zotero/storage/265CUHTN/.zotero-ft-cache

PHYSICAL REVIEW A

VOLUME 45, NUMBER 1
ARTICLES

1 JANUARY 1992

Coherence and spectral filtering in neutron interferometry
H. Kaiser, R. Clothier, * and S. A. Werner Physics Department and Research Reactor Faciit'ty, University ofMissouri Col—urnbia, Columbia, Missouri 65211
H. Rauch and H. Wolwitsch
Atominstitut der Osterreichischen Universitaten, A-1020 Vienna, Austria (Received 24 June 1991)
The coherence length of a neutron wave packet depends on the spectral width of the distribution of wave vectors that make up the packet. The coherent overlap of the wave packets traversing the two beam paths in a perfect-silicon-crystal neutron interferometer (NI) is altered by placing a material with a neutron-nuclear optical potential in one of the beam paths in the NI. If the optical potential is positive, it causes a delay of the wave packet and a loss of fringe visibility. By use of an analyzer crystal, we narrow the spectral distribution after the mixing and interference has occurred in the last crystal slab of the NI. This increases the coherence length and restores some of the fringe visibility.
PACS number(s): 03.65.Bz, 42.50.—p

I. INTRODUCTION

In a recent paper, we described an experiment in which
the fringe visibility in a neutron interferometry experi-
ment is restored by means of a neutron-phase-echo effect
[I]. In this paper, we describe a related experiment,
where the beam coherence can apparently be improved
by the use of an analyzer crystal placed immediately in front of the detector. A preliminary discussion of this ex-
periment has already been published [2].
The superposition principle is perhaps the most important assumption of quantum mechanics. It allows us to
construct a localized neutron wave packet by a Fourier
sum of plane waves, having a (complex) spectrum a(k):

%(r, t)= Ja(k)e

" dk .

The phases of the component waves are correlated by
a(k), so that they add constructively only in some localized region of space time. Consider a neutron wave pack-
et 0; incident on a perfect-Si-crystal neutron interferom-
eter (NI), as shown in Fig. I [3,4]. The first crystal blade
of the NI splits the amplitude of 4; into two parts, 4',
and 4'2, which separate and traverse paths I and II, re-
spectively. Both beams pass through a polished, homo-
geneous aluminum slab, which can be rotated through
" small angles a about a vertical axis. This slab, called the
"phase rotator, induces a variable phase difference
P (a) between 4, and %2, given by

P

(a)=N

b

1d

sin 01 sina
cos 01 —sin

o.

X, b, where k is the neutron wavelength,

and d are, re-

spectively, the atom density, nuclear scattering length,

45

and thickness of the phase rotator, and 81 is the Bragg angle of the interferometer. The beam %2 also passes through a plane-parallel slab of some material, perpendic-
ular to the beam, with atom density N, neutron-nuclear

SLIT 'Pl

He OEfECTQR

S.3 cm
SKEW-SYMMETRIC
~UIIIO4 4lE %RtaalER

FIG. 1. Schematic diagram of a perfect Si crystal skew-

symmetric neutron interferometer. The beam from the mono-

chromator,

of nominal wavelength

A,

=2.35

0
A,

passes

through

a

8X6 mm slit immediately in front of the NI. The circles

represent the coherence length of the wave packets resulting

from one neutron entering the NI and traversing the two beam

paths. The beam on path II passes through one or two bismuth

slabs, which spatially delay the wave packet due to the positive

optical potential of Bi. The packets recombine, interfere, and

are detected by one of the 'He detectors. The C3 beam is

detected after reflection by a Si analyzer crystal. The interfer-

ence contrast is measured by rotating the aluminum phase
shifter in steps a about a vertical axis.

31

1992 The American Physical Society

32

KAISER, CLOTHIER, WERNER, RAUCH, AND WOLWITSCH

45

" scattering length b, and thickness D. This slab, which we
call the "sample, induces a phase shift

y (D)=kbl=-

2mNbD k

= —NbD A,

and a spatial delay

bl=—DV,
2E

(4)

in 4'2, where E is the neutron's kinetic energy and the op-
tical potential is

V, =2M Nblm,

(5)

where m is the neutron mass. We assume for now that
the sample does not attenuate the beam. The two waves recombine in the last perfect crystal blade of the NI, and form two exit waves, which are linear combinations of 4',
and %'2. If the optical path lengths of beams I and II are
nearly equal, the two packets overlap and coherently interfere, giving rise to interference effects.
As discussed in our recent paper (Ref. [1]), and elsewhere [5], one can show that if a detector is placed in an exit beam, the time-averaged intensity it will measure is

f I(D, a)=A

~a(k)~ dk

f +B ~a(k)~ cos[$0+Pp(a)+gz(D)]dk,

mentally, we find CC3(0)=50%%uo. It is the relative con-
trast, Cz(D)= C—(D) IC (0), that is of interest in these ex-
periments. The relative contrast can be calculated from the magnitude of the complex mutual coherence function
I (D, a) [6]:

C(D) ~r(D, a)j
c(o) ~r(0, 0)~

(8)

+, where
I (D, a) = ()II")(0))p2(b,l ) }

1

2 ([yo+yp(a)+r&(D))

(9)

&2n.

Or, if the displacement b l is in the longitudinal direction
(parallel to k), we can replace ~a(k)~ by the wavelength
spectrum g(A, ) to calculate I (D, a), namely

— f r(D )

1

+ (g) (40+4P( )+rS (D))dg

&2m.

