Neutron resonance spin echo, bootstrap method for increasing the effective magnetic field

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Neutron resonance spin echo, bootstrap method for increasing the effective magnetic field

R. Gähler, R. Golub

To cite this version:

R. Gähler, R. Golub. Neutron resonance spin echo, bootstrap method for increasing the effective magnetic field. Journal de Physique, 1988, 49 (7), pp.1195-1202. �10.1051/jphys:019880049070119500�.

�jpa-00210801�

Neutron resonance spin echo, bootstrap method for increasing ^the

effective magnetic ^field

R. Gähler (1) ^{and R. Golub} (2)

(1) Fakultät für Physik, ^{E 21,} Technische Universität München, ⁸⁰⁴⁶ Garching, ^F.R.G.

(2) Technische Universität Berlin, ^{F. R. G.}

(Requ le 15 d6cembre 1987, ^{révisé et} accepté ^{le ler}

1988)

Résumé.

Nous montrons qu’il ^est possible d’utiliser la technique ^du bootstrap pour accroître l’angle ^de précession ^et ainsi la résolution de l’instrument pour

champ magnétique maximum donné

spectrométrie

d’echos de spins ^{de neutrons.} Les limites théoriques ^et techniques à l’amplification ^du champ ^{effectif sont}

discutées.

Abstract.

We show that it is possible ^to

bootstrap technique ^to increase the precession angle, ^and

hence the instrument resolution for

given ^maximum magnetic ^field intensity in Neutron Resonance Spin

Echo (NRSE) spectrometry. ^The limits, theoretical

well

technical, ^to this effective field amplification

discussed.

Classification

Physics ^Abstracts

29.30

Introduction.

Neutron spin ^echo spectrometry, first introduced by

Mezei in 1972 [1] ^{has been} developed ^{into an} extremely productive ^and sophisticated technique

for high resolution neutron spectroscopy [2-7].

In previous ^{work we} have shown that the introduc- tion of magnetic ^resonance spin flippers ^{can lead to}

the elimination of the requirement ^for large magnetic

fields over large ^{distances in} quasi-elastic [8] ^and

inelastic [9] ^neutron spin-echo spectrometers. ^{In the}

latter work we showed that one can obtain a factor of 2 increase in the precession angle ^{of the neutron} spins ^{for a} given magnetic ^{field and that this}

translates into a factor of 2 improvement ^{in the}

relation between instrument resolution and maxi-

mum available field in comparison ^{with neutron} spin

echo (NSE). ^{We also} briefly ^{mentioned the} possibili-

ty of further increases in this factor so that the factor of 2 can in principle ^{be increased to} 2 N. In the present ^{work we} give ^{a detailed} description ^{of this} bootstrap method for increasing ^the instrumental

performance ^and discuss the theoretical and techni- cal limits to the size of N.

The basic principle ^{of an NSE} spectrometer ^{can be} understood in terms of the following simplified

argument [1]. ^Neutrons polarized perpendicular ^{to a}

constant magnetic ^field BNSE undergo ^a precession through ^an angle

on travelling ^a distance L through ^{the field} BNSE ^with velocity v l. ^{y is the neutron} gyromagnetic ^ratio (y

² 03BC /h, 03BC

^neutron magnetic moment). ^After scattering (neutron velocity changed ^to V2) ^the

neutrons travel through ^{a similar} region ^{with rever-}

sed magnetic ^{field and} undergo ^a precession through

an angle - p2’ The net precession angle ^{is thus :}

where m is the neutron mass, líwn ^{is the neutron}

energy transfer and TNSE is called the spin-echo ^time.

cp is measured by ^{means of a static} spin flipper,

followed by ^a polarization analyser ^{and a neutron}

detector. The detector signal ^is given by

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019880049070119500

1196

For an energy transfer spectrum S ^the signal

will be

containing ^the Fourier transform of the _energy transfer spectrum S(w,,), ^{i. e. the time} correlation function of the fluctuations in the scattering system.

