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The constant Q spectrometer resolution function
Nuclear Instruments and Methods 171 (1980) 115-125 © North-Holland Publishing Company
115
THE CONSTANT Q SPECTROMETER RESOLUTION FUNCTION C.G. WINDSOR and R.K. HEENAN * Materials Physics Division, A E R E Harwell, Oxon. 0 X l l ORA, UK
Received 15 October 1979
The formalism of Cooper and Nathans is used to discuss the resolution characteristics of the Constant Q Spectrometer for neutron inelastic scattering described in a previous paper. Expressions are derived for the parameters of the ellipsoid describing the resolution function. The theory is compared with measurements of the resolution function made on a prototype spectrometer on the old Harwell linac. An appendix derives the resolution function analytically for the case of a perfect analyser crystal.
1. Introduction The resolution function o f a neutron scattering spectrometer may be defined as the probability R ( Q , co) of measuring a neutron with m o m e n t u m Q and frequency co when the spectrometer has been set up to measure the desired values Qo, 6% [1]. The calculation of the resolution function is now a common practice, especially in triple-axis spectrometry, for deducing accurate line positions and widths from measured data and for evaluating the spectrometer configuration giving minimum instrumental broadening. The most widely used formalism has been that of Cooper and Nathans [1] who showed that, with the assumption that all deviations from exact geometry t o o k the form of Gaussian distributions, the resolution function o f a triple-axis spectrometer takes the form o f ellipsoidal contours with Gaussian sections. The surface with 50% o f the peak intensity they termed the resolution ellipsoid, which is conveniently described by the lengths o f its major axes and their spatial orientation. Computer programs based on this formalism have been in use for many years and there is now ample evidence that the predicted distributions are reproduced experimentally to at least a 5% accuracy [2]. The method has also been applied to a double-rotor spectrometer [3] and to a crystal monochromator, rotor spectrometer [4]. Here we apply the method to the constant Q time-of-flight spectrometer discussed in detail in a previous paper [5]. The spectrometer is sketched in fig. 1. It operates * Present Address: Chemistry Department, Reading University, Reading, RG6 2AD, UK.
by using the known angle of the analyser 0 A and its d spacing d A to calculate the scattered wave-vector kF from k F = rr/d A sin 0 A .
(1)
The scattered neutron flight time /'2 can than be calculated knowing the scattered neutron flight path L2
Moderator?
Time
Spreadrm
Collimation~1,/~1 Image
f'-~'~
I
\ I
Sample // ~ ~
"I I / ' \ Crystal ~ h m a t i o n
Analyser
ThicknesstA Mosaicr/A
RadiusR
rR~ Mosaic~1s
-'5
"~
~2flZ Collimation%'~3
Detector
"~Y~',L
Fig. 1. A Schematic diagram of the Constant Q Spectrometer showing the contributions to the resolution.
116
C.G. Windsor, R.K. tteenan / Constant Q spectrometer
from T: -
mL2_ mL2dA sin 0 A
hk F
(2)
h~
The time-of-flight spectrum gives the total flight time T = T~ + T2 so that the incident flight time T1 may be used to deduce the incident wave-vector k~ from mL 1
kl -
hT1
-
mL l hT-
(3)
m L 2 d A sin OA/n "
If k I and kv are defined as the incident and scattered wave-vectors with tile highest probability, the desired scattering vector and frequency (Qo, COo) will be given by the reciprocal lattice diagram of fig. 2 where
(4)
Q o = k l -- kV
and h =
-
(s)
The scattering angle 20 s is always equal to the counter angle 0 A in the constant Q geometry. The angle q5 in the scattering triangle is determined by the angle of the sample crystal to the incident beam.
2. Causes of imperfect resolution Figure 1 also suggests several causes of imperfect resolution. There will be other less probable neutron
Qy
~
qb*2o
11
wave-vectors k i and kf giving new values (Q, co) differing by small amounts AQ = Q Qo and Aw = co COo from the desired setting (Qo, COo). We consider the principle causes in turn, defining for each parameter, A, a Gaussian spread a such that the probability of observing a neutron with a value A + 6A is given by P(A + 6A) = P(A) e x p [ - ½ ( f A / a ) 2 ] .
(6)
a for a Gaussian distribution is equal to 1/2.354 of the full width at half height. 2.1. The moderator time broadening
We denote the difference in neutron production time from the time of maximum intensity by ATm and the corresponding Gaussian time spread by Tm. For a pulse source where the pulse widths are primarily determined by the neutron moderation time, rm will be generally inversely proportional to the incident wave-vector. For thin, poisoned or cooled moderators above the Maxwellian energy, "/'m is given approximately by 7"m = A / k I ; A = 44/Is A -
1
(7)
but the coefficient A rises to around twice this value in the Maxwellian region. The moderation pulse shape takes an asymmetric form with a Gaussian leading edge and centre and an exponential trailing edge [6, 7]. However the ratio of half-widths from the trailing and leading edges is only about two [6] so that the Gaussian approximation should be reasonable. For pulsed reactor sources and pulsed beams produced by chopping a steady beam with a rotor, the pulse spread would be more likely to be of Gaussian form with a spread Zm independent of wavelength. 2.2. The sample size time broadening
J2 t.z
For any scattering process at finite scattering angles of with inelasticity there will be a contribution to the time uncertainty AT s depending on the size and shape of the sample. We denote the corresponding broadening by rs. For a cylindrical sample of radius R we may evaluate AT s from a portion of the sample defined by polar co-ordinates (r, (p) as in fig. 1; - m r (cosq~ + c o s ( ~ + 20s) ) Ars(r, ~b) = ~ - - \ .ki kF .
AQzc Fig. 2. The scatteringdiagramin reciprocalspace showingthe three sets of axes usedin the computation.
(8)
We may now ask for the effective Gaussian spread having the same root-mean-square deviation as the
C. G. Windsor, R.K. Heenan / Constant Q spectrometer
broadening at the detector position rd will be the convolution of the two components
actual distribution Ts2
117
: fl-~2f f /"r:(r, *)r dr d¢ -~1
mR( 1 rs = ~ \ ~
1 + k2F
v'
2 cos 20 )1/2 k,k~
"
I-G2 +(mtd]271/2
L a
',WF/
_J
(11)
"
(9) 2.5. The angular collimations
Similar expressions may be derived for other geometries.
2.3. The analyser thickness time broadening Neutrons relfected from the front and back surfaces of the analyser crystal have different flight times depending on the analyser thickness t A and the scattering angle 0 A. The effective root-mean-square deviation being given by rA
1 msinO A tAkv
X/~/~
(10)
These are suggested in fig. 1 in the now conventional notation of Cooper and Nathans [ 1] : a for the effective Gaussian width of the horizontal collimations, /3 for the vertical collimations, region 1 up to the sample, region 2 between sample and analyser and region 3 between analyser and detector. In principle all these might be defined independently of other parameters by soller collimators. In practice it is more likely that in the present case they will be determined by the effective beam apertures. Thus al is determined by the moderator width Wro and the sample size Ws
In practice for the analyser in most common use pyrolytic graphite - this broadening is quite negligible since the thickness t A is only a few ram. We therefore ignore this effect, especially since the exact theory appreciably complicates the algebra and the effect may be included approximately by replacing r s by (r2s + T2A)1/2.
~, ~-- (W~ + W2s)l/2/L, •
2.4. The detector time broadening
In the case of a position sensitive detector of spatial resolution Wps set up parallel to the incident beam, the resolution would also be a function of the scattering angle 0 A = 20 s
Two effects must be considered. Firstly, the timeof-flight analysis is conventionally performed with rectangular time gates giving unit probability of detection within gates of width G d. The distribution in the detection time shift ATd may be approximated by a Gaussian spread X/~Gd having the same standard deviation. The replacement of a rectangular distribution by a Gaussian one is clearly a poor approximation, but the error is unlikely to be significant in practice since it is generally straightforward to select a gate width small compared with the other time broadenings. The second contribution to the time broadening is the effective counter thickness t d. This thickness will correspond to the actual thickness at short wavelengths when the counter has relatively low efficiency. For longer wavelengths when the counter is more nearly "black" there may be a shorter effective thickness. For a rectangular section the effective Gaussian time broadening will be x/q~v2mtd/hkF. For a circular counter of diameter D d, the effective Gaussian broadening will be ~mDd/hkv. The overall time
(12)
The scattered beam collimation is determined by the sample size and the image of the counter seen in the analyser crystal after broadening by the mosaic width
Wd ~ ~ (W: + W~)I/2/L2 .
a2 = (W2s + W~s sin2 20s)'121L2 .
