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Resolution in deep inelastic neutron scattering using pulsed neutron sources
Nuclear Instruments and Methods in Physics Research North-Holland, Amsterdam
A276 (1989) 297-305
297
RESOLUTION IN DEEP INELASTIC NEUTRON SCATTERING USING PULSED NEUTRON SOURCES C. ANDREANI ", G. BACIOCCO a), R.S . HOLT
3)
and J. MAYERS
3)
"Dipartimento di Fisica, Università di Tor Vergata, Via O. Raimondo, 00173 Roma, Italy `' Dipartimento di Fisica, Università "La Sapienza", P. le A . Moro 2, 00185 Roma, Italy 3' Rutherford Appleton Laboratory, Neutron Science Division, Chilton, Didcot, Oxfordshire, UK
Received
30
September 1988
The principal components of the resolution function for deep inelastic neutron scattering experiments on pulsed neutron sources have been calculated directly in atomic momentum space. Analytical expressions for the relative contributions from the energy, angular and time resolutions are presented for both direct and indirect geometry spectrometers. The general trend in the behaviour of the resolution as a function of neutron energy and atomic mass is presented and the results of numerical calculations for recoil scattering from hydrogen, helium and beryllium using the ISIS spectrometers HET and eVS, are also given. It is shown that the resolution difference between HET and eVS is significantly reduced when compared in atomic momentum space rather than in energy space. Moreover, the contribution from the angular resolution term is only significant for atomic masses < 4 au. 1. Introduction In recent years there has been an increasing interest in the experimental determination of atomic momentum distributions in condensed matter by deep inelastic neutron scattering (also called recoil scattering and neutron Compton scattering) at large momentum transfers hq [1]. The basis of the technique rests on the validity of the so-called "impulse approximation" which is asymptotically exact as q tends to infinity [2]. The impulse limit is only reached by utilising large incident neutron energies generally in excess of 1 eV, which are now readily available on spallation neutron sources. These pulsed sources, like the ISIS facility at the Rutherford Appleton Laboratory, UK, have a much higher intensity in this energy regime than that produced from reactor sources. Instruments which can operate in this energy regime and analyse impulsive recoil scattering spectra are therefore likely to become important new sources of information on the motion of atoms in condensed matter . Of the instruments on ISIS both inverse and direct geometry spectrometers, e.g . the ElectronVolt Spectrometer "eVS", the High Energy Transfer spectrometer "HET" and the MultiAngle Rotor Instrument "MARI" (currently under construction), will feature prominently in this new area of science. In most of the published literature on inelastic neutron scattering the data have been presented in terms of S(q, o)) at constant q, where S(q, co) is the neutron scattering function [3], hq is the momentum transferred from the neutron to the sample and h to is the energy 0168-9002/89/$03 .50 C Elsevier Science Publishers B .V . (North-Holland Physics Publishing Division)
transfer . S(q, w) at constant q is usually the property of most theoretical interest ; triple-axis instruments on reactor sources are ideally suited to its experimental determination . In impulsive neutron scattering, however, the property of direct interest is the atomic momentum distribution, n(p), where p is the atomic momentum . It is therefore more beneficial to present and compare the data in terms of measured n(p) rather than S(q, ca). Such a procedure is relatively easy to undertake and, moreover, offers several distinct advantages over a comparison in time-of-flight or energy transfer . For an isotropic system n (p) is a one-dimensional function, even though S(q, w) is a function of two variables. The link between S(q, w) and n (p) is provided by the property of "y scaling" [2] where every point (q, w) corresponds to a unique point in atomic momentum space (hereafter called p-space) . Consequently any scan in (q, w) space which crosses the recoil line gives a measurement of n ( p ). It is generally inefficient for time-of-flight instruments to scan along a particular line in (q, w) space but this is not necessary in impulse scattering experiments if the data are presented in terms of n ( p) . Another advantage of analysing data in this way is that for a multidetector instrument the detectors can be grouped together in p-space (this is only true for detectors which give a sufficiently large q that the impulsive limit is reached) . There is the additional advantage that the limit at which the impulsive regime is reached could be estimated by comparing the distribution in p-space
298
C. Andreani et al. / Resolution in deep inelastic neutron scattering
provided by detectors at different angles . The theoretical criteria for deciding when the impulse regime is reached are still a subject of debate (see ref. [4]) and experimental information on this question would be valuable. The method also allows a comparison of the performance of different instruments directly in p-space. Such a comparison has been made on the basis of the resolution capability of the two ISIS instruments, eVS and HET. Moreover, the various contributions to the resolution, i.e . energy, angular and time components, have been analysed independently to provide an indication as to how these can be optimised in a particular experiment . 2. Theory The impulse approximation (IA) relates S(q, ci) to the atomic momentum distribution n (p) via [2]
S (q, w)
= (Ilh) n(p)8{w-hq 2 /2M-hp-glM) dp, ƒ
where h w and hq are respectively the energy and momentum transfers in the scattering process, hp is the atomic momentum and M the atomic mass . For simplicity we use units such that h = 1 in the rest of this paper. The factors of h in the equations can easily be found by dimensional arguments if required . If we take the z axis along q eq . (1) reduces to . S(q , w) = Mlg J 8 (y - pJn(p) dp,
(2)
y = Mlq(w - q 2 /2M) = Mlq(W - WR) ,
(3)
Eq . (3) provides the relationship between a point in (q, w) space and the corresponding point, y, in p-space. For an isotropic system it can be easily shown that [2] where
J(y) =2,rr
fIrl~n(p)p dp .
(4)
(5)
The function J(y) is analogous to the well known Compton profile in photon scattering . It gives the probability that an atom has the momentum component y along the direction of q . From eqs. (4) and (5) it follows that n(y) = -(1/21Ty) dJ(y)ldy .
(6) As an example of this procedure we consider an isotropic harmonic system . In the impulse regime [3] Xexp[-(w- W R ) 2/44O R K B T *],
T* =1/K
BJ
WZ(w) coth(w/K B T) dw .
(8)
Z(w) is the normalised density of states . From eqs . (3), (4) and (7) J(y) =(1/21TMKB
T*) 1/2
exp { -(y 2/2MK B T *)), (9)
and from eq . (6) n(y) = ( 1 /2mMKBT
* ) 3/2
exp(y2/2MK B T*)~ .
(10)
Eq . (10) can of course be derived directly from the harmonic model without using S(q, co) . It is identical to the three-dimensional momentum distribution of an ideal gas, but with T replaced by T* . 3. Resolution in p-space We now consider the resolution in p-space, at the point _y determined by eq . (3), for both direct and indirect geometry spectrometers on a pulsed source . It is important that this quantity is known so that the feasibility of the proposed momentum measurements can be determined and the data analysed correctly. It is assumed that the variation in resolution across the recoil peak is small . This has been verified in a number of examples . From eq . (3) y = Mwlq - q/2,
(11)
so
where
S(q, w)=(Mlq)J(y),
where K B is Boltzmann's constant and the "effective temperature" T * is given by
(ay/aw) q =Mlq,
(12)
and (aylaq)w = - (Mlg 2 )(W+ W K ),
(13)
(aylaq) w =-1 .
