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The resolution function for a pulsed-source TOF neutron spectrometer
ARTICLE IN PRESS
Nuclear Instruments and Methods in Physics Research A 513 (2003) 511–523
The resolution function for a pulsed-source TOF neutron spectrometer I. Ionita* Institute for Nuclear Research Pitesti, P.O. Box 078, Pitesti 0300, Romania Received 9 December 2002; received in revised form 25 June 2003; accepted 25 June 2003
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. All the corresponding relevant configurations are considered. For crystal and polycrystalline filter as 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 2003 Elsevier B.V. All rights reserved. PACS: 61.12.eX Keywords: Resolution function; Normal approximation; Matrix method; Pulsed-source TOF neutron spectrometer
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 by Cooper and Nathans, [1], is suitable only for rather simple configurations as the conventional double and triple axis spectrometers are. The procedure is no longer appropriate when spatial effects are important as is the case of configurations using focusing effects and curved monochromators, [7], or for TOF instruments. The matrix method, [2], has proved to be suitable both for conventional, focusing or TOF instruments. This computing technique has been successfully used for different configurations as are the TOF diffractometers with pulsed source, [3], or with steady-state source, [5], the crystal diffractometers, [6], or the three-axis spectrometers, [4]. During the last few years the Monte Carlo procedure, [8–11], has begun to be used quite extensively to evaluate the resolution and intensity properties of the neutron spectrometers. The application of the MC technique requires rather powerful computers with increased computing speed, *Tel.: +40-248-213400x579; fax: +40-248-262449. E-mail addresses: [email protected], [email protected] (I. Ionita). 0168-9002/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0168-9002(03)02051-5
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while for the matrix method a 486 PC is quite suitable with computing times of 1–2 s: 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.
2. The general theory The computational procedure involves several steps. The first one is to choose the initial variables of the problems defining the neutron trajectory between the source and detector. The normal approximation of the probability distribution of the initial variables, spatial, angular and time variables defining the vector R, can be computed for a particular experimental setting, giving the corresponding matrix S; the transmission matrix of its components. Throughout the manuscript the notation of Ref. [5] is used. The next step is to characterize the influence of the Soller collimators, neutron guides (defined as a Soller collimator with a wavelength-dependent angular divergence given by the total reflex critical angle), coarse collimators or slits (defined as a coarse collimator of zero length). The effect of the presence of a coarse collimator or slit, characterized by the transmission matrix Ti ; is to modify S to X ð1Þ S-S þ Ti : To characterize the influence of the Soller collimators or neutron guides one has to introduce the vector U of the angular variables, related to R as U ¼ CR:
ð2Þ
Its transmission matrix V is defined by the divergences of the Soller collimators. The effect of the presence of the Soller collimators is to modify S to X S-S þ Ti þ C T VC: ð3Þ Owing to the existence of the constraints, as the Bragg constraints are, the initial variables are not linearly independent and, therefore, the initial set R has to be reduced to a linearly independent subset R0 : The initial variables can be expressed through R0 variables as R ¼ DR0 :
ð4Þ
The transmission matrix for the subset R0 is S0 ¼ DT SD:
ð5Þ
The final step is to write the relation between X, the resolution function variables ðX ¼ Q Q0 ; DE DE0 ; Q ¼ ki kf ; DE ¼ Ei Ef and Q0 ; DE0 defines, respectively, the most probable value of Q, DEÞ and R0 X ¼ AR0 :
ð6Þ
The resolution matrix M is given by M ¼ ½AS 01
AT 1 :
ð7Þ
The Gaussian approximation of the resolution function is W ðXÞ ¼ W0 ð2pÞ2 ðdet MÞ1=2 expðXT MX=2Þ
ð8Þ
with W0 being the normalization factor. To obtain the line widths for different types of scans it is necessary only to define the scan variables Z, generally not linearly independent, related to X as Z ¼ HX:
ð9Þ
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The scan variables covariance matrix is /zi zj S ¼ HM 1 H T
ð10Þ
and therefore the linewidth for the scan ‘‘i’’ is wi ¼ ð8 ln 2/z2i SÞ1=2 :
ð11Þ
3. The resolution function for a pulsed-source TOF neutron spectrometer with crystal monochromator 3.1. 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 ¼ ðl0 ; g0 ; lm ; gm ; ls ; gs ; ld ; gd ; xm ; t0 ; tf Þ; RV ¼ ðz0 ; zm ; zs ; zd ; x0m Þ: 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 ; x0m 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 UH ¼ CH RH ;
UV ¼ CV RV :
ð12Þ
The matrices CH ; CV are given by the expressions of the angular variables, according to 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
ld
l0 χ 0
lm
ls
χs
χd Detector
L2
2θs
χm L1 2θm
Sample
L0
Source
Monochromator
Fig. 1. The experimental set-up for all pulsed-source TOF spectrometers with crystal monochromator, direct geometry.