(13)
(14)
Similar expressions apply to the vertical components 131 and t32 which can generally be rather more relaxed since they do not contribute to the energy uncertainty. The components a3 and/33 may be defined for a MARC analyser by "noses" extending in front of the counters.
2. 6. The analyser mosaic width Analyser crystals have a divergence in the direction of the normals to the scattering planes. We denote the effective Gaussian broadening by ~A. In general beam intensities increase as r?A is increased but at the expense of both Q and co resolution. It is generally sensible therefore to choose an analyser crystal with a mosaic comparable with the angular resolution a2.
C.G. Windsor,R.K. Heenan/ ConstantQ spectrometer
118
3. Derivation of the resolution function
where
We need to calculate the overall probability of detection of a neutron path in reciprocal space deviating from the most probable path. Referring again to fig. 2, consider a neutron whose wave vectors deviate by small components xl along ki, and y~, perpendicular to kb and by x2 along k F and by Y2 perpendicular to k F. A probability PT = Po exp(-~ ZT) from uncertainties in the timing is obtained by multiplying the probabilities from the broadenings rm, rs and rd from the moderator, sample and detector. For Gaussian broadenings we may add the arguments to give
Z T = AT~ _-27Tm
= (ATm}2 +(ATs}2 +(ATd) 2
,C-m,
\G-
(15) "
Neutron times at the sample AT s are a sum of the deviation in initial moderation time AT m and the deviation in time over the initial flight path L
ATI=
xl-
hk~ x l .
(16)
Similarly the deviation in arrival time at the detector AT d is a sum of ATm, AT I and a similar term for the deviation in flight time over the scattered flight path -m Z2 AT F = ~ kTX2.
(17)
Thus eqn. (1 5) reduces to a function of Akb Akv and ATm
\ "rrn
+2ATIATF
--
7-2 2 2
(21)
(22)
Thus we have reduced the expression for the probability of neutron transmission caused by time errors to a function of the deviations xl and x 2 in initial and final wave-vectors. We next consider the probability of transmission caused by the angular terms in the horizontal plane, Pa = Po exp(-~ Z~). The expression contains terms exactly analogous to those derived by Cooper and Nathans. They are expressed in terms of the deviations in wave-vector x 1, Y 1, x2 and Y2 by Y~ Y~ Z a = ,_-WU2+ 2-------7 klal kFa2
+(_x2tanOA-Y2) 2 (2x2tanOA-Y2) 2 kFT/A
+ \
~ F ~ -3
+ (ATm + ATI + ATF)
e = P}P~ = Po e x p ( - ~ Z ) where 2
2
+a2x2
2 2 2 2 + 2a~xlx2 +a4Yl +a6y2
We may now integrate over the unknown ATm, effectively over a neutron pulse, by using the standard formula [ 1],
mL1 r ( ~ s al hk~ Tm a2
1 ) 1/2
"r2d
mL2 r ( I 1 +_s1 hk2F Td \rm rs
' ~1/2
)
m T 1 a3 = , - 7 - 7 (L1L2) 1/2 - ;an kioq "/'dTm n/CIKF
+Bx+C)] dx (19)
as -
This gives for the new integrated probability a7 1 t P'(T) = f P(T)dTm =/'ot exp(--~ZT)
(24)
and (18)
"fd
= exp [--~(C - B 2/4A)] .
. (23)
The overall probability (excluding the vertical terms in qz) may therefore be expressed as a quadratic in the four unknowns x l , Y l , x2 andy2
+ (asx 2 + aTy 2)2 + (asx 2 + a9Y2 )2 2
+ _-~ Ts
and 1 1 1 1 - g 2+ _-27 + r-~d" T2 m Ts
Ts
exp[-~(Ax 2
+
Td Tm
2 2 Z=alXl
ZT=(AT____~m)2 + (f~TmLATI) 2
+
(20)
tan 0 A kF1/A 1 - kFr/A
1 a9 -
kF~3
; ,
a6 a8
1
kFa2 2 tan0 A kFa3
(25)
119
C G. Windsor, R.K. Heenan / Constant Q spectrometer
The next step in the calculation involves expressing three of the unknowns y~, x2 and Yz in terms of the chosen variables of the resolution function kQx, AQy and Aco. The algebra is long but identical with Cooper and Nathans [I] except that the incident wave-vector angle ¢ is now defined with respect to the crystal reciprocal lattice rather than with respect to Q. We also define ¢ and 20 as positive in the conventional anti-clockwise sense• Rotating the co-ordinate systems i~, 1~ clockwise by ~ to bring them into line with the reciprocal lattice co-ordinates, and similarly rotating i2,]2 by q5+ 20 s we find AQx = +xa cos q~ - Y l sin
-m
I cos 20 s
tz k v sin 20 s
/71 = 0 ,
h2 = 0 ,
-m
h4-
1
h kv
(26)
AQy = +x i sin q~+ y i cos ~b
We now substitute eqs. (29) within eq. (24) and thus derive the overall transmission probability as a quadratic function of the one unknown xa, and AQx, AQy and/xco. It may therefore be written Z=A'x~ +B'xl +C'
(31)
where
B ' = 2 [bx CD + b2EF + b3GH + b4(GF + E H ) + b 6 H ] , C' = b i D 2 + b2F 2 + b3H 2 + 2b4FH,
- x z sin(~b + 20s) - Y z cos(4~ + 20s).
(27)
The frequency shift is given by (28)
It is straightforward to invert these three equations to give expressions for Yl, Y2 and x2 in terms of the remaining unknown Xl and the new variables AQx, AQy and Aco. Following Cooper and Nathans [ 1] put
(32)
where bl = a ~ ,
A ~ = (h/m) ( x l k I - X 2 k F ) .
b2 : a 2 + a g + a g ,
b4=asa7+a8ag,
bs = a ~ ,
b3 : a ~ +a~ + a ~ , b6=a~.
(33)
Equation (19) may now be used to integrate over xa and produce an expression for neutron probability in terms of AQx, AQy and Aw. We now write this as 1.
RH
t
exp(-~Z ) = exp(-~
y~ = Cx~ +D ;
~
MklXkXl)
(34)
k,l= 1,2,4
where
D = dl AQx + d2 A Q y + d4 A w ,
(35)
Z ' = C' - B'2 /4A '
Y2 =Ex1 + F ;
and Xk denotes the four component vector AQx, AQy, AQz , Aco. Comparing the terms on the two sides of eq (34) shows that for k, l = 1, 2 or 4
F = f l AQx +f2 AQy + f4 Aco , x2 = Gx~ + H ; H = hi AQx + h2 A Q y + h4 Aco ,
when we may deduce
(29)
Mkl = godkdt + gafkfl + gz hkh 1
+ 1 [g3 (fkht + flhk ) + g4 (dkft + dtfk) (36)
+ gs (dkht + dthk)]
C=(~-~lF
(30)
A ' = bl C 2 + b z E 2 + b3 G2 + 2b4GE + bs + 2b6G ,
xz cos(q~ + 20s) +Y2 sin(~ + 20s)
-
f4-
cos 20s)/sin 20 s, where go = bl - b~ CZ /A ' ,
E=
kFC°S20 s-1
/sin20s,
g, = b2 - (b2E + b4G)2 /A ' ,
G-k~ g2 = b3 - (b3G + b4E + b6)2/A ' ,
dl
COS(~b+ 20s), sin 20 s
-
-m 6 4 ~-/~
d2 - sin(~ + 20s) sin 20 s
1 k F sin 20 s '
fl = cos C/sin 20 s , f2 = sin C/sin 20 s ,
g3 = 2b4 - 2(b2E + b4G) (b3G + b4E + b6)/A' , (37) g4 = - 2 b l C ( b z E + b4G)/A' , gs = -2b~ C(b3G + b4E + b6)/A' .
(37)
The vertical terms in the resolution are very easily evaluated as they have no correlations with AQx , AQy A(.o so that the remaining elements M m are or
C.G. Windsor,R.K. Heenan /Constant Q spectrometer
120
zero except for M33. The theory is identical to that of Cooper and Nathans [1] as later corrected by Dorner [8] with the monochromator and "in pile" terms deleted. R v : e x p ( - I M 3 3X~ ) where 2 2 alOall
M33 - a20 + a211 and
alo = 1/{31kl and a,,-
_(
1 +(4sin2OA~2A+(J~)k~]l
~1/2
Thus all the elements of the resolution matrix Mkl may be calculated. Since it is symmetric, it has 7 distinct non-zero elements. These are most graphically presented by the four principle axes and the three orientation angles of the resolution ellipsoid. These quantities were evaluated by a FORTRAN computer code written by the authors.
