Nuclear Instruments and Methods 198 (1982) 497-513 497 North-Holland Publishing Company

N E U T R O N S P E C T R A L M O D U L A T I O N A S A N E W T H E R M A L N E U T R O N S C A T T E R I N G

T E C H N I Q U E

I. General theory

Yuji ITO, M a s a k a z u N I S H I a n d K i y o i c h i r o M O T O Y A

Institute for Solid State Physics, University of Tokyo, Roppongi, Minato-ku, Tol~yo 106, Japan

Received 9 June 1981 and in revised form 31 August 1981

A thermal neutron scattering technique is presented based on a new idea of labelling each neutron in its spectral position as well as in time, through the scattering process. The method makes possible the simultaneous determination of both the accurate dispersion relation and its broadening by utilizing the resolution cancellation property of zero-crossing points in the cross-correlated time spectrum together with the Fourier transform scheme of the neutron spin echo without resorting to the echoing. The channel Fourier transform applied to the present method also makes possible the determination of the accurate direct energy scan profile of the scattering function with a rather broad incident neutron wavelength distribution. Therefore the intensity sacrifice for attaining high accuracy is minimized. The technique is used with either a polarized or unpolarized beam at the sample position, with no precautions against beam depolarization at the sample for the latter case. Relative time accuracy of the order of 10 3 to 10 4 may be obtained for the general dispersion relation and for the quasi-elastic energy transfers using correspondingly the relative incident neutron wavelength spread of 10 to 1% around an incident neutron energy of a few meV.

1. Introduction

High energy resolution in inelastic neutron scattering gains more and more importance as the application of the technique encompasses wider fields such as polymer and macromolecular dynamics in addition to the con- ventional studies of elementary excitations in solids. In the latter the determination of highly accurate disper- sion relations together with their lifetime broadening remains always of fundamental importance for under- standing the microscopic states of solids.

The difficulty for attaining a better energy resolution than 10 2 with respect to the incoming neutron energy is generally associated with resolution when utilizing the conventional neutron scattering techniques - triple axis or time-of-flight spectrometry including pseudo-statisti- cal correlation method. In either case, improving the resolution to attain a high accuracy necessitates the loss of neutron intensity.

The two known high resolution inelastic neutron scattering methods, backscattering [1] and neutron spin echo (NSE) [2], have been developed in the last decade. The former backscattering technique in a sense pushes the resolution to the limit, whereby the resultant inten- sity loss is won back by employing large detector solid angles. This necessitates the method being suitable mostly for the momentum-independent scattering processes.

0167-5087/82/0000-0000/$02.75 © 1982 North-Holland

On the other hand, the NSE method first proposed by Mezei succeeded in decoupling tile incident neutron wavelength spread from the corresponding resolution by making use of the spin echo focussing property.

For the investigation of non-dispersive scattering effects, the above two methods are in many respects complementary. It is generally said that for studies of spectra with structures or peaks within a small energy range, the direct energy scan in the backscattering method is much preferable to the time Fourier trans~ form scan of NSE, whereas for studies of line shapes the time domain NSE Fourier method is better with its far larger dynamic range [2].

As for the investigation of the dispersive elementary excitation, NSE needs to know the information concern- ing the dispersion relation from the conventional back- ground spectrometer it is incorporated to, therefore simultaneous accurate determination of both dispersion relation and line broadening is impossible. The loss of information on the dispersion relation comes from the very nature of spin echoing, which eliminates the effect of incident wave vectors except in so far as they receive energy changes due to quasi-elastic or line broadening effects.

This situation can be altered if all the incoming neutrons are labelled in their spectral positions and in time, and are followed through the scattering process. As a natural consequence of such spectral labelling, we

498 Y. Ito et al. / Neutron spectral modulation 1

can combine both characteristics of the backscattering method and NSE for the quasi-elastic scattering prob- lems also.

In this paper we present a new thermal neutron scattering technique based on such a concept, whereby both features of backscattering (direct energy scan) and NSE (time Fourier transform) for the non-dispersive scattering case can be covered, and both the accurate dispersion relation and its width for the dispersive scattering process can be determined.

In section 2, the time and spectral labelling of each incoming neutron will be discussed as neutron spectral modulation (NSM) after which the present technique is named. Then the general theory of NSM is presented in section 3; there the similarity and difference from the neutron spin echo technique (NSE) will be discussed. Section 4 deals with the Fourier method (with respect to time-channel) to regain the response related to the di- rect energy scan of backscattering type. The results of computer test experiments for the channel Fourier method will be presented in section 5.

The resolution problem of NSM is deferred and will be fully discussed in the companion paper II, where it is shown that the incident wavelength spread can be effec- tively decoupled from the final instrumental resolution without resorting to the spin echo focussing. It will also be shown that cancellation of instrumental resolution effects at special zero-crossing points in the cross-corre- lated time spectrum assures determination of highly accurate dispersion relations with their true widths. Some results of NSM experiments will also be included in paper II.

We also mention here that portions of the present work and the resolution problem have already appeared in a prepublication form [3].

2. Neutron spectral modulation (NSM)

The main aim of the new method is to label each incoming neutron such that it can be followed through scattering processes at the sample to its final detection. For this labelling we utilize the neutron spin as was first done by the neutron spin echo (NSE) method [2]. It is not the spin echo focussing property to be used here however, but the one to one correspondence the neutron velocity has with respect to its spin phase angle, which is used to label a particular neutron at its spectral position.

Let us consider the instrumental set-up shown in fig. 1. We have an incident polarized neutron beam initially polarized along the z-direction. The polariza- tion may be created by either one of several available methods such as Bragg reflection from polarizing crystals or reflection from super-mirrors. The beam can have a broad incident wavelength distribution. The polarized

Incident polar ized X: Z neutron beam

OP=Po~ y

I I *----~90°spin turn coil

- T '-

< spin precessing field

Do 0 H

, - -gO°spin turn coil

Q P=PxZ t=O

I *~-- - polarization I =0 modulation x(t)

, - --Analyzer

Fig. 1. A schematic diagram of a neutron spin phase selector for NSM.

beam with the initial polarization P0 travels along the y-direction. The effect of divergence will be taken up in paper ! I as stated in the introduction.

At the position of the 1st 90 ° spin turn coil, neutron spins turn the direction from z to x direction as they leave the exit surface of the coil and start to precess around the precessing field direction (z-direction). Spins will then fan out in the precessing guide field H as fast neutrons lag in the neutron phase angles with respect to those of the slower speed. At the entrance face of the 2nd 90 ° spin turn coil a distance D 0 apart from the 1st coil, the x-component of the neutron's polarization can be written as

P,- = P0 cos B~', (1)

where

B = Y foD°H( y ) d y , "r ---- l / v . ( la)

"), is the gyromagnetic ratio of the neutron, H ( y ) is the precessing field at position y. Thus B is the product of the Larmor precessing frequency with the field line integral. • is the inverse of the neutron velocity, and defines the time a neutron traverses a unit distance, thus a time-of-flight variable.

Y. Ito et al. / Neutron spectral modulation I 499

After the second spin turn coil the polarization of the neutron along the guide field direction z is Pc of eq. (I) and this polarization is held thereafter. It should be noticed that the beam polarization after the second spin turn coil is modulated with respect to ~', or conversely neutrons with a particular z are labelled through the polarization as in eq. (1). Here it is assumed that the efficiency of the spin turn coil is 100% and does not depend upon the neutron wavelength. This is an ideal- ized situation, and in the practical instrument, the wave- length dependence of the efficiency must be taken into account. However the modification is of minor impor- tance in the general discussion.

