- Email: [email protected]
The resolution function for a pulsed-source TOF neutron spectrometer with crystal monochromator
ARTICLE IN PRESS
Physica B 350 (2004) e831–e835
The resolution function for a pulsed-source TOF neutron spectrometer with crystal monochromator I. Ionita* Institute for Nuclear Research Pitesti, P.O. Box 078, Pitesti 0300, Romania
Abstract The matrix procedure to compute the resolution function for a given experimental configuration is briefly given followed by its application to a particular one, a pulsed source TOF neutron spectrometer with crystal monochromator. Both the direct and the inverse geometry are considered. As for the matrix procedure a normal 486 PC is quite suited with computing times of 1–2 s in comparison with the Monte Carlo computing technique for which special computer configurations are needed, the matrix procedure should be preferred when the normal approximation is still valid and if a precise description of the line profile is not required. r 2004 Elsevier B.V. All rights reserved. PACS: 61.12.Ex Keywords: Matrix computational method; Resolution function; Pulsed-source TOF configurations
1. Introduction Any attempt to optimize an experimental set-up requires a suitable computing procedure to evaluate the corresponding resolution and intensity. The computational method given in Ref. [1], is suited only for simple configuration, such as the conventional double and triple axis spectrometers. The matrix method [2], has been successfully used for different configurations, for example the TOF diffractometers with pulsed source [3], or with steady state source [4], the three-axis spectrometers [5], or the crystal diffractometers [6]. The matrix procedure is a convenient one and should be preferred if the normal approximation is still valid and a detailed line profile description is not required, otherwise a Monte Carlo procedure should be used. Computations involving both the MC and matrix procedures will be presented in a forthcoming paper. The general procedure, described in Refs. [2–6], will be followed bellow. The matrix method is appropriate for the case of configurations using focussing effects and curved monochromators [7], or for TOF instruments.
*Tel.: +4-048-213400; fax: +4-048-262449. E-mail addresses: [email protected], [email protected] (I. Ionita). 0921-4526/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2004.03.216
ARTICLE IN PRESS I. Ionita / Physica B 350 (2004) e831–e835
e832
2. The resolution function for a pulsed-source TOF neutron spectrometer with crystal monochromator 2.1. The direct geometry The experimental set-up is given in Fig. 1. As, in the first-order approximation there is no correlation between the horizontal (scattering) plane variables and the vertical variables, the corresponding computations should be performed separately and put together at the end. Therefore, the first step is to define the initial variables vectors RH=(l0, g0, lm, gm, ls, gs, ld, gd, xm, t0, tf ), RV=(z0, zm, zs, zd, x0 m). The coordinates g, l are respectively along the thickness and width, t0, tf define the neutron emission respectively the detection and the subscript 0, s, d refer respectively to source, sample and detector; xm, x0 m are the reflectivity curve variable, for the horizontal and vertical plane, respectively. All the coordinates represent the deviation from the corresponding most probable value. For a given set-up the transmission matrices SH, SV can be computed in the normal approximation. The angular variables vectors UH ¼ ðg0 ; g1 ; g2 Þ; UV ¼ ðd0 ; d1 ; d2 Þ are related to RH respectively RV as: U H ¼ CH RH
U V ¼ CV RV :
ð1Þ
The matrices CH, CV are given by the expressions of the angular variables, according the set-up geometry: g0 ¼
lm cosðym þ wm Þ þ gm sinðym þ wm Þ l0 cos w0 g0 sin w0 L0
g1 ¼
ls cosðys þ ws Þ þ gm sinðys þ ws Þ lm cosðym wm Þ gm sinðym wm Þ L1
g2 ¼
ld cos wd þ gd sin wd ls cosðys ws Þ gs sinðys ws Þ L2
d0 ¼
zm z0 L0
d1 ¼
zs zm L1
d2 ¼
zd zs : L2
ð2Þ
The subscripts 0, 1, 2 refer, respectively, to source-crystal, crystal-sample and sample-detector regions. The variables w0, ws, wd represents the moderator, sample and detector orientation angles while wm is the crystal cutting angle, all measured in the trigonometric sense; 2ys is the scattering angle. The effect of the presence of a coarse collimator or slit, characterized by the transmission matrix Ti, or of the Soller
Fig. 1. The experimental setup, direct geometry.