ð13Þ
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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
ð14Þ
Subscripts 0, 1, 2 refer, respectively, to source–crystal, crystal–sample and sample–detector regions. The variables w0 ; ws ; wd represent 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. According to Eqn. (3) the matrices SH ; SV are modified as X T THi þ CH VH CH S1H ¼ SH þ S1V ¼ SV þ
X
TVi þ CVT VV CV :
ð15Þ
The Bragg constraints, in the horizontal plane, are given g0 þ g1 ¼ 2Cm xm þ 2rm lm þ 2rm Bm gm Dki g g1 þ bm xm rm Am gm : ¼ cot ym 0 ki0 2
ð16Þ
The Bragg constraints, in the vertical plane, are given by 0 0 d0 d1 ¼ 2 sin ym ðCm xm þ zm r0m Þ:
ð160 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 five vertical variables can be reduced to a set of four independent variables, RVin ¼ ðzm ; zs ; zd ; x0m Þ: The relation between RHin ; RH and RVin ; RV are RH ¼ DiH RHin
ð17Þ
RV ¼ DiV RVin :
ð170 Þ
The matrix DiH is obtained from first relation (16) and (13), 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
ð18Þ
and recalling that lm ; gm ; ls ; gs ; ld ; gd ; xm ; t0 ; tf are common variables for RH ; RHin : The expressions for l0 ; g0 ; defining DiH ; are L0 l0 ¼ DLi sin w0 þ lm cos w0 2 tan w0 cos ym sin wm þ cosðym þ wm Þ cosðym wm Þ 2L0 rm L1 L0 gm cos w0 2 tan w0 sin ym sin wm þ sinðym þ wm Þ þ sinðym wm Þ þ 2L0 rm Bm L1 L0 2Cm L0 cos w0 xm þ ls sin w0 sinðys þ ws Þ þ cos w0 cosðys þ ws Þ L1 L0 þ gs sin w0 cosðys þ ws Þ þ cos w0 sinðys þ ws Þ ð19Þ L1
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g0 ¼ DLi cos w0 þ lm sin w0 2 cot w0 cos ym sin wm þ cosðym þ wm Þ L0 cosðym wm Þ 2L0 rm L1 L0 gm sin w0 2 cot w0 sin ym sin wm þ sinðym þ wm Þ þ sinðym wm Þ þ 2L0 rm Bm L1 2Cm L0 sin w0 xm L0 þ ls cos w0 sinðys þ ws Þ þ sin w0 cosðys þ ws Þ L1 L0 þ gs cos w0 cosðys þ ws Þ þ sin w0 sinðys þ ws Þ : L1 The matrix DiV is given by L0 L0 z0 ¼ zm 1 þ 2 sin ym r0m zs L1 L1 0 2L0 Cm sin ym x0m
ð20Þ
as zm ; zs ; zd ; x0m are common variables for RVin ; RV : To obtain Eq. (20) relations (14) and (160 ) are used. 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 :
ð21Þ
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 YH ¼ DH RHin ;
YV ¼ DV RVin :
ð22Þ
The matrix DH is given by Dki cosðym wm Þ ¼ lm cot ym rm þ þ gm cot ym rm ðBm Am tan ym Þ L1 ki0 sinðym wm Þ cosðys þ ws Þ sinðys þ ws Þ þ gs cot ym þ xm ðCm cot ym þ bm Þ ls cot ym L1 L1 L1 Dkf DLi kf 0 _kf 0 L0 þ L1 kf 0 cosðym wm Þ ¼ þ ðt0 tf Þ lm cot ym rm þ L1 kf 0 L2 ki0 mL2 L2 ki0 L0 þ L1 kf 0 L0 þ L1 kf 0 ðCm cot ym þ bm Þ gm cot ym rm ðBm Am tan ym Þ xm L2 ki0 L2 ki0 sinðym wm Þ þ L1 kf 0 ld gd ls L0 sin wd þ cos wd þ cot ym 1þ cosðys þ ws Þ L2 L2 L2 ki0 L1 kf 0 gs L0 þ sinðys ws Þ þ cot ym 1þ sinðys þ ws Þ cosðys ws Þ L2 ki0 L1
ð23Þ
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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 :
ð24Þ
To these relations Eq. (13) must to be added. First relation (23) is obtained from Eqs. (16) and (13). Second relation (23) is obtained from the relation giving the spread of the total time-of-flight, tf t0 Lf Dkf _ DLi Li Dki DLf ðtf t0 Þ ¼ þ : ð25Þ m ki0 ki0 ki0 kf 0 kf 0 kf 0 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 DLf ¼ ls sinðys ws Þ gs cosðys ws Þ ld sin wd þ gd cos wd :
ð26Þ
In relation (24) 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 of reaching the detector. The covariance matrix for the YH variables is given by 1 1 T NH ¼ DH S1Hi DH :
ð27Þ
The DV matrix is given by relation (14) taking account of the fact 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 : ð28Þ The final step is to relate YH to the XH ¼ ðX1 ; X2 ; X4 Þ vector XH ¼ AYH :
ð29Þ
The A matrix is given by X1 ¼ cos jDki þ ki0 sin jg0 cos j0 Dkf kf 0 sin j0 g1 X2 ¼ sin jDki þ ki0 cos jg0 þ sin j0 Dkf kf 0 cos j0 g1 X4 ¼
ð30Þ
_ ðki0 Dki kf 0 Dkf Þ m
with tan j ¼
kf 0 sinð2ys Þ ; ki0 kf 0 cosð2ys Þ
j0 ¼ j 2ys :
The resolution matrix M is 1 T M 1 ¼ ANH A :
ð31Þ
ð32Þ
1 given by We have only to complete the M 1 with the element M33 1 2 1 1 1 M33 ¼ ki0 NV11 þ kf20 NV22 2ki0 kf 0 NV12 :
ð33Þ
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3.2. The 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 variable vectors UH ¼ ðg0 ; g1 ; g2 Þ; UV ¼ ðd0 ; d1 ; d2 Þ are related to RH ; respectively, RV according to Eq. (12). For the inverse geometry Li ¼ L0 ; Lf ¼ L1 þ L2 while last two relations (26) giving DLi ; DLf are still valid. Relations (13) should be modified according to the considered geometry: g0 ¼
ls cosðys þ ws Þ þ gs sinðys þ ws Þ l0 cos w0 g0 sin w0 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
ð34Þ
Also, relation (14) should be modified as d0 ¼
zs z0 ; L0
d1 ¼
zm zs ; L1
d2 ¼
zd zm : L2
ð35Þ
Relations (15) and (16) 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 five vertical variables can be reduced to a set of four independent variables, RVin ¼ ðz0 ; zs ; zm ; x0m Þ: Relations (17) are also still valid. The matrix DiH is obtained from first relations (16) and (34), taking into account the expression of DLf ; the last of relations (26), and recalling that lm ; gm ; ls ; gs ; l0 ; g0 ; xm ; t0 ; tf are common variables for RH ; RHin : The expressions for ld ; gd ; defining DiH ; are
χd ld
l0 χ 0
L2
lm χm ls
Detector
2θm
χs L1 2θs
Monochromator
L0
Source
Sample
Fig. 2. The experimental set-up for all pulsed-source TOF spectrometers with crystal monochromator, inverse geometry.