Since the number of neutrons with spin direction parallel ( + ) and antiparallel ( - ) to the guide field direction (z) is related to the beam polarization P as

U ± = ½N0(l --+ P) , (2)

where N 0 is the total number of neutrons in the beam, the number of neutrons with a time-of-flight r and with the spin direction parallel or antiparallel to z is

N + ( r ) = ½1(r)[1 -+ P ( r ) ] , (3)

where P ( r ) is given by eq. (1) and l ( r ) is the spectral distribution of the incident polarized neutron beam. Eqs. (3) and (1) tell us that the number of neutrons with a particular spectral position and a spin direction can be controlled through B.

It now remains to separate each neutron group Of given r from one another. The separation can best be done by the time-of-flight technique. For it we use the polarization modulation method by pseudo-randomly reversing the polarization direction at the position of a polarization modulator [4]. This gives rise to the starting point for the time-of-flight measurement and the result- ing time dependence of the neutron number in the beam with the spin direction parallel to z, say, can be written a s

N ' ( r , t ) : ½ [ l + x ( t ) ] N + ( r ) + ½ [ l - x ( t ) ] N - ( r )

(4a)

: ½[N + ( r ) -- N - (~')] x ( t )

+ ½ [ U + ( r ) + U ( r ) ] (4b)

= ½I(~) P( 'r ) x ( t ) + ½ I ( r ) . (4c)

Here x( t ) is a pseudo-random time sequence of (1, - 1 ), and we chose here a shift register binary sequence of 50% duty cycle. Other pseudo-random sequences with different duty cycles will modify the above equa- tions accordingly.

We place a polarization analyzer after the modula- tor. The analyzer selects only those neutrons with spins parallel to z with an appropriate reflectivity r ( r ) depen- dent upon the wavelength. Here again the analyzer is assumed to reject all neutrons with spin direction anti- parallel to z. To incorporate a practical analyzer of

finite polarizability is an easy matter. For neutrons reflected by the analyzer, the polarization modulation both in spectrum and in time is transformed to a beam intensity modulation as

~ ( r , t , l ) = N T ( r , t - - I t ) r ( ' r )

: ~ (~) x ( t - !r) + g(r) , (Sa)

where

g ( r ) = ½I( r ) r ( r ) and ~(I") = g ( r ) P(~'). (58)

l is the distance measured from the modulator position.

3. General theory of N S M technique

As we have succeeded in labelling each incoming neutron in spectral position as well as in time, all we need to do is to follow the label through its scattering process to its detection in time. We place our sample in the modulated beam at the position ! = l~. Immediately after the scattering into the direction of the scattering angle 28, the intensity of the scattered beam will be- come [5]

~ ( r ' , t , l , ) = a ( r , T') ~ d r , (6)

where ~ is given by eq. (5a) with l = lx and o(r , r ' ) represents a general scattering cross-section. Neutrons with a certain range of r are now transferred to r ' upon scattering, and the distribution of r ' at a detector posi- tion at a distance la from the scatterer becomes

( r ' , t , / d ) = ~ ( r' , t - - ldr', l~). .(7)

However, the detector cannot differentiate neutrons with different time-of-flight values. The counted neutrons at time t at the detector are given by

z ( t ) = f o ~ (r ' , t, ld) d r ' (8a)

oo

fro(z, r') +(~) x( t - r) d r d r ' 0

f f o(r, r,) g(,r) dr dr ' , (8b) + 0

where T = l~r + ld'r' is the total flight time. As the purpose of any scattering experiment is to

gather information on o(r , r ') , we take a cross-correla- tion between the observed intensity z ( t ) and the input modulation pattern x ( t ) as

K( s ) = for"'z( t ) x ( s -- t) dt (9a)

= f f o ( r , r ' ) e ( r ) 0

× ~ , ~ [ s - T ( r , r ' ) ] d r d r ' + C, (9b)

500 Y. I to et al. / Neutron spectral modulation 1

where

• ,,(s)--foTmx(t) x(s-- t) dt (9c)

is the auto-correlation of the input pseudo-random se- quence, T,,, is the total measuring time and C is a constant giving a background independent of time channels. We can neglect the effect of C without loss of generality. In practice, the integrals in eqs. (9a) (9c) are summations over available time channels, x(t) being a pseudo-random binary sequence (PRBS) of a finite length, the auto-correlation function can be expressed a s

• , ( s ) = f ~ ( s ' ) 6 ( s - s ' ) ds I (9d)

with a time window function ~(s). Then eq. (9b) can be rewritten as (neglecting C)

I) 0

XS(s - s ' - - T) d r d r ' d s ' (9e)

- (9f)

where

R(u)= ffo~a(9-,r ')~(r)8(u-- T)drdg- '. (10)

We are assuming an idealized experiment, in which any total flight time uncertainty other than the time resolution of the chopping spread represented by ~(s) may be neglected. This implies negligible angular di- vergence both before and after the scattering, negligible flight path differences in sample and detector as well as negligible field inhomogeneity in the precessing field and negligible mosaic spread in the sample. The effect of these on the resolution will be taken up in the subsequent paper. In the ideal experiment, the total flight time T observed at a particular time channel u defines a group of neutrons with time-of-flight 9- -+- 8r to be scattered to 9-' ±gr' such that

T=l~r+ld'r', ( l l a )

and

8T = (~89- + ld~'r' = 0. (11b)

In the usual time-of-flight method, the incident neu- tron time-of-flight is fixed at a given monochromatic value % and hence the total flight time measurement automatically gives the other scattered time-of-flight distribution 0(%, 9-'). Extension of this to the case of a polychromatic beam is seen for the two-dimensional correlation method [6], where two statistical choppers placed before the sample at a definite distance from one another modulate the beam. The total flight time is measured from the first chopping point, whereas the second chopper gives the time origin for -r'. This essen-

tially puts labels on the neutrons with r scattered to 9-'. However, in this way the labelling on ~" must be

coarse enough to allow sufficient neutrons to be scattered to 9-' to be detected. In the present NSM, all the incident neutrons are already labelled, so a single mea- surement of T suffices to determine both 9- and 9-', and the finer the labelling, the better is the resolution without detriment to intensity.

The 9-, 9-' which are involved in scattering from elementary excitations in a sample must satisfy both the energy and momentum conservation relation,

D I - - 3 " f

~ = 5 ~ ( 9 " 2 ~), (lZa) Q=k i - k f , (12b)

o r

i Q l ~ = ( m / h ) 2 ( 9 - z + 9 , 2 2T 'T' ' c o s 2 0 ) , (12c)

with ki, kf the incident and scattered neutron wave vectors, respectively.

The quantity normally related to the Q, ~o excitation in the scattering process is the scattering law S(Q, w), in terms of which the cross-section o(9-, 9-f) given in eq. (6) can be expressed as [7]

_ _ O a /'?! 9 "

a(9-,9-') 4rr~ ~ 4 s ( e , w ) , (13)

where o~, is taken to be either the incoherent bound atom scattering cross section or the coherent bound atom scattering cross section depending upon whether the scattering process is incoherent or coherent. Since it is just a number, we neglect the aa/4~r factor for sim- plicity in the discussions to follow.