ARTICLE IN PRESS I. Ionita / Physica B 350 (2004) e831–e835
e833
collimators or neutron guides, characterized by the matrices V, is to modify S to: T S1H ¼ SH þ STHi þ CH VH CH ;
S1V ¼ SV þ STVi þ CVT VV CV :
ð3Þ
The Bragg constraints, in the horizontal plane and the vertical plane, are given, respectively, by: g0 þ g1 ¼ 2Cm xm þ 2rm lm þ 2rm Bm gm ¼ cos ym
Dki ki0
g0 g1 þ bm xm rm Am gm ; 2 0
0
d0 d1 ¼ 2 sin ym ðCm xm þ zm r0m Þ:
ð4Þ ð40 Þ
Owing to the existence of the Bragg constraints, the initial set of 11 horizontal variables can be reduced to a set of 10 independent ones, RHin ¼ ðDLi ; lm ; gm ; ls ; gs ; ld ; gd ; xm ; t0 ; tf Þ; while the initial set of 5 vertical variables can be reduced to a set of 4 independent variables, RVin ¼ ðzm ; zs ; zd ; x0m Þ: The relation between RHin, RH and RVin, RV are: RH ¼ DiH RHin
RV ¼ DiV RVin :
ð5Þ
The matrix DiH is obtained from the first relation (2) and (4) taking into account the expression of DLi: DLi ¼ l0 sin w0 g0 cos w0 lm sinðym þ wm Þ þ gm cosðym þ wm Þ
ð6Þ
and remembering that lm, gm, ls, gs, ld, gd, xm, t0, tf are common variables for RH, RHin. The expressions for l0, g0, defining DiH, are obtained using the relations (2) and (40 ). The matrix DiV is given by: L0 L0 2 sin ym r0m zs 2L0 Cm sin ym xm ð7Þ z0 ¼ zm 1 þ L1 L1 as zm, zs, zd, x0 m are common variables for RVin, RV. The expression for the transmission matrices of the above mentioned independent sets of variables are: T S1Hi ¼ DiH S1H DiH
S1Vi ¼ DiVT S1V DiV :
ð8Þ
The next step is to define the mixed vectors YH ¼ ðDki =ki0 ; g1 ; Dkf =kf 0 ; g2; lseff ; gseff ; DtÞ; YV ¼ ðd1 ; d2 ; zs Þ related to RHin respectively RVin as: Y H ¼ DH RHin
Y V ¼ DV RVin :
ð9Þ
The relation giving Dki/ki0, defining the matrix DH, is obtained from (2) and (4), to which have to be added the relation (2) and those giving lseff, gseff, Dt lseff ¼ ls cosðys þ ws Þ þ gs sinðys þ ws Þ gseff ¼ ls sinðys þ ws Þ þ gs cosðys þ ws Þ;
Dt ¼ tf t0 ;
where lseff, gseff are, respectively, the effective length and thickness of the sample, dimensions characterizing that part of the sample both illuminated by the incident beam and scattering neutrons capable to reach the detector. The relation giving Dkf/kf0, also defining DH, is obtained from the relation giving the spread of the total time-of-flight, tft0: Lf Dkf _ DLi Li Dki DLf ðtf t0 Þ ¼ þ : m ki0 ki0 ki0 kf 0 kf 0 kf 0
ð10Þ
ARTICLE IN PRESS I. Ionita / Physica B 350 (2004) e831–e835
e834
For the direct geometry Li ¼ L0 þ L1
Lf ¼ L2
DLi ¼ ls sinðys þ ws Þ þ gs cosðys þ ws Þ þ l0 sin w0 g0 cos w0
ð11Þ
DLf ¼ ls sinðys ws Þ gs cosðys ws Þ ld sin wd þ gd cos wd : The covariance matrix for the YH variables is given by: 1 1 T NH ¼ DH S1Hi DH :
ð12Þ
The DV matrix is given by the relation (2) taking account that zs is a common variable for the two vectors, YH, RVin. The covariance matrix for YV is: 1 T NV1 ¼ DV S1Vi DV :
ð13Þ
The next step is to relate YH to the XH ¼ ðX1 ; X2 ; X4 Þ vector: X H ¼ AY H :