We see from (4) ^that larger ^values of available TNSE imply sensitivity ^{to smaller values} of Cù n (hence higher ^effective energy resolution) ^{and that the}

maximum available _{TNSE is} determined by ^{the maxi-}

mum field and the proportionality ^{constant between} precession angle ^{and field} strength (Eq. (2)).

Amplification ^of precession angle by ^the bootstrap.

1. OPERATION OF 1TCOILS.

We consider a neutron beam initially polarized along ^{the x axis,} travelling ⁱⁿ

the y direction through ^{a series of}

coils whose D-C

magnetic ^fields (magnitude Bo) ^{are either} parallel ^or anti-parallel ^{to the z axis.} (Note ^the change ^of

coordinate system compared ^{to reference} [9]. ^Each

7r

coil contains an oscillating magnetic ^field Bf, pointing along ^{a direction} in the x-y plane (see Fig. 1). As is well known [10] ^the oscillating ^{field can}

be regarded ^{as a} superposition ^{of 2} counter-rotating fields, ^{each with a} magnitude Bf

Bf /2, ^{one of}

which (depending ^{on the direction of} Bo) ^{will be in}

resonance with the neutron precession if the fre- quency of the oscillating ^field, too, satisfies

The other counter-rotating component ^has only ^a negligible ^{effect on} the behaviour of the neutron

spin. ^The operation ^{of the}

^{coils was} explained ⁱⁿ

reference [9]. ^{It was} shown that if a neutron enters a

coil at time to ^{with its} spin making ^an angle

cp (to ) ^with the x axis it will leave the

coil, ^whose length ^{is taken} as f, ^{at a} _times

l / v later with its

spin making ^an angle cp’ ^with the x axis :

where

is the angle between the rotating field and the x axis at the time that the neutron enters the

coil. We

assume that the entry ^{and exit from the}

^{coils is}

rapid enough ^that boundary ^effects play ^{no role.}

2. SIMPLE MODEL OF THE BOOTSTRAP. - In order to illustrate the physical principles ^{involved we will}

first discuss a simplified model of the system ⁱⁿ which we neglect ^{the term in} t, ⁱⁿ equation (6). ^In figure ^{2 we show the} position ^{of the neutron} spin,

~°, ^{at the} time, to, ^of entry ^{into a}

coil whose D-C field is in the positive ^direction (i.e. ^{that direction}

which involves a counter-clockwise precession ⁱⁿ Fig. 2).

In the

coil the neutron spin precesses through

an angle

^about Bl+ ) ^{so that it} returns to the x-y

plane ^{at a} position cp 1 whose angle ^{with the x-axis is} given by (6). ^{If the neutron now leaves the first}

7T

coil and enters a second with reversed Bo, ^{so that}

the effective rotating component Bf(-) ^rotates

clockwise, ^the rotating component Bf(- ) ^{will now be}

at an angle - cp f (t° ) ^{and the} precession through

7T

about this component will result in the neutron

spin reaching ^an angle

with the x axis (a

cP f(tO) - ^cp °). Passage through ^a

Fig. ^1.

Operation ^of

^{coils :} (a) decomposition ^of

oscillating field into two counter-rotating fields ; (b)

^neutron

spin making

angle ^{cp with} respect ^to the x axis

entering ^the coil, leaves the coil at

angle cp’.

Fig. ^2.

Simple model of the bootstrap : (a) overall view of

bootstrap ^coil system ; (b) ^coil C,

^{A neutron} spin, initially ^at

angle ’P 0 with respect ^{to the}

axis makes

angle cp

when leaving ^the

coil; (c) ^Coil C2

^the angle between the spin and x axis is transformed from _(Pl

cp2 ; (d) (e) ^Coils C3, C4

Each coil further increases the angle between the spin and the x axis.

further

coil (with positive Bo) ^{will result in a}

further precession ^about Bf(+ ) ^{and an} angle ^{with the}

x axis of

By passing through ^further

coils this bootstrap

action will continue so that after N-03C0 coils we will have

Taking ^{into account} (7) ^and (5) ^{we see that} equation (10) represents ^a gain ^{of a factor 2 N with} respect ^to equation (1) and this will translate directly

into an improved instrument performance according

to equations (2) ^and (3).