(10)

For this experiment ys &&Pp, so that the phase rotator has a negligible effect on the contrast. Thus, for a Gaussian distribution ~a(k) ~, the relative contrast falls off as a function of the sample thickness D according to the for-
mula
—(XbDa) ) /2

(6)

where ()()0 is some constant, initial phase (under ideal con-

ditions, $0=0). We explicitly note the dependence of the

time-averaged counting rate I(D, a) on the thickness of the sample D and the angle a of the phase rotator, since

these are the parameters that are varied in the experi-
ment. The parameters A and B are constants that de-

pend on which exit beam, C2 or C3, is being considered,

and also on how well the interferometer is working. In

the C3 beam, we usually find B&3=(0.5) AC3, and in the

C2 beam,
c3= Bcq,

Bbuc&t —=A(o0q.=1(72).A7C)2A. C3T. o

conserve

neutrons,

The first integral term in Eq. (6) is a constant. The

second term, however, oscillates as we vary Pp(a). For

example, if the spectral distribution ~(2(k)~ is a Gaussian of standard deviation o k, then Eq. (6) yields an intensity

I(D, a) = A +B cos[Po+Pp(a)+gz(D)]e —( NbDo )2g2

(7)

where o &=2mo k/k is the width of the spectral distribu-
tion in terms of wavelength.
As we vary Pp(a), we trace out a sinusoidal intensity
pattern I(D, a), called an interferogram. The contrast
(fringe visibility) of the interferogram is defined to be the amplitude of the oscillation divided by the mean value of the pattern. As Eq. (7) shows, the contrast C(D) diminishes as the sample thickness D increases; its maximum
value C(0) occurs when D =0, that is, when there is no
sample in the beam. In principle, the maximum contrast
Cc3(0) in the C3 beam may vary from 0% to 100%, de-
pending on how well the interferometer performs; experi-

Such relative contrast curves were first measured for neu-
I tron interferometry by Kaiser, Werner, and George [7].
Note that Eq. (6) for the intensity (D, a) depends only on the magnitude ~o(k) of the spectrum a(k) [8]. There
~
are therefore two interpretations as to why the relative
contrast decreases with increasing sample thickness; both
are equally valid and experimentally indistinguishable [9,10]. First, we know that a wave packet spreads as it propagates, with the faster wavelength components tend-
ing toward the leading end of the packet, and the slower components toward the trailing end. The sample delays the packet on path II with respect to the packet on path
I, so that when they reach the recombination region, the packets are spatially displaced by a distance b l(D), given by Eq. (4). Faster components within one packet then
overlap with slower components within the other. The result is that the interference amplitude "washes out" when summed over the packet, which reduces the con-
trast of the interferogram [11]. If the two packets are dis-
placed by a distance greater than the coherence length,
then the contrast disappears.
If one views the neutron beam as consisting of an incoherent superposition of plane waves, then a second
conceptual interpretation can be applied. Each plane-
wave k component gives rise to its own intensity pattern
I(k, D, a); the overall intensity I(D, a) is the integral of I(k, D, a) over the intensity distribution g(k). On pass-
ing through the sample, each component experiences a
different phase shift yz(k, D). The overall intensity is thus a sum of sinusoidal patterns of differing phases,
which leads to a reduction in the interference amplitude. The beam is said to become "dephased" as the sample
thickness increases, and the interferogram contrast even-

45

COHERENCE AND SPECTRAL FILTERING IN NEUTRON. . .

33

tually approaches zero.

In this experiment, we wish to measure changes in in-

terferogram contrast due to the coherence properties of a

neutron beam. However, there are other effects that also

reduce the contrast. These must be understood and

corrected for if we wish to isolate the dephasing effects.

First, we have to take into account the beam attenuation

due to the sample, which we have not discussed so far. If

an
of

attenuating sample is placed
the wave 4» is attenuated by

in
a

beam factor

IeIx,pt(he—agm),pldiutuedeto

the absorption and scattering cross sections of the materi-

al. As discussed in our phase-echo paper (Ref. [1]), this

attenuation leads to a loss in contrast with increasing

sample thickness,

„a, C(D)

e

C(0)

+e ~a„

(12)

where a, and a&& are the fractional beam intensities on paths I and II, respectively (at+a&&=1). Thus, if we
„, measure a contrast C in our experiment, and wish to
deduce what the contrast C would be without the
„, sample's attenuation, we merely multiply C by an at-
f, tenuation correction factor « ..

Cmeasf a«

(13)

where

f,«=e (at+e a») .

(14)

This gives us a way to measure the contrast C corrected for the effects of attenuation. The intensity fractions a&
and a» and attenuation parameters g were measured ex-
perimentally, and from this information we calculated the
f, attenuation correction factors « for each sample used.
The effect of attenuation on the contrast is different for C2 and C3, and therefore the C2 and C3 data must be corrected separately. The details and results of this
correction process are given in an appendix in Ref. [1].
In addition to attenuation, there are other nonideal
effects that reduce the interferogram contrast, such as imperfections in the machining of the crystal, vibrations
and rotations, thermal gradients, and gravitational warp-
ing of the crystal. These effects are also discussed in Ref. [1]. In light of these effects, it is not surprising that we cannot attain 100% contrast with our interferometer.
We must content ourselves with a maximum contrast C, of order 50% in the C3 beam.

II. THEORY OF CRYSTAL-ANALYZED
COHERENCE MEASUREMENTS

A. Gaussian spectrum
Consider a longitudinal coherence neutron interferometry experiment, as shown in Fig. 1. We place a NI in the neutron beam, and it diffracts neutrons with a distribution of wavelengths gp(A. ). Let us assume for now that the spectrum is Gaussian, with standard deviation
0.