(14)
at the recoil frequency, w = ca R = hq 2 /2 M so Denoting the errors in experimental parameters by Ox ; and assuming that errors can be added in quadrature, then on the recoil line i ),2=1: [(ay/aw)(aw/axr) r
+(aylaq)(aglax,)]2 Ax 12
=
E [Mlq(aWlax ;) - (aglax,)l2
AX 12 .
(15)
Several inferences can be made from eq . (15) . Firstly, we expect that for large atomic masses the energy resolution will be the dominant factor determining the p-resolution ; since w is independent of angle the angu-
C. Andreani et al. / Resolution in deep inelastic neutron scattering lar resolution will be unimportant for large masses . Secondly, the FWHM of the momentum distribution given by eq . (10) is (DY)FwHM = 2 .345(MK BT *
)1/2
=
0 .388(MT* )
1/2
, (16)
where Ay is in X -1 , M in amu and T* in K. For a harmonic solid, T * is given by eq. (8). For liquid He, T * is approximately determined by the zero point kinetic energy. Thus, at values of M>4, where the p-resolution is dominated by the energy resolution, we expect from eq. (15) that approximately (17)
DY/(Ay)FWHM a M 112 ,
i .e . the relative energy resolution becomes progressively poorer as the mass increases . This behaviour is illustrated in table 1 where (AY)FWHM is listed for a series of different elements . ('Y)FWHM was calculated from the Debye model using the Debye temperatures given in table 1. The resolutions for HET and eVS were calculated by the method described in section 4 . Finally we note that the contribution to A y arising from the energy resolution is inversely proportional to q. Thus measurements at large q values, with poor energy resolution, may still provide good resolution in p-space. The analytic form of the resolution in y space for direct and inverse geometry spectrometers is given in the appendix. In the case of chopper spectrometers, such as HET and MARI, the most important contributions to the resolution arise from At, At . and DO, where t c is the time at which a detected neutron passes through the chopper, t m is the time at which the neutron leaves the moderator and O is the scattering angle. On the recoil line conservation of energy and momentum dictates that the initial and final neutron velocities, vo and vl, are related by (taking neutron mass, m = 1) v l /vo =[cos O+(M 2 -sin 20) 1/21 /(M+1) .
(18)
As a consequence, for a given O and M, vl and q are
299
both proportional to vo and from eqs. (46)-(48) : ay /at~ a Eo,
(19)
ay/at,,, a Eo ,
(20)
ay/a0 a E 1 / 2 ,
(21)
where Eo is the incident energy . The time uncertainty in t m is approximately determined by the moderator depth S m At, =
Sm/vo Ce
Eo
1/2 .
(22)
The chopper is usually matched to the moderator pulse width so that Ot~ - t1 t m .
(23)
From eqs. (19)-(23) it can be seen that all three contributions to the resolution are proportional to Eó 2 for given values of M and O. On the other hand for an inverse geometry spectrometer eqs. (55)-(57) show that on the recoil line, where eq . (18) is satisfied, ay /at m a El ,
(24)
ay /aE, cc El 1 / 2 ,
(25)
and ay/aO cz E;/2 .
(26)
For scattering from samples other than hydrogen, the dominant contribution to the resolution comes from the energy term (25) . In these cases the experimental uncertainty in y is inversely proportional to q, for a given energy resolution AE, 4. Numerical results For ease of computation the resolutions have been calculated numerically rather than analytically . For example, the uncertainty in y due to an error in angle of FWHM DO was calculated as I y(x i , O + (1/2)i10) y (x,, O- (1/2)t10) 1 . Numerical computations of the components of the resolution have been made for two ISIS instruments, HET and eVS. HET is a direct geometry chopper spectrometer [5] and eVS an inverse geom-
Table 1 Resolution and atomic momentum FWHM values for some selected elements calculated for HET (E o =1000 meV, 20 =180 °) and eVS (E,=4280 meV, 20=180 ° ) Element Li Be C Mg V Ta
Op
400 1000 1000 318 390 225
M
7 9 12 24 51 181
T * (77 K) 158 376 376 132 154 107
0p (HET) 1 .04 1 .47 2.11 4.71 10 .6 48 .7
0p (eVS) 3.51 4.4 5.65 10 .8 22 .4 78 .1
(DY)FWHM
11 .2 19 .6 22 .7 19 .1 30 .0 47 .0
300
C. Andreani et al . / Resolution in deep inelastic neutron scattering
Table 2 Some of the instrument parameters for the direct (HET : High Energy Transfer spectrometer) and indirect (eVS : ElectronVolt . Spectrometer) geometry spectrometers on ISIS Incident flight path (L O ) Scattered flight path (L,) Angular uncertainty (AO) Moderator depth (SR,)
11 .8 m 2.5 m 0.3 , 4 .0 cm
12 .6 m 1 .0 m 0.75 ° 3.0 cm
etry spectrometer [6] in which the energy of the scattered neutron is determined using either the resonance detector [7] or the resonance filter method [8]. A summary of the parameters for both instruments is listed in table 2 . For HET the results are presented for two incident energies : E, = 1000 and 2000 meV . For eVS two possible final energies have been selected and these are determined by the strong nuclear resonances in Sm and Ta . These energies, together with their respective resonance widths at room temperature (FWHM) are: E t =870 meV (AE,=100 meV) and Et = 4280 meV (.1 Et = 150 meV). In terms of the mass of the recoiling particle three distinct materials have been chosen as representative of the changes in resolution which can be expected . Firstly scattering from hydrogen has been considered in some detail : this is in many ways a special case and is of considerable experimental interest . Secondly, and perhaps the system of greatest interest from an impulsive scattering point of view . is the scattering from 4 He . Hohenberg and Platzman [9] were the first to suggest
that deep inelastic neutron scattering could be used as a method for determining the Bose-condensate fraction in the superfluid phase. Since then both reactor and pulsed source experiments have been performed on this quantum fluid [7]. Finally we consider the high mass regime (in recoil scattering terms) by studying the resolution contributions in beryllium. 4.1 . Scattering front hydrogen In hydrogen VI = 1 and eq . (16) reduces to t'L/tb = cos 0 .
(27)
From this equation it should be noted that scattering from free hydrogen atoms cannot occur for 0 > 90 ° . The scattering vector q is given by q = ni ( có + t, i - 2 r, )c, cos O )
2
.
(28)
where m is the neutron mass . From eqs. (27) and (28) (29)
q = mr o sin 0 = mv, tan 0.
Substituting eqs . (27) and (29) into eqs. (46)-(48) we obtain for a chopper spectrometer a r/at~ _ -ni( t,~/L o )(cos = 0/sin 0 )
x [l + (L~/L,) cos -0] .
av/at~,=rat ( c,,/Lo j cos 2 0/sin 0, a,/a0=-nil") cos 0.
(30) (31) (32)
Figs . l a and lb show- the individual resolution contributions and the total p-resolution calculated for HET
0 E
C E T A
Y.