ð14Þ
The A matrix is given by: X1 ¼ cos jDki þ ki0 sin jg0 cos jDkf kf 0 sin jg1 X2 ¼ sin jDki þ ki0 cos jg0 þ sin jDkf kf 0 cos jg1 X4 ¼
_ ðki0 Dki kf 0 Dkf Þ m
with: tan j ¼
kf 0 sinð2ys Þ ki0 kf 0 cosð2ys Þ
j0 ¼ j 2ys :
ð15Þ
The resolution matrix M is: 1 T M 1 ¼ ANH A :
ð16Þ
We have only to complete the M1 with the element M1 33 given by: 1 2 1 1 1 M33 ¼ ki0 NV11 þ kf20 NV22 2ki0 kf 0 NV12 :
ð17Þ
2.2. Inverse geometry The experimental set-up is given in Fig. 2. The initial variables for this geometry are: RH ¼ ðl0 ; g0 ; ls ; gs ; lm ; gm ; ld ; gd ; xm ; t0 ; tf Þ; RV ¼ ðz0 ; zs ; zm ; zd ; x0m Þ: The angular variables vectors UH ¼ ðg0 ; g1 ; g2 Þ; UV ¼ ðd0 ; d1 ; d2 Þ are related to RH respectively RV accordingly to Eq. (1). For the inverse geometry Li ¼ L0 ; Lf ¼ L1 þ L2 while the last two relations (11) giving DLi, DLf are still valid. The relations (2) should be modified accordingly to the considered geometry: ls cosðys þ ws Þ þ gs sinðys þ ws Þ l0 cos w0 g0 sin w0 g0 ¼ L0 g1 ¼
lm cosðym þ wm Þ þ gm sinðym þ wm Þ ls cosðys ws Þ gs sinðys ws Þ L1
g2 ¼
ld cos wd þ gd sin wd lm cosðym wm Þ gm sinðym wm Þ L2
ARTICLE IN PRESS I. Ionita / Physica B 350 (2004) e831–e835
e835
Fig. 2. The experimental setup, inverse geometry.
d0 ¼
zs z0 L0
d1 ¼
zm zs L1
d2 ¼
zd zm : L2
ð18Þ
Relations (3) and (4) are still valid. Again, owing to the existence of the Bragg constraints, the initial set of 11 horizontal variables can be reduced to a set of 10 independent ones, RHin ¼ ðDLf ; l0 ; g0 ; lm ; gm ; ls ; gs ; xm ; t0 ; tf Þ; while the initial set of 5 vertical variables can be reduced to a set of 4 independent variables, RVin ¼ ðz0 ; zs ; zm ; x0m Þ: Relations (5) are still valid also. The matrix DiH is obtained from the first relation (4) and (18), taking into account the expression of DLf , the last of the relations (11), and remembering that lm, gm, ls, gs, l0, g0, xm, t0, tf are common variables for RH, RHin. The expressions for ld, gd, obtained like above, define DiH. The matrix DiV is given by: L2 L2 0 0 zd ¼ zm 1 þ 2 sin ym rm zs 2L2 Cm sin ym x0m ð19Þ L1 L1 as zm, zs, z0, x0 m are common variables for RVin, RV. To obtain (19) Relations (18) and (40 ) are used. The next step is to define the mixed vectors YH ¼ ðDki =ki0 ; g0; Dkf =kf 0 ; g1; lseff ; gseff ; DtÞ; YV ¼ ðd0 ; d1 ; zs Þ: From this point, the computing procedure is the same as above. Only the relations giving Dki/ki0, Dkf/kf0, must be modified.
References [1] [2] [3] [4] [5] [6] [7]
M.J. Cooper, R. Nathans, Acta Crystallogr. 23 (1967) 357. A.D. Stoica, Acta Crystallogr. A 31 (1975) 189. A.D. Stoica, Acta Crystallogr. A 31 (1975) 193. I. Ionita, J. Appl. Crystallogr. 34 (2001) 252–257. M. Popovici, A.D. Stoica, I. Ionita, J. Appl. Crystallogr. 20 (1987) 90. I. Ionita, A.D. Stoica, J. Appl. Crystallogr. 33 (2000) 1067. I. Ionita, A.D. Stoica, M. Popovici, N.C. Popa, Nucl. Instrum. Methods A 431 (1999) 509.
