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Choice of collimators for neutron powder diffractometry
NLICLEAR
INSTRUMENTS
AND
36 (1965) I79-I80;
METHODS
© NORTH-HOLLAND
PUBLISHING
CO.
C H O I C E OF C O L L I M A T O R S F O R N E U T R O N P O W D E R D I F F R A C T O M E T R Y
M. POPOVICI Institute of Atomic Physics, Bucharest, Rumania Received 26 February 1965
The angular divergences of collimators, assuring the maximum of luminosity for a given resolution for a crystal spectrometer for neutron powder diffraction are determined. 1. Introduction The study of the resolution and luminosity of the experimental arrangements for neutron powder diffractometry has been the subject of a paper by Caglioti et al.~). Applying the method proposed by Sailor et al.2), the aforementioned authors obtained the following expressions for the full width at half maximum of the diffraction peaks A~ and for the area of these peaks L': I
A~ =
2
2 +
2
2
2
2
2
ct2(cq + 4fl ) - 4act2(~ ~ + 2fl ) 2 2 ~2 " + ~1 + ~2 + 4fl 2
2 2
2 2
2 2
4a (~l~Z +cqfl +c~2fl)] + ~12 + ~22 + 4fl J L' = P "
~1(Z2~3fl
+
~-
P'L
½
The purpose of this paper, is to determine analytically the divergences of collimators, assuring the maximum of luminosity for a given full width at half maximum of diffraction peaks. 2. Determination of the optimum divergences of the collimators The values cq that assure the maximum of L' for A~ = const, must be determined. Writing Lagrange's function in the form of F - - A~ + 2L' and eliminating 2 from the conditions 8 F / ~ t = 0 and OF/&% = 0, one obtains the first relation for the determination of the optimum divergences as follows:
c~2 = 2
[~2(2a - 1) + 2aft2] 2 (=I +
(2)
+
where ~, horizontal angular divergence of the i-th collimator, fl, mosaic spread of the monochromating crystal, a - tan 0a/tan 0M, 0~ and 0u being the Bragg angles of the sample and of the monochromating crystal respectively. The notation is from ref. 1). The expression (1) can be directly compared with the experiment and it is found that (1) is in satisfactory agreement with the measured half widths3). The experimental verification of the relation (2) is conditioned by the correct theoretical determination of the factor P, connecting the actual luminosity L' with the magnitude L, the instrumental luminosity, as it was called in ref. ~). This factor depends, among several others, on the parameter a and on the mosaic spread ft. An argument supporting the validity of eq. (2) is the goodness of fit between the measurements of the instrumental luminosities ~) and the relations derived for the more general case of the single crystal diffraction 3"5). By means of the relations (1) and (2) the problem of the choice of the divergences cti can be studied in order to achieve a good compromise between luminosity and resolution. In the aforementioned paper x) this problem was tackled, but the chosen method (that of numerical calculations for sets of the values of ~i) allowed to obtain only some general indications.
+
(3)
+ 4 2)"
If this relation is satisfied, one obtains from (1) for the half width A}: A~
LZc% z+
4 ~2f12([:~a)~] ~ c~22+ 4,82 ] "
(la)
From the conditions OF/&q = 0 and OF/O~2 = 0 another connection between the optimum divergences is obtained: 4(a - 1)2fla =
--
(4)
Using the relations (3), (la) and (4) one can determine the optimum divergences el for given values of parameters a, fl and A½. In fig. 1 the optimum divergences, expressed in unities of A~, vs a=tan0B/tan0M, are represented for some values of the ratio A~/fl. The curves were calculated by taking the value ~z/fl as a parameter. In the case ~2 = oo only the relations (3) and (la) were used. 3. Discussion One can observe from fig. 1 that the optimum divergences depend strongly on a, so that a given set of collimators can assure the optimum relation between the luminosity and the half width for a given diffraction peak only.
179
180
15,4,
M. POPOVICI
15
luminosities given by an optimum arrangement and by a conventional one with ~ = 0(2 = 0(3. In fig. 2 the values of the ratio L/A~ (that can be called relative instrumental luminosity) are represented vs A~/~ for three typical values of the parameter a (a < 1 ; a = 1 ; a > 1). This ratio was calculated for the arrangements with collimators, having optimum divergences (full lines) and for the conventional arrangements with
'
0(1 =
05A~ 1/
} ~2
.