30
4
5C
SCATTT.JJG AUCL.E
60 (DEGREE)
70 >CA7TE2NG A\k-E
(DEGREES;
Fig. 1 . The total resolution ( ) and the energy (---). angular (-- -) and timing ( ) contributions to the resolution in hydrogen ()VI=1 amu) shown as a function of scattering angle and momentum transfer for HET at (a) Eo =1000 meV and (b) E = 2000 meV .
301
C . Andreani et al. / Resolution in deep inelastic neutron scattering
at incident energies 1000 meV and 2000 meV as a function of angle. The q values corresponding to the different scattering angles are also given. As mentioned previously all resolution contributions are proportional to E'' 12 and this behaviour can clearly be seen by comparing the two figures. The resolution at both 1000 meV and 2000 meV is governed primarily by the energy term except for angles in excess of - 60 ° where the angular component begins to dominate . The uncertainties associated with the timing are comparable with the angular term for scattering angles greater than - 30' . For an inverse geometry spectrometer eqs . (27), (29) and (55)-(57) give ay1at m = -m(eiILO ) sin O,
(33)
ay/aE,=-(1/v, sin O)[cos0-(L,/LD)],
(34)
a y /a0 = -mu, .
(35)
Figs. 2a and 2b show the resolution contributions on eVS, calculated for E, = 872.0 meV, AE, = 100 meV and E, = 4280 meV, AE, = 150 meV, respectively . The abscissa gives O and q values . In this case the angular term makes a significant contribution to the overall resolution, particularly for the higher energy resonance where again it is the dominant term for angles in excess of 60 ° . Both figs . 1 and 2 illustrate the overall improvement in p-space resolution as the scattering angle is increased due to the angular variation of ay/aE, . Table 3 summarises the resolution results for both HET and eVS (for O = 60 ° ) indicating that the resonance system
MOMENTUM TRANSFER
Table 3 Selected total resolution results for recoil scattering form hydrogen, helium and beryllium using the chopper (HET) and resonance neutron absorption (eVS) spectrometers on ISIS Mass 1, 6) = 60 °
Instrument HET HET eVS eVS
Energy [meV] 1000
2000 872 4280
Resolution 0 .09 0 .14 0 .82 0 .80
Mass 4, O = 160 °
Instrument HET HET eVS eVS
Energy [meV] 1000
Resolution
Instrument HET HET eVS eVS
Energy [meV] 1000
Resolution
2000 872 4280
0 .46 0 .65 3 .40 2 .30
Mass 9, O = 160'
2000 872 4280
1 .47 2 .07 6 .60 4.48
is only a factor of 6-9 worse than the chopper spectrometer . This is significantly lower than might be anticipated from the energy resolution of the two systems. The energy resolution of HET is - 7 meV at 1000 meV and - 14 meV at 2000 meV. Thus the energy resolution
(DJVERSE ANGSTROMS)
SCATTT7IING ANGLE (DEGREES)
SCATTERING ANGLE (DEGREES)
) and the energy (---), angular ( .--- .) and timing ( . . . . . . ) contributions to the resolution Fig. 2 . The total resolution ( in hydrogen (M =1 amu) shown as a function of scattering angle and momentum transfer for eVS at (a) E l = 872 meV and (b) E, = 4280 meV.
302
C. Andreani et al. / Resolution in deep inelastic neutron scattering MOMENTUM TRANSFER
D E L T A Y
(AVERSE ANGSTROMS)
MOMENTUM TRANSFER
(ANESSE ANGSTROMS)
8
6
4
2
0
Fig. 3. The HET results for helium (M= 4 amu) . of HET is between 10-20 times better than that of eVS. It should be pointed out that much higher q values can be accessed on eVS. It is again clear that the angular resolution is very important and significant improvements in the resolution can be achieved by changes in the collimation of the scattered neutrons . An alternative method of improving the resolution is by arranging the experimental geometry for "time focussing" . This has been discussed for resonance detector spectrometers by Carpenter and MOMENTUM TRANSFER 14 .5
22 .0
29 .7
D E L T A Y
44 .1
49 .7
EIS
E
..
..
2, ~.
~.
When considering the recoil scattering in helium the simplifications introduced in section 4.1 no longer apply and the full expressions for the resolution given in the appendix must be used . Figs . 3a and 3b show the
53 .4
M=4 AU = 872
AE
40
4.2 . Scattering from helium
MOMENTUM TRANSFER
(INVET~SE ANGSTROMS)
37 .2
Watanabe [10] and Rauh et al . [11] . They also point out the dominant effects of angular contributions to the resolution for recoil scattering from low atomic masses .
. . ._ . _ .
' .. r fó
140
=
. ..
meV
100 meV
16U
=
180
SCATTEfUNG ANGLE (DEGREES)
Fig. 4 . The eVS results for helium (M = 4 amu) .
(INVERSE ANGSTROMS)
C . Andreani et al. / Resolution in deep inelastic neutron scattering MOMENTUM TRANSFER
MOMENTUM TRANSFER
(INVERSE ANGSTROMS) 10 .8
20
D E L T A Y
D E L T 17.5 A Y
21 .0
30 .3
38 .4
303
(AVERSE ANGSIRO S) 45 .0
HET
25
50 .1
53 .7
55 .8
M=
9
AU
E
=
2000
meV
15
12.5 15 10
7.5
10
b
5
2.5
0
60
EC
100
SCATTERING ANGLE
120
20
140
(DEGREES)
40
60
80
100
SCATTERING ANGLE
120
140
160
180
(DEGREES)
Fig. 5. The HET results for beryllium (M = 9 amu) . resolution components obtained on HET and figs . 4a and 4b those obtained on eVS. For comparison the FWHM of the atomic momentum distribution in 4 He is - 1.9 l-1-1 . At M = 4 we are already approaching the regime where the p-resolution is dominated by the energy resolution of the spectrometers (see eq . (15) and succeeding comments). We find that the resolution on the chopper spectrometer is better by a factor of 4-7
(for O = 160 °, see table 2) than that for an indirect geometry instrument based on resonance neutron absorption . On eVS the energy resolution dominates; the angular and timing components effectively make a similar and small contribution over the entire scattering angle region of interest . Improvements of the order of 20% or more could be achieved by implementing the so-called double difference method of collecting data
D E L T A Y
SCATTERING ANGLE (DEGREES)
SCATTERING ANGLE (DEGREES)
Fig. 6. The eVS results for beryllium (M = 9 amu) .
304
C. Andreani et al l Resolution in deep inelastic neutron scattering
[12] or alternatively by cooling the analysing foil to very low temperatures [13] to improve the energy resolution component.
tion could be made on both instruments by implementing better collimation of the scattered beam, or by increasing the secondary flight path .