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ld ¼ DLf sin wd þ lm cos wd 2 tan wd cos ym sin wm þ cosðym wm Þ L2 cosðym þ wm Þ þ 2L2 rm þ gd cos wd 2 tan wd sin ym sin wm þ sinðym wm Þ L1 L2 L2 sinðym þ wm Þ þ 2L2 rm Bm þ 2Cm L2 cos wd xm þ ls sin wd sinðys ws Þ þ cos wd cosðys ws Þ L1 L1 L2 þ gs sin wd cosðys ws Þ þ cos wd sinðys ws Þ ð36Þ L1 L2 gd ¼ DLf cos wd þ lm sin wd 2 cot wd cos ym sin wm þ cosðym wm Þ cosðym þ wm Þ þ 2L2 rm L1 L2 þ gm sin wd 2 cot wd sin ym sin wm þ sinðym wm Þ sinðym þ wm Þ þ 2L2 rm Bm L1 L2 þ 2Cm L2 sin wd xm þ ls cos wd sinðys ws Þ þ sin wd cosðys ws Þ L1 L2 þ gs cos wd cosðys ws Þ þ sin wd sinðys ws Þ : ð37Þ L1 The matrix DiV is given by L2 L2 0 0 2 sin ym rm zs 2L2 Cm sin ym x0m zd ¼ zm 1 þ L1 L1
ð38Þ
as zm ; zs ; z0 ; x0m are common variables for RVin ; RV : To obtain Eq. (37) relations (35) and (160 ) 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 relations (23) giving Dki =ki0 ; Dkf =kf 0 ; must be replaced with Dki cosðym þ wm Þ sinðym þ wm Þ ¼ lm cot ym rm þ þ gm cot ym rm ðBm þ Am tan ym Þ þ L1 L1 ki0 cosðys ws Þ sinðys ws Þ gs cot ym L1 L1 Dkf DLfi _kf 0 L0 kf 0 cosðym þ wm Þ ðt0 tf Þ ¼ þ lm cot ym rm þ L1 kf 0 L1 þ L2 mðL1 þ L2 Þ L1 þ L2 ki0 L 0 kf 0 L0 kf 0 ðCm cot ym þ bm Þ gm cot ym rm ðBm þ Am tan ym Þ xm L1 þ L2 ki0 L1 þ L2 ki0 sinðym wm Þ þ L1 kf 0 kf 0 g0 kf 0 L0 l0 ls þ sin w0 cos w0 þ cot ym cosðys ws Þ sinðys þ ws Þ ki0 L1 þ L2 ki0 L1 þ L2 L1 þ L2 ki0 L1 kf 0 L0 gs þ cot ym sinðys ws Þ þ cosðys þ ws Þ : ð39Þ L1 þ L2 ki0 L1 þ xm ðCm cot ym þ bm Þ ls cot ym
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4. The resolution function for a pulsed-source TOF neutron spectrometer with polycrystalline filter as monochromator 4.1. The direct geometry The experimental set-up is given in Fig. 3. 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 ¼ ðDki =ki0 ; l0 ; g0 ; ls ; gs ; ld ; gd ; t0 ; tf Þ; RV ¼ ðz0 ; zs ; zd Þ: The coordinates g; l are, respectively, along the thickness and width, t0 ; tf define the neutron emission, respectively, detection and the subscript 0, s, d refer, respectively, to source, sample and detector. 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. In particular, for a given cutoff wavelength and for a given energy distribution of the emitted neutrons the term /ðDki =ki0 Þ2 S can be computed. The angular variables vectors UH ¼ ðg; g1 Þ; UV ¼ ðd0 ; d1 Þ are related to RH ; respectively, RV as UH ¼ CH RH ;
UV ¼ CV RV :
ð40Þ
The matrices CH ; CV are given by the expressions of the angular variables, according to the set-up geometry g0 ¼
ls cosðys þ ws Þ þ gs sinðys þ ws Þ l0 cos w0 gs sin w0 L0 þ L1
g1 ¼
ld cos wd gd sin wd ls cosðys ws Þ gs sinðys ws Þ L2
ð41Þ
d0 ¼
zs z0 ; L0 þ L1
ð42Þ
d1 ¼
zd zs : L2
The subscripts 0, 1, 2 refer, respectively, to source–filter, filter–sample and sample–detector regions. The variables w0 ; ws ; wd represent the moderator, sample and detector orientation angles, all measured in the trigonometric sense while 2ys is the scattering angle. For the angular variables the subscripts 0, 1 refer to source–sample, respectively, sample–detector regions. As above the matrices SH ; SV are modified according to Eqn. (15). 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 related to RH ; receptively, RV as YH ¼ DH RH ;
YV ¼ DV RV :
ð43Þ
ls
χ0
χd
ld
Filter l0
χs L2
L0
θs
Detector
θs
L1
Source Sample
Fig. 3. The experimental set-up for a pulsed-source TOF spectrometer with polycrystalline filter as monochromator, direct geometry.
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The matrix DH is given by Dkf kf 0 l0 kf 0 g0 k f 0 ls ¼ sin w0 cos w0 þ sinðys þ ws Þ þ sinðys ws Þ kf 0 L2 ki0 L2 ki0 L2 ki0 gs k f 0 L0 þ L1 kf 0 Dki _kf 0 þ cosðys þ ws Þ cosðys ws Þ þ ðt0 tf Þ L2 ki0 L2 ki0 ki0 mL2 ld gd sin wd þ cos wd : L2 L2
ð44Þ
To these relations Eqs. (41) and (24) must be added. The term Dki =ki0 is a common variable for RH ; YH : Relation (44) is obtained from relation (25) giving the spread of the total time-of-flight, tf t0 ; and (26). From this point the computing procedure is the same as in Section 3.1. 4.2. The inverse geometry The experimental set-up is given in Figs. 4 and 5. For this geometry Li ¼ L0 ; Lf ¼ L1 þ L2 and the computing procedure, including the corresponding relations, remain the same as above, provided L2 is replaced with L1 þ L2 and L0 þ L1 is replaced with L0 :
5. The resolution function for a pulsed-source TOF neutron spectrometer with mechanical monochromator The first step is to define the initial variables vectors RH ¼ ðDki =ki0 ; l0 ; g0 ; lch Þ; R0H ¼ ðlch ; ls ; gs ; ld ; gd ; t0 ; tf Þ; RV ¼ ðz0 ; zch Þ; R0V ¼ ðzch ; zs ; zd Þ: The coordinates g; l are, respectively, along the thickness and width, t0 ; tf define the neutron emission, respectively, detection and the subscript 0, s, d refer, respectively, to source, sample and detector. All the coordinates represent the deviation from the corresponding most probable value. For a given set-up the transmission matrices SH ; S1H ; SV ; S1V can be computed in the normal approximation. In particular, for a given mechanical monochromator the term /ðDki =ki0 Þ2 S can be computed. The angular variables vectors UH ¼ ðg0 ; g1 ; g2 Þ; UV ¼ ðd0 ; d1 ; d2 Þ are related to RH ; R0H ; respectively, RV ; 0 RV as 0 UH ¼ CH RH ; UH ¼ CH R0H
UV ¼ CV RV ; UV ¼ CV0 R0V :
ð47Þ ld
Filter
χd Detector
l0
χ0
ls
L2
χs L1
L0
θs
θs
Source Sample Fig. 4. The experimental set-up for a pulsed-source TOF spectrometer with polycrystalline filter as monochromator, inverse geometry.