When we place samples of varying thickness D in beam II
inside the interferometer, the relative contrast is given by Eq. (11},with o & =crz. When the sample thickness D becomes large enough, the beam becomes dephased, and the

contrast falls essentially to zero. Consider now placing an analyzer crystal in the C3

exit beam produced by the arrangement described above.
This crystal has a mosaic width gz, and reflects out of
the exit beam a distribution of wavelengths Gaussian in
shape and of spectral width o &. The setup for such an ex-

periment in the antiparallel configuration is shown in Fig. 2. The width of the reflected spectrum may be considerably less than that of the overall exit C3 beam cr&. It

would seem to follow that the contrast in the analyzed

beam could be much higher than in the overall beam.

It is a straightforward matter to calculate the spectral

width

o. &

that

is reflected

out of the exit beam

by an

analyzer crystal of mosaic width gz. By a geometric ar-

gument [11],it can be shown that the analyzer accepts a

Gaussian window W(A, ) of wavelengths

=e W(A, )

— — (A, A,o) /2(cr~ )

of width 0&,
cos(8I ) cos(8„)
8„) A ~o1~ sin(8I —

8„ where 01 is the interferometer Bragg angle. The analyzer

Bragg angle

is positive for the parallel scattering

„, configuration, and negative for the antiparallel
configuration. We are free to use crystals with different mosaic widths g and can reflect neutrons off of various

Bragg planes within a given crystal. In other words, an

analyzer window W(k} can be created with almost any

width o & that we wish.

The spectrum g(A, ) reflected by the analyzer is the

product of the analyzer window W(A, ) and the incident

spectrum gp(A ):

Skl'newte-Sryfm..
Samp le

Oi rect
CZ B'earn,

C'&

Sl i t'

Oet.

S'earn

CAonnAtCfinirpagyalusyrtrzaaaeltr,lileonl, g~ &0

EnTchleor smuarel

SHetoarpn

Analyzed
v„ a„& CD &Seam,

Casse t te

FIG. 2. Setup for analyzing the C3 exit beam with an
analyzer crystal in the antiparallel configuration. The nominal-
=1. ly monochromatic beam (AA, /A, 2%) is produced by Bragg
reflection from a pyrolytic graphite monochromator (20~ =41 ). The Bragg angle of the (220} reflection in the silicon NI is 0 =37.7'.

— ', 34

KAISER, CLOTHIER, WERNER, RAUCH, AND WOLWITSCH

45

}= g(A, )=g (A, }W(A,

e

)„ 2&2m o

(18)

But what about the analyzed beam? We cannot measure the spectrum in the analyzed beam unless we were to

add a second analyzer arm and crystal, which would be

where

the width

o. &

is

given

by

dif5cult. For lack of a better option, we decided to model

the analyzed spectrum with a single Gaussian. We fit the

0

~
((

0)2+(

A)2)1/2

(19) incident spectrum go(A, ) to a single Gaussian to determine a spectral width o&, and from this calculated the

The form of this equation insures that a& & o &. Figure 3

shows the analyzed

spectral

width

o. &

versus

the analyzer

g„. Bragg angle 8~, calculated for several values of the mosa-
ic width

Now, consider what this means in terms of our contrast measurements. The overall or "direct" C3 beam

has a spectral width 0&, as we vary the sample thickness

D, its relative contrast C~;,(D) is given by Eq. (11), with

0&=o &. However, if we place an analyzer crystal in the C3 beam, we reflect out only a fraction oz/cr), of the

analyzed spectral widths cr) using Eq. (19). We feel justified in doing this because the spectruID is restricted
by the mosaic analyzer crystal. If the analyzer has a
Gaussian mosaic distribution, and if the window W(A, ) has a narrow width e)"„,then the reflected spectrum g(A, ) will also be nearly Gaussian. This is not precisely correct, but the approximation is nevertheless a very good one for this experiment, as we shall see.
C. Phase difFerence between the C2 and analyzed C3 beams

overall spectral width, and find that the analyzed relative
contrast C„(D}falls off at a slower rate:

It is a well-known fact that the interferogram patterns in the C2 and C3 exit beams differ in phase by 180', that

—(NbD0' )2/2

1s,

(20)

IC3 A( 3+B cos(hP)

Beyond a given sample thickness, the contrast in the direct C3 beam is nearly zero, indicating that the beam is

), IC2= AC2+B cos(b, (t)+n

(21)

apparently incoherent. Yet, with the analyzer, we pick out of the "incoherent" beam a certain portion of the neutrons that still yields contrast. The analyzer crystal
thus restores the contrast by reducing the wavelength spectrum of the beam after the interference has taken place. In terms of wave packets, we say that the analyzer
increases the coherence length bx by narrowing the
momentum spread hp.

where hg=go+PI, (a)+gz(D}. This is necessary so that
the neutron current is conserved. As the phase b,P changes, the intensity is swapped back and forth between the two exit beams. In the analyzed beam, however, we find that the phase difference between the C2 beam interferogram and the analyzed C3 beam interferogram is not
necessarily 180' due to the non-Gaussian nature of the
wavelength spectrum.

8. ESects of a non-Gaussian spectrum

To see why this is so, consider a non-Gaussian spectrum modeled by the sum of two equivalent Gaussians:

The discussion in the preceding section assumed, for convenience, that the spectrum was Gaussian. This is nearly, but not exactly, the case in our experiment, as we shall see. From Eq. (10), we see that this affects the behavior of the contrast curves. However, if the incident spectrum go(A, ) is measured, Eq. (10) can be used to predict the shape of the contrast curve Cz;, (D).
0.015

—()j.—k, () /2u&

— — (A. k2) /2m&

(22)

&2ncr„

Although this is an oversimplification, the physics of the
following derivation is the same for this spectrum as for
the real spectrum of our experiment. If this spectrum is
]- placed in Eq. (10), the resulting intensity oscillates with
increasing D as
I(D) = A +B cos[go+Pz(a) Nb, DA,

O
~ Q)
R

0.010 0.005

O

C

&C

0.000—45 -30 —15 0 15 30 45 60

Analyzer Angle e„(deg)

8„ FIG. 3. Spectral width crz of the analyzed beam vs the
analyzer Bragg angle for several values of the mosaic width
g „(inrad).

