Abstract

The rotation method for the absolute testing of three flats by the evaluation of four interference patterns of pairs of these flats is developed further. Least-squares methods for determining and minimizing the effect of random measuring errors are fully applied. This application makes an optimal resolution in depth and an enhanced lateral resolution possible. The computational effort mainly consists of a repeated solution of a linear equation system with 3N unknowns if N diameters of each flat are to be evaluated. The rms error of determining a flatness deviation is calculated as a function of the rms measuring error, the desired lateral resolution, and the position on the surface. The algorithm is extended to the case of using square-grid detector arrays by a special interpolation method.

© 1992 Optical Society of America

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References

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  1. G. Schulz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics XIII, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Chap. IV.
    [CrossRef]
  2. J. Schwider, “Ein Interferenzverfahren zur Absolutprüfung von Planflächennormalen. II,” Opt. Acta 14, 389–400 (1967).
    [CrossRef]
  3. G. Schulz, “Ein Interferenzverfahren zur absoluten Ebenheitsprüfung längs beliebiger Zentralschnitte,” Opt. Acta 14, 375–388 (1967).
    [CrossRef]
  4. J. Grzanna, G. Schulz, “Absolute testing of flatness standards at square-grid points,” Opt. Commun. 77, 107–112 (1990).
    [CrossRef]
  5. G. Schulz, J. Schwider, C. Hiller, B. Kicker, “Establishing an optical flatness standard,” Appl. Opt. 10, 929–934 (1971).
    [CrossRef] [PubMed]
  6. B. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23, 379–383 (1984).
  7. D. Malacara, Optical Shop Testing (Wiley, New York, 1978), App. 2.
  8. J. Y. Wang, D. E. Silva, “Wave-front interpretation with Zernike polynomials,” Appl. Opt. 19, 1510–1518 (1980).
    [CrossRef] [PubMed]
  9. In Refs. 1 (p. 123), 3, and 5, ν is the number of a diameter. In contrast, in our paper ν is the number of a half-diameter with ρ > 0 (Fig. 1). This definition simplifies the description if measuring errors are to be balanced fully. Equations (1) and (2) are valid in either case.
  10. The derivations of Eqs. (39.0)–(39.2) in the combination AB and the derivations of the analogous equations in BC and CA are the same. In ABΦ, however, because of the rotation, a primary measuring point of flat B generally is no primary measuring point of B in AB. In principle this fact results in a small positional difference of the B reference plane between AB and ABΦ, but this difference vanishes with an increasing number of measuring points. In practice it can be neglected, as confirmed by computer simulations.

1990

J. Grzanna, G. Schulz, “Absolute testing of flatness standards at square-grid points,” Opt. Commun. 77, 107–112 (1990).
[CrossRef]

1984

B. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23, 379–383 (1984).

1980

1971

1967

J. Schwider, “Ein Interferenzverfahren zur Absolutprüfung von Planflächennormalen. II,” Opt. Acta 14, 389–400 (1967).
[CrossRef]

G. Schulz, “Ein Interferenzverfahren zur absoluten Ebenheitsprüfung längs beliebiger Zentralschnitte,” Opt. Acta 14, 375–388 (1967).
[CrossRef]

Fritz, B.

B. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23, 379–383 (1984).

Grzanna, J.

J. Grzanna, G. Schulz, “Absolute testing of flatness standards at square-grid points,” Opt. Commun. 77, 107–112 (1990).
[CrossRef]

Hiller, C.

Kicker, B.

Malacara, D.

D. Malacara, Optical Shop Testing (Wiley, New York, 1978), App. 2.

Schulz, G.

J. Grzanna, G. Schulz, “Absolute testing of flatness standards at square-grid points,” Opt. Commun. 77, 107–112 (1990).
[CrossRef]

G. Schulz, J. Schwider, C. Hiller, B. Kicker, “Establishing an optical flatness standard,” Appl. Opt. 10, 929–934 (1971).
[CrossRef] [PubMed]

G. Schulz, “Ein Interferenzverfahren zur absoluten Ebenheitsprüfung längs beliebiger Zentralschnitte,” Opt. Acta 14, 375–388 (1967).
[CrossRef]

G. Schulz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics XIII, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Chap. IV.
[CrossRef]

Schwider, J.