ARTICLE IN PRESS I. Ionita / Nuclear Instruments and Methods in Physics Research A 513 (2003) 511–523 Mechanical Monochromator ls χs
l0
χ0
χd
ld
L2 L0
521
θs
Detector
θs
L1
Source Sample
Fig. 5. The experimental set-up for a pulsed-source TOF spectrometer with mechanical monochromator. 0 The matrices CH ; CH ; CV ; CV0 are given by the expressions of the angular variables, according to the setup geometry
lch l0 cos w0 gs sin w0 L0 ls cosðys þ ws Þ þ gs sinðys þ ws Þ lch g1 ¼ L1 ld cos wd þ gd sin wd ls cosðys ws Þ gs sinðys ws Þ g2 ¼ L2
g0 ¼
d0 ¼
zch z0 ; L0
d1 ¼
zs zch ; L1
d1 ¼
zd zs : L2
ð48Þ ð49Þ
Subscripts 0, 1, 2 refer, respectively, to source–monochromator, monochromator–sample and sample– detector regions. The variables w0 ; ws ; wd represent the moderator, sample and detector orientation angles, all measured in the trigonometric sense while 2ys is the scattering angle. For the angular variables subscripts 0, 1, 2 refer to source–monochromator, monochromator–sample, respectively, sample–detector regions. Again, the matrices SH ; SV are modified according to Eqs. (3) and (15). To relations (15) should be added X 0T 0 S1Hm ¼ S1H þ THi þ CH VH CH X S1Vm ¼ S1V þ TVi þ CV0T VV CV0 : ð150 Þ The correlation introduced by the mechanical monochromator is characterized by g0 ¼ g1 :
ð50Þ
To take this correlation into account we have to introduce the vector RHint ¼ ðDki =ki0 ; g0 ; lch Þ related to RH as RHint ¼ BH RH
ð51Þ
where the matrix BH is given by first relation (41), Dki =ki0 ; lch being common elements for RHint ; RH : The covariant matrix for RHint is given by 1 SHint ¼ BH SHm BTH :
ð52Þ
The presence of the mechanical monochromator also introduces a correlation between RHint and R0Hm ¼ ðDki =ki0 ; lch ; ls ; gs ; ld ; gd ; t0 ; tf Þ RHint ¼ B0H R0Hm : is given by ls cosðys þ ws Þ þ gs sinðys þ ws Þ lch g0 ¼ g1 ¼ L1
The matrix
ð53Þ
B0H
ð54Þ
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as Dki =ki0 ; lch are common elements for RHint and R0Hm : The transmission matrix for R0Hm is 0 SH1f ¼ B0T H SHint BH þ S1Hm :
ð55Þ
In Eq. (55) the summation of matrix of different dimensions are performed after having added ‘‘0’’ elements for the matrix lines and columns corresponding to the unconsidered variables. The correlation introduced by the mechanical monochromator is characterized, in the vertical plane, by d0 ¼ d1 :
ð56Þ
To take into account this correlation we have to introduce the vector RVint ¼ ðd0 ; zch Þ related to RV as RVint ¼ BV RV
ð57Þ
where the matrix BV is given by first relation (42), zch being a common element for RVint ; RV : The covariant matrix for RVint is given by 1 ¼ BV SVm BTH : SVint
ð58Þ
The presence of the mechanical monochromator also introduces a correlation between RVint and R0V RVint ¼ B0V R0V :
ð59Þ
The matrix B0V is given by zs zch d0 ¼ d1 ¼ L1
ð60Þ
as zch is a common element for RVint and R0V : The transmission matrix for R0V is 0 SV1f ¼ B0T V SVint BV þ S1Vm :
ð61Þ
In Eq. (61) the summation of matrix of different dimensions arc performed after having added ‘‘0’’ elements for the matrix lines and columns corresponding to the unconsidered variables. 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 related to ½RH ; respectively RV as YH ¼ DH RH ;
YV ¼ DV RV :
ð62Þ
The matrix DH is given by Dkf l0 kf 0 g0 k f 0 ¼ sin w0 cos w0 kf 0 L2 ki0 L2 ki0 kf 0 ls þ sinðys þ ws Þ þ sinðys ws Þ L2 ki0 gs k f 0 þ cosðys þ ws Þ cosðys ws Þ L2 ki0 L0 þ L1 kf 0 Dki _kf 0 þ ðt0 tf Þ L2 ki0 ki0 mL2 ld gd sin wd þ cos wd : L2 L2
ð63Þ
To these relations Eqs. (41) and (24) must to be added. The term Dki =ki0 is a common variable for RH ; YH : Relation (63) is obtained from relation (25) giving the spread of the total time-of-flight, tf t0 ; and Eq. (26). As above 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 of reaching the detector. From this point the computing procedure is the same as in Section 3.1.
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6. Conclusions The matrix procedure to compute the resolution function for a given experimental configuration is applied for a particular one, a pulsed source TOF neutron spectrometer. All the corresponding relevant configurations are considered. The matrix method has proved to be suitable both for conventional focusing or TOF instruments. The matrix procedure described above can be used to optimize an experimental set-up or to evaluate its properties. 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.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]
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. M. Popovici, A.D. Stoica, I. Ionita, J. Appl. Crystallogr. 20 (1987) 90. I. Ionita, J. Appl. Crystallogr. 34 (2001) 252. I. Ionita, A.D. Stoica, J. Appl. Crystallogr. 33 (2000) 1067. I. Ionita, A.D. Stoica, M. Popovici, N.C. Popa, Nucl. Instr. and Meth. A 431 (1999) 509. S.V. Roth, A. Zirkel, J. Bossy, J. Neuhauss, J. Peters, H. Schober, W. Petry, Appl. Phys. A 74 (2002) S1449. R. Schedler, M. Rotter, M. Loewenhaupt, N.M. Pyka, Appl. Phys. A 74 (2002) S1468. H. Edwards, K. Lefmann, B. Lake, K. Nielsen, P. Shaarup, Appl. Phys. A 74 (2002) S1492. W.T. Lee, X.L. Wang, J.L. Robertson, F. Klose, Ch. Rehm, Appl. Phys. A 74 (2002) S1502.