Eq. (10) now becomes

m f f g- K ( u ) = - h - ~Tgs(o,o~)O('r)8(u-T)dg-dg-',

(10a)

where the lower integration limit is extended to -- ~ for convenience with q,(~') = 0 for 9- < 0. To proceed further than the above equation we must consider each scatter- ing process separately.

3.1. Elastic scattering

In this case the scattering function becomes

S(Q, ~) = S(Q) 8(~), (14)

and the two 8-functions in eq. (10a) can be calculated as shown in appendix 1 to give

P0 /~ (u ) = I S ( Q u ) ~ ( % ) =~S(Q, , ) g(%) cos Bg-.,

(15)

where L = I s + l a is the total flight path length from the

Y. Ito et al. / Neutron spectral modulation 1 501

modulator position to the detector, and % = u/L and

m 2 s i n 0 = h _ ~ s i n 0 ' (15a)

For an incoherent scatterer S(Q,)= constant, then eq. (15) simply states that the time spectrum observed by the detector time channels reflects the incident spec- tral modulation of the beam.

3.2. Quasi-elastic scattering

We consider the elastic scattering discussed above as a reference process, and measure any • from % = u/L as z = % + 3,. Upon quasi-elastic scattering, ~" changes to 1-' = , , + 8 , ' , where 3, , 8 , ' satisfy eq. (1 lb). A small quasi-elastic energy transfer can then be written from eq. (12a) as

m 1 ( 3 , _ 3 , , ) = m 1 £ 3 , . (16) to h .2 h , : l d

The basic eq. (10a) is now approximated to first order in 3 , / , , as

l F l((u) ~-~ _ooS(Q,, to) 0 ( ' ) dto, (17)

with I Q.I given by eq. (15a). This reduces to eq. (15) for the elastic case as it should.

We next expand the incident time-of-flight distribu- tion function g( , ) with respect to the small variable 3 , at %.

g ( , ) ~-g(%) + g ' (%) a , , (18)

where g' = dg/d , . Assuming an even function of to for the quasi-elastic scattering law S(Q, to)(taking the de- tailed balance factor as effectively unity), we get,

I ( ( u ) .Z ( Po/L )[ g( % ) cos B,uF( ~',, B)

- -g ' (%) sin B%G(%, B)], (19a)

" 2 F(%, ) = f S ( Q , , t o ) cos[tot(z,)] dto, (19b)

a(.., B) =,,. S(Q.,to) osin[tot(..)ldto, (19c) oo

where

ct ,=-- a n d t ( % ) = B a u . (19d) r n

Usually a quasi-elastic S(Q, co) is a smoothly varying function of to with maximum near to ~-0, and going to zero for larger to. Hence in general G << F. With this approximation we get

. e o K( u) ~-- z g ( % ) cos B%I( Q,,, t)

1 = ~q~(%) I (Q, , t), (19e)

where the intermediate scattering function I(Qu, t) is

given by eq. (19b), as a cosine Fourier transform of S(Q,, o).

Let us consider the meaning of eq. (19e). From eq. (15a) each time channel u corresponds to a specific Q-value. For a fixed B, which d~termines the t-value together with the particular time channel we are consid- ering, we can deduce I(Q,, t) since ~(%) in eq. (19e) can be determined from the separate elastic incoherent scattering measurements. This should be compared with the situation in NSE, where the beam polarization which is essentially the K(u) integrated over % is what is observed. Different time channels will give different (Q, t) 's with very fine steps, whereas wide mapping can be done by changing B for t and 2 0 for Q. This is precisely the point discussed by Mezei, when he combined the NSE method with time-of-flight measurements [8]. However, with the NSM method, these features come out naturally without echoing.

Following Hayter and Penfold [9], we now take the specific example of scattering from a system undergoing self-diffusion, for which the scattering law is Lorentzian

DQ 2 S(Q'to)=~r[(DQ2)2 + to:] ' (20)

with D the diffusion coefficient. Performing the to in- tegration at a fixed Q. (that is at a particular channel u), we get

I ( e , , t) = ¢xp[-DQ]t(%)]. (21)

The exponent can be rewritten as

DQ2,t( % ) = DflB%,

where

(2 sin 0 ) 2 ~ m

f l = ~-- . ( 2 2 )

Eq. (19e) then becomes

/ ( ( u ) = ~_ g(%)P0 cos B,, exp( - ORB%). (23)

Eq. (23) can be used to describe the difference be- tween NSM and NSE. Let us fix at any time channel u 0 = L , °, for example at ,o giving a maximum g(,O); then B , ° = 2~rN 0 is also fixed, N O being the total num- ber of Larmor precessions for a particular neutron group ,o, and changing B to vary N O gives the symmet- ric spectrometer configuration of NSE. On the other hand, variation of N from N O as a function of % (that is of u) is the orthogonal asymmetric scan. Thus NSM always gives the asymmetric type scan as a function of % by a single time-of-flight correlation measurment. Always the raw spectral modulation of the incident spectrum, not a transform, appears directly in the time spectrum.

It should also be noticed that in NSM there is no echoing and no need to keep AN 0 = 0 for the N O scan.

502 Y. Ito et a L / Neutron spectral modulation 1

Each time-channel corresponds to a different Q as mentioned above with a differing D value. The ad- vantage of using the spin echo focussing property will become apparent only after considering the resolution effect of beam divergence as will be discussed in the companion paper II.

3.3. Sharp excitations with dispersion w(Q)

In this case we have definite dispersion relation with no life time broadening effects, so the scattering law may be expressed as

S( Q, oa ) = S ( Q ) 8( oa -- OaQ ). (24)

Let us examine the meaning of WQ. The excitation under consideration has a definite dispersion relation, which we write generally as f (Q). Since Q is defined by eq. (12b), for the fixed scattering angle 20 and the crystal orientation ~ in the reciprocal space as given in fig. 2, f ( Q ) becomes a function of (r , r ' ) as

f ( Q ) = f Q ( r , r ' ) lee,,tixed. (25)

Thus in the three-dimensional space of w, r and r ' , oa =fQ(r, r') gives a surface representing the dispersion. The energy conservation relation of eq. (12a) gives another surface in this space as shown in fig. 3. The crossing of the two above surfaces gives rise to a curve 09Q also depicted in fig. 3. A projection of the WQ curve upon the ( r , r ' ) plane gives r ' as a function of r, which we write as o( r ) . o ( r ) thus contains the same informa- tion as the scattering surface in k-space. In fig. 3, we also show the T = u curve in the (r , r ' ) plane, whose projection onto the energy surface gives ~, . The cross- ing point of £0Q and o~. [or equivalently T = u and p(r)] is denoted by f.. The bar over r. is needed to differenti- ate the present case from the one discussed for the elastic case.

Now we have wQ as a function of r only as

~ ' e = ~ r 2 0~(r)

& "2

¢,

bl

Fig. 2. The relation between k i, k f and Q in the reciprocal space spanned by b I and b 2. ,# is the angle between k i and -- b I, whereas 4' is the one between ki and R* which is a fixed reciprocal lattice vector to define q.

Ito

"C'

~" = ~:,

Fig. 3. A schematic illustration of the energy transfer surface in (w, r, r') space. Only the portion for w~0 is depicted for clarity. For the meanings of ~q, w., o~', p(r) and "~,, see the text.

for fixed 20 and 4,. Combining eqs. (10a), (24) and (26), we have for K(u)

1% I~ ( u ) = ~7 ~ S( Q . l ck( ¢.)

