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On the resolution function of crystal spectrometers for neutron scattering
Nuclear Instruments and Methods in Physics Research 219 (1984) 553-557 North-Holland, Amsterdam
553
O N T H E R E S O L U T I O N F U N C T I O N OF CRYSTAL S P E C T R O M E T E R S F O R N E U T R O N SCATTERING H..GRIMM
Kernforsehungsanlage Jalich, P.O. Box 1913, D-5170 Ji~lich, Fed. Rep. Germany Received 4 July 1983 and in revised form 8 September 1983
Within the Gaussian approximation for the wave vector distribution an exact formulation of the resolution function of a crystal spectrometer is given. It is unlimited in the range of momentum transfers and takes into account the non-linear relation of energy and momentum for neutrons. The connection to the formulation of M.J. Cooper and R. Nathans is discussed.
1. Introduction
This leads to the following integral to be solved
In the pioneering works of Stedman and Nilsson (1966), Cooper and Nathans (C&N) (1967) and Nielsen and Bjerrum Meller (1969) [1] the properties of the resolution function for a triple axis spectrometer are determined for the usual range of energy and momentum transfers. This is done with the assumption that the variance of certain wave vectors is small compared to their mean values so that the energy conservation can be linearized. For small angle scattering the mean value of the momentum transfer is of the same order or even much smaller than its varianc so that in certain situations the linearization of the energy conservation may introduce errors. Thus it seems helpful to reformulate the resolution function without these assumptions.
R,< o, to)= f dk,dk,n[ to
2. Formulation of the problem A spectrometer offers to the sample a certain " b u n die" of incident wave vectors k i and accepts a certain bundle of final wave vectors kf from the sample. Representing these bundles by distribution functions Pi (ki) and Pf(kf), the resolution function in (ki, k f ) space is given by the product
)]
×q$3(Q-ki+kf)R6(ki,kf).
(1)
The projection is nonlinear in to in contrast to the case of X-rays. The integral can be solved analytically if one keeps the assumption that the bundless of k i and kf vectors are normally distributed, i.e. (indices i, f suppressed)
P ( k ) = *r-3/2[ det (M)] 1/2 e x p ( - a k M a k ) , where the matrix M (positive definite, symmetric) describes the variance of k and Ak --- k - ( k ) denotes the deviation from the mean value of k. The distributions are normalized to 1 for convenience (see ref. 2 for the normalization of the resolution function). It is generally agreed that this Gaussian approximation is realistic for the devices which define ki, kf bundles (collimators, mosaic- and perfect crystals). Moreover a non-Gaussian transmission of such a device is rapidly suppressed by the successive application of several devices as stated by the central limit theorem. The creation of M by a sequence of these beam devices is described in the appendix.
R 6 ( k i, kf) = e i ( k i ) P f ( k f ) . Since the sample properties are described in the four dimensional space of momentum transfer Q and energy transfer to, one has the task to project this six-dimensional function into the (Q, to) space. The projection law is given by the conservation of energy and momentum for neutrons. * to = l ( k 2 - k f 2 ) ,
Q=ki-k
f.
* h / m = 1 throughout this work. This corresponds closely to units THz, ,A-1 for energy and momentum. 0167-5087/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics P~blishing Division)
3. Integration of R6(ki, kf) Since C & N linearize the energy conservation relation, the projection from R 6 to R 4 gives again a Gaussian distribution. Although this linearization is justified for most practical purposes, it is not necessary for an analytical solution of eq. (1). The linear transformation to the new variable x Ak i = x + M=IMfAQ,
Akf =
x = M+IMiAQ,
(2)
554
H. Grimm/ Resolution functton of crystal spectrometers
automatically fulfills the momentum conservation and the energy conservation becomes
co = Qx + QK-~QM~1M AQ.
2K=(ki)+~kf
AQ=Q-Q0,
),
Aco=co-co0.
With transformation (2) the integral (1) becomes
R4(Q,,~) = - 3 [
det(Mi) det(M~)]1/2
× exp(-AQGAQ) ×
fdx exp( - x M + x ) B ( a
-
Qx),
(3)
with G =
MiM+IMf
or
G -1
= M i -1 +
a=ACO-KAQ+½QM+IM
M f 1,
AQ.