Physica B 350 (2004) e831–e835
The resolution function for a pulsed-source TOF neutron spectrometer with crystal monochromator I. Ionita* Institute for Nuclear Research Pitesti, P.O. Box 078, Pitesti 0300, Romania
Abstract The matrix procedure to compute the resolution function for a given experimental configuration is briefly given followed by its application to a particular one, a pulsed source TOF neutron spectrometer with crystal monochromator. Both the direct and the inverse geometry are considered. As for the matrix procedure a normal 486 PC is quite suited with computing times of 1–2 s in comparison with the Monte Carlo computing technique for which special computer configurations are needed, the matrix procedure should be preferred when the normal approximation is still valid and if a precise description of the line profile is not required. r 2004 Elsevier B.V. All rights reserved. PACS: 61.12.Ex Keywords: Matrix computational method; Resolution function; Pulsed-source TOF configurations
1. Introduction Any attempt to optimize an experimental set-up requires a suitable computing procedure to evaluate the corresponding resolution and intensity. The computational method given in Ref. [1], is suited only for simple configuration, such as the conventional double and triple axis spectrometers. The matrix method [2], has been successfully used for different configurations, for example the TOF diffractometers with pulsed source [3], or with steady state source [4], the three-axis spectrometers [5], or the crystal diffractometers [6]. The matrix procedure is a convenient one and should be preferred if the normal approximation is still valid and a detailed line profile description is not required, otherwise a Monte Carlo procedure should be used. Computations involving both the MC and matrix procedures will be presented in a forthcoming paper. The general procedure, described in Refs. [2–6], will be followed bellow. The matrix method is appropriate for the case of configurations using focussing effects and curved monochromators [7], or for TOF instruments.
*Tel.: +4-048-213400; fax: +4-048-262449. E-mail addresses: [email protected], [email protected] (I. Ionita). 0921-4526/$ - see front matter r 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2004.03.216
ARTICLE IN PRESS I. Ionita / Physica B 350 (2004) e831–e835
e832
2. The resolution function for a pulsed-source TOF neutron spectrometer with crystal monochromator 2.1. The direct geometry The experimental set-up is given in Fig. 1. As, in the first-order approximation there is no correlation between the horizontal (scattering) plane variables and the vertical variables, the corresponding computations should be performed separately and put together at the end. Therefore, the first step is to define the initial variables vectors RH=(l0, g0, lm, gm, ls, gs, ld, gd, xm, t0, tf ), RV=(z0, zm, zs, zd, x0 m). The coordinates g, l are respectively along the thickness and width, t0, tf define the neutron emission respectively the detection and the subscript 0, s, d refer respectively to source, sample and detector; xm, x0 m are the reflectivity curve variable, for the horizontal and vertical plane, respectively. All the coordinates represent the deviation from the corresponding most probable value. For a given set-up the transmission matrices SH, SV can be computed in the normal approximation. The angular variables vectors UH ¼ ðg0 ; g1 ; g2 Þ; UV ¼ ðd0 ; d1 ; d2 Þ are related to RH respectively RV as: U H ¼ CH RH
U V ¼ CV RV :
ð1Þ
The matrices CH, CV are given by the expressions of the angular variables, according the set-up geometry: g0 ¼
lm cosðym þ wm Þ þ gm sinðym þ wm Þ l0 cos w0 g0 sin w0 L0
g1 ¼
ls cosðys þ ws Þ þ gm sinðys þ ws Þ lm cosðym wm Þ gm sinðym wm Þ L1
g2 ¼
ld cos wd þ gd sin wd ls cosðys ws Þ gs sinðys ws Þ L2
d0 ¼
zm z0 L0
d1 ¼
zs zm L1
d2 ¼
zd zs : L2
ð2Þ
The subscripts 0, 1, 2 refer, respectively, to source-crystal, crystal-sample and sample-detector regions. The variables w0, ws, wd represents the moderator, sample and detector orientation angles while wm is the crystal cutting angle, all measured in the trigonometric sense; 2ys is the scattering angle. The effect of the presence of a coarse collimator or slit, characterized by the transmission matrix Ti, or of the Soller
Fig. 1. The experimental setup, direct geometry.