_ Po "~. S ( Q . ) g ( f . ) cos Bf . , (27) L' p.

where

0,,=o(~.) L' = I~ + pu'ld and O.' = do~dr I . . . . (28)

The detailed derivation is given in appendix 2. The example of p ( r ) is given in the figure of appendix 3 for the spin wave dispersion of f ( Q ) = Dq 2 for illustration.

Eq. (27) gives the essential feature of NSM to de- termine w(Q). Although intensity and profile of the time spectrum are modified on scattering, it always carries the incident spectral modulation pattern as a function of ~., which is the solution of u = lse u + ldP(?u).

This point is illustrated in fig. 4. The incident neu- trons are labelled by spectral modulation as shown in the elastic time spectrum. From the inelastic time spec- trum we can alway decide, at least for a few points in the spectrum, the u which corresponds to a particular ~. in the original incident pattern. For this correspondence zero-crossing points (ZCP) are utilized. Once "~. is de- termined, the difference of the time channels is given by

u - Le, = la(p . - f , ) , (29)

which gives Q and WQ from eqs. (12b), (12c) and (26). We have the means to control the spectral pattern by changing B, so in principle, we can scan the whole time spectrum using the ZCP as a scanning parameter.

Y. ho el al. / Neutron spectral modulation I 503

¢

R(u) I /"i'/'. ~ijii",,, C(u)l__

, h(,g,

Elastic time spectrum

time channel

Inelastic time spectrum

T=uO~i'"..

d(/'~, - ~-'~(2))i

Fig. 4. An illustrative example for determining ~,,, P, and hence Q, and w~2 as explained in the text.

3. 4. Dispersive excitations having a finite width

')

~ ....... " ~

Fig. 5. The projection of the energy transfer surface (fig. 3) onto the (w, r) plane. The relation between the effective energy width h(~') and the true width /~(w') is shown schematically. Always the true width can be obtained for the quasi-elastic case.

Instead of eq. (24), we approximate the scattering law for the present case as

S( Q, ~o ) : S( Q ) h ( w - ¢OQ), (30)

so that h(w'), which represents the line shape function of the dispersion with w' = co - ¢oQ, is independent of Q for the range of w under consideration. With eq. (30), eq. (lOa) can be written as

oo mfff, K(u) =--~ _ ~ S ( Q ) h ( w ' ) 8 ( w ' - w +,.,O)

× B ( u - T) ~b(r) d~" d~"d o~'. (31)

In view of eq. ( l l a ) for T, the integration over ~-' gives for ~ with the second B-function in eq. (31), and with eq. (12a),

~o=w - - ~ - , ~ , ~'~=~(u-/~,). (31a) " r u !

The first &function then gives for w'

w ' = w --wO. (31b)

This is shown schematically in fig. 3 for r = ~, - Br for illustration. The width of ~ ( Q ) lies in the w energy surface and is measured here from the peak denoted by WQ as given by the above equation.

This is further shown in the (w, z) plane in fig. 5. It is obvious there that the true width can be obtained only for cases when WQ is flat with respect to the z axis, or, equivalently, it is non-dispersive locally around r - - ~,. This condition is always fulfilled for quasi-elastic scattering as indicated in the figure. In the general case treated here, what is observed as h(w') is the effective

linewidth and it is related to the true width/~(w') as

1 h ( w ' ) -- h (w ' ) ¢(1 + p 2 ) ' (32)

where

P = d7 ,=e. h - P" , (32a)

Pu, P~ given as in eq. (28).

As discussed in section 3.3, f, , p, and therefore O" are all measurable quantities by making use of the scanning scheme of the zero-crossing points. We can thus de- termine the true width of the dispersion from eq. (32) provided the effective width h(w') is obtained.

For that purpose, we solve eq. (31) to the first order in 8~/~-, as shown in appendix 4 to give

I ( (u) =-£7 ~ S ( Q , , ) f h ( w ' ) ~( ' r ) dw' , (33a)

with

" r = e , , + B ' r a n d w ' - m 1 L' h 03 I d B'r. (33b)

L' is given by eq. (28). For the quasi-elastic case, the above equation is naturally reduced to eq. (17).

We follow the same discussion given for eq. (17), and expand the g(~-) with respect to 8~- at f,. Assuming again an even function of w' for h (~ ' ) we get similar to eq. (19a)

/'of. K ( u ) ' ~ 7 ~ S ( Q , ) ( g ( f , ) cos Bf , F(?,,, B)

-g ' (?,) sin Bf, G(f , , B ) } , (34a)

504 Y. ho et al. / Neutron spectral modulation I

F(%, B) =f_~ h(oJ') cos[ ~ ' t ' (?~)] d~ ' , (34b)

f? G(e. B ) - ' ~ ' s in [ ' ' - (34e) , - a . h(w') w t (7.)] dw',

where

h l o 3 ' = - - = Ba , (34d) a . m ~7#~ and t'

Generally G << F holds as before and we get corre- sponding to eq. (19e)

R(u)-L--P° ~. - ~ S ( Q . ) g(rr.) cos BZr~ H ( t ' ) (35a)

_ 1 ~ S ( Q ~ ) 4~(~) H ( t ' ) , (35b)

where H(t ' ) given by eq. (34b) is the cosine Fourier transform of the lineshape function h(~0'). Comparing the above equation with eq. (27), we have the procedure to observe both the dispersion relation ~(Q) and its Fourier transformed effective width H(t ' ) in the follow- ing way.

First with the zero-crossing point (ZCP) at a particu- lar u, we determine Q and ,.,0 as before. Using the ZCP as a scanning parameter, L' and a~, can be determined. Next at the fixed u-channel we measure the intensity change of/~(u) as a function of B (that is the precessing field strength). H(t ' ) is then obtained from the ampli- tude modulation (damping) of the cos Bq~ oscillation. Finally from eq. (32) we obtain the Fourier transformed true width function as

1 (35c) t?l(t ') = " ( t ' ) ¢(1 +p2)'

where p is given by eq. (32a). We emphasize here that in NSM, eq. (35a) holds for

any dispersion relation for which 0,' is finite and L' is non-zero. (Even when L' = 0 for a certain u-channel, it is always made non-zero by changing the ls / l d ratio.) This is because NSM does not require the focussing property, whereas it is essential for NSE. The latter method requires both a special tilt magnet arrangement of the triple axis spectrometer to measure H(t ' ) and a rather precise knowledge of ~0(Q) beforehand for pre- selection [ 10].

It should be noticed that eq. (35a) [and eq. (19e)] holds 4vithout condition G << P (and G << F) for those f,'s where g ' ( f , ) = 0 , for example at maxima of the g(r) distribution. However, for these equations to be valid, the other condition which states that h(~0') is even with respect to ~o' [S(Q~, ~0) is an even function of to] must be met independently. If that is not the case, eqs. (35a) and (19e) must be modified as

e 0 f , . . f:( u) = Z7 ~ s( o_~) g(fg[cos 8f~ H(t')

- s i n B~, H~(t')], (35a)'

P0 Pt") = ~-g(%)[cos B% I ( Q . t) i . quas i - e l .

--sin B'GI~(Q,, t )] , (19e)'

where //,(t ') and l~(Qu , t) are the sine Fourier trans- form of h(oF) and S ( Q , , o~) respectively.