Choosing the orthogonal basis e i with x = xle I + x2e 2 + x3e 3 such that ellIQ, one gets for the remaining integral in eq. (3) 7r [ ( M + ' ) n det(M+)] -1/2 ×
fdxl e x p [ - x~/(MT. '),1] a ( a -
Qxl).
The result is a function of Q2(M+ l)11. The geometrical meaning of ((M ~ 1)11)1/2 is the extension of the ellipsoid x M + x = const, projected onto the el-direction. Therefore, the scalar quantity Q 2(M + x)al in this special coordinate system may be written in any coordinate system as QMT.]Q. Thus one has the final result, independent of a reference system and valid for any Q R4(Q,co ) = 7r-3/2 [ det(G)] 1/2 e x p ( - A Q G A Q )
×¢r_I/2(QM+XQ)
×
=
d 2,
(5)
V ( Q o ) = Vo.
In this case the non-quadratic terms in the exponent vanish and one gets the familiar 4 × 4 resolution matrix built up by G and the contributions from
(aco- VoaQ)2/d~.
~oo=KQo,
Q0 = ( k i ) - ( k f ) ,
QM+IQ ~ Qo M;1Q0
V(Q)-,
The notation in terms of the mean values and the variances of the ki, f bundles is: M + = M i + M f,
of C&N:
1/2
(4)
with
a = Aco - VAQ,
Under normal conditions for triple axis spectrometers the approximation (5) is very well justified. Remember that (QM+]Q)]/2 means the extension of the ellipsoid x M + x = const, projected onto the Q vector. The replacement (5) is no longer justified as soon as the magnitude and direction of Q varies appreciably within the resolution function. The situation becomes more severe if in addition the ellipsoid (M+) is highly anisotropic. The former is the case for small angle scattering and the latter e.g. by using perfect crystals as beam devices. On the other hand it is useful to consider the orders of magnitude of the terms in eq. (4) which represent the non-linear energy conservation (5). With "normal" collimation and crystai-mosaicity one gets a variance of k i and kf of the order of one percent. With k~,r being about 1, the elements of Mis are then of the order of 10 4, Going to "small" angle scattering with Q, AQ---10 -2 one has
QM+lQ --, 10 s V(Q)=K-½(M+IM )To --~ 1 -- ½(10 -4 X 104)10 -2 --~ K,
~r-,/2(QM÷,Q)-,/z
e x p [ - a 2 / ( Q M + ' Q ) ] ---,6( a ).
Thus one has the - at first glance - surprising result that in the small angle scattering regime the linearization of the energy-momentum relation becomes justified again although AQ = Q. V(Q) approaches the constant vector K and QM+IQ is absorbed in the &function. The dimensionality is consequently reduced to three variables and the resolution can be approximated by the 3 × 3 matrix G of the Gaussian distribution and the &function:
V( Q ) = K - ½(MTIM_ )rQ. Note that the second exponent is quadratic in co but not in Q.
4. Discussion The difference between this result and the corresponding expression of C&N is the replacement of Q by Q0 in eq. (4) wherever Q appears explicitly. This means that the following replacement leads to the result
R 3 ( Q , w ) = ~.-3/2 [ det(G)],/z × exp( - A Q G ~ Q ) 8 ( A w
- KaQ).
This limit is less surprising if one gives the energy conservation the form co = k Q with 2k = k i + kf. In terms of the deviations from the mean values one has:
A~ = K A Q + Q A k , with
A k = k - K = ½( Aki + A k f ).
H. Grimm/ Resolutionfunction of crystal spectrometers .
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.a~, momentum transfer Q
.001
momentum transfer Q
18
Fig. 1. Contour maps of R2(Q, to) for Qo =1, 0.1, 0.01 with - 0.09 ~ ~ ~<0.09; units ~.- 1 and THz; the outermost contour line corresponds to 20% of the maximum, the stepwidth is 10%.