In NSE the resolution is sensitive to the precision

of the line integral ^{of the} magnetic ^field between the

flippers. ^This integral ^{must be constant} (in ^{time and} space) ^{for all neutron} trajectories ^{within a} precision

better than the required ^relative energy resolution.

In NRSE (with ^{or without} bootstrap) ^the require-

ment of precision ^is transfered to the rf-frequencies (easy ^to maintain) ^{and to} the distance between the

rf-flipper ^coils, whose variations for all trajectories

must be constant within the relative resolution.

3. DETAILED ANALYSIS OF THE BOOTSTRAP. - We consider a series of

coils : A1, A2

An ... AN

each with its Bo ^field reversed relative to the

preceding ^one. Applying equation (6) ^{we find that}

after leaving ^the nth

coil the neutron phase ^will

make an angle cp n with ^{the x axis :}

where tA ^{is the time of} entry ^{into the first it-coil.}

By forming ^{the sum}

using (11) ^and cancelling ^{terms on} both sides we

obtain the solution to the recursion relation (11) :

which is equation (19) ^{of reference} [9] ^{if we take} cp ° = 0. ^{The factor} 2 N in the first term of

equation (13) represents ^the gain factor due to the

bootstrap.

It is amusing ^{to note that the same} bootstrap technique ^of increasing ^{an initial} angle by ^alternate precessions ^{around two different axes has been} applied ^{to the} problem ^of searching ^{for neutron-}

antineutron (n - n ) oscillations [11] although ^we only ^{noticed this} analogy ^{after the fact. In the}

n - n work a recursion relation similar to

equation (11) ^{was solved} by ^{the same} technique ^as

used above.

4. NRSE SYSTEM WITH BOOTSTRAP. - In order to

use the bootstrap ^{in an NRSE} spectrometer ^{we will} need 4 sets of coils (each ^set consisting ^{of N} single

7r

coils) ; which will be designated A, B, C, ^D.

tA, 8.., ^{will represent} the times that a neutron enters each set of coils. Set A will have frequency

w 1, ^set B, w 1= ± w 1, ^as required, while C and D will have _{w 2} and Cù2 ^{= ±} W 2 (see Fig. 3). ^{As in}

NRSE without bootstrap ^{we can assume the drift}

regions between the coil sets to be field-free. Any

1198

Fig. ^3.

Outline of

spectrometer showing ^{the sets of} bootstrap ^{coils A, B,} C, ^D. Li, L2

^{the drift} regions.

guide ^{fields used will have} only ^{a trivial effect on the} operation of the instrument. At the entry ^{to coil-set} A the neutrons will be polarized along the x direction

so we apply equation (13) ^with cp °

^{0 and obtain}

Now for set B, cp NA represents the initial phase ^so applying (13) ^and using (14) ^{we find :}

and choosing w 1

(- l)N + 1 . úJ l’ ^{the N Z terms}

cancel and

Similarly ^{for C and D} taking w2

(- l)N + 1 . w 2 ^we find

Hence by choosing úJ l’ úJ 2 to have opposite signs ^we

have

with T2

tD - tc ^and T,

tB - tA ^{and the} magni-

tudes of w l, 2 ^{can be chosen to} give ^the spin-echo focussing condition for either quasi-elastic ^{or inelas-}

tic scattering [4]. In the above discussion all angles

are measured with respect ^{to the x-axis} (fixed ^{in the}

lab. frame) ^{and the} spin ^{returns to} the x-y plane ^at

the exit of each coil (perfect

coils). ^After travelling through ^a polarization analyser ^directed along ^{the x-}

axis the rate of neutrons reaching the detector will

be given by (3) ^{with cp} replaced by cp ND ^of

equation (18). ^{We thus see that it is} possible ^to

obtain a 2 N times improvement ^{in the} relationship

between maximum field strength and instrument resolution. This can be used to design ^an apparatus for higher resolution, ^{or to} work with lower field

strengths, ^{or some} combination according ^{to the} particular application.