4. Measurement of the elastic resolution function In the special case of elastic scattering it is straightforward to measure the resolution function and so check our computations. This is because the Bragg scattering from a single crystal provides a good approximation to a 6 function in both Q and co so that the measured scattering gives the resolution function R(Q, co) directly. The experiments were performed using the prototype constant Q spectrometer described in our earlier paper [5]. A large iron single crystal of mosaic spread of order 1° was used as the sample. Studies were made around the 200 lattice point which corresponds to a scattering vector Q = 4.392 A -~. The graphite 004 plane having a d spacing d A = 0.838 ,~ was used as analyser leading to an incident wave-vector k I = 4.25 A -1, and the scattering diagram shown in fig. 3. Some 41 time-of-flight spectra were recorded at the calculated scattering angle 20 s = 62.9 °, and at A20 s = -+0.5°, +1 °, -+2° and +3 °. At each scattering angle around five spectra were recorded at crystal angles ~b varying by -+1° or +-2° from the angle where maximum intensity was observed. The spectra were of varying duration from one minute to one hour but
Oy(~, -1 )
I = z, 25
.&-~
/ e =62 87
\I.,\ \~\~
\kf\\x Fig. 3. The reciprocal lattice diagram corresponding to the experimental measurement of the resolution function. The measured contours at h ~ = 0.0 and 0.8 m e V are shown. The inset shows the area near the reciprocal latticepoint magnified 12~ times.
C G. Windsor, R.K. Heenan / Constant Q spectrometer
were all normalized by a fission chamber in the incident beam. Count rates were often low because the neutron beam available was only the straight-through beam of another spectrometer. For each time channel of each spectrum the values of Q and co were calculated together with a normalized intensity R and its statistical error ~2xR. The resulting set of some 1253 values of R(Q, co) can be presented in various ways. For example in fig. 4 we show sections through the resolution function as contour maps of R(Q, w) for two relatively narrow energy bands (a)hey = (0-+ 0.1) meV and (b) h a ) = ( 0 . 8 + 0.1) meV. At the expense of an even greater restriction in the amount of data presented we may represent, as in fig. 5, some principle lines through the ellipsoid. These curves confirm that the resolution function has the nearly Gaussian form we have postulated. We performed a least squares fit describing the measured resolution function in the form given by eq. 34. Ten adjustable parameters were used; the first six representing the six inequivalent parameters of the (Qx, Qy, w) section of the resolution ellipsoid, the seventh the normalization and the remaining three small shifts in the centre of the ellipsoid with respect to the initially assumed centre of the Bragg reciprocal lattice point. The
100
i
i
50
'/
f,, Z' 100
-005
0
121 I
i
[
i
"~1 ~"
0"05 -0'1 qx 1~-11
i
I
~'~
0
I
0"1 qy (A-1)"
. "
"E
5O
0 !
-1
I • *
0 hw ( m e w
Fig. 5. Sections through the resolution ellipsoid along the
qx, qy and co axes. The full line shows the best-fitting ellipsoid and the dashed line the computed ellipsoid.
w 5
19
l~"
14
z4 / ~
31
11
12 9
,
:,%
~ -0"1
I
7
Ji
0"1 1 qx(]C )
77\
9
26 i
-01
15
6
45
27
9
4~
35
I
'k
6
0!I ,/~-1, qx t I
9
5~ -0'1
i\';'i-.".", '~44~5J
8
42
h2 t5~, ?,-' \ zo
_'.,
5 11
5
t : 5 0,
01
1 10
",,19 ~
\ -,, ~,i',9~_ \ \ p4 4
,
~\\[ 1519
6
18
14
7
21 25
Fig. 4. Contour map of the scattering intensity in the qx, qy plane within the energy range: (a) (0 -+ 0.1) meV; (b) (0.8 ± 0.1) rneV. The full line is the best fitted ellipsoid and the dashed line the best calculated ellipsoid.
122
C.G. Windsor, R.K. Heenan / Constant Q spectrometer
Table 1 Parameters of the resolution ellipsoids obtained by fitting to experiment and by calculation, and the measured and fitted instrumental parameters Experiment Ellipsoid elements Ml 1 (A2) M12 (A2) M14 (A meV -1) M22 (A2) 3/24 (A meV-1) M44 (meV -2) Principle axis L 1 (A, meV units) L2 L4 Slope X44/(X214 + X24 ) (meV A) Angle of longest axis on XY plane tan -I (X24/X14) (°) Angle of longest axis of energy section X Y plane tan-1 (Xz2/XI2) (o) Instrumental parameters A (Us A-1) c~1 (tad) ¢x2 ce3 r~A r~s
3544 851 -114.5 322 -14.3 6.49
Calculation
-+21 2643 -+ 6 1654 -+ 0.6 109 • 2 1208 -+ 0.2 91.7 • 0.04 8.80
0.0192 0.111 1.103
0.0193 0.105 1.113
7.78
7.01
117.6
107.8
103.7
123.6
100 0.010 0.027 0.027 0.014 0.001
99.5 0.001 0.023 0.020 0.029 0.003
parameters are given in table 1, and the solid lines in figs. 4 and 5 shows the fit obtained. In fig. 3 we have drawn some contours of the resolution function on the scattering diagram. The longest axis of the ellipsoid lies close to the incident wavevector ki, and the long axis of the constant energy sections also lies approximately along k I. The resolution function thus has a very narrow width perpendicular to k I - a shape unfamiliar in triple axis spectrometry. A numerical diagonalization of the measured resolution ellipsoid is given in table 1 showing the lengths of the three principle axes, the slope of the longest axis, the direction of the longest axis in the X Y plane, and the direction of the longer axis of the constant energy sections in the X Y plane. 5. Comparison of the measured resolution function with calculation The formalism developed in paragraph 3 was used to calculate a resolution ellipsoid to compare with the
measured ellipsoid. Since many of the instrumental parameters were not well known some fitting was performed to obtain the angular divergences. The results of this procedure are shown in the right hand side of table 1, in fig. 6a and by the dashed lines in figs. 4 and 5. The three principle axes, and the slope and direction of the longest axis are all well given. There is however a discrepancy in the direction of the longer axis in the constant energy sections. The fitted instrumental parameters are given in table 1. They agree moderately well with values measured independently from the neutron pulse length, crystal rocking curves and from geometrical considerations. The calculated incident beam collimation c~1 is certainly less than would have been expected from geometrical considerations. This might have arisen from some miss-alignment of the spectrometer. The more important angular collimations of the scattered beam are well fitted by our calculation. In order to understand the relative importance of the various terms in the expression for the resolution, model calculations were performed, some of the results being shown in fig. 6. Section (b) represents a simplified configuration with negligible sample size, detector width and scattered neutron flight path L2. Comparison with our original calculation (a) shows that these terms have a negligible effect in our configuration. Section (c) shows that the effect of sample mosaic r~s is to rotate the ellipsoid towards the qy axis. (The change in al had a small effect.) Section (e) shows a highly simplified ellipsoid which still presents a useful approximation to the observed ellipsoid. r/a has been put equal to zero (a3 is then irrelevant), and the effect is to make all broadenings perpendicular to k0 vanish. The parameters of this ellipsoid may be evaluated analytically. Lastly in sections (d) and (f) we show two effects which worsen the resolution. In (d) the analyser is set to scatter in the same sense as the sample (FOC = +1). In (f) we have increased c~ causing the major broadening to be perpendicular to k0 rather than along it. In conclusion it is worth recalling that the ellipsoids in fig. 6 show a qualitative difference from triple-axis resolution sections which tend to lie along qx at a given energy transfer and slope towards qy as the energy increases. The fact that with good incident collimation but relaxed scattered collimation, as in section (c), the ellipsoid lies along a given line could be remarkably useful for focussing certain excitations. The authors are grateful to Martyn Cooper for
C.G. Windsor, R.K. Heenan / Constant Q spectrometer
123
0.1
0"1 - ~
(c).rts,--,,O. ~m-,,O {
N
\ ~ N" ~ 0
\\ \
",,
\
\k 'Xx \
T
[
\
---\\ -
0"II
\
'lhw=08meV
(d)~="FOC 1 "<' \x \\:\\"~\"\\
\ \
~
I
~
I
0"1-I} qx(~
L
"x
"" "2?,,
\, \\\ \\ \ 0IF-~--x\,, \ \ ~
01 _ PJ,, q,(~-l)
0!1
qx(~_iI
~(el°~A " "\\ "-" "xj 0\\''~c(3"--~1~
I
k\
~x(°~1-, _ PJ,,
/
/ /
\ -\,,
/
\\ Fig. 6. Some typical 0.0 and 0.8 meV sections of the resolution ellipsoid: (a) is the ellipsoid whose parameters are given in table 1; (b) is a simplified version having L 2 = 0, R = 0, al = 0, ~2 = a3 = 0.02, 77A = 0.03, r/s = 0.003. (c) is the simplified fit with r/s and a 1 put to zero. (d) is the full fit with FOC = +1, (e) is as (c) but with a3 large, and r/A put to zero. (f) is the simplified fit with al = 0.03 and */s and r/A put to zero.
several discussions and comments, and to Roger Sinclair for his perserverance in questioning which lead to the successful check at the end of the appendix.
cally. Differentiating the expression for Bragg's law gives AkF = cot OAk F A0A • In our notation A k F = x 2 and kF A0A =Y2, so that
Appendix: a Simplified theory for a perfect analyser The general theory simplifies considerably for the practical case of importance when the analyser crystal has a low mosaic angle compared with the scattered collimation (r/A ~ 0). The scattered wave-vector deviations along and perpendicular to kv, x2 and Y2 are then coupled together by Bragg's law. There are only three independent variables x l , y l , and say x2 which may be expressed in terms of the deviations AQx , AQy and /xw. In the special case of the Constant Q geometry the resolution ellipsoid has principle axes whose lengths and slopes may be calculated analyti-
Y2 =x2 tan 0 A .