X

cos(NbDEA,

—(NbDcr )e

) /2

(23)

where bk, =(A, , —Az)/2, and A, =(A)+A&}/2. The
difference between this expression and Eq. (7) is the term

cos(Nb, Db, A, }, which is a slowly oscillating amplitude

envelope. The other cosine term is rapidly oscillating. In a phase rotator scan, we measure the phase of the rapidly oscillating curve. The relative contrast of this intensity

pattern is

C„(D)=)cos(NbDbA, )(e —( NbDo )2/2

(24)

Figure 4 shows a representative intensity pattern generated from Eq. (23). At the point labeled Q, the oscillation pattern goes through a node; the amplitude envelope
cos(NBDbA, )=0, so that NbDbl. =m/2 (modulus 2m).

COHERENCE AND SPECTRAL FILTERING IN NEUTRON. . .

35

270

N

180

C

'~

8- 90—

oA

I
8O- 0

~'

I

I

~

I

I

l

I

I

I

I

C
Q)
C

Node,

r

Contrast

J
~'

Minimum,

180 Shift

01 D2

Dg

Sample Thickness D (or b. uni t s )

I

I

I

I

I

FIG. 6. The phase difference between the C2 beam and ana-

2

4

6

8

10

12 lyzed C3 beam, based on Fig. 5.

Sample Thickness D (arb. units)

FIG. 4. A representative neutron intensity oscillation pat-
tern, from Eq. (23).

In this derivation, we have simplified matters by as-
surning that the spectrum could be modeled by the sum
of two equivalent Gaussian peaks. This actually turns

out to be quite a good approximation for the spectrum

so- At point P, slightly to the left of Q, NbDEA, =n l2 5,

used in this experiment. Even if this were not the case, we would still expect the phase of the pattern to shift by

that the intensity at P is

180' when the contrast curve goes through a minimum.

I(DP)= A+B sin(5) cos[Po+Pp(a) —NbDX]
Xe —(NbDo )2l2

This should happen in both the C2 beam and the overall (direct) C3 beam.
Consider what happens, however, if we analyze the C3 (25) beam, and compare it to the overall C2 beam. We com-

At point R slightly to the right of Q, NbDEA, =n /2+5,
so that the intensity at R is
I(DR ) = A +B[ —sin(5)] cos[$0+Pp(a) NbDX]—

pare these two beams because they were monitored sirnul-
taneously in the experiment. The relative contrast in the analyzed C3 beam should fall off more slowly than in the C2 beam, as shown in Fig. 5. When no sample is in the

Xe —(,NbDcr&) l2
= A +B sin(5) cos[$0+ P~(a)
Xe —(NbDcr~) l2

]- + (26)
NbD A, n

The phase of the function at points I' and R differs by m.
Thus, when the oscillation goes through a node, or the contrast goes through a minimum, we expect the phase of the intensity pattern to shift of 180'.

interferometer, the two beams are 180' out of phase. At
some thickness D, the C2 beam (and the direct C3 beam)
goes through a contrast rninirnum, and its phase is shifted by 180'. At this point, however, the analyzed C3 beam has not reached a contrast minimum, so its phase is not
shifted. Thus, for a range of sample thicknesses between
D& and D3, the C2 and analyzed C3 beams are in phase with each other. At some greater thickness D3, one of the beams reaches another contrast minimum, and experiences another 180' phase shift, so that the two beams are again 180' out of phase. Figure 6 shows the difference

in phase between the C2 and analyzed C3 beams, based

100

Beam

on the curves in Fig. 5.

80

Although this effect makes sense in a mathematical

60

way, it runs contrary to our normal experience in neutron

180'

interferometry. For a certain range of sample

40

Phose Shift

thicknesses, the C2 and analyzed C3 beams are in phase.

20

That is, if the counts go up in the C2 detector as we ro-

00
100
80 0 60 CL 40

tate the phase shifter, they will also go up in the analyzed C3 beam. This seems to contradict the conservation of
neutrons, but it does not, because the counts in the
ouerall (direct) C3 beam must go down. The mean phase of the neutrons in the analyzed C3 beam is different than in the C3 beam as a whole.

20

D)

D2

D~

Sample Thickness D (orb. units)

III. EXPERIMENT
A. Apparatus

FIG. 5. Representative relative contrast curves in the C2

To begin the experiment, we had to select some

beam and in the analyzed C3 beam.

analyzer crystals to work with and decide which Bragg

36

KAISER, CLOTHIER, ORNER, RAUCH, AND W'OLWITSCH

reflections to use. We had at our disposal two silicon
crystals, one nearly perfect [68=0.02' full width at half
maximum (FWHM), rI„=00.0015 rad], and the other a

pressed silicon crystal with a relatively broad mosaic

rl„=0. width [b,8=0.47' (FWHM),

0035 rad]. We la-

" beled these crystals "NP" and "PR, for "nearly perfect"

" and "pressed silicon. We decided to work with three

different analyzer configurations. We used the PR

analyzer in both the (111) parallel and (111) antiparallel

configurations, and used the NP analyzer in the (111)an-

8„= tiparallel configuration. With A, =2.35 A, the Bragg an-

gle of these reflections is

+22.0', where the plus sign

is for the parallel configuration, and the minus sign is for

the antiparallel. The interferometer Bragg angle for the
(220) reflection was 8t=37.7'. From this, we calculated

the analyzed spectral widths 0& expected from the crys-
tals in each configuration; the values are given in Table I.