G. Schulz, J. Schwider, C. Hiller, B. Kicker, “Establishing an optical flatness standard,” Appl. Opt. 10, 929–934 (1971).
[CrossRef] [PubMed]

J. Schwider, “Ein Interferenzverfahren zur Absolutprüfung von Planflächennormalen. II,” Opt. Acta 14, 389–400 (1967).
[CrossRef]

G. Schulz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics XIII, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Chap. IV.
[CrossRef]

Silva, D. E.

Wang, J. Y.

Appl. Opt.

Opt. Acta

J. Schwider, “Ein Interferenzverfahren zur Absolutprüfung von Planflächennormalen. II,” Opt. Acta 14, 389–400 (1967).
[CrossRef]

G. Schulz, “Ein Interferenzverfahren zur absoluten Ebenheitsprüfung längs beliebiger Zentralschnitte,” Opt. Acta 14, 375–388 (1967).
[CrossRef]

Opt. Commun.

J. Grzanna, G. Schulz, “Absolute testing of flatness standards at square-grid points,” Opt. Commun. 77, 107–112 (1990).
[CrossRef]

Opt. Eng.

B. Fritz, “Absolute calibration of an optical flat,” Opt. Eng. 23, 379–383 (1984).

Other

D. Malacara, Optical Shop Testing (Wiley, New York, 1978), App. 2.

In Refs. 1 (p. 123), 3, and 5, ν is the number of a diameter. In contrast, in our paper ν is the number of a half-diameter with ρ > 0 (Fig. 1). This definition simplifies the description if measuring errors are to be balanced fully. Equations (1) and (2) are valid in either case.

The derivations of Eqs. (39.0)–(39.2) in the combination AB and the derivations of the analogous equations in BC and CA are the same. In ABΦ, however, because of the rotation, a primary measuring point of flat B generally is no primary measuring point of B in AB. In principle this fact results in a small positional difference of the B reference plane between AB and ABΦ, but this difference vanishes with an increasing number of measuring points. In practice it can be neglected, as confirmed by computer simulations.

G. Schulz, J. Schwider, “Interferometric testing of smooth surfaces,” in Progress in Optics XIII, E. Wolf, ed. (North-Holland, Amsterdam, 1976), Chap. IV.
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

One of the three flats (A, B, or C) in (a) top view and (b) side view. The polar coordinate system (ρ, ϑ) is appropriate here. The unknown deviations of the optical surface from the reference plane (dashed line) are to be determined.

Fig. 2
Fig. 2

Combination AB in side view. The surface distance dν is known by an interference measurement, i.e., it is known apart from an additive constant, which is not of interest. The distance between the reference planes is Dν. The unknown flatness deviations xν and y−ν are to be determined. For each value of ν (Fig. 1), the azimuth ϑ = νΦ/2 of flat A coincides with the azimuth ϑ = −νΦ/2 of flat B if the coordinate systems are oriented correspondingly. Functions of the position are dν, Dν, xν, and y−ν.

Fig. 3
Fig. 3

Four positional combinations of the rotation method in top view. In the fourth combination, ABΦ, flat B has been rotated from its position in the first combination, AB, by the angle Φ. The coordinate systems of all three surfaces are oriented so that in the first three combinations, which are the basic combinations, the azimuths ϑ = 0 of both flats coincide (coincidence is at ν = 0). Then in the fourth combination, which is the rotational combination, the azimuths ϑ = Φ/2 coincide (coincidence is at ν = 1).

Fig. 4
Fig. 4

Matrix equation (10). The rectangular matrix H consists of twelve quadratic submatrices with 2N × 2N elements each. Four of these submatrices are null matrices and four are unit matrices. Of the remaining four submatrices, three are obtained from a unit matrix by a reflection with respect to its vertical midline and by a subsequent transposition of the first column behind the last one. One submatrix is obtained in the same way, but by a transposition of the last column before the first one. The azimuth ϑ belonging to an element of X or M is obtained from Eq. (3), i.e., by multiplying the subscript of the element by Φ/2.