(A1)
Referring to eq. (23) we see that this condition removes t h e third term which the condition r/A ~ 0, would cause to diverge. If the analyser is perfect we may, without loss of generality, put the counter collimation a3 = ~o so that the last term in eq. (23) also vanishes. We also suppose that the three timing errors rm, 7"s and rd from moderator, sample and detector respectively may be convoluted to give a single overall timing error r. We then have no need for the complicated discussion leading to eq. (21). We denote the probability of counting a neutron having incident and
C.G. Windsor, R.K. Heenan / Constant Q spectrometer
124
scattered wave-vector changes xx and x2 as PT = exp(~ZT) where,
al
(alxl
+ a2x2) 2
(A2)
roLl 1 mL2 1 - and a2 hk~ r hk~F r
(A3)
ZT =
=--
where G = kl/k v as before and H = - m w / h k v . Then
AQx
-Yl
2 2 2 2 Z = ( a l x 2 + a 2 x 2 ) 2 + a4Yl + a 6 y 2
(A4)
where 1
a4 =
0A)
sin 4) - H cos 4)/cos 0A ,
H sin 4)/cos 0 A .
(A9)
We now solve f o r x l a n d y l to give XI
= (AQ x cos 4) + AQy sin 4) + / / / c o s
0A)/(1
Yl = - A Q x sin 4) + AQy cos 4) .
kF°~2
This is a function of only three variables Xl, x2 and y~ and so may be expressed directly in terms of the three variables AQx , AQy and co. Using condition A1 the eqs. (26) and (27) may be written
AQ± = - A Q x sin 4) + AQy cos 4),
AQ x = x 1 cos 4) - y 1 sin 4)
x t = (AQ//+ H/cos 0A)/(1 -- G/cos 0 A ) ,
tan 0 A cos(4)+ 20s).
AQ//= AQx cos 4) + AQy sin 4) , (A11)
then
(A12)
The equation for x2 is equally simple
AQy = x 1 sin 4) + y i cos 4) - x2 sin(4) + 20s) (A5)
G AQ//+ GH/cos Oa +H(1 - G/cos 0A) X2 =
These may be written
AQ x = x l c o s 4 ) - y l
(A10)
Yx = AQ± .
- x2 cos(4) + 20s) + x2 tan 0 A sin(4) + 20s)
-x2
G/cos 0A)
--
We now see that appropriate axes for the resolution ellipsoid in the Qx, Qy plane are along k I at an angle 4) to Qx. We define as in fig. 7
1
kict I and t/6
COS 4)(1 -- G/cos
AQy =xl sin 4)(1 - G/cos 0 a ) + y l c o s 4) --
The full expression for the probability of counting a neutron is then the product of PT and the remaining angular terms P~ to give an overall probability P = exP(½Z) where
=X 1
(1 - G/cos 0 A )
G AQ//+H sin4)---
X2
COS 0 A
cos(4) + 20 s + 0A)
- 1 - G/cos 0 A '
(AI3)
Equation (A4) then becomes X2
AQy = xx sin 4) +Yl cos 4) - - -
cos 0 A
sin(4) + 20 s + 0 A ) . (A6)
We now suppose the constant Q geometry 0 A = - 2 0 s. This causes the terms in xz to have the same angular dependence as those containing x~. Physically the wave-vector change from the analyser acceptance is perpendicular to the analyser d spacing, and so parallel to the incident wave-vector.
Z = [aI(AQ//+ H/cos 0A) + a2(G AQ//+ 14)12/r2 + a62 tan20A(G AQII +H)2/Y 2 + a] AQ~, where Y = 1 - G/cos 0 A .
Z = MIx AQ] (A7)
(A15)
Expressing the resolution function as the general quadratic in AQ//, AQI and H,
AQx = cos 4)(xl - x 2 / c o s 0A) --Yx sin 4), AQy = sin 4)(xl - x2/cos 0A) --Yx Cos 4).
(A14)
+
2Mr4 AQIIH + M44H 2 + M22 AQj2 , (A16)
we have M l l = [(al +a2G)2 + a 2 t a n 20A G2]/Y2 ,
The two uncertainties x~ and x 2 are coupled by the energy transfer
ktx l X2 -
kF
m Aw hkv -=Gxl +H
M14 = [(al +a2G)(al/cOSOA +a2)+a~ tanEOAG]/r: M44 = [(al/cos 0 A + a2)2 + a~ tan z 0A] / I(2 ,
(A8)
M22 = a~ .
(A17)
C.G. Windsor, R.K. Heenan / Constant Q spectrometer
125
energy of a Bragg reflection) is
4/
[1 - kt/(k F cos 0 A ) ]
W=2.354--
// > Oy
(A22) The full width at half height of the whole ellipsoid, irrespective of Q, (i.e., the width of a vanadium scan) may be found by differentiating Z with respect to Q. This gives
W=2.354(
M_M_x~_
),/2
\M1 IM44 -- M 2 4
¢ ..,~ ~ . . / - ~ o~,
o; :2.,o~
~
= 2.354/~kF( [1 +(a2/a~)G] 2 G2] 112 m a62 tan20A + a~ !
o.
~
-..-
= 2.354 ~-m
/
1 + -- ~ ~2k4 Lx k F) 2 F c o t 2 0 A + \ m L ~ ]
/ /
(A23)
Fig. 7. The resolution ellipsoid in the special case treated in the appendix having contributions only from the angular divergences cq, a 2 and the moderator time spread, r m. The first principle axis of the ellipsoid has a projection on the Qx, Qy plane which lies along k I.
It is instructive to check this equation by the simple method of evaluating the uncertainty in the energy transfer given by eqs. (1), (3) and (5).
w=
f i T - mL2d A sin 0A/n
d A sin 0 A
(A24) These four equations thus determine the resolution ellipsoid in this special case. The ellipsoid is already diagonal in the AQ± axis, and has a principle axis full width at half height W± -
2.354 a4
AO) = m \
{k s L2 cot 0 A I ~1
kF
)
+ k} cot 0A _ A0 a .
(A25)
Differentiating with respect to T gives - 2.354 kith .
(AI8)
In the AQ//, Ac~ plane, the ellipse may be diagonalized by rotation by an angle ~ where
tan 2 ~
Differentiating this respect to 0 A gives
2/]414 MI 1 - M44
Aw
= (h \mII ~-~IAT.
(A26)
Putting A0 A ~-- a2 and A T ~ r confirms eq. (A23) for the overall width.
(A19)
References the principle axes lengths are then 2.354 W1 - - (Mll cos2ff +M14 sin 2ff +M44 sin 2 ~))112 (A20)
W2-
2.354 (M11 sin 2 ff - M14 sin 2ff + M44 COS2 ~)1[2
(a21) The full width at half height of the section of the ellipsoid along the energy axis, (i.e., the width in
[1] M.J. Cooper and R. Nathans, Acta. Cryst. 23 (1967) 357. [2] B.C. Haywood, Acta. Cryst. A27 (1971) 408. f3J S. Komura and M.J. Cooper, Jap. J. Appl. Phys. 9 (1970)
866. [4] O. Steinsvoll, Nucl. Instr. and Meth. 106 (1973) 453. [5] C.G. Windsor, R.K. Heenan, B. Boland and D. Mildner, Nucl. Instr. and Meth. 151 (1978) 477. [6] D.H. Day and R.N. Sinclair, Nucl. Instr. and Meth. 72 (1969) 237. [7] C.G. Windsor and R.N. Sinclair, Acta. Cryst. A32 (1976) 395. [8] B. Dorner, Acta. Cryst. A28 (1972) 319,
115
THE CONSTANT Q SPECTROMETER RESOLUTION FUNCTION C.G. WINDSOR and R.K. HEENAN * Materials Physics Division, A E R E Harwell, Oxon. 0 X l l ORA, UK
Received 15 October 1979
The formalism of Cooper and Nathans is used to discuss the resolution characteristics of the Constant Q Spectrometer for neutron inelastic scattering described in a previous paper. Expressions are derived for the parameters of the ellipsoid describing the resolution function. The theory is compared with measurements of the resolution function made on a prototype spectrometer on the old Harwell linac. An appendix derives the resolution function analytically for the case of a perfect analyser crystal.
