 + Δ , J 2 ) ,             ( X 1 + 2 Δ , J 3 ) ,             ( X 1 + 3 Δ , J 4 ) .
| 1 J 1 J 2 t J 2 J 3 t 2 J 3 J 4 | = 0.
b 1 = - 1 Δ log e t 1 , b 2 = - 1 Δ log e t 2 , a 1 = e b 1 X 1 ( J 2 - J 1 t 2 t 1 - t 2 ) , a 2 = e - b 2 X 1 ( J 1 - a 1 e - b 1 X 1 ) .
b 1 = - 1 Δ log e t 1 , b 2 = - 1 Δ log e t 2 , a 1 = J 2 - t 2 t 1 - t ˙ 2 , a 2 = 1 - a 1 .
J ¯ ( x ) = a 1 exp ( - b 1 x 1 2 ) + a 2 exp ( - b 2 x 1 2 )
J 3 2 J 2 < J 4 < J 3 - J 2 2 - J 3 2 + J 2 J 3 1 - J 2 .
J 1 = 1 when x = 0 , J 2 = 0.324 when x = Δ = 0.39 , J 3 = 0.151 when x = 2 Δ = 0.78.
0.070 < J 4 < 0.107.
b n = - 1 / Δ log e t n ,             ( n = 1 , 2 , 3 ) ,
i = 1 3 a i t i n - 1 = J n ,             ( n = 1 , 2 , 3 ; J 1 = 1 ) .
f ( λ ) ~ 1 / λ 2             ( for     large     λ ) .
μ ~ λ n ,             n > 1             ( for     large     λ ) .
ϕ ( μ ) = f ( λ ) d λ d μ ~ μ - ( n + 1 / n ) ( for     large     μ ) .
z Φ ( z ) ~ z - 1 / n             ( for     large     z ) .
J ( x ) ~ x - 1 + 1 / n             ( n > 1 ; x small ) ;