ARTICLE IN PRESS I. Ionita / Physica B 350 (2004) e831–e835
e833
collimators or neutron guides, characterized by the matrices V, is to modify S to: T S1H ¼ SH þ STHi þ CH VH CH ;
S1V ¼ SV þ STVi þ CVT VV CV :
ð3Þ
The Bragg constraints, in the horizontal plane and the vertical plane, are given, respectively, by: g0 þ g1 ¼ 2Cm xm þ 2rm lm þ 2rm Bm gm ¼ cos ym
Dki ki0
g0 g1 þ bm xm rm Am gm ; 2 0
0
d0 d1 ¼ 2 sin ym ðCm xm þ zm r0m Þ:
ð4Þ ð40 Þ
Owing to the existence of the Bragg constraints, the initial set of 11 horizontal variables can be reduced to a set of 10 independent ones, RHin ¼ ðDLi ; lm ; gm ; ls ; gs ; ld ; gd ; xm ; t0 ; tf Þ; while the initial set of 5 vertical variables can be reduced to a set of 4 independent variables, RVin ¼ ðzm ; zs ; zd ; x0m Þ: The relation between RHin, RH and RVin, RV are: RH ¼ DiH RHin
RV ¼ DiV RVin :
ð5Þ
The matrix DiH is obtained from the first relation (2) and (4) taking into account the expression of DLi: DLi ¼ l0 sin w0 g0 cos w0 lm sinðym þ wm Þ þ gm cosðym þ wm Þ
ð6Þ
and remembering that lm, gm, ls, gs, ld, gd, xm, t0, tf are common variables for RH, RHin. The expressions for l0, g0, defining DiH, are obtained using the relations (2) and (40 ). The matrix DiV is given by: L0 L0 2 sin ym r0m zs 2L0 Cm sin ym xm ð7Þ z0 ¼ zm 1 þ L1 L1 as zm, zs, zd, x0 m are common variables for RVin, RV. The expression for the transmission matrices of the above mentioned independent sets of variables are: T S1Hi ¼ DiH S1H DiH
S1Vi ¼ DiVT S1V DiV :
ð8Þ
The next step is to define the mixed vectors YH ¼ ðDki =ki0 ; g1 ; Dkf =kf 0 ; g2; lseff ; gseff ; DtÞ; YV ¼ ðd1 ; d2 ; zs Þ related to RHin respectively RVin as: Y H ¼ DH RHin
Y V ¼ DV RVin :
ð9Þ
The relation giving Dki/ki0, defining the matrix DH, is obtained from (2) and (4), to which have to be added the relation (2) and those giving lseff, gseff, Dt lseff ¼ ls cosðys þ ws Þ þ gs sinðys þ ws Þ gseff ¼ ls sinðys þ ws Þ þ gs cosðys þ ws Þ;
Dt ¼ tf t0 ;
where lseff, gseff are, respectively, the effective length and thickness of the sample, dimensions characterizing that part of the sample both illuminated by the incident beam and scattering neutrons capable to reach the detector. The relation giving Dkf/kf0, also defining DH, is obtained from the relation giving the spread of the total time-of-flight, tft0: Lf Dkf _ DLi Li Dki DLf ðtf t0 Þ ¼ þ : m ki0 ki0 ki0 kf 0 kf 0 kf 0
ð10Þ
ARTICLE IN PRESS I. Ionita / Physica B 350 (2004) e831–e835
e834
For the direct geometry Li ¼ L0 þ L1
Lf ¼ L2
DLi ¼ ls sinðys þ ws Þ þ gs cosðys þ ws Þ þ l0 sin w0 g0 cos w0
ð11Þ
DLf ¼ ls sinðys ws Þ gs cosðys ws Þ ld sin wd þ gd cos wd : The covariance matrix for the YH variables is given by: 1 1 T NH ¼ DH S1Hi DH :
ð12Þ
The DV matrix is given by the relation (2) taking account that zs is a common variable for the two vectors, YH, RVin. The covariance matrix for YV is: 1 T NV1 ¼ DV S1Vi DV :
ð13Þ
The next step is to relate YH to the XH ¼ ðX1 ; X2 ; X4 Þ vector: X H ¼ AY H :