Although it is possible to separately obtain H(I ) and Hs(Is) by eliminating alternately the cosine or sine term with variable B, this is far more complicated since the simple correspondence of ZCP loses its meaning. In this respect the Fourier transformed time domain measure- ment is not suited for the scattering function which may not be approximated to have a symmetric line profile. And this point holds true also for the NSE method.

Another method of data analysis is unique for NSM, which avoids the above difficulty and also provides the direct energy scan profile corresponding to the impulse response such as the one observed in the backscattering method. This will be taken up in the next section.

4. Channel Fourier transform of N S M data

In the previous sections we developed the NSM theory, which in many respects goes parallel with the NSE method. In this section we give another treatment of NSM data, which is basically an approximate treat- ment. However, with a judicious choice of parameters, it can provide an almost correct impulse response of the direct energy scan giving access for NSM to such excita- tion spectra that possess structures in the narrow energy range.

4.1. Elastic incoherent scattering

We start from the elastic incoherent scattering as the reference spectrum. The intensity at each time channel u is given by eq. (15), where now S ( Q , ) = C O (constant). In practice the u comprises a sequence of equidistant N channels.

Let us Fourier transform /~(u) with respect to u. Denoting the Fourier transformed function by underlin- ing, we have

l~eJ(v ) _ 1 e iuv d u = CoPo[g ], (36) - ¢ ( 2 ~ )

where

1 I-g] -- ~/(2~') ('ru) eiuv du

= ½[g(Lv + B ) + g ( L v -- B)], (36a)

and

1 f~_oog(~ ) e i~" d~. (36b) _g(v) -- 4(2~r1

%= u / L is used. g is the Fourier transform of the

K lto et al. / Neutron spectral modulation I 505

incident neutron time-of-flight distribution. The spectral modulat ion essentially shifts the g func-

tional form along the Lv axis in both positiqe and negative directions by an amount B, Of course in the actual calculation, the integral transform must be re- placed by a Fourier series of N terms as will be dis- cussed later.

4. 2. Quasi-elastic scattering

The equation we deal with here is eq. (17) for the quasi-elastic case. From the outset we consider such quasi-elastic excitations that have little Q dependence around the energy transfer region of interest. Therefore we denote S(Q,,, ~ ) = S ( 0 ~ ) in eq. (17). The Fourier transform of eq. (17) becomes

1 f f s ( ,~)~( . .+3~. )e ,U~.d ,odu, / ( ( v ) - ¢ ( 2 ~ - ) ~ o~

(37)

where % = u / L as before and 3% is given by eq. (16). We rewrite 3% to show explicitly the u dependence as

3% - h ld r~w = d.¢0. (37a) m L

This dependence of &. on u coming from the non-linear dependence of ¢0 on the time-of-flight variable com- plicates eq. (37). We employed ~ . = - a . of eq. (19d) for convenience.

Conversion of eq. (37) to a convolution form can be done by expanding % at the point r ° , which gives the maximum for g(z~). From eq. (37a) we have for rea- sonable values of A = r. -- ~o

8% = ~0,~ (1 + A / , o ) ' _~ a0,~ (1 + 3 a / V ) , (37b)

where

~0=a.(V). Inserting eq. (37b) into eq. (37), we get to a good approximation, as shown in apendix 5,

× exp [ -- i Lv O~oW { d~,

(38)

where [8] is given by eq. (36a) and 8~ "° = aow. The fact that [8] is in the integration indicates that it is still related to the u-integration through A as discussed in the appendix.

Only when the last two terms in the curly brackets are neglecte d compared to 1, eq. (38) is reduced to the product of Fourier transforms as

K( v ) = PoS_*( Lvao) . [g ] , (39)

where the asterisk means the complex conjugate. Now we compare eq. (36) with the above equation.

Normalizat ion aside, at each Lv value we can determine the corresponding channel Fourier component of S(w) as S*(Lvao) by simple division. The absolute intensity calibration may be effected by the known standard scatterer separately. The range of any Lv values can be covered by the spectral modulat ion B. The narrow incident wavelength distribution gives the correspond- ing broad g and a wide range of Lv can be covered by few discrete changes of B. On the other hand if a broad incident spectral distribution is used, we have a narrow peaked g and many steps of B are needed to obtain a complete channel Fourier set for S*.

Once S* is obtained for N channels of v, it is a simple matter to invert the Fourier transform to regain the S(w) as

L°t° f ~ c S * ( L v a o ) eiL . . . . dv . s(,~) =Tg (40)

Here we have the procedure to obtain any S(w) as a function of w, to a very good approximation, provided eq. (38) is reduced to eq. (39). In observing the struct- ural details of S(w), we employed the incident neutron wavelength spread (m), which is much larger than the energy transfer width (~0) , where the bar indicates the measure of, e.g., half-width. This circumstance is made possible only by using B modulation. The illustration of the above argument will be made with concrete exam- ples by computer experiment in section 5.

Let us go back to eq. (38) to examine the condition for which it is linearized to give eq. (39). As we are mostly interested in cases when the energy transfer ~ is rather small, i.e. h w / E o ~ 10 -3 with E 0 the incident neutron energy, we can put the last term in eq. (38) as

3"r ° 10 -3 and can safely neglect it relative to the h term. LA is the channel shift relative to the central channel u 0 = L r ° where the incident neutron group of the most probable time-of-flight (wavelength) arrives. How far away the A must go from the center is dictated by the z distribution of the incident beam, since only by the adequate coverage of g ( r ) by the A, we can have good g values.

For the broad g(¢), the correction term 3 A f t ° may become non-negligible. Adding to the many steppings of B variation, the resultant distortion of S(w) may become unacceptable. A narrower gO') is preferable from the point of view of both better approximation to arrive at eq. (39) and fewer B variation steps. However, for sufficiently narrow g ( r ) of say ~ < ~ 0 , we have the situation when the backscattering method prevails and there is no merit in employing NSM. We have the intermediate case where A f t ° is few percent, which is the most usual one for conventional spectrometers.

It should thus be noticed that with the NSM method we can reproduce reasonable S ( ~ ) structural details

506 Y. Ito et a L / Neutron spectral modulation I

with an energy resolution better than 10 -3 by sensing as much high frequency Fourier components of S* as B permits with rather a relaxed incident neutron wave- length spread. How far in the high frequency range one can reach by varying B depends on the instrumental resolution other than the incident wavelength spread, and this is the problem to be discussed in the paper II. However, for the sake of computer experiment we men- tion that one of the limiting instrumental factors comes from the elementary pulse width of the pseudo-random modulat ion or, equivalently, from the spread of the auto-correlation width. The effect of A on the S(~0) deformation will also be studied in section 5.