We next had to have a suitable set of phase shifting

samples. We chose to make the samples out of bismuth,

m, bbe=cau8s.e533

Bi
fm,

has a an atom

large, density

positive scattering
N =2.82 X 10

length
and a

relatively low absorption
o, = 9. 156 b). Initially,

cross
five

section (o, =0.0388 b
slab-shaped samples

and
of

bismuth and titanium were machined, polished, and

etched, and then epoxied onto aluminum mounting

brackets. The samples were made so that their nominal

thicknesses were multiples of 4 mm (4, 8, 12, 16, and 20

mm). The exact thicknesses of the samples are given in
Table III. In addition, we made a sixth sample that was

thinner than the others (2.1 rnm thick). Since this sample
'." was about half as thick as sample 1, it was dubbed "sam-
ple —, This sample was used in combination with larger

samples to give finer increments in thickness. For exam-

ple, bismuth

samples

1 and

'
—,

could

be

used

simultaneous-

'." ly to place 6.10 mm of Bi in the beam; we called such a
combination "bismuth sample 1—, The thin samples in

" " set

'
—,

could

also

be

combined

with

the other samples

to

create "sample 2—',, sample 3—',, and so on. This

effectively doubled the number of data points that we

could obtain.

The basic setup for the experiment is shown schemati-

-7 cally in Fig. 2. A slit, 8 mm wide by 6 mm high, was

mounted

cm in front of the interferometer to restrict

the beam size. When no sample was in the beam, this

— produced a counting rate of 1030 counts/min with — -46%%uo contrast in the C3 exit beam, and a rate of 3430 — counts/min with 16%%uo contrast in the C2 detector.
The samples were affixed by means of their mounting brackets to an aluminum bar, which held the samples at the right height and position to intercept the beam. The bar was attached to a translation table, so that the samples could be driven in and out of beam II by a
computer-controlled stepping motor. The analyzer crystal was mounted on a goniometer,
whose tilt and rotation were controlled by stepping mo-
tors. This assembly, called the analyzer table, was
mounted -60 cm downstream from the interferometer
on a movable arm, as indicated in Fig. 2. The analyzer
was placed in the C3 beam, and its height and translation were optimized so that it was centered in the beam.
The neutrons were detected by —'-,in. -diameter, cylindri-
cal, 20-atm He detectors, which are essentially black to thermal neutrons. The C2 detector was mounted verti-
cally -20 cm past the NI. It was surrounded by a casing
made of 84C powder suspended in epoxy, which absorbed any background neutrons. The beam was allowed to
enter only through an opening, —, in. wide and 1 in. high,
defined by a cadmium tube several inches long. The C3 detector was mounted on a rotatable arm attached to the
analyzer table. Its position was also controlled by a step-
ping motor, so that it could be driven into the analyzed beam direction. At this distance (60 cm), the beam was too broad to be accepted by a —'-, in. -wide vertical detector. For that reason, the C3 "detector" actually consisted of
three 3He detectors (all of which fed into the same elec-
tronic counting chain), mounted horizontally in a
neutron-shielding 84C-epoxy cassette with a 1X1 in.
collimated opening. This arrangement also had the ad-
vantage of making the detection probability constant in the horizontal plane. The wavelength of a neutron ray
reflected by the monochromator depends on the reflection angle, and therefore varies in the horizontal
plane. To detect all parts of the spectrum equally, the C3
detectors were mounted horizontally, so that all com-
ponents saw the same detector thickness. The vast majority of neutrons entering the inteferome-
ter fall outside the Darwin acceptance width and are not diffracted. This so-called "direct beam" also has a divergence to it, and would spill into the C3 detector and

TABLE I. Spectral widths expected in the direct and ana-
lyzed beams.

Beam

g „(rad)

Direct
Analyzed,
PR crystal, (111) parallel
Analyzed,
PR crystal, (111) antipar.
Analyzed, NP crystal,
(111) antipar.

0.0035 0.0035 0.000 15

0.0120 0.0106 0.0070 0.000 30

1.20% 1.06% 0.70% 0.03%

TABLE II. Mean signal and background counting rates per
minute, and mean counting time per point.

Beam

Signal (min ')

Background (min ')

Time per point (min)

C2, all configs. Direct C3
C3, PR analyzer (111) parallel
C3, PR analyzer (111) antipar.
C3, NP analyzer (111) antipar.

3430 1030 400

207. 5+0.7 49. 1+1.2 37.621.1

(same as C3)
5 12

230

58.2+0.4

20

32

61.2+1.1

40

45

COHERENCE AND SPECTRAL FILTERING IN NEUTRON. . .

37

overwhelm the relatively weak C3 beam. To prevent this from happening, the direct beam was blocked with a B4C-epoxy block directly after the NI.
B. Data collection and analysis
Even when there is no sample in the beam, a neutron interferometer operates at contrast Co that is less than
100%. The coherence loss due to the dephasing effect
manifests itself in a relative drop in contrast from this
starting value Co to a new value C(D) when a sample was placed in the beam. It was therefore important to
measure the "sample-out" contrast Co as well as the
"sample-in" contrast C(D) for each data set. The basic procedure is described in detail in Ref. [1],
and consists of taking simultaneous "sample-out" and "sample-in" interferograms. A sample is rn.ounted on the translation bar, but driven to a position where it does not intercept the beam; we call this the "sample-out" posi-
tion. With the phase rotator set to a certain position a, ,
we collect data until our monitor counter reaches a fixed preset number corresponding to a (nearly) constant detection time per point. This is the "sample-out" data. The
sample is then driven into the beam on leg II of the inter-
ferometer, and we collect data again for a period approximately four times longer than the sample-out collection time. This is the "sample-in" data. The sample is then driven back out of the beam, the phase shifter is moved to a new position az, and the counting process is repeated. After 30 or so data points, the result is a pair of interferograms, one with sample-out, and one with sample-in.
It is important to take these interferograms simultane-
ously, because the absolute contrast Co drifts a small amount with time due to environmental factors. Over the course of a day, the sample-out contrast Co might
vary +3% or so; however, the relative loss of contrast
due to dephasing will not drift, as long as the sample-in and sample-out curves have the same "starting" contrast