1. Introduction The resolution function o f a neutron scattering spectrometer may be defined as the probability R ( Q , co) of measuring a neutron with m o m e n t u m Q and frequency co when the spectrometer has been set up to measure the desired values Qo, 6% [1]. The calculation of the resolution function is now a common practice, especially in triple-axis spectrometry, for deducing accurate line positions and widths from measured data and for evaluating the spectrometer configuration giving minimum instrumental broadening. The most widely used formalism has been that of Cooper and Nathans [1] who showed that, with the assumption that all deviations from exact geometry t o o k the form of Gaussian distributions, the resolution function o f a triple-axis spectrometer takes the form o f ellipsoidal contours with Gaussian sections. The surface with 50% o f the peak intensity they termed the resolution ellipsoid, which is conveniently described by the lengths o f its major axes and their spatial orientation. Computer programs based on this formalism have been in use for many years and there is now ample evidence that the predicted distributions are reproduced experimentally to at least a 5% accuracy [2]. The method has also been applied to a double-rotor spectrometer [3] and to a crystal monochromator, rotor spectrometer [4]. Here we apply the method to the constant Q time-of-flight spectrometer discussed in detail in a previous paper [5]. The spectrometer is sketched in fig. 1. It operates * Present Address: Chemistry Department, Reading University, Reading, RG6 2AD, UK.
by using the known angle of the analyser 0 A and its d spacing d A to calculate the scattered wave-vector kF from k F = rr/d A sin 0 A .
(1)
The scattered neutron flight time /'2 can than be calculated knowing the scattered neutron flight path L2
Moderator?
Time
Spreadrm
Collimation~1,/~1 Image
f'-~'~
I
\ I
Sample // ~ ~
"I I / ' \ Crystal ~ h m a t i o n
Analyser
ThicknesstA Mosaicr/A
RadiusR
rR~ Mosaic~1s
-'5
"~
~2flZ Collimation%'~3
Detector
"~Y~',L
Fig. 1. A Schematic diagram of the Constant Q Spectrometer showing the contributions to the resolution.
116
C.G. Windsor, R.K. tteenan / Constant Q spectrometer
from T: -
mL2_ mL2dA sin 0 A
hk F
(2)
h~
The time-of-flight spectrum gives the total flight time T = T~ + T2 so that the incident flight time T1 may be used to deduce the incident wave-vector k~ from mL 1
kl -
hT1
-
mL l hT-
(3)
m L 2 d A sin OA/n "
If k I and kv are defined as the incident and scattered wave-vectors with tile highest probability, the desired scattering vector and frequency (Qo, COo) will be given by the reciprocal lattice diagram of fig. 2 where
(4)
Q o = k l -- kV
and h =
-
(s)
The scattering angle 20 s is always equal to the counter angle 0 A in the constant Q geometry. The angle q5 in the scattering triangle is determined by the angle of the sample crystal to the incident beam.
2. Causes of imperfect resolution Figure 1 also suggests several causes of imperfect resolution. There will be other less probable neutron
Qy
~
qb*2o
11
wave-vectors k i and kf giving new values (Q, co) differing by small amounts AQ = Q Qo and Aw = co COo from the desired setting (Qo, COo). We consider the principle causes in turn, defining for each parameter, A, a Gaussian spread a such that the probability of observing a neutron with a value A + 6A is given by P(A + 6A) = P(A) e x p [ - ½ ( f A / a ) 2 ] .
(6)
a for a Gaussian distribution is equal to 1/2.354 of the full width at half height. 2.1. The moderator time broadening
We denote the difference in neutron production time from the time of maximum intensity by ATm and the corresponding Gaussian time spread by Tm. For a pulse source where the pulse widths are primarily determined by the neutron moderation time, rm will be generally inversely proportional to the incident wave-vector. For thin, poisoned or cooled moderators above the Maxwellian energy, "/'m is given approximately by 7"m = A / k I ; A = 44/Is A -
1
(7)
but the coefficient A rises to around twice this value in the Maxwellian region. The moderation pulse shape takes an asymmetric form with a Gaussian leading edge and centre and an exponential trailing edge [6, 7]. However the ratio of half-widths from the trailing and leading edges is only about two [6] so that the Gaussian approximation should be reasonable. For pulsed reactor sources and pulsed beams produced by chopping a steady beam with a rotor, the pulse spread would be more likely to be of Gaussian form with a spread Zm independent of wavelength. 2.2. The sample size time broadening
J2 t.z
For any scattering process at finite scattering angles of with inelasticity there will be a contribution to the time uncertainty AT s depending on the size and shape of the sample. We denote the corresponding broadening by rs. For a cylindrical sample of radius R we may evaluate AT s from a portion of the sample defined by polar co-ordinates (r, (p) as in fig. 1; - m r (cosq~ + c o s ( ~ + 20s) ) Ars(r, ~b) = ~ - - \ .ki kF .
AQzc Fig. 2. The scatteringdiagramin reciprocalspace showingthe three sets of axes usedin the computation.
(8)
We may now ask for the effective Gaussian spread having the same root-mean-square deviation as the
C. G. Windsor, R.K. Heenan / Constant Q spectrometer
broadening at the detector position rd will be the convolution of the two components
actual distribution Ts2
117
: fl-~2f f /"r:(r, *)r dr d¢ -~1
mR( 1 rs = ~ \ ~
1 + k2F
v'
2 cos 20 )1/2 k,k~
"
I-G2 +(mtd]271/2
L a
',WF/
_J
(11)
"
(9) 2.5. The angular collimations
Similar expressions may be derived for other geometries.
2.3. The analyser thickness time broadening Neutrons relfected from the front and back surfaces of the analyser crystal have different flight times depending on the analyser thickness t A and the scattering angle 0 A. The effective root-mean-square deviation being given by rA
1 msinO A tAkv
X/~/~
(10)
These are suggested in fig. 1 in the now conventional notation of Cooper and Nathans [ 1] : a for the effective Gaussian width of the horizontal collimations, /3 for the vertical collimations, region 1 up to the sample, region 2 between sample and analyser and region 3 between analyser and detector. In principle all these might be defined independently of other parameters by soller collimators. In practice it is more likely that in the present case they will be determined by the effective beam apertures. Thus al is determined by the moderator width Wro and the sample size Ws
In practice for the analyser in most common use pyrolytic graphite - this broadening is quite negligible since the thickness t A is only a few ram. We therefore ignore this effect, especially since the exact theory appreciably complicates the algebra and the effect may be included approximately by replacing r s by (r2s + T2A)1/2.
~, ~-- (W~ + W2s)l/2/L, •
2.4. The detector time broadening
In the case of a position sensitive detector of spatial resolution Wps set up parallel to the incident beam, the resolution would also be a function of the scattering angle 0 A = 20 s
Two effects must be considered. Firstly, the timeof-flight analysis is conventionally performed with rectangular time gates giving unit probability of detection within gates of width G d. The distribution in the detection time shift ATd may be approximated by a Gaussian spread X/~Gd having the same standard deviation. The replacement of a rectangular distribution by a Gaussian one is clearly a poor approximation, but the error is unlikely to be significant in practice since it is generally straightforward to select a gate width small compared with the other time broadenings. The second contribution to the time broadening is the effective counter thickness t d. This thickness will correspond to the actual thickness at short wavelengths when the counter has relatively low efficiency. For longer wavelengths when the counter is more nearly "black" there may be a shorter effective thickness. For a rectangular section the effective Gaussian time broadening will be x/q~v2mtd/hkF. For a circular counter of diameter D d, the effective Gaussian broadening will be ~mDd/hkv. The overall time
(12)
The scattered beam collimation is determined by the sample size and the image of the counter seen in the analyser crystal after broadening by the mosaic width
Wd ~ ~ (W: + W~)I/2/L2 .
a2 = (W2s + W~s sin2 20s)'121L2 .
(13)
(14)
Similar expressions apply to the vertical components 131 and t32 which can generally be rather more relaxed since they do not contribute to the energy uncertainty. The components a3 and/33 may be defined for a MARC analyser by "noses" extending in front of the counters.