4. 3. Sharp dispersive excitations having structures

We are now in a position to treat a much more general case of sharp dispersive excitations which have fine structures in a narrow energy range. An example would be two close dispersion curves near their crossing point. Again we assume from the outset that the Q dependence around the energy transfer region of inter- est is of a smooth nature, so that the scattering law can be approximated as S( Q, , ~ ) = S( Q °, ~o ), where Q0 = Qu(e ° ) and T ° is the most probable incident neutron time-of-flight labelled by the ZCP procedure discussed in section 3.3. In the same spirit we approximate eq. (33a) as

l, ,°s(oo ) fh( o,) (41) po

where a / is given by eq. (33b), and L~ = L ' (¢°) , P0 = p(,ro). What is different from the previous case of quasi-

elastic scattering is that the channel variable u is now non-linear with respect to ~,, so it must be linearized first. This is done as follows

u = u o + L~A', (42)

where

Up = ;~fo + lap0 ' A' = % -- ¢ . (42a)

With these, 8% can be expanded as for eq. (37b) to give

po PO o ] A'j k

= a'o*o' 1 + , (43)

where

h pS ° la 1 _ 3p; " la (43a) - - - - t , t a'° = m Lo 71o Po Po Lo

and

d:Pl (43b) P ; ' = d r 2 [,=~.o

From eqs. (42a) and (43) we write for ~ ( r )

q~(r) = 0(¢u + 6%) ---- q~ + a ' o w ' + A ' 1 + . 31o

(44)

Taking the channel Fourier transform of eq. (41) for appropriate channel intervals of say N ' time channels covering all the re levant /£(u) ' s around Po, we get

1 ¢o

. ( o) f f / ( ( v ) = 7 - S Q~ h( to ' ) ¢ (2~) L~ Po oc

X q~(,'r) e i''~ d o / d u , (45)

Putting eq. (44) into eq. (45), and taking the same change of variable as previously done for eq. (38), we get to a good approximation

,~0 oo

,'o h ]'

Xexp[-iL'oa'oVW'(l+l(A'+3r°)}]dto ', (46)

where

~o = Id(Po -- PfrO~) and &r~ ° = a~0' . (46a)

The prime in [g]' indicates that the argument of Lv in [g] in eq. (36a)-should now be replaced by L'oV: For the q~asi-elastic case, the above equation reduces to eq. (38).

We note that in deriving eq. (46) [and eq. (43)], P('~u) is smoothly varying over the region of interest at .~o, therefore ]p;'] << 1. Only when the term 1 / % ( A ' + 8r ° ) is neglected relative to 1, we again arrive at the product of two Fourier transforms as

l ¢ ( v ) ~--Po~°S(Q °) ei~°h*(L'ova'o) .[g] ' . (47) - - P0

N o w it is more difficult to obtain the desired infor- mation h* from eq. (47), since simple division by the elastic incoherent scattering as calibration is not valid here for two reasons. The first is the change of the scale from Lv to L;v , and this may be corrected for with proper scaling of [g] to [g]'. The second is due to the change of the instrffmental-resolution effect and for this there is no simple solution and this point will be dis- cusse in paper II. From the ZCP procedure we can get a fair estimate of 77o from experiment, and with the known value of ~70, the amount of the correction term ( A ' + 8 r ° ) / r /0 which introduces a distortion for h* may be estimated. For energy transfers 8r°<<A '. i-t will not generally introduce any serious error for A'/~/0 being of the order of a few percent as shown in the next section.

Y. lto et aL / Neutron spectral modulation 1 507

5. Computer test experiment of NSM channel Fourier method

For the computer experiment we consider an asym- metrical S(to) which is composed of an elastic and two close side-band (inelastic) peaks, each made up of a &function as

S ( t o ) = e 0 6 ( t o ) + c , 8 ( w + t o 0 ) + c _ , 8 ( t o - t o 0 ) , (48)

where c 0, c~, c ~ and too are appropriate constants. In terms of the channel number variable n, the above

equation can be rewritten with the help of the time window function ~5(n) introduced in eq. (9d) as

S . = ~ ~ ( n - m ) S m o 1 - - - - o o

= e o ~ ( n ) + c , ~ ( n + s o ) + C , ~ ( n - s o ) , (49)

where S,. is the S(to) converted to the time channel variable as

S,, = Co8,..o + c~Sm.-~o + c ,rm,so, (49a)

with a0to 0 = s 0, and ~(n) is given by triangle function having fwhm of A A channels. The S, depicted in fig. 6 thus constitutes the fictitious impulse response of an ideal experiment with the finite auto-correlation time window A A = 3 channels.

The purpose of the computer experiment is to see how well the S, is retrieved by the NSM channel Four- ier method under the assumed condition which mimics the real experimental situation.

o

io

o

o

tad T -

O O o

o

-~25. O0 -IS. O0 -5.00 5.00 15.00 25.00 OIIEGA

Fig. 6. A fictitious impulse response of S,,(~o) in time channels under the ideal experimental condition having a finite auto-cor- relation represented by triangle function with a time window A A of 3 channels.

5.1. Test experiment

For the assumed experimental condition, we take the incident neutron beam having the most probable time- of-flight ~'0 -- 600 # s / m (roughly E 0 --~ 14.5 meV) with a relative spread of 1%. The total flight path length L was chosen as 5 m, and the unit of a time channel is 1/~s. The s o being 6 as shown in the figure then corresponds to the relative time shift of 2 × l0 -3, or equivalently to one fifth of the incident wavelength spread. This choice was made since it is within the convenient access to the PANSI spectrometer [1 l] available to us.

First we construct the g,(~-) representing the spectral modulated incident beam. One example is shown by crosses in fig. 7, again with the time window function convoluted. The solid curve in the figure represents the incident time-of-flight distribution g(~') here approxi- mated as Gaussian. The modulat ion B is chosen at will as 0.3/channel , which corresponds to 0.82 × 104 Oe . cm or some 143 precessions for r o neutrons in the precess- ing field.

Following section4.2, we calculate the channel Four- ier transform K ( v ) [ is taken out since the /~ (u ) is now convoluted with the auto-correlation to give K(s) as in eq. (90] similar to eq. (37) as

j = O m = - - e ¢

where M 0 = % L = 3 × 103 and N = 189 channels and fixed to make use of our F F T program.

The results of the calculation are shown for illus- tration in fig. 8a, b and c by crosses for B values of 0.1, 0.5 and 0 .9/channel respectively. The solid curves rep-

o

O

X o

o

~ l ÷ ÷ TAU ÷ ÷

I'L-mo j ÷÷~ ÷÷

?

Fig. 7. An example of the spectral-modulated incident beam is shown by crosses. The modulation B is taken as 0.3/channel. The fine solid curve represents the incident time-of-fright distri- bution g(,Q approximated as Gaussian.

508 Y. lto et aL / Neutron spectral modulation 1

,5

o o

T

x~ (5

w

r,:l o

L °0. O0 10. O0 20. O0 30. O0 4,0. O0

v

o

(b)

, . . . . i . . i i , . . . , . . . . v . . . . r . . . . i

c~. O0 tO. O0 20. O0 30. O0 ¢0. O0

v

o

o

{ D

dO. O0 tO. O0

(c)

20.00 30.00 tO. O0 v

Fig. 8. Examples of the channel Fourier transform K(v) as a function of channel variable v are shown by crosses for B values of 0.1/channel (a), 0.5/channel (b) and 0.9/channel (c), respectively. Fine solid curves represent the Fourier transforms of the spectral modulated incident beam such as the one shown in fig. 7 for the respective B value multiplied by the Fourier transform of the auto-correlation time window function. A certain modulation B sees only a portion of K(v). Only magni- tudes of K(v) are plotted for simplicity.

resent [g] of eq. (36a); that is the Fourier transform of i f ( r ) given for example by fig. 7, properly convoluted by the window function. The change of B obviously scans the portion of the _S*(Vk), the undistorted part of which directly obtained from S,, of eq. (49) is shown for comparison in fig. 9. The envelope function given in the figure by the solid curve is the Fourier transform of the auto-correlation time window function. Notice the change of scale of time channels. Notice also that only the magnitudes are plotted for the Fourier transforms for the sake of simplicity.