ð14Þ
The A matrix is given by: X1 ¼ cos jDki þ ki0 sin jg0 cos jDkf kf 0 sin jg1 X2 ¼ sin jDki þ ki0 cos jg0 þ sin jDkf kf 0 cos jg1 X4 ¼
_ ðki0 Dki kf 0 Dkf Þ m
with: tan j ¼
kf 0 sinð2ys Þ ki0 kf 0 cosð2ys Þ
j0 ¼ j 2ys :
ð15Þ
The resolution matrix M is: 1 T M 1 ¼ ANH A :
ð16Þ
We have only to complete the M1 with the element M1 33 given by: 1 2 1 1 1 M33 ¼ ki0 NV11 þ kf20 NV22 2ki0 kf 0 NV12 :
ð17Þ
2.2. Inverse geometry The experimental set-up is given in Fig. 2. The initial variables for this geometry are: RH ¼ ðl0 ; g0 ; ls ; gs ; lm ; gm ; ld ; gd ; xm ; t0 ; tf Þ; RV ¼ ðz0 ; zs ; zm ; zd ; x0m Þ: The angular variables vectors UH ¼ ðg0 ; g1 ; g2 Þ; UV ¼ ðd0 ; d1 ; d2 Þ are related to RH respectively RV accordingly to Eq. (1). For the inverse geometry Li ¼ L0 ; Lf ¼ L1 þ L2 while the last two relations (11) giving DLi, DLf are still valid. The relations (2) should be modified accordingly to the considered geometry: ls cosðys þ ws Þ þ gs sinðys þ ws Þ l0 cos w0 g0 sin w0 g0 ¼ L0 g1 ¼
lm cosðym þ wm Þ þ gm sinðym þ wm Þ ls cosðys ws Þ gs sinðys ws Þ L1
g2 ¼
ld cos wd þ gd sin wd lm cosðym wm Þ gm sinðym wm Þ L2
ARTICLE IN PRESS I. Ionita / Physica B 350 (2004) e831–e835
e835
Fig. 2. The experimental setup, inverse geometry.
d0 ¼
zs z0 L0
d1 ¼
zm zs L1
d2 ¼
zd zm : L2
ð18Þ
Relations (3) and (4) are still valid. Again, owing to the existence of the Bragg constraints, the initial set of 11 horizontal variables can be reduced to a set of 10 independent ones, RHin ¼ ðDLf ; l0 ; g0 ; lm ; gm ; ls ; gs ; xm ; t0 ; tf Þ; while the initial set of 5 vertical variables can be reduced to a set of 4 independent variables, RVin ¼ ðz0 ; zs ; zm ; x0m Þ: Relations (5) are still valid also. The matrix DiH is obtained from the first relation (4) and (18), taking into account the expression of DLf , the last of the relations (11), and remembering that lm, gm, ls, gs, l0, g0, xm, t0, tf are common variables for RH, RHin. The expressions for ld, gd, obtained like above, define DiH. The matrix DiV is given by: L2 L2 0 0 zd ¼ zm 1 þ 2 sin ym rm zs 2L2 Cm sin ym x0m ð19Þ L1 L1 as zm, zs, z0, x0 m are common variables for RVin, RV. To obtain (19) Relations (18) and (40 ) are used. The next step is to define the mixed vectors YH ¼ ðDki =ki0 ; g0; Dkf =kf 0 ; g1; lseff ; gseff ; DtÞ; YV ¼ ðd0 ; d1 ; zs Þ: From this point, the computing procedure is the same as above. Only the relations giving Dki/ki0, Dkf/kf0, must be modified.
References [1] [2] [3] [4] [5] [6] [7]
M.J. Cooper, R. Nathans, Acta Crystallogr. 23 (1967) 357. A.D. Stoica, Acta Crystallogr. A 31 (1975) 189. A.D. Stoica, Acta Crystallogr. A 31 (1975) 193. I. Ionita, J. Appl. Crystallogr. 34 (2001) 252–257. M. Popovici, A.D. Stoica, I. Ionita, J. Appl. Crystallogr. 20 (1987) 90. I. Ionita, A.D. Stoica, J. Appl. Crystallogr. 33 (2000) 1067. I. Ionita, A.D. Stoica, M. Popovici, N.C. Popa, Nucl. Instrum. Methods A 431 (1999) 509.