1.0-

—2 —Gauss Fit
--- 1 —Gouss Fit

0.6-
O
04
0.2-

0.0

I

I

2.32

2.34

2.36

2.38

Wavelength 'A (A)

FIG. 7. The measured wavelength spectrum g(A, ) for the ex-
periment. The single-Gaussian fit parameters to the data are
A.0=2.3456 A, cr&=0.0120 A, a =0.974. The double-Gaussian
fit parameters to the data are A, 1=2.3366 A og1=0.00798 A,
al =0.606; A2=2. 3524 A, og2=0. 00800 A, aq =0.793.

Typical Sample —Out Data

Analyzed C3
3000

NC

c E 20PO)

1000-

0

p—1.0

.

I
-0.5

.

I
0.0

.

I
0,5

1.0

n (deg. )

Direct C2
25000
20000—
15000'
1000—0 1.0 -0.5 0.0 0.5 1.0
a (deg. )

Sample —In Data

Analyzed C3
6000 . D=2mrn
4000—

Direct C2
50000 . D=2mrn 40000-

2000—

30000-

p

.I .I .I

c

. D=4mrn

4000-

2000-

T

p

~

I

~

I

~

I

4000- D = 6mrn

0 2000'

p

.I

~

I

~

I

4000- D=8mm

I

~

I

~

I

C

~ D=4mm

E 40000-

0 30000-

~

I

~

I

~

I

. D=6mm
40000-

C

30000-

I

~

I

40000- D =8rnm

2000-

30000-

-0. p—1.0 ~

I

~

I

~

I

5 0.0 0.5

1.0

a (deg )

20000—1.0 -0.5 0.0 0.5 1.0 n (deg )

(b)
Sample —In Data, continued

Analyzed C3
6000
4000- D=10mrn

Direct C2
50000
40000- D=10rnm

2000-

30000-

4000- D=12mm
2000-

40000- D=12rnm
30000-

' D=14mm
E 2000-
4000- D=16mrn
C
2000O
4ppp D 1 8 m m
2000-

C

D=14rnm

30000-

~

I

~

I

40000-
30000O

rn

30000-

. 4000-

D=20mm

D = 20mrn

2000-

30000-

-05 00 10 05 10 p—

.

I

~

I

~

I

~

a (deg )

20000—1.0 —0.5 0.0 0.5 1.0
n (deg )

FIG. 8. Interferograms in the C2 and analyzed C3 beams vs

phase rotator angle cx (see Fig. 1) normalized to the same count-

ing time per point (10 min) for the PR analyzer crystal in the

(111)antiparallel configuration, with various sample thicknesses

D.

38
Bi Sample No.
1
2 1
1 —' 2 22
32 4 41 5

KAISER, CLOTHIER, WERNER, RAUCH, AND WOLW'ITSCH

TABLE III. Relative contrast results in the direct C3 beam.

D (mm)
2.09 4.01 6.10 7.99 10.08 12.26 14.35 16.15 18.24 20.08

61 (A)
—44.2 ——18248..97 ——211638..08 ——235093..21 ——338451..42 —424.3

4f (rad) —118 ——232475 ——455721 — —689142 ——1901324 —1135

Cout
46. 3+0.5 46. 9+0.5 43. 5+0.5 48.0+0.5 47. 8+0.5 45.2+0.5
45.0%0.5
41.5%0.5 46.6+0.5 48.4+0.5

C;„(%)
40. 8+0.4 26. 8+0.4 7.7+0.4
1.3+0.4 4.2%0.4 3.6+0.4 1.8+0.4 0.7+0.3 1.3+0.3 1.4+0.3

45
C„(%)
88. 1+1.3 57.3+1.0 17.8+0.9 2. 2+0.8 8.7+0.8 8.0+0.8 3.9+0.9 1.8+0.8 2. 8+0.7 2.9+0.6

Cp. To insure that this is the case, we have to take the due to the sample, as discussed in Sec. I. The fit program

two data sets at the same time.

also computed a statistical error for each fit parameter.

Collecting two such interferograms for a given sample
" constituted one data "run. The process was repeated for '" all Bi samples individually (including the paired "—, sam-

Using standard statistical methods, we calculated error
bars for our results. If the sample-out contrast is Cp+0'0,
and the sample-in contrast is C;„+or;„th,en the relative

ples). The counting time per point depended on the contrast Cz koz is given by

analyzer crystal and configuration being used; see Table

II. In selecting out a small portion (crzicrz) of the wave-
length spectrum, we also selected out only a fraction of

Cln
Cz C0

(27)

the available neutron intensity, which led to long count- where

ing times and greater statistical error. The counting times listed below are average values; for high contrast

0'p

' 2 1/2

runs with thin samples, we used shorter counting times;

Cp

C;„

(28)

with thicker samples we increased the counting time to

more accurately measure the expected small contrast.

The direct beam scans took about 6 h apiece; the PR analyzer scans lasted about 13 h; and the NP analyzer

C. Results

scans required over 36 h to complete.