4.3. Scattering from high-mass systems In the case of a high-mass system, e.g . beryllium, figs . 5 and 6 show the components of the resolution for the two ISIS spectrometers . For eVS the p-resolution comes almost entirely from the energy resolution contribution and this would also be true in the case of higher-mass particles. With regard to HET the energy resolution term also dominates the resolution over the entire angular range. From eq . (15) we expect the presolution to be roughly proportional to M for M > 4. For such masses the resolution of the HET chopper spectrometer is only a factor of 2-4 better than that on eVS (for O = 160 °, see table 2) despite the large difference in energy resolution of the two instruments . Due to the limitation on the chopper energies available it is doubtful whether q values in excess of 50 A - ' can be reached. In a harmonic solid the impulse approximation becomes valid when w R >> c,w D, where ca D is the Debye energy [l4] . Since w R cc 1/M large q values are particularly necessary for the IA to be reached in highmass systems. 5. Summary and conclusions Calculations have been presented of the resolution in atomic momentum space (p-space), for both direct and inverse geometry instruments on the pulsed neutron source ISIS . From the general analytic forms for the resolution in momentum space (eq. (15)) and from the numerical calculations it is shown that for large masses the energy resolution contribution is the dominant term and the angular resolution is relatively unimportant . Moreover, the relative resolution (i .e . resolution/ width of distribution) deteriorates as the atomic mass increases. As expected the resolution on the chopper instrument was significantly better than that for the indirect geometry spectrometer based on neutron absorption resonances . However, the resolution when analysed in terms of p-space was better than expected on eVS because of the larger q values accessed . Moreover, these larger q values mean that the impulse approximation is approached more closely on eVS than on HET, which is limited to momentum transfers up to 50 f\ - ', and renders a relatively simpler data analysis essentially free from final state effects . Calculations of the resolution components in recoil scattering from hydrogen reveals that the angular term makes a significant contribution to the resolution on both HET and eVS . Significant improvements in resolu-
Acknowledgements We would like to thank the European Economic Community Stimulation Action scheme for funding this collaboration and Dr . M.P. Paoli and Dr . A.D . Taylor for stimulating discussions.
Appendix p-space resolution for a chopper spectrometer For simplicity we assume that the distance between the chopper and the sample is negligible . This approximation should not affect the results significantly. The largest experimental errors arise from uncertainties in : (1) the time the neutron leaves the moderator, t , ; (2) the time the neutron passes through the chopper, tc; (3) the scattering angle, O. It is convenient to work in terms of vo, the incident neutron velocity, and v l, the scattered neutron velocity, then vO -L Ol( t c - tm) ,
(36)
v, =Lil( 1 d - l ) .
(37)
Lo and L, are the incident and scattered flight paths and td is the time at which the neutron is detected . With units such that h = 1 w = 1/2m ( vó - v2 ),
(38)
q2= m 2 [ vó+v 2 -2vov l cos 19] .
(39)
m is the neutron mass . From eqs. (36)-(39)
I aw/atc = -m(vólL,)[1 + (v /v o)3(Lo/L , )j,
(40)
aglat~ = M 21 Iq(v03ILO) X {(tl/Vo) 3 ( LO/Ll)[ 1 - (volvl) Cos 01 +(vl/vo) cos O- l} '
(41)
a w/at m = mvó ILO,
(42)
ag/atm = (m 2vO)l(Loq)[ 1- v,lvo cos O],
(43)
aw/ao = 0,
(44)
agla 0= (m 2vov1 sin 9)/q .
(45)
C. Andreani et al. / Resolution in deep inelastic neutron scattering From eqs. (15) and (40), (41) :
aylöE, = (mlq)((volul) 3 (Lt/Lo)[ 1 - (Mlm)] - 1
ayl a t = -(m2ó)/(Loq) X { M/m
- (M/m) + (vo/VI) cos 0
[1 + (VIIVO)3(LOILI)l
X[(volvt)(LO/Lt) -1 ]} ,
+ (UI/UO) 3 ( L O/ L t)[ 1 - (VOlvt) cos 0] + [1 - ( v,/vo ) cos O ] } .
ay/a0= -(m2VOVl sin 0)/q .
(47)
Finally
(48)
[21 [3] [4]
p-space resolution of an inverse geometry spectrometer In this case the experimental uncertainties are i10,
At . and i1 El , where EI is the energy of the scattered neutrons . We obtain
aca/at m = -mvó ILO, (49) 2 aq/atm= - (m VO)/(Loq)[ 1 - (Vtlvo) cos O], (50) aw/aE, = - [1 + (vo/Vi) 3 (LI/LO)]
[61
[7]
(51)
aglaE, = (mlq)[ 1 - (vo/vi) 3 (LI/LO) +(VO /vl)2(LI/Lo) cos O
[8]
01 ,
(52)
a q/a0=(m 2vo v l sin 9)1q .
(54)
- ( V,/v, ) cos aw/ao = 0,
(53)
Then from eqs. (15) and (48)-(53)
[9] [101 [111 [121
aylatm = (m2VO)/(Loq) X[1-(M/m)-(v i /vo )cos0 ],
(57)
References
a y /at m = (m 2 V°)l(Loq)
ay/a0= -(m 2 VOUt sin O)/q .
(56)
(46)
Similarly, using eqs . (42) and (43)
X[(M/m)-1+(v l /vo )cos0 ] .
305
(55)
[13] [14]
B. Tanatar, G.C . Lefever and H.R . Glyde, Physica B136 (1986) 187 and references therein . V.F . Sears, Phys. Rev. B30 (1984) 44 . S.W . Lovesey, Theory of Neutron Scattering from Condensed Matter, vol. 1 (Oxford University Press, 1987). J. Mayers, C. Andreani and G. Baciocco, submitted to Phys . Rev. B. A.D. Taylor, B.C . Boland, Z.A . Bowden and T.J .L . Jones, Rutherford Appleton Laboratory Report RAL-87-012 (1987) . R.J . Newport, M.P . Paoli, V.T . Pugh, R.N . Sinclair, A.D . Taylor and W.G . Williams, Proc. 8th Meeting Int. Coll . on Advanced Neutron Sources (ICANS VIII), Rutherford Appleton Laboratory (1984) p. 562; R.S. Holt and M.P . Paoli, submitted to Nucl . Instr. and Meth. J.M . Carpenter, N. Watanabe, S. Ikeda, Y . Masuda and S. Sato, Proc . 6th Meeting Int. Coll. on Advanced Neutron Sources (ICANS VI), Argonne National Lab. (1982) ; also R.S . Holt, L.M . Needham and M.P. Paoli, Phys . Lett . A126 (1988) 373. R.J. Newport, P.A . Seeger and W.G. Williams, Nucl . Instr. and Meth. A238 (1985) 177. P.C . Hohenberg and P.M . Platzman, Phys . Rev. 152 (1966) 198. J.M. Carpenter and N . Watanabe, Nucl . Instr. and Meth . 213 (1983) 311. H. Rauh, S. Ikeda and N. Watanabe, Nucl . Instr. and Meth . 224 (1984) 469. P.A . Seeger, A.D . Taylor and R.M . Brugger, Nucl . Instr. and Meth . A240 (1985) 98 . H. Rauh and N. Watanabe, Nucl . Instr. and Meth. A228 (1984) 147. M.S. Nelkin and D.E . Parks, Phys . Rev. 119 (1960) 1060 .