Nuclear Instruments and Methods in Physics Research A 513 (2003) 511–523
The resolution function for a pulsed-source TOF neutron spectrometer I. Ionita* Institute for Nuclear Research Pitesti, P.O. Box 078, Pitesti 0300, Romania Received 9 December 2002; received in revised form 25 June 2003; accepted 25 June 2003
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. All the corresponding relevant configurations are considered. For crystal and polycrystalline filter as 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 2003 Elsevier B.V. All rights reserved. PACS: 61.12.eX Keywords: Resolution function; Normal approximation; Matrix method; Pulsed-source TOF neutron spectrometer
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 by Cooper and Nathans, [1], is suitable only for rather simple configurations as the conventional double and triple axis spectrometers are. The procedure is no longer appropriate when spatial effects are important as is the case of configurations using focusing effects and curved monochromators, [7], or for TOF instruments. The matrix method, [2], has proved to be suitable both for conventional, focusing or TOF instruments. This computing technique has been successfully used for different configurations as are the TOF diffractometers with pulsed source, [3], or with steady-state source, [5], the crystal diffractometers, [6], or the three-axis spectrometers, [4]. During the last few years the Monte Carlo procedure, [8–11], has begun to be used quite extensively to evaluate the resolution and intensity properties of the neutron spectrometers. The application of the MC technique requires rather powerful computers with increased computing speed, *Tel.: +40-248-213400x579; fax: +40-248-262449. E-mail addresses: [email protected], [email protected] (I. Ionita). 0168-9002/$ - see front matter r 2003 Elsevier B.V. All rights reserved. doi:10.1016/S0168-9002(03)02051-5
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while for the matrix method a 486 PC is quite suitable with computing times of 1–2 s: 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.
2. The general theory The computational procedure involves several steps. The first one is to choose the initial variables of the problems defining the neutron trajectory between the source and detector. The normal approximation of the probability distribution of the initial variables, spatial, angular and time variables defining the vector R, can be computed for a particular experimental setting, giving the corresponding matrix S; the transmission matrix of its components. Throughout the manuscript the notation of Ref. [5] is used. The next step is to characterize the influence of the Soller collimators, neutron guides (defined as a Soller collimator with a wavelength-dependent angular divergence given by the total reflex critical angle), coarse collimators or slits (defined as a coarse collimator of zero length). The effect of the presence of a coarse collimator or slit, characterized by the transmission matrix Ti ; is to modify S to X ð1Þ S-S þ Ti : To characterize the influence of the Soller collimators or neutron guides one has to introduce the vector U of the angular variables, related to R as U ¼ CR:
ð2Þ
Its transmission matrix V is defined by the divergences of the Soller collimators. The effect of the presence of the Soller collimators is to modify S to X S-S þ Ti þ C T VC: ð3Þ Owing to the existence of the constraints, as the Bragg constraints are, the initial variables are not linearly independent and, therefore, the initial set R has to be reduced to a linearly independent subset R0 : The initial variables can be expressed through R0 variables as R ¼ DR0 :
ð4Þ
The transmission matrix for the subset R0 is S0 ¼ DT SD:
ð5Þ
The final step is to write the relation between X, the resolution function variables ðX ¼ Q Q0 ; DE DE0 ; Q ¼ ki kf ; DE ¼ Ei Ef and Q0 ; DE0 defines, respectively, the most probable value of Q, DEÞ and R0 X ¼ AR0 :
ð6Þ
The resolution matrix M is given by M ¼ ½AS 01
AT 1 :
ð7Þ
The Gaussian approximation of the resolution function is W ðXÞ ¼ W0 ð2pÞ2 ðdet MÞ1=2 expðXT MX=2Þ
ð8Þ
with W0 being the normalization factor. To obtain the line widths for different types of scans it is necessary only to define the scan variables Z, generally not linearly independent, related to X as Z ¼ HX:
ð9Þ
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The scan variables covariance matrix is /zi zj S ¼ HM 1 H T
ð10Þ
and therefore the linewidth for the scan ‘‘i’’ is wi ¼ ð8 ln 2/z2i SÞ1=2 :
ð11Þ
3. The resolution function for a pulsed-source TOF neutron spectrometer with crystal monochromator 3.1. 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 ¼ ðl0 ; g0 ; lm ; gm ; ls ; gs ; ld ; gd ; xm ; t0 ; tf Þ; RV ¼ ðz0 ; zm ; zs ; zd ; x0m Þ: 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 ; x0m 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 UH ¼ CH RH ;
UV ¼ CV RV :
ð12Þ
The matrices CH ; CV are given by the expressions of the angular variables, according to 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
ld
l0 χ 0
lm
ls
χs
χd Detector
L2
2θs
χm L1 2θm
Sample
L0
Source
Monochromator
Fig. 1. The experimental set-up for all pulsed-source TOF spectrometers with crystal monochromator, direct geometry.