Since the contrast results depend on the shape of g(A, ),

„, After completion, each interferogram was fit to a
cosine function by a nonlinear least-squares-fit routine. We used the fit parameters to calculate the contrast C

we first need to know the spectrum to be able to predict the outcome of the experiment. The measurement tech-
nique is described in Ref. Il], and consists of using a

of the interferogram (see Ref. [I]). This was done sepa- (nearly) perfect silicon crystal to do a 8-28 scan in the C3

rately for the data in both detector channels C2 and C3 beam, and converting this to a wavelength spectrum.

for both the sample-out and sample-in conditions. The The resulting spectrum, arbitrarily normalized to 1, is

sample-in contrast had to be corrected for attenuation shown in Fig. 7, along with the one- and two-Gaussian

TABLE IV. Relative contrast results in the analyzed C3 beam, PR analyzer crystal, (111)antiparal-
lel configuration.

Bi Sample No.
1
2 1 1I
2
2-
3
3-
4 41
5

D (mm)
2.09 4.01 6.10 7.99 10.08 12.26 14.35 16.15 18.24 20.08

„, C, (%%uo)
47. 9+0.8
49.9+0.8 50. 1+0.8
51.3+0.8 50.9+0.8
51.2+70.8 51.8+0.8 50.7+0.8 53.0+0.8
49.6+0.8

C;„(%)
47. 8+0.6 38.8+0.6 31.6+0.5 21.9+0.5 15.6+0.4 9.9+0.4 6. 3+0.4 3.9+0.4 3.8+0.4
1.4+0.3

Cg (%%uo)
99.8+2.0 77. 8+1.7 61.1+1.5 42. 6+1.1 30.6+1.0 19.2+0.9 12.2+0.8 7.7+0.7 7.0+0.7 2. 8+0.6

COHERENCE AND SPECTRAL FILTERING IN NEUTRON. . .

39

TABLE V. Relative contrast results in the analyzed C3 beam, PR analyzer crystal, (111) parallel
configuration.

Bi Sample No.
1
2 1
11
2
2-
3
3-
4 42
5

2.09 4.01 6.10 7.99 10.08 12.26 14.35 16.15 10.24 20.08

„, C, (%)
48. 2+0.7 49. 5+0.7 48. 3+0.7
49.0%0.7
48.620.7
49.7%0.7
48.0+0.7 46. 8+0.7 47.4+0.7
49. 3%0.7

C;„(%)
44. 3+0.5 33.6+0.5 22. 2+0.4 7.4+0.4 1.0+0.3 2.9+0.3 1.8+0.3 0. 5+0.3 0. 5+0.3 1.0+0.3

Cg (%)
91.8+1.7 67.9+1.4 45.9+1~ 1 15. 1+0.8 2.0+0.7 5.9+0.7 3.8+0.6 1.1+0.7 1.1+0.6 1.9+0.6

fits to the data. Note that the two-Gaussian fit gives two

peaks that are nearly equivalent in width and amplitude;
thus, our approximation in Sec. IIC was a fairly good

one.

We then began our data collection, beginning with the

contrast scans in the direct C3 beam. The analyzer crys-

tal was rotated to a position perpendicular to the beam,

so that there would be no Bragg re6ection and minimal

absorption. The detector cassette was moved into the

direct C3 beam and its position was optimized with a

rocking curve. We then took a series of contrast scans

using different sample thicknesses. The relative contrast
results of these scans are given in Table III.

We next mounted the pressed silicon (PR) analyzer

crystal in the C3 beam, and rotated it and the detector

(8„= arm

u—nti2l 2.

we
0').

found the (111) antiparallel The positions of the analyzer

reflection and detec-

tor were optimized with a series of rocking curves. Once

maximized, the analyzer and detector remained fixed.

We then took a second set of contrast scans with different

bismuth samples. The contrast results of these scans are

given in Table IV.

Plots of the PR antiparallel interferograms are shown

in Fig. 8, along with the fits to the data. The sample-out

curves did not vary much from run to run, so only one

representative set is shown. The plots show that the con-

trast does fall off more slowly in the analyzed beam than

in the direct beam as anticipated. The relative contrast

curve in the C2 beam is identical to that in the direct C3

beam because both have the same spectral width cr&.

Comparing the graphs, we see that the overall C2 con-

trast decayed quickly, reaching (nearly) zero by a sample
thickness of D =8 rnm; the analyzed C3 beam, on the

= other hand, had visible contrast up to a thickness of
D 14 mm or so.

We next rotated the PR analyzer and detector arm into
the (111)parallel direction (8„=+220.'), and optimized

their positions with a series of rocking curves. This ar-

rangernent was used to take the third set of contrast scans

for various sample thicknesses D. The contrast results of

the PR (111)parallel scans are given in Table V. For the

fourth and final data set, we replace the PR analyzer with

the nearly perfect (NP) silicon crystal. The analyzer and

(8„= detector positions
configuration

we—re2o2p. 0tim'),izaendd

in the (111)antiparallel another set of contrast

scans using various sample thicknesses was performed

with this arrangement. The contrast results calculated

for the NP (111) antiparallel runs are given in Table VI.

Due to the long collection time, there were fewer data

runs taken in this configuration.

The relative contrast results of all four data sets are shown in Fig. 9. Also shown are the curves predicted by

our model using Eq. (10) and the fit parameters listed in

the caption to Fig. 7. As Fig. 9 shows, there was an in-

crease in contrast in the analyzed beam compared to the

direct beam. The contrast in the direct C3 beam fell

C„=2. rather rapidly to a minimum of

2% at D =8 mm,

improved a bit, fell to another minimum at D =16 mm,

TABLE VI. Relative contrast results in the analyzed C3 beam, NP analyzer crystal, (111)antiparal-
lel configuration.

Bi Sample No.