This exact relation is linear as soon as the term Qzlk can be neglected. The conclusion to be drawn is that there exists in general an upper and a lower limit for the Q-range, within which the nonlinear energy conservation shows up. This fact could be demonstrated quantitatively in many different ways. Its relevance or irrelevance depends on the kind of beam devices which determine the shape of M i and M f. Moreover, the differences between the linearized and the exact version may be enhanced or suppressed by the folding with the scattering function of the sample. A reasonable compromise could be as follows: the frequently used equipment with pyrolytic graphite crystals [reflection (002), mosaic = 0.4 °] together with
Fig. 2. Contour maps of R2(Q,~0) for Q0 = 0.4; ~0-range, units, contour steps are the same as in fig. 1. (a) Non-linarized. (b) Linearized energy conservation relation. (c) = (b)-(a) (contour step = 10% of max/min-distance).
tight horizontal (0.2 ° ) and relaxed vertical collimation (4°). The further choices are: ( k ) = 2.66, to = 0 , Wconfiguration. The sample is assumed to be isotropic. In this case the two-dimensional resolution function R 2 (Q, to) applies: RE(Q,to)
= f dOeO2R4( O.,to).
The two-dimensional integration over the direction of Q (one-dimensional in the small angle limit) is performed numerically. The accuracy of the numerical integration is checked by the normalization condition
f n2 ( Q,to )dQ dto = l. Contour maps of R2(Q,to ) are shown in fig. 1 for three values of Q0 = 1, 0.1, 0.01. The non-elliptical shape is mainly caused by the integration over the solid angle and not by the non-quadratic terms in the exponent of
H. Grimm/Resolution function of crystal spectrometers
556
I thank F. Hossfeld and W. Meyer of the Institute of Applied Mathematics of our Research Center for fruitful discussions which revealed that the mathematics of this work are standard in statistics. The very efficient help of Mrs. L. Schaetzler in programming is gratefully acknowledged.
eq. (4). At Q0 = 0.1 one recognizes the tail at large Q which becomes important, e.g. for magnon determination in amorphous systems [4]. For Q0 = 0.01 the resolution is compressed into the sector I~[ ~< QK whose boundary corresponds to zero scattering angle. A comparison of the exact and the linearized version is shown in fig. 2 for Q0 = 0.4 together with a contour map of the difference. The normalized absolute value of this difference A R = 0.02, 0.12, 0.06 for Q0 = 0.1, 0.4, 1, respectively. Note that A R is not a monotonic function of Q0. For no overlap of both versions one would get AR=2.
Appendix
How does one create M,,/? An account of this question has been given in an internal report [3]. For completeness the essential results are summarized in this appendix. A beam device of a spectrometer transforms an incoming wave vector distribution into an outgoing one. The distribution shall be characterized by a 3 × 3 matrix M at each point from the source to the detector. The matrix M is represented in an orthogonal, right-hapded coordinate system with the 1-axis (anti)parallel to the mean neutron path (after) before the sample. The 3-axis points upward for a horizontal mean scattering plane. Then M has block-diagonal form.
5. Summary
Within the Gaussian approximation for the distribution of incident and final wave vectors of a spectrometer an exact formulation of the resolution function has been given, i.e. without linearization of energy conservation for neutrons. Inspection of the result shows that for both large and small m o m e n t u m transfers this linearization ( C & N ) gives a good approximation. Unfortunately C & N did not formulate their result in tensor notation. The disadvantages are less flexibility in the beam device arrangement and a less obvious transition to the 3 × 3 matrix Mi(M i + M f ) - l M t for small angle scattering. Between both limits of m o m e n t u m transfers there may arise situations (M+ anisotropic, M_ "large", anisotropic scattering function, etc.) where the exact formulation is useful, A computer program is available from the author for the determination of the various forms (R 2 to R4) of the resolution function in both the linearized and the non-linearized approximation.
m:0
The transformation for the horizontal part is
moo , = sTmin$ + dhC.
(6a)
N o rotation of matrices is necessary using the representation of s and c as given in table 1 below (see also for the remaining definitions). For the vertical part one gets P ( m 33)oot = (m33)iP. + dvP .