A276 (1989) 297-305
297
RESOLUTION IN DEEP INELASTIC NEUTRON SCATTERING USING PULSED NEUTRON SOURCES C. ANDREANI ", G. BACIOCCO a), R.S . HOLT
3)
and J. MAYERS
3)
"Dipartimento di Fisica, Università di Tor Vergata, Via O. Raimondo, 00173 Roma, Italy `' Dipartimento di Fisica, Università "La Sapienza", P. le A . Moro 2, 00185 Roma, Italy 3' Rutherford Appleton Laboratory, Neutron Science Division, Chilton, Didcot, Oxfordshire, UK
Received
30
September 1988
The principal components of the resolution function for deep inelastic neutron scattering experiments on pulsed neutron sources have been calculated directly in atomic momentum space. Analytical expressions for the relative contributions from the energy, angular and time resolutions are presented for both direct and indirect geometry spectrometers. The general trend in the behaviour of the resolution as a function of neutron energy and atomic mass is presented and the results of numerical calculations for recoil scattering from hydrogen, helium and beryllium using the ISIS spectrometers HET and eVS, are also given. It is shown that the resolution difference between HET and eVS is significantly reduced when compared in atomic momentum space rather than in energy space. Moreover, the contribution from the angular resolution term is only significant for atomic masses < 4 au. 1. Introduction In recent years there has been an increasing interest in the experimental determination of atomic momentum distributions in condensed matter by deep inelastic neutron scattering (also called recoil scattering and neutron Compton scattering) at large momentum transfers hq [1]. The basis of the technique rests on the validity of the so-called "impulse approximation" which is asymptotically exact as q tends to infinity [2]. The impulse limit is only reached by utilising large incident neutron energies generally in excess of 1 eV, which are now readily available on spallation neutron sources. These pulsed sources, like the ISIS facility at the Rutherford Appleton Laboratory, UK, have a much higher intensity in this energy regime than that produced from reactor sources. Instruments which can operate in this energy regime and analyse impulsive recoil scattering spectra are therefore likely to become important new sources of information on the motion of atoms in condensed matter . Of the instruments on ISIS both inverse and direct geometry spectrometers, e.g . the ElectronVolt Spectrometer "eVS", the High Energy Transfer spectrometer "HET" and the MultiAngle Rotor Instrument "MARI" (currently under construction), will feature prominently in this new area of science. In most of the published literature on inelastic neutron scattering the data have been presented in terms of S(q, o)) at constant q, where S(q, co) is the neutron scattering function [3], hq is the momentum transferred from the neutron to the sample and h to is the energy 0168-9002/89/$03 .50 C Elsevier Science Publishers B .V . (North-Holland Physics Publishing Division)
transfer . S(q, w) at constant q is usually the property of most theoretical interest ; triple-axis instruments on reactor sources are ideally suited to its experimental determination . In impulsive neutron scattering, however, the property of direct interest is the atomic momentum distribution, n(p), where p is the atomic momentum . It is therefore more beneficial to present and compare the data in terms of measured n(p) rather than S(q, ca). Such a procedure is relatively easy to undertake and, moreover, offers several distinct advantages over a comparison in time-of-flight or energy transfer . For an isotropic system n (p) is a one-dimensional function, even though S(q, w) is a function of two variables. The link between S(q, w) and n (p) is provided by the property of "y scaling" [2] where every point (q, w) corresponds to a unique point in atomic momentum space (hereafter called p-space) . Consequently any scan in (q, w) space which crosses the recoil line gives a measurement of n ( p ). It is generally inefficient for time-of-flight instruments to scan along a particular line in (q, w) space but this is not necessary in impulse scattering experiments if the data are presented in terms of n ( p) . Another advantage of analysing data in this way is that for a multidetector instrument the detectors can be grouped together in p-space (this is only true for detectors which give a sufficiently large q that the impulsive limit is reached) . There is the additional advantage that the limit at which the impulsive regime is reached could be estimated by comparing the distribution in p-space
298
C. Andreani et al. / Resolution in deep inelastic neutron scattering
provided by detectors at different angles . The theoretical criteria for deciding when the impulse regime is reached are still a subject of debate (see ref. [4]) and experimental information on this question would be valuable. The method also allows a comparison of the performance of different instruments directly in p-space. Such a comparison has been made on the basis of the resolution capability of the two ISIS instruments, eVS and HET. Moreover, the various contributions to the resolution, i.e . energy, angular and time components, have been analysed independently to provide an indication as to how these can be optimised in a particular experiment . 2. Theory The impulse approximation (IA) relates S(q, ci) to the atomic momentum distribution n (p) via [2]
S (q, w)
= (Ilh) n(p)8{w-hq 2 /2M-hp-glM) dp, ƒ
where h w and hq are respectively the energy and momentum transfers in the scattering process, hp is the atomic momentum and M the atomic mass . For simplicity we use units such that h = 1 in the rest of this paper. The factors of h in the equations can easily be found by dimensional arguments if required . If we take the z axis along q eq . (1) reduces to . S(q , w) = Mlg J 8 (y - pJn(p) dp,
(2)
y = Mlq(w - q 2 /2M) = Mlq(W - WR) ,
(3)
Eq . (3) provides the relationship between a point in (q, w) space and the corresponding point, y, in p-space. For an isotropic system it can be easily shown that [2] where
J(y) =2,rr
fIrl~n(p)p dp .
(4)
(5)
The function J(y) is analogous to the well known Compton profile in photon scattering . It gives the probability that an atom has the momentum component y along the direction of q . From eqs. (4) and (5) it follows that n(y) = -(1/21Ty) dJ(y)ldy .
(6) As an example of this procedure we consider an isotropic harmonic system . In the impulse regime [3] Xexp[-(w- W R ) 2/44O R K B T *],
T* =1/K
BJ
WZ(w) coth(w/K B T) dw .
(8)
Z(w) is the normalised density of states . From eqs . (3), (4) and (7) J(y) =(1/21TMKB
T*) 1/2
exp { -(y 2/2MK B T *)), (9)
and from eq . (6) n(y) = ( 1 /2mMKBT
* ) 3/2
exp(y2/2MK B T*)~ .
(10)
Eq . (10) can of course be derived directly from the harmonic model without using S(q, co) . It is identical to the three-dimensional momentum distribution of an ideal gas, but with T replaced by T* . 3. Resolution in p-space We now consider the resolution in p-space, at the point _y determined by eq . (3), for both direct and indirect geometry spectrometers on a pulsed source . It is important that this quantity is known so that the feasibility of the proposed momentum measurements can be determined and the data analysed correctly. It is assumed that the variation in resolution across the recoil peak is small . This has been verified in a number of examples . From eq . (3) y = Mwlq - q/2,
(11)
so
where
S(q, w)=(Mlq)J(y),
where K B is Boltzmann's constant and the "effective temperature" T * is given by
(ay/aw) q =Mlq,
(12)
and (aylaq)w = - (Mlg 2 )(W+ W K ),
(13)
(aylaq) w =-1 .