ð13Þ
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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
ð14Þ
Subscripts 0, 1, 2 refer, respectively, to source–crystal, crystal–sample and sample–detector regions. The variables w0 ; ws ; wd represent 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. According to Eqn. (3) the matrices SH ; SV are modified as X T THi þ CH VH CH S1H ¼ SH þ S1V ¼ SV þ
X
TVi þ CVT VV CV :
ð15Þ
The Bragg constraints, in the horizontal plane, are given g0 þ g1 ¼ 2Cm xm þ 2rm lm þ 2rm Bm gm Dki g g1 þ bm xm rm Am gm : ¼ cot ym 0 ki0 2
ð16Þ
The Bragg constraints, in the vertical plane, are given by 0 0 d0 d1 ¼ 2 sin ym ðCm xm þ zm r0m Þ:
ð160 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 five vertical variables can be reduced to a set of four independent variables, RVin ¼ ðzm ; zs ; zd ; x0m Þ: The relation between RHin ; RH and RVin ; RV are RH ¼ DiH RHin
ð17Þ
RV ¼ DiV RVin :
ð170 Þ
The matrix DiH is obtained from first relation (16) and (13), 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
ð18Þ
and recalling that lm ; gm ; ls ; gs ; ld ; gd ; xm ; t0 ; tf are common variables for RH ; RHin : The expressions for l0 ; g0 ; defining DiH ; are L0 l0 ¼ DLi sin w0 þ lm cos w0 2 tan w0 cos ym sin wm þ cosðym þ wm Þ cosðym wm Þ 2L0 rm L1 L0 gm cos w0 2 tan w0 sin ym sin wm þ sinðym þ wm Þ þ sinðym wm Þ þ 2L0 rm Bm L1 L0 2Cm L0 cos w0 xm þ ls sin w0 sinðys þ ws Þ þ cos w0 cosðys þ ws Þ L1 L0 þ gs sin w0 cosðys þ ws Þ þ cos w0 sinðys þ ws Þ ð19Þ L1
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g0 ¼ DLi cos w0 þ lm sin w0 2 cot w0 cos ym sin wm þ cosðym þ wm Þ L0 cosðym wm Þ 2L0 rm L1 L0 gm sin w0 2 cot w0 sin ym sin wm þ sinðym þ wm Þ þ sinðym wm Þ þ 2L0 rm Bm L1 2Cm L0 sin w0 xm L0 þ ls cos w0 sinðys þ ws Þ þ sin w0 cosðys þ ws Þ L1 L0 þ gs cos w0 cosðys þ ws Þ þ sin w0 sinðys þ ws Þ : L1 The matrix DiV is given by L0 L0 z0 ¼ zm 1 þ 2 sin ym r0m zs L1 L1 0 2L0 Cm sin ym x0m
ð20Þ
as zm ; zs ; zd ; x0m are common variables for RVin ; RV : To obtain Eq. (20) relations (14) and (160 ) are used. 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 :
ð21Þ
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 YH ¼ DH RHin ;
YV ¼ DV RVin :
ð22Þ
The matrix DH is given by Dki cosðym wm Þ ¼ lm cot ym rm þ þ gm cot ym rm ðBm Am tan ym Þ L1 ki0 sinðym wm Þ cosðys þ ws Þ sinðys þ ws Þ þ gs cot ym þ xm ðCm cot ym þ bm Þ ls cot ym L1 L1 L1 Dkf DLi kf 0 _kf 0 L0 þ L1 kf 0 cosðym wm Þ ¼ þ ðt0 tf Þ lm cot ym rm þ L1 kf 0 L2 ki0 mL2 L2 ki0 L0 þ L1 kf 0 L0 þ L1 kf 0 ðCm cot ym þ bm Þ gm cot ym rm ðBm Am tan ym Þ xm L2 ki0 L2 ki0 sinðym wm Þ þ L1 kf 0 ld gd ls L0 sin wd þ cos wd þ cot ym 1þ cosðys þ ws Þ L2 L2 L2 ki0 L1 kf 0 gs L0 þ sinðys ws Þ þ cot ym 1þ sinðys þ ws Þ cosðys ws Þ L2 ki0 L1
ð23Þ
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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 :
ð24Þ
To these relations Eq. (13) must to be added. First relation (23) is obtained from Eqs. (16) and (13). Second relation (23) is obtained from the relation giving the spread of the total time-of-flight, tf t0 Lf Dkf _ DLi Li Dki DLf ðtf t0 Þ ¼ þ : ð25Þ m ki0 ki0 ki0 kf 0 kf 0 kf 0 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 DLf ¼ ls sinðys ws Þ gs cosðys ws Þ ld sin wd þ gd cos wd :
ð26Þ
In relation (24) 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 of reaching the detector. The covariance matrix for the YH variables is given by 1 1 T NH ¼ DH S1Hi DH :
ð27Þ
The DV matrix is given by relation (14) taking account of the fact 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 : ð28Þ The final step is to relate YH to the XH ¼ ðX1 ; X2 ; X4 Þ vector XH ¼ AYH :
ð29Þ
The A matrix is given by X1 ¼ cos jDki þ ki0 sin jg0 cos j0 Dkf kf 0 sin j0 g1 X2 ¼ sin jDki þ ki0 cos jg0 þ sin j0 Dkf kf 0 cos j0 g1 X4 ¼
ð30Þ
_ ðki0 Dki kf 0 Dkf Þ m
with tan j ¼
kf 0 sinð2ys Þ ; ki0 kf 0 cosð2ys Þ
j0 ¼ j 2ys :
The resolution matrix M is 1 T M 1 ¼ ANH A :
ð31Þ
ð32Þ
1 given by We have only to complete the M 1 with the element M33 1 2 1 1 1 M33 ¼ ki0 NV11 þ kf20 NV22 2ki0 kf 0 NV12 :
ð33Þ
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3.2. The 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 variable vectors UH ¼ ðg0 ; g1 ; g2 Þ; UV ¼ ðd0 ; d1 ; d2 Þ are related to RH ; respectively, RV according to Eq. (12). For the inverse geometry Li ¼ L0 ; Lf ¼ L1 þ L2 while last two relations (26) giving DLi ; DLf are still valid. Relations (13) should be modified according to the considered geometry: g0 ¼
ls cosðys þ ws Þ þ gs sinðys þ ws Þ l0 cos w0 g0 sin w0 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
ð34Þ
Also, relation (14) should be modified as d0 ¼
zs z0 ; L0
d1 ¼
zm zs ; L1
d2 ¼
zd zm : L2
ð35Þ
Relations (15) and (16) 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 five vertical variables can be reduced to a set of four independent variables, RVin ¼ ðz0 ; zs ; zm ; x0m Þ: Relations (17) are also still valid. The matrix DiH is obtained from first relations (16) and (34), taking into account the expression of DLf ; the last of relations (26), and recalling that lm ; gm ; ls ; gs ; l0 ; g0 ; xm ; t0 ; tf are common variables for RH ; RHin : The expressions for ld ; gd ; defining DiH ; are
χd ld
l0 χ 0
L2
lm χm ls
Detector
2θm
χs L1 2θs
Monochromator
L0
Source
Sample
Fig. 2. The experimental set-up for all pulsed-source TOF spectrometers with crystal monochromator, inverse geometry.