}~ 0
I
[
l
2
0C2 =
0(3'
Fig. 2 illustrates the gain of intensity one can obtain through appropriate choice of collimators. This gain can be particularly large in the case o f a = 1, when for optimum divergences =~ =0(3; 0(2 = oo, the half width A~ does not depend on ~, see eq. (la), while the luminosity increases with ~. The choice of the parameter A~/~ influences the relative luminosity rather strongly at small values of a, as one can see from fig. 2. With the aid of the curves of fig. 2 one can determine the optimum value of the mosaic spread ~, if the dependence of the factor P in relation (2) upon/~ is known.
I
o=tonOBlton8~.,
Fig. 1. Angular divergences of the collimators for which the luminosity L' is maximum at given values o f A~/fl and a = tan0B/tan 0M. Figures indicate the values o f A~/fl.
The author wishes to thank Dr. D. Bally for the interest shown in this work and Mr. Z. Gheorghiu for helpful discussions.
At relative large values of a, where the choice of optimum divergences is particularly interesting, one finds again the result indicated by Caglioti et al.~), that the divergences ~i must satisfy the inequalities c% > ~2 > ~ . In the range of relative small values of a it is necessary to eliminate the collimator between the monochromating crystal and the sample (0(2 = 00) and to satisfy the inequalities ~3 > cq (if a > I) or 0~3 < 0( 1 (if a
References 1) G. Caglioti, A. Paoleni and F. P. Ricci, Nucl. Instr. 3 (1958) 223. 2) V. L. Sailor, H. L. Foote, H. H. Landon and R. E. Wood, Rev. Sci. Instr. 27 (1956) 26. 3) G. Caglioti and F. P. Ricci, Nucl. Instr. and Meth. 15 (1962) 155. 4) G. Caglioti, Acta Cryst. 17 (1964) 1902. 5) G. Caglioti, A. Paoletti and F. P. Ricci, Nucl. Instr. and Meth. 9 (1960) 195. L
L
L
A ~,
A~ : 2 V j t - o ) 075
05
0.25
07!
075
0
05
02
025
. . . . . .
ct=3
(3=I
0 = 075 I
71
T
x SO
I
2
l
A~
9-
0
i
z ~
Fig. 2. Relation between instrumental luminosity and full width at half maximum of the diffraction peaks for the optimum divergences (full lines) and the divergences ~l = ~2 = =(3 (dashed lines).
INSTRUMENTS
AND
36 (1965) I79-I80;
METHODS
© NORTH-HOLLAND
PUBLISHING
CO.
C H O I C E OF C O L L I M A T O R S F O R N E U T R O N P O W D E R D I F F R A C T O M E T R Y
M. POPOVICI Institute of Atomic Physics, Bucharest, Rumania Received 26 February 1965
The angular divergences of collimators, assuring the maximum of luminosity for a given resolution for a crystal spectrometer for neutron powder diffraction are determined. 1. Introduction The study of the resolution and luminosity of the experimental arrangements for neutron powder diffractometry has been the subject of a paper by Caglioti et al.~). Applying the method proposed by Sailor et al.2), the aforementioned authors obtained the following expressions for the full width at half maximum of the diffraction peaks A~ and for the area of these peaks L': I
A~ =
2
2 +
2
2
2
2
2
ct2(cq + 4fl ) - 4act2(~ ~ + 2fl ) 2 2 ~2 " + ~1 + ~2 + 4fl 2
2 2
2 2
2 2
4a (~l~Z +cqfl +c~2fl)] + ~12 + ~22 + 4fl J L' = P "
~1(Z2~3fl
+
~-
P'L
½
The purpose of this paper, is to determine analytically the divergences of collimators, assuring the maximum of luminosity for a given full width at half maximum of diffraction peaks. 2. Determination of the optimum divergences of the collimators The values cq that assure the maximum of L' for A~ = const, must be determined. Writing Lagrange's function in the form of F - - A~ + 2L' and eliminating 2 from the conditions 8 F / ~ t = 0 and OF/&% = 0, one obtains the first relation for the determination of the optimum divergences as follows:
c~2 = 2
[~2(2a - 1) + 2aft2] 2 (=I +
(2)
+
where ~, horizontal angular divergence of the i-th collimator, fl, mosaic spread of the monochromating crystal, a - tan 0a/tan 0M, 0~ and 0u being the Bragg angles of the sample and of the monochromating crystal respectively. The notation is from ref. 1). The expression (1) can be directly compared with the experiment and it is found that (1) is in satisfactory agreement with the measured half widths3). The experimental verification of the relation (2) is conditioned by the correct theoretical determination of the factor P, connecting the actual luminosity L' with the magnitude L, the instrumental luminosity, as it was called in ref. ~). This factor depends, among several others, on the parameter a and on the mosaic spread ft. An argument supporting the validity of eq. (2) is the goodness of fit between the measurements of the instrumental luminosities ~) and the relations derived for the more general case of the single crystal diffraction 3"5). By means of the relations (1) and (2) the problem of the choice of the divergences cti can be studied in order to achieve a good compromise between luminosity and resolution. In the aforementioned paper x) this problem was tackled, but the chosen method (that of numerical calculations for sets of the values of ~i) allowed to obtain only some general indications.
