Abstract

Equations for calculation of the crossing point determination error and the signal width error in level-crossing experiments are generalized when the normalized second-order factorial moment of the fluorescence obtained is measured. To minimize the magnitude of these errors we study the behavior of these values on varying the magnetic field sweep interval, the counting time, and the mean fluorescence intensity.

© 1983 Optical Society of America

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References

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  1. M. A. Rebolledo, J. J. Sanz, Appl. Opt. 19, 2533 (1980).
    [CrossRef] [PubMed]

1980 (1)

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Figures (8)

Fig. 1
Fig. 1

Periodic triangular sweep of the magnetic field in a level-crossing experiment with a Lorentzian fluorescence profile.

Fig. 2
Fig. 2

As for Fig. 1 with a dispersion profile.

Fig. 3
Fig. 3

Values of eΓ for a dispersion profile when M/Ī = 0.1, T/Γ = 0.1, and Ī = 105 pulses/sec.

Fig. 4
Fig. 4

Values of eto for a dispersion profile when M/Ī = 0.1, T/Γ = 0.1, and Ī = 105 pulses/sec.

Fig. 5
Fig. 5

Values of eΓ and eto for a dispersion profile when M/Ī = 0.1, T/Γ = 0.01, to = 2Γ, and Ī = 105 pulses/sec.

Fig. 6
Fig. 6

Values of eΓ for a Lorentzian profile when M/Ī = 0.1, T/Γ = 0.1, and Ī = 105 pulses/sec.

Fig. 7
Fig. 7

Values of eto for a Lorentzian profile when M/Ī = 0.1, T/Γ = 0.1, and Ī = 105 pulses/sec.

Fig. 8
Fig. 8

Values of eΓ and eto for a Lorentzian profile when M/Ī = 0.1, T/Γ = 0.01, to = 2Γ, and Ī = 105 pulses/sec.

Tables (2)

Tables Icon

Table I Dispersion Profile: eΓ and eto Values for a Total Measurement Time of 1 sec; M/Ī = 0.1, to = 2Γ, and P = 6.5Γ; the Minimum Errors for Each Value of Ī are Underlined

Tables Icon

Table II Lorentzian Profile: eΓ and eto Values for a Total Measurement Time of 1 sec; M/Ī = 0.1, to = 2Γ, and P = 8Γ; the Minimum Errors for Each Value of Ī are Underlined

Equations (19)

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Var α = [ n ( 2 ) ( T ) α ] - 2 · Var n ( 2 ) ( T ) ,
Var n ( 2 ) ( T ) = 1 N { n ( 4 ) + 4 n ( 3 ) [ 1 / n ¯ - n ( 2 ) ] + 2 n ( 2 ) / n ¯ 2 - [ n ( 2 ) ] 2 ( 4 / n ¯ + 1 ) + 4 [ n ( 2 ) ] 3 } ,
n ( r ) = l = 0 r ( r l ) ( M / I ¯ ) l F l ,
F l = 1 P T l 0 P Y l ( t , T ) d t ,
Y ( t , T ) = t t + T F ( t ) d t ,
F l = 1 2 ( l - 1 ) i 1 = 1 i l - 1 = 1 i l A i 1 A i l G ( w i 1 ) G ( w i l ) ,
G ( w l ) = sin ( w l T / 2 ) / ( w l T / 2 ) ;             w l = 2 l π / P ;
w i l { j = 1 l - 1 ( - 1 ) m j w j m j = 1 , 2 ;             j = 1 l - 1 m j > l - 1 } .
K 1 = { 0 for a Lorentzian profile , 1 for a dispersion profile ;
K 2 = { 1 if t [ - P / 2 , 0 ] , - 1 if t [ 0 , P / 2 ] ;
C = { 1 + [ Γ / 2 t o ] 2 if K 1 = 0 and P 4 t o , 1 + [ Γ / ( P - 2 t o ) ] 2 if K 1 = 0 and P 4 t o , 1 if K 1 = 1 ;
F ( t ) = { C ( Γ / 2 ) 2 ( t + K 2 t o ) 2 + ( Γ / 2 ) 2 if K 1 = 0 , - C K 2 ( Γ / 2 ) ( t + K 2 t o ) ( t + K 2 t o ) 2 + ( Γ / 2 ) 2 if K 1 = 1.
A l = ( 2 π C Γ / P ) cos ( x l + π K 1 / 2 ) exp ( - l π Γ / P ) + ( 2 C / l π ) n = 1 ( l π Γ / P ) 2 n - K 1 × { [ F n ( x l ) + F n ( l π - x l ) ] cos ( x l - π K 1 / 2 ) + [ G n ( x l ) - G n ( l π - x l ) ] sin ( x l + π K 1 / 2 ) } ,
x l = w l t o ,
F n ( x ) = 1 ( 2 n - 1 - K 1 ) ! × [ f n ( x ) cos x - ( - 1 ) K 1 g n ( x ) sin x - s i ( x ) ] ,
G n ( x ) = 1 ( 2 n - 1 - K 1 ) ! × [ g n ( x ) cos x + ( - 1 ) K 1 f n ( x ) sin x + ( - 1 ) K 1 c i ( x ) ] ,
f n ( x ) = k = 0 n - 1 - K 1 ( - 1 ) k + 1 ( 2 k ) ! x 2 k + 1 ,
g n ( x ) = k = 0 n - 2 ( - 1 ) k + K 1 ( 2 k + 1 ) ! x 2 ( k + 1 ) .
Γ < P - 2 t o if P 4 t o , Γ < 2 t o if P 4 t o .

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