2. 6. The analyser mosaic width Analyser crystals have a divergence in the direction of the normals to the scattering planes. We denote the effective Gaussian broadening by ~A. In general beam intensities increase as r?A is increased but at the expense of both Q and co resolution. It is generally sensible therefore to choose an analyser crystal with a mosaic comparable with the angular resolution a2.
C.G. Windsor,R.K. Heenan/ ConstantQ spectrometer
118
3. Derivation of the resolution function
where
We need to calculate the overall probability of detection of a neutron path in reciprocal space deviating from the most probable path. Referring again to fig. 2, consider a neutron whose wave vectors deviate by small components xl along ki, and y~, perpendicular to kb and by x2 along k F and by Y2 perpendicular to k F. A probability PT = Po exp(-~ ZT) from uncertainties in the timing is obtained by multiplying the probabilities from the broadenings rm, rs and rd from the moderator, sample and detector. For Gaussian broadenings we may add the arguments to give
Z T = AT~ _-27Tm
= (ATm}2 +(ATs}2 +(ATd) 2
,C-m,
\G-
(15) "
Neutron times at the sample AT s are a sum of the deviation in initial moderation time AT m and the deviation in time over the initial flight path L
ATI=
xl-
hk~ x l .
(16)
Similarly the deviation in arrival time at the detector AT d is a sum of ATm, AT I and a similar term for the deviation in flight time over the scattered flight path -m Z2 AT F = ~ kTX2.
(17)
Thus eqn. (1 5) reduces to a function of Akb Akv and ATm
\ "rrn
+2ATIATF
--
7-2 2 2
(21)
(22)
Thus we have reduced the expression for the probability of neutron transmission caused by time errors to a function of the deviations xl and x 2 in initial and final wave-vectors. We next consider the probability of transmission caused by the angular terms in the horizontal plane, Pa = Po exp(-~ Z~). The expression contains terms exactly analogous to those derived by Cooper and Nathans. They are expressed in terms of the deviations in wave-vector x 1, Y 1, x2 and Y2 by Y~ Y~ Z a = ,_-WU2+ 2-------7 klal kFa2
+(_x2tanOA-Y2) 2 (2x2tanOA-Y2) 2 kFT/A
+ \
~ F ~ -3
+ (ATm + ATI + ATF)
e = P}P~ = Po e x p ( - ~ Z ) where 2
2
+a2x2
2 2 2 2 + 2a~xlx2 +a4Yl +a6y2
We may now integrate over the unknown ATm, effectively over a neutron pulse, by using the standard formula [ 1],
mL1 r ( ~ s al hk~ Tm a2
1 ) 1/2
"r2d
mL2 r ( I 1 +_s1 hk2F Td \rm rs
' ~1/2
)
m T 1 a3 = , - 7 - 7 (L1L2) 1/2 - ;an kioq "/'dTm n/CIKF
+Bx+C)] dx (19)
as -
This gives for the new integrated probability a7 1 t P'(T) = f P(T)dTm =/'ot exp(--~ZT)
(24)
and (18)
"fd
= exp [--~(C - B 2/4A)] .
. (23)
The overall probability (excluding the vertical terms in qz) may therefore be expressed as a quadratic in the four unknowns x l , Y l , x2 andy2
+ (asx 2 + aTy 2)2 + (asx 2 + a9Y2 )2 2
+ _-~ Ts
and 1 1 1 1 - g 2+ _-27 + r-~d" T2 m Ts
Ts
exp[-~(Ax 2
+
Td Tm
2 2 Z=alXl
ZT=(AT____~m)2 + (f~TmLATI) 2
+
(20)
tan 0 A kF1/A 1 - kFr/A
1 a9 -
kF~3
; ,
a6 a8
1
kFa2 2 tan0 A kFa3
(25)
119
C G. Windsor, R.K. Heenan / Constant Q spectrometer
The next step in the calculation involves expressing three of the unknowns y~, x2 and Yz in terms of the chosen variables of the resolution function kQx, AQy and Aco. The algebra is long but identical with Cooper and Nathans [I] except that the incident wave-vector angle ¢ is now defined with respect to the crystal reciprocal lattice rather than with respect to Q. We also define ¢ and 20 as positive in the conventional anti-clockwise sense• Rotating the co-ordinate systems i~, 1~ clockwise by ~ to bring them into line with the reciprocal lattice co-ordinates, and similarly rotating i2,]2 by q5+ 20 s we find AQx = +xa cos q~ - Y l sin
-m
I cos 20 s
tz k v sin 20 s
/71 = 0 ,
h2 = 0 ,
-m
h4-
1
h kv
(26)
AQy = +x i sin q~+ y i cos ~b
We now substitute eqs. (29) within eq. (24) and thus derive the overall transmission probability as a quadratic function of the one unknown xa, and AQx, AQy and/xco. It may therefore be written Z=A'x~ +B'xl +C'
(31)
where
B ' = 2 [bx CD + b2EF + b3GH + b4(GF + E H ) + b 6 H ] , C' = b i D 2 + b2F 2 + b3H 2 + 2b4FH,
- x z sin(~b + 20s) - Y z cos(4~ + 20s).
(27)
The frequency shift is given by (28)
It is straightforward to invert these three equations to give expressions for Yl, Y2 and x2 in terms of the remaining unknown Xl and the new variables AQx, AQy and Aco. Following Cooper and Nathans [ 1] put
(32)
where bl = a ~ ,
A ~ = (h/m) ( x l k I - X 2 k F ) .
b2 : a 2 + a g + a g ,
b4=asa7+a8ag,
bs = a ~ ,
b3 : a ~ +a~ + a ~ , b6=a~.
(33)
Equation (19) may now be used to integrate over xa and produce an expression for neutron probability in terms of AQx, AQy and Aw. We now write this as 1.
RH
t
exp(-~Z ) = exp(-~
y~ = Cx~ +D ;
~
MklXkXl)
(34)
k,l= 1,2,4
where
D = dl AQx + d2 A Q y + d4 A w ,
(35)
Z ' = C' - B'2 /4A '
Y2 =Ex1 + F ;
and Xk denotes the four component vector AQx, AQy, AQz , Aco. Comparing the terms on the two sides of eq (34) shows that for k, l = 1, 2 or 4
F = f l AQx +f2 AQy + f4 Aco , x2 = Gx~ + H ; H = hi AQx + h2 A Q y + h4 Aco ,
when we may deduce
(29)
Mkl = godkdt + gafkfl + gz hkh 1
+ 1 [g3 (fkht + flhk ) + g4 (dkft + dtfk) (36)
+ gs (dkht + dthk)]
C=(~-~lF
(30)
A ' = bl C 2 + b z E 2 + b3 G2 + 2b4GE + bs + 2b6G ,
xz cos(q~ + 20s) +Y2 sin(~ + 20s)
-
f4-
cos 20s)/sin 20 s, where go = bl - b~ CZ /A ' ,
E=
kFC°S20 s-1
/sin20s,
g, = b2 - (b2E + b4G)2 /A ' ,
G-k~ g2 = b3 - (b3G + b4E + b6)2/A ' ,
dl
COS(~b+ 20s), sin 20 s
-
-m 6 4 ~-/~
d2 - sin(~ + 20s) sin 20 s
1 k F sin 20 s '
fl = cos C/sin 20 s , f2 = sin C/sin 20 s ,
g3 = 2b4 - 2(b2E + b4G) (b3G + b4E + b6)/A' , (37) g4 = - 2 b l C ( b z E + b4G)/A' , gs = -2b~ C(b3G + b4E + b6)/A' .
(37)
The vertical terms in the resolution are very easily evaluated as they have no correlations with AQx , AQy A(.o so that the remaining elements M m are or
C.G. Windsor,R.K. Heenan /Constant Q spectrometer
120
zero except for M33. The theory is identical to that of Cooper and Nathans [1] as later corrected by Dorner [8] with the monochromator and "in pile" terms deleted. R v : e x p ( - I M 3 3X~ ) where 2 2 alOall
M33 - a20 + a211 and
alo = 1/{31kl and a,,-
_(
1 +(4sin2OA~2A+(J~)k~]l
~1/2
Thus all the elements of the resolution matrix Mkl may be calculated. Since it is symmetric, it has 7 distinct non-zero elements. These are most graphically presented by the four principle axes and the three orientation angles of the resolution ellipsoid. These quantities were evaluated by a FORTRAN computer code written by the authors.