Fig. 5
Fig. 5

Matrix equation (15) or, if the superscript tilde is replaced by a superscript double tilde, matrix equation (20). Because changes of subscripts by ±N are irrelevant, we have added N or 2N to nonpositive subscripts. The matrix h is the same as the matrix H of Fig. 4 with the following exceptions. First, instead of 2N × 2N elements, each of the twelve submatrices has only N × N elements, and one of these submatrices (below) is obtained from a unit matrix by a reflection with respect to its vertical midline without a subsequent transposition of a column. The azimuth ϑ belonging to an element of x ˜ or m ˜ is obtained from Eqs. (3) and (13), i.e., by multiplying the subscript of the element by Φ. The azimuth belonging to an element of x ˜ ˜ or m ˜ ˜ is obtained from Eqs. (3) and (17) or (18), e.g., the azimuth belonging to z ˜ ˜ τ is (−1 − 2τ) Φ/2.

Fig. 6
Fig. 6

Error-propagation factor fν(x) (N) for the mean error of xν according to Eqs. (22) and (31). The figure shows the dependence of fν(x) (N) on the azimuth ϑ = 180° × ν/N according to Eqs. (3) and (1) for M = 1 (compare Fig. 1);ν = −N, −N + 1, … N; and N = 3, 4, … 8. The rms error of a specific result xν or yν is σfν(x) (N). The rms error of the interferometer is σ. The figure shows that for N = 8, 0.74 ≲ fν(x)(8) ≲ 1.10. Correspondingly, for N = 30, we obtain 0.74 ≲ fν(x) (30) ≲ 1.99.

Fig. 7
Fig. 7

Quadratic means of the error-propagation factors according to Eqs. (32) and (33). We can regard fmain(x) (N) as the mean error of all results xν or yν and fmean(z) (N) of all results zν, using one set of four interference patterns.

Fig. 8
Fig. 8

Square grid whose nodes Pkl are the primary measuring points with the Cartesian coordinates k, l = 0, ±1, ±2, … (1 is the lateral unit of length and is equal to the grid spacing). A secondary measuring point with the polar coordinates ρ = ρμ, ϑ = ϑν [Eq. (3)] and the Cartesian coordinates k + ξ, l + η (ξ and η between 0 and 1) is Sμν.

Fig. 9
Fig. 9

Second-order interpolation of a deviation sum along the line GL, which is a grid line or a line parallel to it.

Fig. 10
Fig. 10

Surface deviations along the line ν = 0 at some stages of the simulation procedure. Stage I, ideal flatness deviations of surface A according to Eq. (52) with c = 1.85; stage II (bars), deviation sums of the interference pattern of combination AB with individual measuring errors (algorithm step 1 and algorithm step 2, case Q); stage III (open circles), flatness deviations of A after evaluation (according to algorithm step 3) of one set of four interference patterns with M = 7; stage IV, errors of results of stage III; stage V, errors of mean results of stage III. (Instead of evaluating one set of interference patterns, for stage IV we evaluated four additional sets with other individual measuring errors and averaged all five results.)

Fig. 11
Fig. 11

Rms error propagation from the measurements to the results of the evaluation of one set of four interference patterns as simulated according to Eqs. (52)(54) with c = 1.6 and M = 1. Stage I, primary mean measuring error σ in the square grid (algorithm step 1); stage II, interpolation from the square grid to the polar grid by Eq. (36) with (35) (algorithm step 2, case Q); stage III, secondary mean measuring error σS in the polar grid; stage IV, absolute determination (algorithm step 3); stage V, rms error of the result in the polar grid; stage VI, back interpolation into the square grid (compare Section VI); stage VII, rms error of the result in the square grid.