(14)
at the recoil frequency, w = ca R = hq 2 /2 M so Denoting the errors in experimental parameters by Ox ; and assuming that errors can be added in quadrature, then on the recoil line i ),2=1: [(ay/aw)(aw/axr) r
+(aylaq)(aglax,)]2 Ax 12
=
E [Mlq(aWlax ;) - (aglax,)l2
AX 12 .
(15)
Several inferences can be made from eq . (15) . Firstly, we expect that for large atomic masses the energy resolution will be the dominant factor determining the p-resolution ; since w is independent of angle the angu-
C. Andreani et al. / Resolution in deep inelastic neutron scattering lar resolution will be unimportant for large masses . Secondly, the FWHM of the momentum distribution given by eq . (10) is (DY)FwHM = 2 .345(MK BT *
)1/2
=
0 .388(MT* )
1/2
, (16)
where Ay is in X -1 , M in amu and T* in K. For a harmonic solid, T * is given by eq. (8). For liquid He, T * is approximately determined by the zero point kinetic energy. Thus, at values of M>4, where the p-resolution is dominated by the energy resolution, we expect from eq. (15) that approximately (17)
DY/(Ay)FWHM a M 112 ,
i .e . the relative energy resolution becomes progressively poorer as the mass increases . This behaviour is illustrated in table 1 where (AY)FWHM is listed for a series of different elements . ('Y)FWHM was calculated from the Debye model using the Debye temperatures given in table 1. The resolutions for HET and eVS were calculated by the method described in section 4 . Finally we note that the contribution to A y arising from the energy resolution is inversely proportional to q. Thus measurements at large q values, with poor energy resolution, may still provide good resolution in p-space. The analytic form of the resolution in y space for direct and inverse geometry spectrometers is given in the appendix. In the case of chopper spectrometers, such as HET and MARI, the most important contributions to the resolution arise from At, At . and DO, where t c is the time at which a detected neutron passes through the chopper, t m is the time at which the neutron leaves the moderator and O is the scattering angle. On the recoil line conservation of energy and momentum dictates that the initial and final neutron velocities, vo and vl, are related by (taking neutron mass, m = 1) v l /vo =[cos O+(M 2 -sin 20) 1/21 /(M+1) .
(18)
As a consequence, for a given O and M, vl and q are
299
both proportional to vo and from eqs. (46)-(48) : ay /at~ a Eo,
(19)
ay/at,,, a Eo ,
(20)
ay/a0 a E 1 / 2 ,
(21)
where Eo is the incident energy . The time uncertainty in t m is approximately determined by the moderator depth S m At, =
Sm/vo Ce
Eo
1/2 .
(22)
The chopper is usually matched to the moderator pulse width so that Ot~ - t1 t m .
(23)
From eqs. (19)-(23) it can be seen that all three contributions to the resolution are proportional to Eó 2 for given values of M and O. On the other hand for an inverse geometry spectrometer eqs. (55)-(57) show that on the recoil line, where eq . (18) is satisfied, ay /at m a El ,
(24)
ay /aE, cc El 1 / 2 ,
(25)
and ay/aO cz E;/2 .
(26)
For scattering from samples other than hydrogen, the dominant contribution to the resolution comes from the energy term (25) . In these cases the experimental uncertainty in y is inversely proportional to q, for a given energy resolution AE, 4. Numerical results For ease of computation the resolutions have been calculated numerically rather than analytically . For example, the uncertainty in y due to an error in angle of FWHM DO was calculated as I y(x i , O + (1/2)i10) y (x,, O- (1/2)t10) 1 . Numerical computations of the components of the resolution have been made for two ISIS instruments, HET and eVS. HET is a direct geometry chopper spectrometer [5] and eVS an inverse geom-
Table 1 Resolution and atomic momentum FWHM values for some selected elements calculated for HET (E o =1000 meV, 20 =180 °) and eVS (E,=4280 meV, 20=180 ° ) Element Li Be C Mg V Ta
Op
400 1000 1000 318 390 225
M
7 9 12 24 51 181
T * (77 K) 158 376 376 132 154 107
0p (HET) 1 .04 1 .47 2.11 4.71 10 .6 48 .7
0p (eVS) 3.51 4.4 5.65 10 .8 22 .4 78 .1
(DY)FWHM
11 .2 19 .6 22 .7 19 .1 30 .0 47 .0
300
C. Andreani et al . / Resolution in deep inelastic neutron scattering
Table 2 Some of the instrument parameters for the direct (HET : High Energy Transfer spectrometer) and indirect (eVS : ElectronVolt . Spectrometer) geometry spectrometers on ISIS Incident flight path (L O ) Scattered flight path (L,) Angular uncertainty (AO) Moderator depth (SR,)
11 .8 m 2.5 m 0.3 , 4 .0 cm
12 .6 m 1 .0 m 0.75 ° 3.0 cm
etry spectrometer [6] in which the energy of the scattered neutron is determined using either the resonance detector [7] or the resonance filter method [8]. A summary of the parameters for both instruments is listed in table 2 . For HET the results are presented for two incident energies : E, = 1000 and 2000 meV . For eVS two possible final energies have been selected and these are determined by the strong nuclear resonances in Sm and Ta . These energies, together with their respective resonance widths at room temperature (FWHM) are: E t =870 meV (AE,=100 meV) and Et = 4280 meV (.1 Et = 150 meV). In terms of the mass of the recoiling particle three distinct materials have been chosen as representative of the changes in resolution which can be expected . Firstly scattering from hydrogen has been considered in some detail : this is in many ways a special case and is of considerable experimental interest . Secondly, and perhaps the system of greatest interest from an impulsive scattering point of view . is the scattering from 4 He . Hohenberg and Platzman [9] were the first to suggest
that deep inelastic neutron scattering could be used as a method for determining the Bose-condensate fraction in the superfluid phase. Since then both reactor and pulsed source experiments have been performed on this quantum fluid [7]. Finally we consider the high mass regime (in recoil scattering terms) by studying the resolution contributions in beryllium. 4.1 . Scattering front hydrogen In hydrogen VI = 1 and eq . (16) reduces to t'L/tb = cos 0 .
(27)
From this equation it should be noted that scattering from free hydrogen atoms cannot occur for 0 > 90 ° . The scattering vector q is given by q = ni ( có + t, i - 2 r, )c, cos O )
2
.
(28)
where m is the neutron mass . From eqs. (27) and (28) (29)
q = mr o sin 0 = mv, tan 0.
Substituting eqs . (27) and (29) into eqs. (46)-(48) we obtain for a chopper spectrometer a r/at~ _ -ni( t,~/L o )(cos = 0/sin 0 )
x [l + (L~/L,) cos -0] .
av/at~,=rat ( c,,/Lo j cos 2 0/sin 0, a,/a0=-nil") cos 0.
(30) (31) (32)
Figs . l a and lb show- the individual resolution contributions and the total p-resolution calculated for HET
0 E
C E T A
Y.