(6b)
Table 1 Definitions a) of matrices. Collimator
Mosaic crystal - 2lt
C (41n2/dh)l/2(k)
p (41n2/dv)]/2/(k)
1
ah 1 av
a) 1 = ( - )1 if the device deflects the beam to the (left) right. t = t a n O , s = s i n O , 0 = Bragg angle. ah, v = horizontal, vertical divergence (fwhm, arc). /7,,v = horizontal, vertical mosaic (fwhm, arc). A d/d = effective "spread" o f lattice spacing. ( k ) = m e a n w a v e vector at device. b) (m33)out ~ (m33)i n for this case.
-It ~h -1 2srtv
Perfect crystal
°t
1
-t
n
_ lt-I
Ad/d --lb) 0 b)
t 2
•
H. Grimm/Resolution function of crystal spectrometers Table 1 gives definitions of the matrices for c o m m o n beam devices. At the source and the detector one may start with M = (0) due to the " w h i t e " and isotropic emission and absorption of neutrons. Approaching the sample from both ends by successive application of the transformation (6) one creates M i,f in the (ki, f)-coordinate systems (1-axis parallel to ( k i ) and antiparallel to ( k f ) ) . Rotation of both matrices to a common coordinate system is of course assumed in the definition of M ±.
557
References [1] R. Stedman and G. Nilsson, Phys: Rev. 145 (1966) 492; M.J. Cooper and R. Nathans, Acta Cryst. 23 (1967) 357; M. Nielsen and H. Bjerrum Moiler, Acta Cryst. A25 (1969) 547. [2] B. Domer, Acta Cost. A28 (1972) 319; N.J. Chesser and J.D. Axe, Acta Cryst. A29 (1973) 160. [3] H. Grimm, Internal Report (ILL Grenoble) 76G198T (1976). [4] J.D. Axe, G. Shirane, T. Mizoguchi and K. Yamauchi, Phys. Rev. B15 (1977) 2763.
553
O N T H E R E S O L U T I O N F U N C T I O N OF CRYSTAL S P E C T R O M E T E R S F O R N E U T R O N SCATTERING H..GRIMM
Kernforsehungsanlage Jalich, P.O. Box 1913, D-5170 Ji~lich, Fed. Rep. Germany Received 4 July 1983 and in revised form 8 September 1983
Within the Gaussian approximation for the wave vector distribution an exact formulation of the resolution function of a crystal spectrometer is given. It is unlimited in the range of momentum transfers and takes into account the non-linear relation of energy and momentum for neutrons. The connection to the formulation of M.J. Cooper and R. Nathans is discussed.
1. Introduction
This leads to the following integral to be solved
In the pioneering works of Stedman and Nilsson (1966), Cooper and Nathans (C&N) (1967) and Nielsen and Bjerrum Meller (1969) [1] the properties of the resolution function for a triple axis spectrometer are determined for the usual range of energy and momentum transfers. This is done with the assumption that the variance of certain wave vectors is small compared to their mean values so that the energy conservation can be linearized. For small angle scattering the mean value of the momentum transfer is of the same order or even much smaller than its varianc so that in certain situations the linearization of the energy conservation may introduce errors. Thus it seems helpful to reformulate the resolution function without these assumptions.
R,< o, to)= f dk,dk,n[ to
2. Formulation of the problem A spectrometer offers to the sample a certain " b u n die" of incident wave vectors k i and accepts a certain bundle of final wave vectors kf from the sample. Representing these bundles by distribution functions Pi (ki) and Pf(kf), the resolution function in (ki, k f ) space is given by the product
)]
×q$3(Q-ki+kf)R6(ki,kf).
(1)
The projection is nonlinear in to in contrast to the case of X-rays. The integral can be solved analytically if one keeps the assumption that the bundless of k i and kf vectors are normally distributed, i.e. (indices i, f suppressed)
P ( k ) = *r-3/2[ det (M)] 1/2 e x p ( - a k M a k ) , where the matrix M (positive definite, symmetric) describes the variance of k and Ak --- k - ( k ) denotes the deviation from the mean value of k. The distributions are normalized to 1 for convenience (see ref. 2 for the normalization of the resolution function). It is generally agreed that this Gaussian approximation is realistic for the devices which define ki, kf bundles (collimators, mosaic- and perfect crystals). Moreover a non-Gaussian transmission of such a device is rapidly suppressed by the successive application of several devices as stated by the central limit theorem. The creation of M by a sequence of these beam devices is described in the appendix.