Limits to the bootstrap gain ^factor.

The question immediately ^{arises as to the limits of}

this process. How large ^{can N in} equation (18) ^{be ?}

The obvious practical ^{limit is} that each unit increase of N requires ^the neutrons to pass through ^an

additional 4 coils and there is a limit to the amount of material the neutrons can traverse. But in addition to this technical limit, ^{which we discuss} below, ^a physical ^{limit to N is set} by ^{the width of the} acceptable velocity spectrum. ^The above discussion has assumed that the precession about B in ^each

w

coil is precisely

^{radians. The} angle ^of precession

about B is given by

and Bf ^{and f are chosen so} that 03BE =

^{for some} velocity vo. ^Then

and the above discussion must be modified to take into account the deviation of 03BE (v ) from the ideal value of

radians. It is _easy to see that the correction will be of second order in x = § (v ) -

and that the error will increase with N.

1. PHYSICAL LIMIT TO N.

In order to study ^this

question ^{in more detail we} have carried out an

analysis ^{of the} system starting ^{with the} Schroedinger equation :

where the Hamiltonian, H, represents ^the interaction of a neutron with a static magnetic ^field, Bo, along

the z axis, ^{and a} rotating magnetic ^field Bf (fib

03BCBf), rotating ^with f requency w ^{in the x,} y

plane.

(0 ) ^{is the} probability amplitude ^{for the} spin

up (down) ^state.

For neutrons entering ^{the coil at} time t1 ^and exiting ^{at a} time t1 ⁺

^{the solution of} equation (21)

can be written [10]

Now for a system ^{of coils} with tA ^{the time of entry of}

the first coil we take ’A (tA) ^as 1/B/2 1 ^{for initial}

polarization ⁱⁿ the x direction and 1 for initial

polarization ^{in the z direction and} multiply by ^a

matrix [C (ti )] ^{for each} coil. Then with the resulting

wave function we calculate .p lux I .p > or qi I cr, I 03C8 ) ^{as the case may be.}

We see that each spin component ^{of the final wave} function with initial x polarization will consist of 4 terms in the case of 2 coils, ^{8 terms for 3} coils and 16

terms for 4 coils. The number of terms in each spin component ^{of the final wave function for the case of} M coils will be 2m and the order of each term in sin 6r and/or cos b T will be M. Thus the output

polarization will be of order 2 M in sin b 7 /cos ^{b 7,}

and the number of terms in the output polarization

will be 22 M, ^an uncomfortably large number for M>_4.

However, ^{most of these terms will be} oscillating

functions of tA, the time of arrival of a particular

neutron at the first coil and hence will not make any contribution to the time average of the output

polarization ^{or detector} counting ^{rate. In addition}

we are interested in the case where (tB - tA ), ^the

time of flight ^{between coils, is much} larger ^{than T,}

the transit time of a single ^{coil so that all terms} containing oscillating functions of (tB - tA ) or (tD - tc ) or ^{the sum of these two} quantities ^{will be} rapidly varying functions of neutron velocity ^{and will}

go ^{to zero} when averaged ^{over the} incoming velocity

distribution. Therefore we need only keep ^terms

depending ^on the difference between (tB - tA ) ^and (tD - tc) ^{and on}

To begin ^{with we have} calculated the case of two

w-coils, ^{the neutrons} entering ^the first coil at time tA, ^and the second coil B at a time tA ^{+ ? + T.}

The result for u x) ^{at the} output of coil B is

after averaging ^{out the terms} depending ^on tA. ^We

see the leading ^term is of order 2 M in sin b T as

expected. ^The x-polarization ^at the exit of B is seen to be a measure of the travel time between A and B with the factor of 2 gain ^{as discussed above for}

N = 1.