30
4
5C
SCATTT.JJG AUCL.E
60 (DEGREE)
70 >CA7TE2NG A\k-E
(DEGREES;
Fig. 1 . The total resolution ( ) and the energy (---). angular (-- -) and timing ( ) contributions to the resolution in hydrogen ()VI=1 amu) shown as a function of scattering angle and momentum transfer for HET at (a) Eo =1000 meV and (b) E = 2000 meV .
301
C . Andreani et al. / Resolution in deep inelastic neutron scattering
at incident energies 1000 meV and 2000 meV as a function of angle. The q values corresponding to the different scattering angles are also given. As mentioned previously all resolution contributions are proportional to E'' 12 and this behaviour can clearly be seen by comparing the two figures. The resolution at both 1000 meV and 2000 meV is governed primarily by the energy term except for angles in excess of - 60 ° where the angular component begins to dominate . The uncertainties associated with the timing are comparable with the angular term for scattering angles greater than - 30' . For an inverse geometry spectrometer eqs . (27), (29) and (55)-(57) give ay1at m = -m(eiILO ) sin O,
(33)
ay/aE,=-(1/v, sin O)[cos0-(L,/LD)],
(34)
a y /a0 = -mu, .
(35)
Figs. 2a and 2b show the resolution contributions on eVS, calculated for E, = 872.0 meV, AE, = 100 meV and E, = 4280 meV, AE, = 150 meV, respectively . The abscissa gives O and q values . In this case the angular term makes a significant contribution to the overall resolution, particularly for the higher energy resonance where again it is the dominant term for angles in excess of 60 ° . Both figs . 1 and 2 illustrate the overall improvement in p-space resolution as the scattering angle is increased due to the angular variation of ay/aE, . Table 3 summarises the resolution results for both HET and eVS (for O = 60 ° ) indicating that the resonance system
MOMENTUM TRANSFER
Table 3 Selected total resolution results for recoil scattering form hydrogen, helium and beryllium using the chopper (HET) and resonance neutron absorption (eVS) spectrometers on ISIS Mass 1, 6) = 60 °
Instrument HET HET eVS eVS
Energy [meV] 1000
2000 872 4280
Resolution 0 .09 0 .14 0 .82 0 .80
Mass 4, O = 160 °
Instrument HET HET eVS eVS
Energy [meV] 1000
Resolution
Instrument HET HET eVS eVS
Energy [meV] 1000
Resolution
2000 872 4280
0 .46 0 .65 3 .40 2 .30
Mass 9, O = 160'
2000 872 4280
1 .47 2 .07 6 .60 4.48
is only a factor of 6-9 worse than the chopper spectrometer . This is significantly lower than might be anticipated from the energy resolution of the two systems. The energy resolution of HET is - 7 meV at 1000 meV and - 14 meV at 2000 meV. Thus the energy resolution
(DJVERSE ANGSTROMS)
SCATTT7IING ANGLE (DEGREES)
SCATTERING ANGLE (DEGREES)
) and the energy (---), angular ( .--- .) and timing ( . . . . . . ) contributions to the resolution Fig. 2 . The total resolution ( in hydrogen (M =1 amu) shown as a function of scattering angle and momentum transfer for eVS at (a) E l = 872 meV and (b) E, = 4280 meV.
302
C. Andreani et al. / Resolution in deep inelastic neutron scattering MOMENTUM TRANSFER
D E L T A Y
(AVERSE ANGSTROMS)
MOMENTUM TRANSFER
(ANESSE ANGSTROMS)
8
6
4
2
0
Fig. 3. The HET results for helium (M= 4 amu) . of HET is between 10-20 times better than that of eVS. It should be pointed out that much higher q values can be accessed on eVS. It is again clear that the angular resolution is very important and significant improvements in the resolution can be achieved by changes in the collimation of the scattered neutrons . An alternative method of improving the resolution is by arranging the experimental geometry for "time focussing" . This has been discussed for resonance detector spectrometers by Carpenter and MOMENTUM TRANSFER 14 .5
22 .0
29 .7
D E L T A Y
44 .1
49 .7
EIS
E
..
..
2, ~.
~.
When considering the recoil scattering in helium the simplifications introduced in section 4.1 no longer apply and the full expressions for the resolution given in the appendix must be used . Figs . 3a and 3b show the
53 .4
M=4 AU = 872
AE
40
4.2 . Scattering from helium
MOMENTUM TRANSFER
(INVET~SE ANGSTROMS)
37 .2
Watanabe [10] and Rauh et al . [11] . They also point out the dominant effects of angular contributions to the resolution for recoil scattering from low atomic masses .
. . ._ . _ .
' .. r fó
140
=
. ..
meV
100 meV
16U
=
180
SCATTEfUNG ANGLE (DEGREES)
Fig. 4 . The eVS results for helium (M = 4 amu) .
(INVERSE ANGSTROMS)
C . Andreani et al. / Resolution in deep inelastic neutron scattering MOMENTUM TRANSFER
MOMENTUM TRANSFER
(INVERSE ANGSTROMS) 10 .8
20
D E L T A Y
D E L T 17.5 A Y
21 .0
30 .3
38 .4
303
(AVERSE ANGSIRO S) 45 .0
HET
25
50 .1
53 .7
55 .8
M=
9
AU
E
=
2000
meV
15
12.5 15 10
7.5
10
b
5
2.5
0
60
EC
100
SCATTERING ANGLE
120
20
140
(DEGREES)
40
60
80
100
SCATTERING ANGLE
120
140
160
180
(DEGREES)
Fig. 5. The HET results for beryllium (M = 9 amu) . resolution components obtained on HET and figs . 4a and 4b those obtained on eVS. For comparison the FWHM of the atomic momentum distribution in 4 He is - 1.9 l-1-1 . At M = 4 we are already approaching the regime where the p-resolution is dominated by the energy resolution of the spectrometers (see eq . (15) and succeeding comments). We find that the resolution on the chopper spectrometer is better by a factor of 4-7
(for O = 160 °, see table 2) than that for an indirect geometry instrument based on resonance neutron absorption . On eVS the energy resolution dominates; the angular and timing components effectively make a similar and small contribution over the entire scattering angle region of interest . Improvements of the order of 20% or more could be achieved by implementing the so-called double difference method of collecting data
D E L T A Y
SCATTERING ANGLE (DEGREES)
SCATTERING ANGLE (DEGREES)
Fig. 6. The eVS results for beryllium (M = 9 amu) .
304
C. Andreani et al l Resolution in deep inelastic neutron scattering
[12] or alternatively by cooling the analysing foil to very low temperatures [13] to improve the energy resolution component.
tion could be made on both instruments by implementing better collimation of the scattered beam, or by increasing the secondary flight path .