4. Measurement of the elastic resolution function In the special case of elastic scattering it is straightforward to measure the resolution function and so check our computations. This is because the Bragg scattering from a single crystal provides a good approximation to a 6 function in both Q and co so that the measured scattering gives the resolution function R(Q, co) directly. The experiments were performed using the prototype constant Q spectrometer described in our earlier paper [5]. A large iron single crystal of mosaic spread of order 1° was used as the sample. Studies were made around the 200 lattice point which corresponds to a scattering vector Q = 4.392 A -~. The graphite 004 plane having a d spacing d A = 0.838 ,~ was used as analyser leading to an incident wave-vector k I = 4.25 A -1, and the scattering diagram shown in fig. 3. Some 41 time-of-flight spectra were recorded at the calculated scattering angle 20 s = 62.9 °, and at A20 s = -+0.5°, +1 °, -+2° and +3 °. At each scattering angle around five spectra were recorded at crystal angles ~b varying by -+1° or +-2° from the angle where maximum intensity was observed. The spectra were of varying duration from one minute to one hour but
Oy(~, -1 )
I = z, 25
.&-~
/ e =62 87
\I.,\ \~\~
\kf\\x Fig. 3. The reciprocal lattice diagram corresponding to the experimental measurement of the resolution function. The measured contours at h ~ = 0.0 and 0.8 m e V are shown. The inset shows the area near the reciprocal latticepoint magnified 12~ times.
C G. Windsor, R.K. Heenan / Constant Q spectrometer
were all normalized by a fission chamber in the incident beam. Count rates were often low because the neutron beam available was only the straight-through beam of another spectrometer. For each time channel of each spectrum the values of Q and co were calculated together with a normalized intensity R and its statistical error ~2xR. The resulting set of some 1253 values of R(Q, co) can be presented in various ways. For example in fig. 4 we show sections through the resolution function as contour maps of R(Q, w) for two relatively narrow energy bands (a)hey = (0-+ 0.1) meV and (b) h a ) = ( 0 . 8 + 0.1) meV. At the expense of an even greater restriction in the amount of data presented we may represent, as in fig. 5, some principle lines through the ellipsoid. These curves confirm that the resolution function has the nearly Gaussian form we have postulated. We performed a least squares fit describing the measured resolution function in the form given by eq. 34. Ten adjustable parameters were used; the first six representing the six inequivalent parameters of the (Qx, Qy, w) section of the resolution ellipsoid, the seventh the normalization and the remaining three small shifts in the centre of the ellipsoid with respect to the initially assumed centre of the Bragg reciprocal lattice point. The
100
i
i
50
'/
f,, Z' 100
-005
0
121 I
i
[
i
"~1 ~"
0"05 -0'1 qx 1~-11
i
I
~'~
0
I
0"1 qy (A-1)"
. "
"E
5O
0 !
-1
I • *
0 hw ( m e w
Fig. 5. Sections through the resolution ellipsoid along the
qx, qy and co axes. The full line shows the best-fitting ellipsoid and the dashed line the computed ellipsoid.
w 5
19
l~"
14
z4 / ~
31
11
12 9
,
:,%
~ -0"1
I
7
Ji
0"1 1 qx(]C )
77\
9
26 i
-01
15
6
45
27
9
4~
35
I
'k
6
0!I ,/~-1, qx t I
9
5~ -0'1
i\';'i-.".", '~44~5J
8
42
h2 t5~, ?,-' \ zo
_'.,
5 11
5
t : 5 0,
01
1 10
",,19 ~
\ -,, ~,i',9~_ \ \ p4 4
,
~\\[ 1519
6
18
14
7
21 25
Fig. 4. Contour map of the scattering intensity in the qx, qy plane within the energy range: (a) (0 -+ 0.1) meV; (b) (0.8 ± 0.1) rneV. The full line is the best fitted ellipsoid and the dashed line the best calculated ellipsoid.
122
C.G. Windsor, R.K. Heenan / Constant Q spectrometer
Table 1 Parameters of the resolution ellipsoids obtained by fitting to experiment and by calculation, and the measured and fitted instrumental parameters Experiment Ellipsoid elements Ml 1 (A2) M12 (A2) M14 (A meV -1) M22 (A2) 3/24 (A meV-1) M44 (meV -2) Principle axis L 1 (A, meV units) L2 L4 Slope X44/(X214 + X24 ) (meV A) Angle of longest axis on XY plane tan -I (X24/X14) (°) Angle of longest axis of energy section X Y plane tan-1 (Xz2/XI2) (o) Instrumental parameters A (Us A-1) c~1 (tad) ¢x2 ce3 r~A r~s
3544 851 -114.5 322 -14.3 6.49
Calculation
-+21 2643 -+ 6 1654 -+ 0.6 109 • 2 1208 -+ 0.2 91.7 • 0.04 8.80
0.0192 0.111 1.103
0.0193 0.105 1.113
7.78
7.01
117.6
107.8
103.7
123.6
100 0.010 0.027 0.027 0.014 0.001
99.5 0.001 0.023 0.020 0.029 0.003
parameters are given in table 1, and the solid lines in figs. 4 and 5 shows the fit obtained. In fig. 3 we have drawn some contours of the resolution function on the scattering diagram. The longest axis of the ellipsoid lies close to the incident wavevector ki, and the long axis of the constant energy sections also lies approximately along k I. The resolution function thus has a very narrow width perpendicular to k I - a shape unfamiliar in triple axis spectrometry. A numerical diagonalization of the measured resolution ellipsoid is given in table 1 showing the lengths of the three principle axes, the slope of the longest axis, the direction of the longest axis in the X Y plane, and the direction of the longer axis of the constant energy sections in the X Y plane. 5. Comparison of the measured resolution function with calculation The formalism developed in paragraph 3 was used to calculate a resolution ellipsoid to compare with the
measured ellipsoid. Since many of the instrumental parameters were not well known some fitting was performed to obtain the angular divergences. The results of this procedure are shown in the right hand side of table 1, in fig. 6a and by the dashed lines in figs. 4 and 5. The three principle axes, and the slope and direction of the longest axis are all well given. There is however a discrepancy in the direction of the longer axis in the constant energy sections. The fitted instrumental parameters are given in table 1. They agree moderately well with values measured independently from the neutron pulse length, crystal rocking curves and from geometrical considerations. The calculated incident beam collimation c~1 is certainly less than would have been expected from geometrical considerations. This might have arisen from some miss-alignment of the spectrometer. The more important angular collimations of the scattered beam are well fitted by our calculation. In order to understand the relative importance of the various terms in the expression for the resolution, model calculations were performed, some of the results being shown in fig. 6. Section (b) represents a simplified configuration with negligible sample size, detector width and scattered neutron flight path L2. Comparison with our original calculation (a) shows that these terms have a negligible effect in our configuration. Section (c) shows that the effect of sample mosaic r~s is to rotate the ellipsoid towards the qy axis. (The change in al had a small effect.) Section (e) shows a highly simplified ellipsoid which still presents a useful approximation to the observed ellipsoid. r/a has been put equal to zero (a3 is then irrelevant), and the effect is to make all broadenings perpendicular to k0 vanish. The parameters of this ellipsoid may be evaluated analytically. Lastly in sections (d) and (f) we show two effects which worsen the resolution. In (d) the analyser is set to scatter in the same sense as the sample (FOC = +1). In (f) we have increased c~ causing the major broadening to be perpendicular to k0 rather than along it. In conclusion it is worth recalling that the ellipsoids in fig. 6 show a qualitative difference from triple-axis resolution sections which tend to lie along qx at a given energy transfer and slope towards qy as the energy increases. The fact that with good incident collimation but relaxed scattered collimation, as in section (c), the ellipsoid lies along a given line could be remarkably useful for focussing certain excitations. The authors are grateful to Martyn Cooper for
C.G. Windsor, R.K. Heenan / Constant Q spectrometer
123
0.1
0"1 - ~
(c).rts,--,,O. ~m-,,O {
N
\ ~ N" ~ 0
\\ \
",,
\
\k 'Xx \
T
[
\
---\\ -
0"II
\
'lhw=08meV
(d)~="FOC 1 "<' \x \\:\\"~\"\\
\ \
~
I
~
I
0"1-I} qx(~
L
"x
"" "2?,,
\, \\\ \\ \ 0IF-~--x\,, \ \ ~
01 _ PJ,, q,(~-l)
0!1
qx(~_iI
~(el°~A " "\\ "-" "xj 0\\''~c(3"--~1~
I
k\
~x(°~1-, _ PJ,,
/
/ /
\ -\,,
/
\\ Fig. 6. Some typical 0.0 and 0.8 meV sections of the resolution ellipsoid: (a) is the ellipsoid whose parameters are given in table 1; (b) is a simplified version having L 2 = 0, R = 0, al = 0, ~2 = a3 = 0.02, 77A = 0.03, r/s = 0.003. (c) is the simplified fit with r/s and a 1 put to zero. (d) is the full fit with FOC = +1, (e) is as (c) but with a3 large, and r/A put to zero. (f) is the simplified fit with al = 0.03 and */s and r/A put to zero.
several discussions and comments, and to Roger Sinclair for his perserverance in questioning which lead to the successful check at the end of the appendix.
cally. Differentiating the expression for Bragg's law gives AkF = cot OAk F A0A • In our notation A k F = x 2 and kF A0A =Y2, so that
Appendix: a Simplified theory for a perfect analyser The general theory simplifies considerably for the practical case of importance when the analyser crystal has a low mosaic angle compared with the scattered collimation (r/A ~ 0). The scattered wave-vector deviations along and perpendicular to kv, x2 and Y2 are then coupled together by Bragg's law. There are only three independent variables x l , y l , and say x2 which may be expressed in terms of the deviations AQx , AQy and /xw. In the special case of the Constant Q geometry the resolution ellipsoid has principle axes whose lengths and slopes may be calculated analyti-
Y2 =x2 tan 0 A .