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ld ¼ DLf sin wd þ lm cos wd 2 tan wd cos ym sin wm þ cosðym wm Þ L2 cosðym þ wm Þ þ 2L2 rm þ gd cos wd 2 tan wd sin ym sin wm þ sinðym wm Þ L1 L2 L2 sinðym þ wm Þ þ 2L2 rm Bm þ 2Cm L2 cos wd xm þ ls sin wd sinðys ws Þ þ cos wd cosðys ws Þ L1 L1 L2 þ gs sin wd cosðys ws Þ þ cos wd sinðys ws Þ ð36Þ L1 L2 gd ¼ DLf cos wd þ lm sin wd 2 cot wd cos ym sin wm þ cosðym wm Þ cosðym þ wm Þ þ 2L2 rm L1 L2 þ gm sin wd 2 cot wd sin ym sin wm þ sinðym wm Þ sinðym þ wm Þ þ 2L2 rm Bm L1 L2 þ 2Cm L2 sin wd xm þ ls cos wd sinðys ws Þ þ sin wd cosðys ws Þ L1 L2 þ gs cos wd cosðys ws Þ þ sin wd sinðys ws Þ : ð37Þ L1 The matrix DiV is given by L2 L2 0 0 2 sin ym rm zs 2L2 Cm sin ym x0m zd ¼ zm 1 þ L1 L1
ð38Þ
as zm ; zs ; z0 ; x0m are common variables for RVin ; RV : To obtain Eq. (37) relations (35) and (160 ) 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 relations (23) giving Dki =ki0 ; Dkf =kf 0 ; must be replaced with Dki cosðym þ wm Þ sinðym þ wm Þ ¼ lm cot ym rm þ þ gm cot ym rm ðBm þ Am tan ym Þ þ L1 L1 ki0 cosðys ws Þ sinðys ws Þ gs cot ym L1 L1 Dkf DLfi _kf 0 L0 kf 0 cosðym þ wm Þ ðt0 tf Þ ¼ þ lm cot ym rm þ L1 kf 0 L1 þ L2 mðL1 þ L2 Þ L1 þ L2 ki0 L 0 kf 0 L0 kf 0 ðCm cot ym þ bm Þ gm cot ym rm ðBm þ Am tan ym Þ xm L1 þ L2 ki0 L1 þ L2 ki0 sinðym wm Þ þ L1 kf 0 kf 0 g0 kf 0 L0 l0 ls þ sin w0 cos w0 þ cot ym cosðys ws Þ sinðys þ ws Þ ki0 L1 þ L2 ki0 L1 þ L2 L1 þ L2 ki0 L1 kf 0 L0 gs þ cot ym sinðys ws Þ þ cosðys þ ws Þ : ð39Þ L1 þ L2 ki0 L1 þ xm ðCm cot ym þ bm Þ ls cot ym
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4. The resolution function for a pulsed-source TOF neutron spectrometer with polycrystalline filter as monochromator 4.1. The direct geometry The experimental set-up is given in Fig. 3. 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 ¼ ðDki =ki0 ; l0 ; g0 ; ls ; gs ; ld ; gd ; t0 ; tf Þ; RV ¼ ðz0 ; zs ; zd Þ: The coordinates g; l are, respectively, along the thickness and width, t0 ; tf define the neutron emission, respectively, detection and the subscript 0, s, d refer, respectively, to source, sample and detector. 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. In particular, for a given cutoff wavelength and for a given energy distribution of the emitted neutrons the term /ðDki =ki0 Þ2 S can be computed. The angular variables vectors UH ¼ ðg; g1 Þ; UV ¼ ðd0 ; d1 Þ are related to RH ; respectively, RV as UH ¼ CH RH ;
UV ¼ CV RV :
ð40Þ
The matrices CH ; CV are given by the expressions of the angular variables, according to the set-up geometry g0 ¼
ls cosðys þ ws Þ þ gs sinðys þ ws Þ l0 cos w0 gs sin w0 L0 þ L1
g1 ¼
ld cos wd gd sin wd ls cosðys ws Þ gs sinðys ws Þ L2
ð41Þ
d0 ¼
zs z0 ; L0 þ L1
ð42Þ
d1 ¼
zd zs : L2
The subscripts 0, 1, 2 refer, respectively, to source–filter, filter–sample and sample–detector regions. The variables w0 ; ws ; wd represent the moderator, sample and detector orientation angles, all measured in the trigonometric sense while 2ys is the scattering angle. For the angular variables the subscripts 0, 1 refer to source–sample, respectively, sample–detector regions. As above the matrices SH ; SV are modified according to Eqn. (15). 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 related to RH ; receptively, RV as YH ¼ DH RH ;
YV ¼ DV RV :
ð43Þ
ls
χ0
χd
ld
Filter l0
χs L2
L0
θs
Detector
θs
L1
Source Sample
Fig. 3. The experimental set-up for a pulsed-source TOF spectrometer with polycrystalline filter as monochromator, direct geometry.
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The matrix DH is given by Dkf kf 0 l0 kf 0 g0 k f 0 ls ¼ sin w0 cos w0 þ sinðys þ ws Þ þ sinðys ws Þ kf 0 L2 ki0 L2 ki0 L2 ki0 gs k f 0 L0 þ L1 kf 0 Dki _kf 0 þ cosðys þ ws Þ cosðys ws Þ þ ðt0 tf Þ L2 ki0 L2 ki0 ki0 mL2 ld gd sin wd þ cos wd : L2 L2
ð44Þ
To these relations Eqs. (41) and (24) must be added. The term Dki =ki0 is a common variable for RH ; YH : Relation (44) is obtained from relation (25) giving the spread of the total time-of-flight, tf t0 ; and (26). From this point the computing procedure is the same as in Section 3.1. 4.2. The inverse geometry The experimental set-up is given in Figs. 4 and 5. For this geometry Li ¼ L0 ; Lf ¼ L1 þ L2 and the computing procedure, including the corresponding relations, remain the same as above, provided L2 is replaced with L1 þ L2 and L0 þ L1 is replaced with L0 :
5. The resolution function for a pulsed-source TOF neutron spectrometer with mechanical monochromator The first step is to define the initial variables vectors RH ¼ ðDki =ki0 ; l0 ; g0 ; lch Þ; R0H ¼ ðlch ; ls ; gs ; ld ; gd ; t0 ; tf Þ; RV ¼ ðz0 ; zch Þ; R0V ¼ ðzch ; zs ; zd Þ: The coordinates g; l are, respectively, along the thickness and width, t0 ; tf define the neutron emission, respectively, detection and the subscript 0, s, d refer, respectively, to source, sample and detector. All the coordinates represent the deviation from the corresponding most probable value. For a given set-up the transmission matrices SH ; S1H ; SV ; S1V can be computed in the normal approximation. In particular, for a given mechanical monochromator the term /ðDki =ki0 Þ2 S can be computed. The angular variables vectors UH ¼ ðg0 ; g1 ; g2 Þ; UV ¼ ðd0 ; d1 ; d2 Þ are related to RH ; R0H ; respectively, RV ; 0 RV as 0 UH ¼ CH RH ; UH ¼ CH R0H
UV ¼ CV RV ; UV ¼ CV0 R0V :
ð47Þ ld
Filter
χd Detector
l0
χ0
ls
L2
χs L1
L0
θs
θs
Source Sample Fig. 4. The experimental set-up for a pulsed-source TOF spectrometer with polycrystalline filter as monochromator, inverse geometry.