R 6 ( k i, kf) = e i ( k i ) P f ( k f ) . Since the sample properties are described in the four dimensional space of momentum transfer Q and energy transfer to, one has the task to project this six-dimensional function into the (Q, to) space. The projection law is given by the conservation of energy and momentum for neutrons. * to = l ( k 2 - k f 2 ) ,
Q=ki-k
f.
* h / m = 1 throughout this work. This corresponds closely to units THz, ,A-1 for energy and momentum. 0167-5087/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics P~blishing Division)
3. Integration of R6(ki, kf) Since C & N linearize the energy conservation relation, the projection from R 6 to R 4 gives again a Gaussian distribution. Although this linearization is justified for most practical purposes, it is not necessary for an analytical solution of eq. (1). The linear transformation to the new variable x Ak i = x + M=IMfAQ,
Akf =
x = M+IMiAQ,
(2)
554
H. Grimm/ Resolution functton of crystal spectrometers
automatically fulfills the momentum conservation and the energy conservation becomes
co = Qx + QK-~QM~1M AQ.
2K=(ki)+~kf
AQ=Q-Q0,
),
Aco=co-co0.
With transformation (2) the integral (1) becomes
R4(Q,,~) = - 3 [
det(Mi) det(M~)]1/2
× exp(-AQGAQ) ×
fdx exp( - x M + x ) B ( a
-
Qx),
(3)
with G =
MiM+IMf
or
G -1
= M i -1 +
a=ACO-KAQ+½QM+IM
M f 1,
AQ.
Choosing the orthogonal basis e i with x = xle I + x2e 2 + x3e 3 such that ellIQ, one gets for the remaining integral in eq. (3) 7r [ ( M + ' ) n det(M+)] -1/2 ×
fdxl e x p [ - x~/(MT. '),1] a ( a -
Qxl).
The result is a function of Q2(M+ l)11. The geometrical meaning of ((M ~ 1)11)1/2 is the extension of the ellipsoid x M + x = const, projected onto the el-direction. Therefore, the scalar quantity Q 2(M + x)al in this special coordinate system may be written in any coordinate system as QMT.]Q. Thus one has the final result, independent of a reference system and valid for any Q R4(Q,co ) = 7r-3/2 [ det(G)] 1/2 e x p ( - A Q G A Q )
×¢r_I/2(QM+XQ)
×
=
d 2,
(5)
V ( Q o ) = Vo.
In this case the non-quadratic terms in the exponent vanish and one gets the familiar 4 × 4 resolution matrix built up by G and the contributions from
(aco- VoaQ)2/d~.
~oo=KQo,
Q0 = ( k i ) - ( k f ) ,
QM+IQ ~ Qo M;1Q0
V(Q)-,
The notation in terms of the mean values and the variances of the ki, f bundles is: M + = M i + M f,
of C&N:
1/2
(4)
with
a = Aco - VAQ,
Under normal conditions for triple axis spectrometers the approximation (5) is very well justified. Remember that (QM+]Q)]/2 means the extension of the ellipsoid x M + x = const, projected onto the Q vector. The replacement (5) is no longer justified as soon as the magnitude and direction of Q varies appreciably within the resolution function. The situation becomes more severe if in addition the ellipsoid (M+) is highly anisotropic. The former is the case for small angle scattering and the latter e.g. by using perfect crystals as beam devices. On the other hand it is useful to consider the orders of magnitude of the terms in eq. (4) which represent the non-linear energy conservation (5). With "normal" collimation and crystai-mosaicity one gets a variance of k i and kf of the order of one percent. With k~,r being about 1, the elements of Mis are then of the order of 10 4, Going to "small" angle scattering with Q, AQ---10 -2 one has
QM+lQ --, 10 s V(Q)=K-½(M+IM )To --~ 1 -- ½(10 -4 X 104)10 -2 --~ K,
~r-,/2(QM÷,Q)-,/z
e x p [ - a 2 / ( Q M + ' Q ) ] ---,6( a ).