Equations (100)

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Φ = 360 ° × M / N ,
x ν + y - ν = D ν - d ν = a ν ,
y ν + z - ν = b ν ,
z ν + x - ν = c ν ,
x ν + y 2 - ν = a ν .
ρ μ ( μ = 1 , 2 , R ) , ϑ ν = ν Φ / 2 ( ν = - N + 1 , - N + 2 , N ) ,
x ν = x ν ( ρ μ ) y ν = y ν ( ρ μ ) z ν = z ν ( ρ μ ) } = v ν ( ρ μ ) ,
μ , ν [ v ν ( ρ μ ) ] 2 = minimum ,
μ , ν v ν ( ρ μ ) = 0 ,
μ , ν v ν ( ρ μ ) ρ μ cos ϑ ν = 0 ,
μ , ν v ν ( ρ μ ) ρ μ sin ϑ ν = 0 ,
D ν = D ν ( ρ μ ) = α 0 + α 1 ρ μ cos ϑ ν + α 2 ρ μ sin ϑ ν ,
α 0 = μ , ν d ν ( ρ μ ) / T ,
α 1 = 2 μ , ν d ν ( ρ μ ) ρ μ cos ϑ ν / ( T ρ 2 ¯ ) ,
α 2 = 2 μ , ν d ν ( ρ μ ) ρ μ sin ϑ ν / ( T ρ 2 ¯ ) .
m o = σ / T ,
m 1 = m 2 = σ 2 / ( T ρ 2 ¯ ) ,
HX = M ,
X = ( H T H ) - 1 H T M .
ν = 2 τ ,
x ˜ τ = x 2 τ , y ˜ τ = y 2 τ , z ˜ τ = z 2 τ , a ˜ τ = a 2 τ , b ˜ τ = b 2 τ , c ˜ τ = c 2 τ , a ˜ τ = a 2 τ .
x ˜ τ + y ˜ - τ = a ˜ τ ,
y ˜ τ + z ˜ - τ = b ˜ τ ,
z ˜ τ + x ˜ - τ = c ˜ τ ,
x ˜ τ + y ˜ 1 - τ = a τ ,
h x ˜ = m ˜ ,
x ˜ = ( h T h ) - 1 h T m ˜ .
x ˜ ˜ τ = x 1 - 2 τ ,             y ˜ ˜ τ = y 1 - 2 τ ,             a ˜ ˜ τ = a 1 - 2 τ ,             b ˜ ˜ τ = b 1 - 2 τ ,             a ˜ ˜ τ = a 1 - 2 τ ( ν = 1 - 2 τ ) ,
z ˜ ˜ τ = z - 1 - 2 τ ,             c ˜ ˜ τ = c - 1 - 2 τ ( ν = - 1 - 2 τ ) .
x ˜ ˜ τ + y ˜ ˜ - τ = a ˜ ˜ τ ,
y ˜ ˜ τ + z ˜ ˜ - τ = b ˜ ˜ τ ,
z ˜ ˜ τ + x ˜ ˜ - τ = c ˜ ˜ τ ,
x ˜ ˜ τ + y ˜ ˜ 1 - τ = a ˜ ˜ τ .
h x ˜ ˜ = m ˜ ˜ ,
x ˜ ˜ = ( h T h ) - 1 h T m ˜ ˜ .
x = [ ( a + a ) / 2 - b + c ] / 2 , y = [ ( a + a ) / 2 + b - c ] / 2 , z = [ - ( a + a ) / 2 + b + c ] / 2.
m ν ( x ) = σ f ν ( x ) ( N ) , m ν ( y ) = σ f ν ( y ) ( N ) , m ν ( z ) = σ f ν ( z ) ( N ) , ( ν = - N + 1 , - N + 2 , N ) ,
a ν + b ν + c ν - ( a - ν + b - ν + c - ν ) = 0             ( ν = 1 , 2 , N - 1 ) ,
b 1 + ν + c - 1 + ν + a 1 + ν - ( b 1 - ν + c - 1 - ν + a 1 - ν ) = 0             ( ν = 1 , 2 , N - 1 ) ,
τ = 1 N a 2 τ - τ = 1 N a 2 τ = 0 ,
τ = 1 N a 2 τ - 1 - τ = 1 N a 2 τ - 1 = 0.
L ν ( ρ μ ) = a ν ( ρ μ ) + b ν ( ρ μ ) + c ν ( ρ μ ) - [ a - ν ( ρ μ ) + b - ν ( ρ μ ) + c - ν ( ρ μ ) ] ,