We now consider the case of four it-coils all with the same applied frequency. ^{The neutrons enter coil}

A at time tA (polarized along ^{the x} direction), ^{coil B}

at time tB

tA ^{+ T +} Tl, ^{coil C} (which ^{has its D-C} magnetic ^field reversed, ^i.e. WC = - W A) at ^time

te = tB ⁺

+ ts ^{and coil D} (also with reversed

magnetic field) ^{at a} time tD = tC ⁺

⁺ T2 (see Fig. 3).

The polarization in the x direction (averaged ^over

tA ^and discarding ^{terms as discussed} above) ^{is found}

to be :

From the definition of b (following Eq. (21)) ^{we see}

that the Larmor frequency around the rotating ^field

is given by WL (rot)

^{2 b so that for one}

^{coil :}

and

where 8 v = v - v o is ^{less than the} half-width of the

incoming velocity spectrum.

Thus for 6v /v = ^{10-1 as} is necessary for the linear relation between spin-echo phase ^{and neutron en-}

ergy change ^{to be valid} (Eq. (2) ^{and Refs.} [1, 2]),

the higher ^{order terms in} (23) ^and (24) ^{are com-} pletely negligible, ^{and the} only important ^contri-

bution comes from the leading ^term, which contains the factor of 2 gain expected ^from equation (18)

the major reduction with respect ^{to the ideal case of} equation (18) being the factor sing bT which is on the order of

In figure ^{4 we} plot the result for the 4

system

(Eq. (24)

^{note that} only ^{the first term is} signifi-

1200

Fig. ^4.

Transform functions for

spin-echo spec- trometer. The horizontal scale is chosen for the 4 w system with TNSE(AO) ^defined

for the classical NSE with the

same

magnetic field value

in the

coils. The period ^of

the fast oscillations is _{(;ù n .} T NSE (À 0)

cant) ^{as a} function of the neutron energy transfer variable _(On, averaged ^{over an} incoming wavelength

distribution which was assumed constant between

A o - ^{8 A and} A o ^{+ 6A for}

6A/Ao

^{3 % and}

6 % taking ^{into account} the À 3 dependence ^of

TNSE (Eq. (2)). ^The separation ^between neighbour- ing maxima and minima is given by (see Eqs. (24)

and (2))

with

defined as in the classical spin ^{echo for a} magnetic

field equal ^to that in the

coils.

In a quasi-elastic scattering experiment (for example) the functions shown in figure ^{4 are the true}

transform functions (for Q independent scattering),

which replace ^{the cos} (On - T NSE function in equation (4), ^when calculating ^the ouput ^{for a} given S(Wn)-

As is usual in spin ^echo the resolution does not

depend ^on

^but only ^on the maximum value of

TNSE (actually ^{2 .} TNSE in ^our present ⁴

case).

However we see that the use of larger ^{values of}

E

results in a reduction of the range of Cl) n which is included in the Fourier transform of S ( w n ) ^measured by the instrument. This is not a problem since it is

usually ^not necessary ^{to measure a} broad frequency

distribution with a high resolution [12]. ^{We see that}

the virtue of the spin-echo ^{method is to obtain the}

desired high resolution with a relatively large

e

(and ^{hence a} high intensity) ^while only paying ^the price ^{of a reduction in} dynamic range which ^is

unimportant ^{when one is} measuring ^narrow peaks.

While the rapid ^{increase of the number of terms in} with ^{the number of coils M} makes further calcu- lation exceedingly ^{difficult we see that this is un-}

necessary, based ^on the preceding discussion we can

conclude that the leading ^{term of} (o,.,) ^{will go as}

with

For reasonable narrow velocity distributions as

above we have

which witch 17 = 0.1 would allow values of N -- 5.