Thus one has the - at first glance - surprising result that in the small angle scattering regime the linearization of the energy-momentum relation becomes justified again although AQ = Q. V(Q) approaches the constant vector K and QM+IQ is absorbed in the &function. The dimensionality is consequently reduced to three variables and the resolution can be approximated by the 3 × 3 matrix G of the Gaussian distribution and the &function:
V( Q ) = K - ½(MTIM_ )rQ. Note that the second exponent is quadratic in co but not in Q.
4. Discussion The difference between this result and the corresponding expression of C&N is the replacement of Q by Q0 in eq. (4) wherever Q appears explicitly. This means that the following replacement leads to the result
R 3 ( Q , w ) = ~.-3/2 [ det(G)],/z × exp( - A Q G ~ Q ) 8 ( A w
- KaQ).
This limit is less surprising if one gives the energy conservation the form co = k Q with 2k = k i + kf. In terms of the deviations from the mean values one has:
A~ = K A Q + Q A k , with
A k = k - K = ½( Aki + A k f ).
H. Grimm/ Resolutionfunction of crystal spectrometers .
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.
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.
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.
.
.
.
.
.
.
.
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.
.
.
.
.
( 1
.
.
.
.
a
,0
i , f
\ .-~J?.-- /
98
101, b
Qo = . l
3
o
.1::
.06
;18
Qo =
~ l _ ~ _
.OI
,
,
,
.
.
.
.
.
,
,
i
.
.
.
.
.
.
.
.
.
.38
~
.
.
.
.
.
.
.
,
,
.a~, momentum transfer Q
.001
momentum transfer Q
18
Fig. 1. Contour maps of R2(Q, to) for Qo =1, 0.1, 0.01 with - 0.09 ~ ~ ~<0.09; units ~.- 1 and THz; the outermost contour line corresponds to 20% of the maximum, the stepwidth is 10%.
This exact relation is linear as soon as the term Qzlk can be neglected. The conclusion to be drawn is that there exists in general an upper and a lower limit for the Q-range, within which the nonlinear energy conservation shows up. This fact could be demonstrated quantitatively in many different ways. Its relevance or irrelevance depends on the kind of beam devices which determine the shape of M i and M f. Moreover, the differences between the linearized and the exact version may be enhanced or suppressed by the folding with the scattering function of the sample. A reasonable compromise could be as follows: the frequently used equipment with pyrolytic graphite crystals [reflection (002), mosaic = 0.4 °] together with
Fig. 2. Contour maps of R2(Q,~0) for Q0 = 0.4; ~0-range, units, contour steps are the same as in fig. 1. (a) Non-linarized. (b) Linearized energy conservation relation. (c) = (b)-(a) (contour step = 10% of max/min-distance).
tight horizontal (0.2 ° ) and relaxed vertical collimation (4°). The further choices are: ( k ) = 2.66, to = 0 , Wconfiguration. The sample is assumed to be isotropic. In this case the two-dimensional resolution function R 2 (Q, to) applies: RE(Q,to)
= f dOeO2R4( O.,to).
The two-dimensional integration over the direction of Q (one-dimensional in the small angle limit) is performed numerically. The accuracy of the numerical integration is checked by the normalization condition
f n2 ( Q,to )dQ dto = l. Contour maps of R2(Q,to ) are shown in fig. 1 for three values of Q0 = 1, 0.1, 0.01. The non-elliptical shape is mainly caused by the integration over the solid angle and not by the non-quadratic terms in the exponent of
H. Grimm/Resolution function of crystal spectrometers
556
I thank F. Hossfeld and W. Meyer of the Institute of Applied Mathematics of our Research Center for fruitful discussions which revealed that the mathematics of this work are standard in statistics. The very efficient help of Mrs. L. Schaetzler in programming is gratefully acknowledged.
eq. (4). At Q0 = 0.1 one recognizes the tail at large Q which becomes important, e.g. for magnon determination in amorphous systems [4]. For Q0 = 0.01 the resolution is compressed into the sector I~[ ~< QK whose boundary corresponds to zero scattering angle. A comparison of the exact and the linearized version is shown in fig. 2 for Q0 = 0.4 together with a contour map of the difference. The normalized absolute value of this difference A R = 0.02, 0.12, 0.06 for Q0 = 0.1, 0.4, 1, respectively. Note that A R is not a monotonic function of Q0. For no overlap of both versions one would get AR=2.