{ μ = 1 R ν = 1 N - 1 [ L ν ( ρ μ ) ] 2 / ( N - 1 ) R } 1 / 2 .
σ = { μ - 1 R ν = 1 N - 1 [ a ν ( ρ μ ) + b ν ( ρ μ ) + c ν ( ρ μ ) - a - ν ( ρ μ ) - b - ν ( ρ μ ) - c - ν ( ρ μ ) ] 2 / 6 ( N - 1 ) R } 1 / 2 .
σ = ( μ = 1 R { ν = 1 2 N [ a ν ( ρ μ ) - a ν ( ρ μ ) ] } 2 / 4 N R ) 1 / 2 .
σ R = | ν = 1 2 N [ a ν ( ρ R ) - a ν ( ρ R ) ] | / 4 N
σ < R = ( μ = 1 R - 1 { ν = 1 2 N [ a ν ( ρ μ ) - a ν ( ρ μ ) ] } 2 / 4 N ( R - 1 ) ) 1 / 2 .
σ right = { μ = 1 R ν = 1 ( N - 1 ) / 2 [ a ν ( ρ μ ) + b ν ( ρ μ ) + c ν ( ρ μ ) - a - ν ( ρ μ ) - b - ν ( ρ μ ) - c - ν ( ρ μ ) ] 2 / 3 ( N - 1 ) R } 1 / 2             ( M = 1 , N = odd ) ,
σ left = { μ = 1 R ν = ( N + 1 ) / 2 N - 1 [ a ν ( ρ μ ) + b ν ( ρ μ ) + c ν ( ρ μ ) - a - ν ( ρ μ ) - b - ν ( ρ μ ) - c - ν ( ρ μ ) ] 2 / 3 ( N - 1 ) R } 1 / 2             ( M = 1 , N = odd ) .
f ν ( x ) ( N ) = f ν ( y ) ( N ) ,
f ν ( x ) ( N ) = Δ ν + N , f ν ( y ) ( N ) = Δ ν + 3 N , f ν ( z ) ( N ) = Δ ν + 5 N ,
f mean ( x ) ( N ) = { ν = - N + 1 N [ f ν ( x ) ( N ) ] 2 / 2 N } 1 / 2 ,
f mean ( z ) ( N ) = { ν = - N + 1 N [ f ν ( z ) ( N ) ] 2 / 2 N } 1 / 2 .
a ν ( ρ μ ) = a k l + ( a k + 1 , l - a k l ) ξ + ( a k , l + 1 - a k l ) η + ( a k l + a k + 1 , l + 1 - a k + 1 , l - a k l + 1 ) ξ η .
q = q { p ( 1 ) , p ( 2 ) , p ( 3 ) , p ( 4 ) ; δ } = p ( 2 ) + { [ - p ( 1 ) - 3 p ( 2 ) + 5 p ( 3 ) - p ( 4 ) ] / 4 } δ + { [ p ( 1 ) - p ( 2 ) - p ( 3 ) + p ( 4 ) ] / 4 } δ 2 ,
a ν ( ρ μ ) = q { q { a k - 1 , l - 1 , a k , l - 1 , a k + 1 , l - 1 , a k + 2 , l - 1 ; ξ } , q { a k - 1 , l , a k l , a k + 1 , l , a k + 2 , l ; ξ } , q { a k - 1 , l + 1 , a k , l + 1 , a k + 1 , l + 1 , a k + 2 , l + 1 ; ξ } , q { a k - 1 , l + 2 , a k , l + 2 , a k + 1 , l + 2 , a k + 2 , l + 2 ; ξ } ; η }
k = 0 , ± 1 , ± 2 , l = 0 , ± 1 , ± 2 , } ( k 2 + l 2 s 2 ρ R 2 ) .
D k l = α 0 + α 1 k + α 2 l ,
α 0 = k , l d k l / k , l 1 ,
α 1 = k , l d k l k / k , l k 2 ,
α 2 = k , l d k l l / k , l l 2 .
x k l + y k , - l = D k l - d k l = a k l ,
a k l + b k l + c k l - ( a k , - l + b k , - l + c k , - l ) = 0 ,