In order to obtain the approximate S*(vk) both its magnitude and phase by dividing _K(vk) by [g]k chan- nel by channel, it is a matter of practical interest to see how far B can be varied without losing too much intensity, since the S_*(Vk) invariably goes to zero at k o = N / A A. k 0 = 6 3 for the present case as seen in fig. 9. In addition to the above resolution restriction, we have the problem: how many times the B modulation has to be repeated to cover the appropriate range of v k set by the above restriction. This problem is certainly related to the ~,idth of [g], which is inversely propor- tional to the incident wavelength spread.

For the first problem we have the practical measure for the range of k channels up to k, , ~ r / ( 8 In 2) k~, introducing the termination errors for the neglect of the rest of the channels, k, , ~ 36 for the present case. The measure is obtained from the fact that the B modulat ion is limited when the oscillation period .seen in fig. 7 becomes as small as the auto-correlation window width. The details of the argument are deferred to the discus- sion of reolution problems taken up in the forthcoming paper II. At k, , the value of the envelope function of fig. 9 is reduced to about 40% of the peak value at 0th channel.

The second problem defies any clear-cut approach except that it must give reasonable values of S*(v~) for all k ' s between the 0th channel and k,,,. For a start we choose the interval of B such as to get well-overlapped

o

x

O31

C~O. O0 20. O0 40. O0 60, O0 80, O0 ! 00. O0 v

Fig. 9. The Fourier transform (magnitude) of the fictitious S,(oa) shown in fig. 6. The solid envelope curve represents the Fourier transform (magnitude) of the auto-correlation triangle function.

Y. lto et al. / Neutron spectral modulation I 509

(D t~

CD o

925.00 -ts.oo -s. oo r i

5.00 Ig. 00 2S. O0

OMEG^

Fig. 10. The reproduced S,,(co) of the original pattern of fig. 6 by the NSM channel Fourier method. The vertical scale is arbitrary. Distortions are mostly brought in by the termination errors.

values, say half of S*(v k) at each successive stepping of B. In the present case of a 1% incident wavelength spread, we have a total of six equal steppings of B to cover the whole 36 channels. With each B setting we thus deduce 6 channels of S*(vk), although the single B setting covers twice as many channels.

The inverse Fourier transform of the S_*(Vk) thus obtained convoluted with the window function is shown as a final result in fig. 10, which should be compared with the original pattern of fig. 6. It is clear from the comparison that all the essential features of the original S(~o) are regained in spite of the approximation dis- cussed in section 4.2 and of the termination errors mentioned above. In fact most of the distortion was brought in not by the approximation for linearization, but from the termination errors of the cut-off effect at the k,, for the present case. This point will be discussed further in the next section.

5.2. Effects of experimental parameters

The above approach will imply that the number of steps of B variation increases proportionally as the incident wavelength distribution g(z) is increased or relatively speaking as the k,, or equivalently N / A A increases for fixed g(r). The latter condition can be realized by increasing either % or L or both for fixed A A. For example, a 10-fold increase of k,, may be effected by % = 1500/zs/m (-~ 6 tk) with L ---- 20 m, giv-

ing a relative time shift of 2 × 10 -4 (AE --~ 1 #eV) for the S(~0) under consideration.

However, to see such details with an incident beam of 1% wavelength spread, we must repeat the B varia- tion step 60 times, covering 6 channels each. With a

. (a)

;:1/1 =1°/I /

411.00 lO. 00 ~ 0 . 0 0 SO.O0 40.00 V

1 , (b)

T

d

"k,o" ..... ' ...................... ~o.oo Io, oo 41o.oo ¥

it (c)

¥

Fig. l l, Examples of the channel Fourier transform K(v) with a relative incident neutron wavelength spread of 2% for B values of 0.1/channel (a), 0.5/channel (b) and 0.9/channel (c) respectively as in fig. 8.

510 Y. Ito et al. / Neutron spectral modulation I

further relaxed incident wavelength spread, correspond- ingly more steps of B are needed. This is rather imprac- tical if not impossible.

We can remedy this to some extent by reducing the number of overlapping channels. We take the same example as in section 5.1 and increase the incident wavelength spread to 2% as shown in fig. I 1. Results are shown for two cases in figs. 12a and 12b. In the former, the same degree of channel overlap as previously ex- amined was used with a consequent proportional in- crease of B steppings to 12 times. In the latter case we reduced the overlap to a minimum with an unchanged repetition time (6 times) for B variation.

qC

,.0

X:

0 09

-02S. O0 - tS . O0 -S. O0 5.00 OMEGA

(a)

15. O0 25. O0

bJ f - c~ o

o 0")

o

g

-~S. O0 -15. O0 -S. O0

I S. O0

OMEGA

(b)

IS. O0 25, O0

Fig. 12. Reproduction of S,,(~o) of the original pattern of fig. 6 by the increased relative incident wavelength spread of 2%, for the B stepping of 12 times (a) and 6 times (b).

The results differ very little as seen in the figure. This may be fortuitous partly due to the very nature of the computer experiment without statistical errors. Never- theless it indicates that a clever choice of B modulation stepping can reduce the necessary repetition time con- siderably.

Finally we mention the distortion of S(¢0) discussed in section 4.2. The condition that eq. (37) or (38) reduces to eq. (39) can be rephrased here such that Im I j / M o in eq. (50) is of the order of one or less for all major m a n d j values. For the present case Iml ~s0 = 6 at most and j / M o is of the order of the incident wavelength spread, i.e. 1%. Clearly the above criteria are well satisfied. In fact for a small relative energy shift of the order of 10 -3 , the incident spread of the order of 10% does not introduce any serious distortion. For the matter of practical interest we can safely say that the distortion of S(~o) brought about by the linearization of eq. (50) can well be neglected.

6. Summa W

Combining spectral labelling of neutrons using the neutron spin phase angle with pseudo-statistical polari- zation time modulation, we have developed a new ther- mal neutron scattering technique called neutron spectral modulat ion (NSM). The new method can claim two essential features as its characteristics among the ex- isting thermal neutron scattering methods including the two high resolution methods of neutron spin echo (NSE) and neutron backscattering.

The first is its capability of simultaneously determin- ing both a highly accurate dispersion relation and its true linewidth due to lifetime broadening. This is accomplished by utilizing the resolution cancellation property of zero-crossing points in the cross-correlated time spectrum together with the Fourier transform scheme of B-modulation similar to NSE without resort- ing to echoing.

The second is the amalgamation of the time Fourier transform scan of NSE with the direct energy scan of the backscattering technique by means of the channel Fourier method, thus making any S(w) of the quasi- elastic region accessible to NSM. This becomes possible since in NSM the Fourier component of S(~0) both in magnitude and phase of every necessary channel can be determined experimentally.

These features are results of NSM in which each time channel carries the information conveyed by the labelled neutrons, whereas the information of NSE is essentially that of NSM integrated over all time chan- nels, or equivalently over all the incident neutrons.

In determining the quasi-elastic lineshape by means of the time Fourier scan, NSE is generally superior both in intensity and in attainable energy resolution. How-

Y. ho et al. / Neutron spectral modulation I 511

ever, even here NSM is more advantageously applied when scattering events are strongly Q-dependent, or when the investigation needs to be done in a strong magnetic field, or if the separation of spin-flip and non-spin-flip scattering is desired, * although in the latter case we need another polarization analyzer in NSM. Further discussion on the NSM resolution will appear in the paper II.

where

d~r ' /d ~- : - - I J l d , L : l~ + I d and "r : "r~ = % : u / L ( 5 5 )

were used to arrive at eq. (54). Also Q , : Q(%, 28) and [Q,[ becomes as in eq. (15a).