+
(3)
+ 4 2)"
If this relation is satisfied, one obtains from (1) for the half width A}: A~
LZc% z+
4 ~2f12([:~a)~] ~ c~22+ 4,82 ] "
(la)
From the conditions OF/&q = 0 and OF/O~2 = 0 another connection between the optimum divergences is obtained: 4(a - 1)2fla =
--
(4)
Using the relations (3), (la) and (4) one can determine the optimum divergences el for given values of parameters a, fl and A½. In fig. 1 the optimum divergences, expressed in unities of A~, vs a=tan0B/tan0M, are represented for some values of the ratio A~/fl. The curves were calculated by taking the value ~z/fl as a parameter. In the case ~2 = oo only the relations (3) and (la) were used. 3. Discussion One can observe from fig. 1 that the optimum divergences depend strongly on a, so that a given set of collimators can assure the optimum relation between the luminosity and the half width for a given diffraction peak only.
179
180
15,4,
M. POPOVICI
15
luminosities given by an optimum arrangement and by a conventional one with ~ = 0(2 = 0(3. In fig. 2 the values of the ratio L/A~ (that can be called relative instrumental luminosity) are represented vs A~/~ for three typical values of the parameter a (a < 1 ; a = 1 ; a > 1). This ratio was calculated for the arrangements with collimators, having optimum divergences (full lines) and for the conventional arrangements with
'
0(1 =
05A~ 1/
} ~2
.
}~ 0
I
[
l
2
0C2 =
0(3'
Fig. 2 illustrates the gain of intensity one can obtain through appropriate choice of collimators. This gain can be particularly large in the case o f a = 1, when for optimum divergences =~ =0(3; 0(2 = oo, the half width A~ does not depend on ~, see eq. (la), while the luminosity increases with ~. The choice of the parameter A~/~ influences the relative luminosity rather strongly at small values of a, as one can see from fig. 2. With the aid of the curves of fig. 2 one can determine the optimum value of the mosaic spread ~, if the dependence of the factor P in relation (2) upon/~ is known.
I
o=tonOBlton8~.,
Fig. 1. Angular divergences of the collimators for which the luminosity L' is maximum at given values o f A~/fl and a = tan0B/tan 0M. Figures indicate the values o f A~/fl.
The author wishes to thank Dr. D. Bally for the interest shown in this work and Mr. Z. Gheorghiu for helpful discussions.
At relative large values of a, where the choice of optimum divergences is particularly interesting, one finds again the result indicated by Caglioti et al.~), that the divergences ~i must satisfy the inequalities c% > ~2 > ~ . In the range of relative small values of a it is necessary to eliminate the collimator between the monochromating crystal and the sample (0(2 = 00) and to satisfy the inequalities ~3 > cq (if a > I) or 0~3 < 0( 1 (if a
References 1) G. Caglioti, A. Paoleni and F. P. Ricci, Nucl. Instr. 3 (1958) 223. 2) V. L. Sailor, H. L. Foote, H. H. Landon and R. E. Wood, Rev. Sci. Instr. 27 (1956) 26. 3) G. Caglioti and F. P. Ricci, Nucl. Instr. and Meth. 15 (1962) 155. 4) G. Caglioti, Acta Cryst. 17 (1964) 1902. 5) G. Caglioti, A. Paoletti and F. P. Ricci, Nucl. Instr. and Meth. 9 (1960) 195. L
L
L
A ~,
A~ : 2 V j t - o ) 075
05
0.25
07!
075
0
05
02
025
. . . . . .
ct=3
(3=I
0 = 075 I
71
T
x SO
I
2
l
A~
9-
0
i
z ~
Fig. 2. Relation between instrumental luminosity and full width at half maximum of the diffraction peaks for the optimum divergences (full lines) and the divergences ~l = ~2 = =(3 (dashed lines).