4.3. Scattering from high-mass systems In the case of a high-mass system, e.g . beryllium, figs . 5 and 6 show the components of the resolution for the two ISIS spectrometers . For eVS the p-resolution comes almost entirely from the energy resolution contribution and this would also be true in the case of higher-mass particles. With regard to HET the energy resolution term also dominates the resolution over the entire angular range. From eq . (15) we expect the presolution to be roughly proportional to M for M > 4. For such masses the resolution of the HET chopper spectrometer is only a factor of 2-4 better than that on eVS (for O = 160 °, see table 2) despite the large difference in energy resolution of the two instruments . Due to the limitation on the chopper energies available it is doubtful whether q values in excess of 50 A - ' can be reached. In a harmonic solid the impulse approximation becomes valid when w R >> c,w D, where ca D is the Debye energy [l4] . Since w R cc 1/M large q values are particularly necessary for the IA to be reached in highmass systems. 5. Summary and conclusions Calculations have been presented of the resolution in atomic momentum space (p-space), for both direct and inverse geometry instruments on the pulsed neutron source ISIS . From the general analytic forms for the resolution in momentum space (eq. (15)) and from the numerical calculations it is shown that for large masses the energy resolution contribution is the dominant term and the angular resolution is relatively unimportant . Moreover, the relative resolution (i .e . resolution/ width of distribution) deteriorates as the atomic mass increases. As expected the resolution on the chopper instrument was significantly better than that for the indirect geometry spectrometer based on neutron absorption resonances . However, the resolution when analysed in terms of p-space was better than expected on eVS because of the larger q values accessed . Moreover, these larger q values mean that the impulse approximation is approached more closely on eVS than on HET, which is limited to momentum transfers up to 50 f\ - ', and renders a relatively simpler data analysis essentially free from final state effects . Calculations of the resolution components in recoil scattering from hydrogen reveals that the angular term makes a significant contribution to the resolution on both HET and eVS . Significant improvements in resolu-
Acknowledgements We would like to thank the European Economic Community Stimulation Action scheme for funding this collaboration and Dr . M.P. Paoli and Dr . A.D . Taylor for stimulating discussions.
Appendix p-space resolution for a chopper spectrometer For simplicity we assume that the distance between the chopper and the sample is negligible . This approximation should not affect the results significantly. The largest experimental errors arise from uncertainties in : (1) the time the neutron leaves the moderator, t , ; (2) the time the neutron passes through the chopper, tc; (3) the scattering angle, O. It is convenient to work in terms of vo, the incident neutron velocity, and v l, the scattered neutron velocity, then vO -L Ol( t c - tm) ,
(36)
v, =Lil( 1 d - l ) .
(37)
Lo and L, are the incident and scattered flight paths and td is the time at which the neutron is detected . With units such that h = 1 w = 1/2m ( vó - v2 ),
(38)
q2= m 2 [ vó+v 2 -2vov l cos 19] .
(39)
m is the neutron mass . From eqs. (36)-(39)
I aw/atc = -m(vólL,)[1 + (v /v o)3(Lo/L , )j,
(40)
aglat~ = M 21 Iq(v03ILO) X {(tl/Vo) 3 ( LO/Ll)[ 1 - (volvl) Cos 01 +(vl/vo) cos O- l} '
(41)
a w/at m = mvó ILO,
(42)
ag/atm = (m 2vO)l(Loq)[ 1- v,lvo cos O],
(43)
aw/ao = 0,
(44)
agla 0= (m 2vov1 sin 9)/q .
(45)
C. Andreani et al. / Resolution in deep inelastic neutron scattering From eqs. (15) and (40), (41) :
aylöE, = (mlq)((volul) 3 (Lt/Lo)[ 1 - (Mlm)] - 1
ayl a t = -(m2ó)/(Loq) X { M/m
- (M/m) + (vo/VI) cos 0
[1 + (VIIVO)3(LOILI)l
X[(volvt)(LO/Lt) -1 ]} ,
+ (UI/UO) 3 ( L O/ L t)[ 1 - (VOlvt) cos 0] + [1 - ( v,/vo ) cos O ] } .
ay/a0= -(m2VOVl sin 0)/q .
(47)
Finally
(48)
[21 [3] [4]
p-space resolution of an inverse geometry spectrometer In this case the experimental uncertainties are i10,
At . and i1 El , where EI is the energy of the scattered neutrons . We obtain
aca/at m = -mvó ILO, (49) 2 aq/atm= - (m VO)/(Loq)[ 1 - (Vtlvo) cos O], (50) aw/aE, = - [1 + (vo/Vi) 3 (LI/LO)]
[61
[7]
(51)
aglaE, = (mlq)[ 1 - (vo/vi) 3 (LI/LO) +(VO /vl)2(LI/Lo) cos O
[8]
01 ,
(52)
a q/a0=(m 2vo v l sin 9)1q .
(54)
- ( V,/v, ) cos aw/ao = 0,
(53)
Then from eqs. (15) and (48)-(53)
[9] [101 [111 [121
aylatm = (m2VO)/(Loq) X[1-(M/m)-(v i /vo )cos0 ],
(57)
References
a y /at m = (m 2 V°)l(Loq)
ay/a0= -(m 2 VOUt sin O)/q .
(56)
(46)
Similarly, using eqs . (42) and (43)
X[(M/m)-1+(v l /vo )cos0 ] .
305
(55)
[13] [14]
B. Tanatar, G.C . Lefever and H.R . Glyde, Physica B136 (1986) 187 and references therein . V.F . Sears, Phys. Rev. B30 (1984) 44 . S.W . Lovesey, Theory of Neutron Scattering from Condensed Matter, vol. 1 (Oxford University Press, 1987). J. Mayers, C. Andreani and G. Baciocco, submitted to Phys . Rev. B. A.D. Taylor, B.C . Boland, Z.A . Bowden and T.J .L . Jones, Rutherford Appleton Laboratory Report RAL-87-012 (1987) . R.J . Newport, M.P . Paoli, V.T . Pugh, R.N . Sinclair, A.D . Taylor and W.G . Williams, Proc. 8th Meeting Int. Coll . on Advanced Neutron Sources (ICANS VIII), Rutherford Appleton Laboratory (1984) p. 562; R.S. Holt and M.P . Paoli, submitted to Nucl . Instr. and Meth. J.M . Carpenter, N. Watanabe, S. Ikeda, Y . Masuda and S. Sato, Proc . 6th Meeting Int. Coll. on Advanced Neutron Sources (ICANS VI), Argonne National Lab. (1982) ; also R.S . Holt, L.M . Needham and M.P. Paoli, Phys . Lett . A126 (1988) 373. R.J. Newport, P.A . Seeger and W.G. Williams, Nucl . Instr. and Meth. A238 (1985) 177. P.C . Hohenberg and P.M . Platzman, Phys . Rev. 152 (1966) 198. J.M. Carpenter and N . Watanabe, Nucl . Instr. and Meth . 213 (1983) 311. H. Rauh, S. Ikeda and N. Watanabe, Nucl . Instr. and Meth . 224 (1984) 469. P.A . Seeger, A.D . Taylor and R.M . Brugger, Nucl . Instr. and Meth . A240 (1985) 98 . H. Rauh and N. Watanabe, Nucl . Instr. and Meth. A228 (1984) 147. M.S. Nelkin and D.E . Parks, Phys . Rev. 119 (1960) 1060 .