Appendix 2

For the development of the NSM theory, we have benefitted from discussions with members of the neu- tron scattering group of ISSP, in particular those with Dr. M. Sato, and also with Prof. Y. Yamada of Osaka University and with Dr. G. Shirane of BNL, to whom we wish to express our acknowledgement. We are also thankful to Dr. C.G. Windsor of Harwell for his kindly correcting the English and making comments on the first draft of the manuscript.

Appendix 1

Derivation of eq. (15) is straightforward in that we substitute eq. (14) into eq. (10a) to give

m ~ - ~ S ( Q ) 8 ( w ) ~ ( ' r ) 8 ( u - T ) d r d , ' , f : ( . ) = ~ _

(51) where T, w and Q are given by eqs. (I l a), (12a) and (12b) respectively. Integration over z ' gives

1 m ~ "r

fm(m l)1 6 ~ .2 " ,2 q , ( r ) d~-, (52)

q ' u

where

1 Q' , r,; = ~ ( u - - l~-), = Q(~', ~',, 20) . (53)

The integration over r in eq. (52) gives

I m S ( Q ' ~ ) R ( u ) - l~ h r . ~3

1 ~(r,;)

r u r=-ru

: I S ( Q , ) f f ( r , ) , (54)

* It should be pointed out that in NSEmhat is measured is the difference of spin-flip and non-spin-flip scattering, whereas in NSM it is the sum.

We insert eq. (24) into the basic equation, eq. (10a), and integrate over r ' as was done for eq. (52). With the notat ion of eq. (53) we have similar to eq. (52),

1 m t ~ 7" ~_

)< " ~ T2 ,2 ,i- u

We remember that wo is given by eq. (26) and it is a function of r only. Subsequent integration over r gives

m K ( u ) = 1 e. -~a h - S ( Q,, ¢ u

1

where e. is the solution of

2h # p( -j

Equivalently it is the crossing point of w o and w, as shown in fig. 3. Since w o is given as a function of r as of eq. (26), we have

dr ~,=~ h ~ o ( ~ ) ~ 0; , (59)

with p~ given in eq. (28), Putting eq. (59) back into eq. (57), and keeping in mind that %' at ~- = ?. is nothing but the 0 ( ~ ) = p ~ , we finally get

/~ (u ) = ~7 ~-~ S ( Q . ) q~(~.), (60)

where L ' = l ~ + p ' l d as in eq. (28), and Q . = Q('r., p., 2 O, if) for fixed 2 0 and q,.

Appendix 3

Consulting fig. 2, we can write q in terms of k i , k f ,

20, qJ and G = I R * I , where Q = R * + q with the known reciprocal lattice vector R*. For example q = [q[ is given a s

q = [ k "z, + k 2 + G 2 - 2 k i G cos

512 Y. Ito et al. / Neutron spectral modulation I

!

-2kr{k icos20-Gcos(~b+ 20))]2. (61)

On the other hand the dispersion relation can be written as a function of q for fixed R* as f(q). For the isotropic spin wave dispersion for example, we have for small q

f (q) : Dq ~. (62)

The energy conservation relation should hold to give

h a 2m ( - k 2 - k ~ ) = f ( q ) i " (63)

The above three equations give the relationship between kf and k i in terms of the known quantities 20, 4' and G. k i, kf and ~-, P ( r ) are related as

m m (64) "r = , p('r) -- hkf

For the dispersion relation given by eq. (62), we have the expression of PO') as a function of ~" as

1

p(~) = h~ ) Gq" cOs + 1 - 2 m---O

. f ( '~) vs. "/7 700 /

/ /

/ 690 /

/

~ 6 8 0

~'_-f (i:) ~ / ~ I

62(]

t / I I I i 1 I I I 612 6'It* 616 618 620 622

z; ( r s / m )

Fig. 13. An example of PO) vs. ~- for the isotropic spin wave dispersion given by f ( q ) = Dq 2. Two p('r)'s are obtained for the neutron energy loss scattering process as explained in the text.

(-) 11"1 - 2 h G~ ' cos4 ,+ 1 - ~ . (65) m

This is calculated for the following experimental param- eters;

D = 1 0 0 m e V . . ~ 2 , G = l . 3 0 A - I , 2 0 = 3 1 . 0 ° , ~ b = 7 0 . 0 °.

In this particular case, we have two widely separate PO') solutions for the neutron energy loss scattering and none for the energy gain process. The p O ) ' s thus calcu- lated are plotted as a function of ~-(#s/m) in fig. 13 for illustration.

Appendix 4

Eq. (31) together with eq. (31a) becomes upon in- tegration over ~-'

m f f ~S(O_ ' )h(w ' ) l~(u) =~ ) < 8 ( w ' - wu + wO) q~(~') d~'dw'. (66)

Remembering that ~ , , 0~ O are both functions of ~" only, whereas w' is the independent variable, we integrate over r to get

'. . . . . ] × d~,d~, dWed~" eff(~" ) dw' . (67)

From the condition w ' = w u - ~Q, we get the relation between w' and 8~" to first order in 8"r/p u as

r o l L ' ,0' - ~ , (68)

h p3 u l d

where 8~- is measured from e, as z = ?, + 8~', and p,, L ' are given in eq. (28); i.e. the solution of w ' = 0 (i.e. ~u = wo) is denoted as ~r u. From eqs. (26) and (31a) we have to first order in cSr/p,

dW~d.: dwq,~,=,~-,~o=~ ~ l ~ d ' r 1 p,[

m L ' l . . . . (69)

h I a p3 .

Y. Ito et al. / Neutron spectral modulation 1 513

Inserting eq. (69) into eq. (67) and retaining the term of first order in 8r/Pu, we finally get

1% / ( ( u ) - - L '

7S(Q.) h(o/) dw' , (70)

with r = e u + 6r, and 8r is related to ¢o' by eq. (68).

In the above equation, the separation of variable is not complete since A is a function of t. However, if we for the moment consider A being a function of ~, and integrate with respect to t first, we can write eq. (73) in the form of eq. (38) with the help of eq. (36a). We must of course remember that [g] in eq. (38) is still a function of ~, and eq. (38) essentially means eq. (73).

Appendix 5

We calculate the channel Fourier transform of eq. (37) using the relation given by eq. (37b) and z, = u/L, A = % - r ° . Since du=Ld%=LdA, we have for the channel Fourier transform

~ o

ff _/~(v) -- e(2rr)

x , 1+ .0 IJ

Xe ic(r°+A)v d ~ d A . (71)

Putting the argument of q, as t for the change of variable, we have

a= {,_(,o +

--~ t - r ° + ao~O (2 -- 3t/r °)

~--t--r°--aooa{ l + 3 (aooa+ A ) }, (72) %

where aoOa/r ° < 1 was used. Inserting eq. (72) into eq. (71) and noticing dA ----- dt we have

/ ~ ( v ) ~ 1 ff:ooS(o~)*(t)eiLtv f(2 ) -

[ { __3 ( a ° w + A ) } l d ~ ° d t " × e x p --iLao~V 1 +

(73)

References

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[4] L. Phi, N. Kro6, P. Pellionisz, F. Szlavik and 1. Vizi, Proc. IAEA Symp. on Neutron inelastic scattering, Copenhagen (1968) Vol. 2, p. 407.

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