Appendix
How does one create M,,/? An account of this question has been given in an internal report [3]. For completeness the essential results are summarized in this appendix. A beam device of a spectrometer transforms an incoming wave vector distribution into an outgoing one. The distribution shall be characterized by a 3 × 3 matrix M at each point from the source to the detector. The matrix M is represented in an orthogonal, right-hapded coordinate system with the 1-axis (anti)parallel to the mean neutron path (after) before the sample. The 3-axis points upward for a horizontal mean scattering plane. Then M has block-diagonal form.
5. Summary
Within the Gaussian approximation for the distribution of incident and final wave vectors of a spectrometer an exact formulation of the resolution function has been given, i.e. without linearization of energy conservation for neutrons. Inspection of the result shows that for both large and small m o m e n t u m transfers this linearization ( C & N ) gives a good approximation. Unfortunately C & N did not formulate their result in tensor notation. The disadvantages are less flexibility in the beam device arrangement and a less obvious transition to the 3 × 3 matrix Mi(M i + M f ) - l M t for small angle scattering. Between both limits of m o m e n t u m transfers there may arise situations (M+ anisotropic, M_ "large", anisotropic scattering function, etc.) where the exact formulation is useful, A computer program is available from the author for the determination of the various forms (R 2 to R4) of the resolution function in both the linearized and the non-linearized approximation.
m:0
The transformation for the horizontal part is
moo , = sTmin$ + dhC.
(6a)
N o rotation of matrices is necessary using the representation of s and c as given in table 1 below (see also for the remaining definitions). For the vertical part one gets P ( m 33)oot = (m33)iP. + dvP .
(6b)
Table 1 Definitions a) of matrices. Collimator
Mosaic crystal - 2lt
C (41n2/dh)l/2(k)
p (41n2/dv)]/2/(k)
1
ah 1 av
a) 1 = ( - )1 if the device deflects the beam to the (left) right. t = t a n O , s = s i n O , 0 = Bragg angle. ah, v = horizontal, vertical divergence (fwhm, arc). /7,,v = horizontal, vertical mosaic (fwhm, arc). A d/d = effective "spread" o f lattice spacing. ( k ) = m e a n w a v e vector at device. b) (m33)out ~ (m33)i n for this case.
-It ~h -1 2srtv
Perfect crystal
°t
1
-t
n
_ lt-I
Ad/d --lb) 0 b)
t 2
•
H. Grimm/Resolution function of crystal spectrometers Table 1 gives definitions of the matrices for c o m m o n beam devices. At the source and the detector one may start with M = (0) due to the " w h i t e " and isotropic emission and absorption of neutrons. Approaching the sample from both ends by successive application of the transformation (6) one creates M i,f in the (ki, f)-coordinate systems (1-axis parallel to ( k i ) and antiparallel to ( k f ) ) . Rotation of both matrices to a common coordinate system is of course assumed in the definition of M ±.
557
References [1] R. Stedman and G. Nilsson, Phys: Rev. 145 (1966) 492; M.J. Cooper and R. Nathans, Acta Cryst. 23 (1967) 357; M. Nielsen and H. Bjerrum Moiler, Acta Cryst. A25 (1969) 547. [2] B. Domer, Acta Cost. A28 (1972) 319; N.J. Chesser and J.D. Axe, Acta Cryst. A29 (1973) 160. [3] H. Grimm, Internal Report (ILL Grenoble) 76G198T (1976). [4] J.D. Axe, G. Shirane, T. Mizoguchi and K. Yamauchi, Phys. Rev. B15 (1977) 2763.