We will see below that the practical ^{limits are more} stringent ^{than this.}

Equation (30) ^can also be derived from a simple

estimate (see Fig. 5) : ^{for the mean neutron} velocity

vo, the rotation angle {3 ^around B f ^is exactly

^within

one single coil. For a velocity distribution vo - Ov, the travel times through ^{the coil} will vary, and ^we get :

Fig. ^5.

^{Motion of}

^neutron spin ^{in the} rotating ^frame.

Neutrons with velocity vo precess through

angle ^{7T about} Brf. ^Neutrons with other velocities precess through

angle ^f3 (Eq. (31)). ^The angle y determines the loss of

polarization ^when f3 #

Now the vector of polarization ^{will be} spread ^around

z

0. For each neutron magnetic ^{moment, the} angular ^{deviation y from the} x-y plane ^{will be} proportional ^{to a,} which is the angle between the neutron polarization ^{and the field} B f ^{at the entrance}

of each coil. There will be no deviation for

a ⁼

0, ^{the maximum value will be for}

90° and the mean value we obtain for a

45°. Thus the rms

value ( y ) ^{becomes :}

For M single ^coils (with ^random distribution of a at each coil), ^the spreads ⁱⁿ y will ^add quadratically ^and

the final polarization P M ^{will be :}

with q taken from equation (25). ^{For small} Av Ivo ^we

obtain :

which is just ^the leading ^term in the above calculation

(Eq. (30)).

For completeness ^{we would like to} present ^the results for an earlier system ^{which we have not} discussed in print [13]. ^{In this system we take the} polarizer ^and analyser ^{in the z} direction, ^coils A(D)

are v/2 coils resonating ^{at a} frequency ^{2 w} (- 2 w )

while coils B(C) ^are

^coils resonating ^{at w} (- w ).

The neutrons enter coil A at time tA, ^{coil B at} tB

tA ^{+ T +} Tl, ^{coil C at} tc

tB ^{+ T} + ts ^{and coil} D at tc ^{+ T +} T2. ^{All coils are taken to have the}

same length, ^{and the} amplitudes ^{of the} rotating

fields are adjusted ^{so that}

The probability Pz ^{for the} polarization ⁱⁿ

^direction

at the output ^{is then}

considering ^the narrowness of the velocity ^distri-

bution as discussed above (following Eq. (24)).

The factor 2 multiplying (ù (Tz - T1) ^{does not}

represent any gain ⁱⁿ performance ^as the maximum field (in ^{coils A +} D) corresponds ^{to 2}

2. TECHNICAL LIMITS TO N.

In order to go from ^a field-free region ^{into the}

^coils (or vice-versa) ^the

neutrons must penetrate ^the conductors carrying ^the

current which produce ^the magnetic ^{field. If we take}

n turns/cm of conductor with thickness t [cm] ^the height ^{of the} conductors is a =1 /n (neglecting ^the

space taken by insulator) ^{and the} magnetic ^{field is}

where I is the current in amperes. The resistance per

cm2 of the coil is

where _p is the resistivity ^{of the} conductor. The power dissipated ^is

using (38) ^and (39), ^and taking Bo

^{300 Gauss} (vL = ¹ MHz) ^and p

2.6 x 10-7 [O.cm] ^{for com-} mercially ^available high-purity ^{aluminium at 80 °K} (ten times smaller than the room temperature value).

Of all the good (normal) conductors Al has the best neutron transmission, ^{the mean free} path being greater ^{than 11} [cm] ^{for neutron} wavelengths A:

4.5 A ^A 11 A and A : 1.8 Å and reaching ^a

maximum of 22 cm for A = 4.8 A.

With a 400 cm2 winding ^{area we obtain}

and taking t

^{0.2 cm} yields ³⁵ W/coil. With M = 12

(gain ^of 6) the total material traversed would be 2 M . t

4.8 cm or a transmission better than 65 %

over the above mentioned wavelength region. ^Work- ing ^{in the} vicinity ^{of A}

⁵ A would allow double the amount of material, ^{but small} angle scattering might

set a more stringent ^limit.