ARTICLE IN PRESS I. Ionita / Nuclear Instruments and Methods in Physics Research A 513 (2003) 511–523 Mechanical Monochromator ls χs
l0
χ0
χd
ld
L2 L0
521
θs
Detector
θs
L1
Source Sample
Fig. 5. The experimental set-up for a pulsed-source TOF spectrometer with mechanical monochromator. 0 The matrices CH ; CH ; CV ; CV0 are given by the expressions of the angular variables, according to the setup geometry
lch l0 cos w0 gs sin w0 L0 ls cosðys þ ws Þ þ gs sinðys þ ws Þ lch g1 ¼ L1 ld cos wd þ gd sin wd ls cosðys ws Þ gs sinðys ws Þ g2 ¼ L2
g0 ¼
d0 ¼
zch z0 ; L0
d1 ¼
zs zch ; L1
d1 ¼
zd zs : L2
ð48Þ ð49Þ
Subscripts 0, 1, 2 refer, respectively, to source–monochromator, monochromator–sample and sample– detector regions. The variables w0 ; ws ; wd represent the moderator, sample and detector orientation angles, all measured in the trigonometric sense while 2ys is the scattering angle. For the angular variables subscripts 0, 1, 2 refer to source–monochromator, monochromator–sample, respectively, sample–detector regions. Again, the matrices SH ; SV are modified according to Eqs. (3) and (15). To relations (15) should be added X 0T 0 S1Hm ¼ S1H þ THi þ CH VH CH X S1Vm ¼ S1V þ TVi þ CV0T VV CV0 : ð150 Þ The correlation introduced by the mechanical monochromator is characterized by g0 ¼ g1 :
ð50Þ
To take this correlation into account we have to introduce the vector RHint ¼ ðDki =ki0 ; g0 ; lch Þ related to RH as RHint ¼ BH RH
ð51Þ
where the matrix BH is given by first relation (41), Dki =ki0 ; lch being common elements for RHint ; RH : The covariant matrix for RHint is given by 1 SHint ¼ BH SHm BTH :
ð52Þ
The presence of the mechanical monochromator also introduces a correlation between RHint and R0Hm ¼ ðDki =ki0 ; lch ; ls ; gs ; ld ; gd ; t0 ; tf Þ RHint ¼ B0H R0Hm : is given by ls cosðys þ ws Þ þ gs sinðys þ ws Þ lch g0 ¼ g1 ¼ L1
The matrix
ð53Þ
B0H
ð54Þ
ARTICLE IN PRESS I. Ionita / Nuclear Instruments and Methods in Physics Research A 513 (2003) 511–523
522
as Dki =ki0 ; lch are common elements for RHint and R0Hm : The transmission matrix for R0Hm is 0 SH1f ¼ B0T H SHint BH þ S1Hm :
ð55Þ
In Eq. (55) the summation of matrix of different dimensions are performed after having added ‘‘0’’ elements for the matrix lines and columns corresponding to the unconsidered variables. The correlation introduced by the mechanical monochromator is characterized, in the vertical plane, by d0 ¼ d1 :
ð56Þ
To take into account this correlation we have to introduce the vector RVint ¼ ðd0 ; zch Þ related to RV as RVint ¼ BV RV
ð57Þ
where the matrix BV is given by first relation (42), zch being a common element for RVint ; RV : The covariant matrix for RVint is given by 1 ¼ BV SVm BTH : SVint
ð58Þ
The presence of the mechanical monochromator also introduces a correlation between RVint and R0V RVint ¼ B0V R0V :
ð59Þ
The matrix B0V is given by zs zch d0 ¼ d1 ¼ L1
ð60Þ
as zch is a common element for RVint and R0V : The transmission matrix for R0V is 0 SV1f ¼ B0T V SVint BV þ S1Vm :
ð61Þ
In Eq. (61) the summation of matrix of different dimensions arc performed after having added ‘‘0’’ elements for the matrix lines and columns corresponding to the unconsidered variables. 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 related to ½RH ; respectively RV as YH ¼ DH RH ;
YV ¼ DV RV :
ð62Þ
The matrix DH is given by Dkf l0 kf 0 g0 k f 0 ¼ sin w0 cos w0 kf 0 L2 ki0 L2 ki0 kf 0 ls þ sinðys þ ws Þ þ sinðys ws Þ L2 ki0 gs k f 0 þ cosðys þ ws Þ cosðys ws Þ L2 ki0 L0 þ L1 kf 0 Dki _kf 0 þ ðt0 tf Þ L2 ki0 ki0 mL2 ld gd sin wd þ cos wd : L2 L2
ð63Þ
To these relations Eqs. (41) and (24) must to be added. The term Dki =ki0 is a common variable for RH ; YH : Relation (63) is obtained from relation (25) giving the spread of the total time-of-flight, tf t0 ; and Eq. (26). As above 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 of reaching the detector. From this point the computing procedure is the same as in Section 3.1.
ARTICLE IN PRESS I. Ionita / Nuclear Instruments and Methods in Physics Research A 513 (2003) 511–523
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6. Conclusions The matrix procedure to compute the resolution function for a given experimental configuration is applied for a particular one, a pulsed source TOF neutron spectrometer. All the corresponding relevant configurations are considered. The matrix method has proved to be suitable both for conventional focusing or TOF instruments. The matrix procedure described above can be used to optimize an experimental set-up or to evaluate its properties. 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.
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