(A1)
Referring to eq. (23) we see that this condition removes t h e third term which the condition r/A ~ 0, would cause to diverge. If the analyser is perfect we may, without loss of generality, put the counter collimation a3 = ~o so that the last term in eq. (23) also vanishes. We also suppose that the three timing errors rm, 7"s and rd from moderator, sample and detector respectively may be convoluted to give a single overall timing error r. We then have no need for the complicated discussion leading to eq. (21). We denote the probability of counting a neutron having incident and
C.G. Windsor, R.K. Heenan / Constant Q spectrometer
124
scattered wave-vector changes xx and x2 as PT = exp(~ZT) where,
al
(alxl
+ a2x2) 2
(A2)
roLl 1 mL2 1 - and a2 hk~ r hk~F r
(A3)
ZT =
=--
where G = kl/k v as before and H = - m w / h k v . Then
AQx
-Yl
2 2 2 2 Z = ( a l x 2 + a 2 x 2 ) 2 + a4Yl + a 6 y 2
(A4)
where 1
a4 =
0A)
sin 4) - H cos 4)/cos 0A ,
H sin 4)/cos 0 A .
(A9)
We now solve f o r x l a n d y l to give XI
= (AQ x cos 4) + AQy sin 4) + / / / c o s
0A)/(1
Yl = - A Q x sin 4) + AQy cos 4) .
kF°~2
This is a function of only three variables Xl, x2 and y~ and so may be expressed directly in terms of the three variables AQx , AQy and co. Using condition A1 the eqs. (26) and (27) may be written
AQ± = - A Q x sin 4) + AQy cos 4),
AQ x = x 1 cos 4) - y 1 sin 4)
x t = (AQ//+ H/cos 0A)/(1 -- G/cos 0 A ) ,
tan 0 A cos(4)+ 20s).
AQ//= AQx cos 4) + AQy sin 4) , (A11)
then
(A12)
The equation for x2 is equally simple
AQy = x 1 sin 4) + y i cos 4) - x2 sin(4) + 20s) (A5)
G AQ//+ GH/cos Oa +H(1 - G/cos 0A) X2 =
These may be written
AQ x = x l c o s 4 ) - y l
(A10)
Yx = AQ± .
- x2 cos(4) + 20s) + x2 tan 0 A sin(4) + 20s)
-x2
G/cos 0A)
--
We now see that appropriate axes for the resolution ellipsoid in the Qx, Qy plane are along k I at an angle 4) to Qx. We define as in fig. 7
1
kict I and t/6
COS 4)(1 -- G/cos
AQy =xl sin 4)(1 - G/cos 0 a ) + y l c o s 4) --
The full expression for the probability of counting a neutron is then the product of PT and the remaining angular terms P~ to give an overall probability P = exP(½Z) where
=X 1
(1 - G/cos 0 A )
G AQ//+H sin4)---
X2
COS 0 A
cos(4) + 20 s + 0A)
- 1 - G/cos 0 A '
(AI3)
Equation (A4) then becomes X2
AQy = xx sin 4) +Yl cos 4) - - -
cos 0 A
sin(4) + 20 s + 0 A ) . (A6)
We now suppose the constant Q geometry 0 A = - 2 0 s. This causes the terms in xz to have the same angular dependence as those containing x~. Physically the wave-vector change from the analyser acceptance is perpendicular to the analyser d spacing, and so parallel to the incident wave-vector.
Z = [aI(AQ//+ H/cos 0A) + a2(G AQ//+ 14)12/r2 + a62 tan20A(G AQII +H)2/Y 2 + a] AQ~, where Y = 1 - G/cos 0 A .
Z = MIx AQ] (A7)
(A15)
Expressing the resolution function as the general quadratic in AQ//, AQI and H,
AQx = cos 4)(xl - x 2 / c o s 0A) --Yx sin 4), AQy = sin 4)(xl - x2/cos 0A) --Yx Cos 4).
(A14)
+
2Mr4 AQIIH + M44H 2 + M22 AQj2 , (A16)
we have M l l = [(al +a2G)2 + a 2 t a n 20A G2]/Y2 ,
The two uncertainties x~ and x 2 are coupled by the energy transfer
ktx l X2 -
kF
m Aw hkv -=Gxl +H
M14 = [(al +a2G)(al/cOSOA +a2)+a~ tanEOAG]/r: M44 = [(al/cos 0 A + a2)2 + a~ tan z 0A] / I(2 ,
(A8)
M22 = a~ .
(A17)
C.G. Windsor, R.K. Heenan / Constant Q spectrometer
125
energy of a Bragg reflection) is
4/
[1 - kt/(k F cos 0 A ) ]
W=2.354--
// > Oy
(A22) The full width at half height of the whole ellipsoid, irrespective of Q, (i.e., the width of a vanadium scan) may be found by differentiating Z with respect to Q. This gives
W=2.354(
M_M_x~_
),/2
\M1 IM44 -- M 2 4
¢ ..,~ ~ . . / - ~ o~,
o; :2.,o~
~
= 2.354/~kF( [1 +(a2/a~)G] 2 G2] 112 m a62 tan20A + a~ !
o.
~
-..-
= 2.354 ~-m
/
1 + -- ~ ~2k4 Lx k F) 2 F c o t 2 0 A + \ m L ~ ]
/ /
(A23)
Fig. 7. The resolution ellipsoid in the special case treated in the appendix having contributions only from the angular divergences cq, a 2 and the moderator time spread, r m. The first principle axis of the ellipsoid has a projection on the Qx, Qy plane which lies along k I.
It is instructive to check this equation by the simple method of evaluating the uncertainty in the energy transfer given by eqs. (1), (3) and (5).
w=
f i T - mL2d A sin 0A/n
d A sin 0 A
(A24) These four equations thus determine the resolution ellipsoid in this special case. The ellipsoid is already diagonal in the AQ± axis, and has a principle axis full width at half height W± -
2.354 a4
AO) = m \
{k s L2 cot 0 A I ~1
kF
)
+ k} cot 0A _ A0 a .
(A25)
Differentiating with respect to T gives - 2.354 kith .
(AI8)
In the AQ//, Ac~ plane, the ellipse may be diagonalized by rotation by an angle ~ where
tan 2 ~
Differentiating this respect to 0 A gives
2/]414 MI 1 - M44
Aw
= (h \mII ~-~IAT.
(A26)
Putting A0 A ~-- a2 and A T ~ r confirms eq. (A23) for the overall width.
(A19)
References the principle axes lengths are then 2.354 W1 - - (Mll cos2ff +M14 sin 2ff +M44 sin 2 ~))112 (A20)
W2-
2.354 (M11 sin 2 ff - M14 sin 2ff + M44 COS2 ~)1[2
(a21) The full width at half height of the section of the ellipsoid along the energy axis, (i.e., the width in
[1] M.J. Cooper and R. Nathans, Acta. Cryst. 23 (1967) 357. [2] B.C. Haywood, Acta. Cryst. A27 (1971) 408. f3J S. Komura and M.J. Cooper, Jap. J. Appl. Phys. 9 (1970)
866. [4] O. Steinsvoll, Nucl. Instr. and Meth. 106 (1973) 453. [5] C.G. Windsor, R.K. Heenan, B. Boland and D. Mildner, Nucl. Instr. and Meth. 151 (1978) 477. [6] D.H. Day and R.N. Sinclair, Nucl. Instr. and Meth. 72 (1969) 237. [7] C.G. Windsor and R.N. Sinclair, Acta. Cryst. A32 (1976) 395. [8] B. Dorner, Acta. Cryst. A28 (1972) 319,