These considerations are only ^{meant to be indica-}

tive of the possibilities using ^a relatively simple technology. ^{The future} availability ^of high Tc super- conductors or the _use, as is already planned by

several groups [14], ^{of classical} superconductors ^can,

of _course, result in an improved instrument.

Conclusions.

We have seen that in addition to allowing ^the

operation ^{of a Neutron} Spin ^Echo spectrometer

without the need for large magnetic ^{fields over} large

distances, ^{an NRSE} spectrometer ^with bootstrap ^can

provide significant ^increases in the resolution avail-

able with a given ^maximum magnetic ^{field. This}

improvement ^{can be shared between} higher ^resol-

1202

ution and lower magnetic fields. In addition the combination of the bootstrap idea with the

generalized ^NRSE system (Sect. 3 of Ref. [9]) ^will

allow the increased resolution together ^with arbitrary

field on the sample ^{and the} reduction of the sensitivi- ty ^to sample ^size.

It may also be possible ^{to use the} bootstrap principle ^to produce ^a spectrometer for rather high

energy ^neutrons (100-200 meV) ^where relative resol- utions of 10-3-10-4 may be of interest [15,16].

While somewhat pedestrian ^{when seen from the} viewpoint ^of present day spin-echo techniques ^{such a}

resolution would represent ^a significant improve-

ment over what is currently possible ^{at these} higher energies.

Another interest of the bootstrap technique ^{lies in}

the possibility ^of extending spectroscopy ^{with meV}

neutrons to relative resolutions approaching ^{10- 5.}

To realize such a spectrometer ^would require ^that

the neutron flight paths between the flipper systems be stable and constant for all neutrons to within

10- 5 of their length ^{or about 10 03BCm in the case of} flight paths in the range of several ^meters. Position-

ing ^and adjustment ^to such accuracy is possible using

well established optical ^bench technology. ^Whether

the flipper ^coils necessary ^for the bootstrap ^{can be} produced ^to this accuracy remains ^to be seen but it is

possible ^{to visualize} using evaporation ^{of aluminium}

structures on silicon and similar micron engineering techniques.

Acknowledgments.

Work of this kind would not be possible without the encouragement ^and support ^{as well as the incisive} criticisms of many of one’s colleagues. ^{We would} particularly ^{like to thank} Profs. D. Dubbers, ^F.

Mezei, ^{W. Glaser and 0.} Scharpf. ^{We are} grateful

to B. Farrago, ^C. Lartigue, ^B. Sarkissian and C.

Zeyen ^for helping ^{us find our} way through ^{some of}

the intricacies of spin ^{echo. C. Skorski, as} always, performed ^{miracles with her} typing.

References

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[2] ^Ed. MEZEI, F., ^Neutron Spin ^{Echo, Lectures notes in} physics (Springer, Berlin) ¹²⁸ (1980).

[3] ^{FARAGO, B. and} MEZEI, F., Physica B, C 136 (1986)

100.

[4] MEZEI, F., ^Neutron Spin Echo and Polarized Neu- trons, Neutron Inelastic Scattering, 1977, IAEA

(Vienna) ^1978, pp. 125-134.

[5] ^{PYNN, R., J.} Phys. ^{E 11} (1978) ^1133.

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[8] GÄHLER, ^{R. and} GOLUB, R., ^Z. Phys. ^{B 65} (1987)

269.

[9] GOLUB, ^{R. and} GÄHLER, R., Phys. Lett. A 123

(1987) ^43.

[10] RAMSEY, N., Molecular Beams (Oxford) ^1956.

[11] YOSHIKI, ^{H. and} GOLUB, R., Ultra-Cold Anti-Neu- trons, KEK internal report ^86-19 KEK, National Lab. for High Energy Physics (Tsukuba, Japan)

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