σ = [ k , l ( 1 > 0 ) ( a k l + b k l + c k l - a k , - l - b k , - l - c k , - l ) 2 / 6 n ] 1 / 2 ,
σ right = [ k , l ( k > 0 , l > 0 ) ( a k l + b k l + c k l - a k , - l - b k , - l - c k , - l ) 2 / 6 n * ] 1 / 2 ,
σ left = [ k , l ( k < 0 , l > 0 ) ( a k l + b k l + c k l - a k , - l - b k , - l - c k , - l ) 2 / 6 n * ] 1 / 2 ,
σ S = g ( σ , σ I ) ,
g ( 0 , σ I ) = σ I .
g ( σ , 0 ) = α ¯ σ
α ¯ = 2 / 3 ,
α ¯ = 91 / 120 3 / 4 ,
[ g ( σ , σ I ) ] 2 = [ g ( 0 , σ I ) ] 2 + [ g ( σ , 0 ) ] 2 ,
σ S = [ σ I 2 + ( α ¯ σ ) 2 ] 1 / 2 ,             α ¯ < 1.
a ν ( ρ μ ) = D ν ( ρ μ ) - d ν ( ρ μ ) .
a k l = D k l - d k l .
F A = sin ( k + l ) × 3 λ / 40 + sin ( c k + c l ) × λ / 40 ,
F B = cos ( k / 2 + l ) × 3 λ / 40 + cos ( c k + c l ) × λ / 40 + cos ( k / 10 ) × 3 λ / 100 ,
F C = sin ( k + l / 2 ) × 3 λ / 40 + cos ( k + c l ) × λ / 40.
0 = μ , ν D ν ( ρ μ ) - μ , ν d ν ( ρ μ ) = μ , ν a ν ( ρ μ )
ν cos ϑ ν = ν sin ϑ ν = 0
0 = μ , ν α 0 - μ , ν d ν ( ρ μ ) ,
0 = μ , ν D ν ( ρ μ ) ρ μ cos ϑ ν - μ , ν d ν ( ρ μ ) ρ μ cos ϑ ν
ν sin ϑ ν cos ϑ ν = 0
0 = α 1 μ , ν ρ μ 2 cos 2 ϑ ν - μ , ν d ν ( ρ μ ) ρ μ cos ϑ ν .
μ ρ μ 2 = ρ 2 ¯ R ,             ν cos 2 ϑ ν = ν 1 2 = N ,             R N = T / 2 ,
μ , ν ρ μ 2 cos 2 ϑ ν = ρ 2 ¯ T / 2
0 = α 1 ρ 2 ¯ T / 2 - μ , ν d ν ( ρ μ ) ρ μ cos ϑ ν ,
m 0 = μ , ν ( σ / T ) 2 ,
m 1 = { μ , ν [ 2 σ ρ μ cos ϑ ν / ( T ρ 2 ¯ ) ] 2 } 1 / 2 = [ 2 σ / ( T ρ 2 ¯ ) ] ( μ , ν ρ μ 2 cos 2 ϑ ν ) 1 / 2 ,
X = AM ,
A = ( H T H ) - 1 H T .
X i = { x ν ( i = ν + N ) y ν ( i = ν + 3 N ) z ν ( i = ν + 5 N ) ,
M k = { a ν ( k = ν + N ) b ν ( k = ν + 3 N ) c ν ( k = ν + 5 N ) a ν ( k = ν + 7 N ) ,
X i / M k = A i k ,
mean error of X i = σ [ k ( X i / M k ) 2 ] 1 / 2 ,
mean error of X i = σ k A i k 2 .
k A i k 2 = Δ i .
mean error of X i = σ Δ i .
[ ( H T H ) - 1 H T ] [ ( H T H ) - 4 H T ] T = [ ( H T H ) - 1 H T ] H [ ( H T H ) - 1 ] T = [ ( H T H ) - 1 ] T = ( H T H ) - 1 ,
m ν ( x ) = σ Δ ν + N , m ν ( y ) = σ Δ ν + 3 N , m ν ( z ) = σ Δ ν + 5 N ,

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