D (mm)

„, C, (%)

C;„(%)

Cg (%)

4.01 2.77 12.26 16.15 20.08

37.2+1.4 41.7+1.4 43. 1+1.5
42. 2+1.8 39.1+1.6

36. 1+1.3 41.2+1.3 38.7+1.4 36.3+1.3 37.3+1.3

97. 1+5.1 99.0+4.8 89.6+4.4 86.0+4.8 95.225.2

KAISER, CLOTHIER, VfERNER, RAUCH, AND WOL%'ITSCH

TABLE VII. Phase difference between the direct C2 and analyzed C3 interferograms for the PR (111)antiparallel analyzer contrast runs.

Bi Sample No.

bc~ (deg)

0c~ (deg)

4C2 PC3 (deg)

2

2.09

19+0.5

202+0.6

177+0.8

1

4.01

216+0.7

39+0.8

177+1.1

1 —'

6.10

110+15.

306+0.9

164+1.7

2

7.99

186+9.8

182+1.1

4+9.8

21

10.08

141+2.8

148+1.5

7+3.2

3

12.26

174+3.2

183+2.4

—9+4.0

3-

14.35

207+9.8

192+3.5

15+10

4

16.15

40+17

294+5.7

106+18

4 —'

18.24

28+9.3

180+5.4

208+11

5

20.08

120+13

287+ 14

193+18

and then tailed off. When the PR analyzer was used in the (111) parallel configuration, the contrast fell a bit
more slowly than in the direct C3 beam, but not substantially so. This is to be expected, since this reAection only
reduced the spectral width by -12%. In the antiparallel
con6guration, the PR analyzed contrast falls o8' much more slowly than in the direct C3 beam. While the direct
beam curve goes through a contrast minimum at D =8
mm, the antiparallel PR data still have 48% relative con-

Longitudinal Displacement hx (A)
0 100 200 300 400

100 i

Direct C3 Beam

50 25

100,
75
50 25 0C 0 100 & O 75 50 25

PR Analyzer
11) Parallel
aI I
zer
nti

. 1 00 ".
755025-

(d) NP Analyzer
(111) Anti

0

I

I

~

I

I

0

4

8

12 16 20

Bi Sample Thickness D (mm)

FIG. 9. The relative contrast results vs sample thickness for
the different analyzer configurations used in the experiment. The solid lines represent the theoretical curves based on our model and spectral measurements.

trast. When the nearly perfect analyzer is used, the contrast hardly declines at all; even with 20 mm of Bi, the
relative contrast remains near 90%. Very good contrast
can thus be extracted from an apparently incoherent
beam. The measured data also agree fairly well with the
predicted curves.
The second goal of the experiment was to calculate the phase di8'erence between the interferograms in the C2 beam and in the analyzed C3 beam. We used the data from the PR (111) antiparallel scans to make the phase comparison. The C2 and analyzed C3 relative contrast curves are plotted jointly in Fig. 10(a). The C2 curve
goes through contrast minima at D =8 and 16 mm, at
which points we expect a 180' shift in the phase of the interferogram. The PR analyzed curve does not go through a contrast minimum (Fig. 9), so it should not ex-

Longitudinal

0

100

(a) 100 $

80

o 60
40 0
20
CL

Displacement ht ()t)
200 300 400

O
180

90
U
0

I

4

8

12 16 20

Eli Sample Thickness D (mm)

FIG. 10. (a) The relative contrast vs sample thickness in the C2 beam and the analyzed C3 beam, using the PR analyzer in the (111) antiparallel configurations. (b) The phase difference
between the C2 and analyzed C3 beam is this configuration.

45

COHERENCE AND SPECTRAL FILTERING IN NEUTRON. . .

41

Analyzed C3
6000
- P=Q mm
4000 i
2000—

Direct C2
50000
FWV~ ~ D=O mm
40000-

0

I' I

- 0=4mm

4000-

2000—

E

0

I

~

I

I

~ D=10 mm

3000 '

2000-
. D=12 mm
3000-

™ 2500-

I

~

I

I

8 2800 D 1

2600

—1.0 -0.5 0.0 0.5 1.0
n (deg )

I

I

I

40000- 0=4 rnm

C

3O0OO-

I

I

I

C) 36ooo, - 0=10 mm

34000-

M

C
0

~

I

I

36ooo - D=12 mm

V

0

I, 34000-

I

~

I

0=18 mrn

33500 -

~~ ~

33000 -

~~ ~~

~

~~

II
~

I

~

I

i

I

—1.0 -0.5 0.0 0,5 1.0

a (deg )

FIG. 11. Several interferograms replotted from Fig. 8, with
the vertical axes scaled so that the phasing is readily apparent.
In the first two panels (D =0 and 4 mm), the C2 and analyzed C3 curves are clearly 180' out of phase; for D =10 and 12 mm, = in the next two panels, the curves are in phases; and in the final
panel, with D 18 mm, the curves are again out of phase.

perience a 180' phase shift at any point.

difTfearbenleceVPIIc2g—ivePsc3thoef

initial phases Pc2 and the C2 and analyzed

Pc3 and the C3 interfero-

grams, sample-in condition, for each sample thickness in

the PR (111)antiparallel added to Pc& to make it

data set. If necessary,
greater than Pc3. The

3P6c20'—wPeQre3

data in Table VII are plotted in Fig. 10(b), and show that

the 180' phase shift does occur. Up to a thickness of 6 mm of Bi, C2 and the analyzed C3 beam are 180' out of

phase. Between 6 and 8 mm of Bi, C2 goes through a

contrast minimum, so that from 8 to 14 mm of Bi, the C2

and analyzed C3 beams are in phase. At about 16 mm of

Bi, C2 goes through another contrast minimum, so that

from 18 to 20 mm of Bi, the two beams are again 180' out

of phase. This effect can be seen directly from the inter-

ferograms shown in Fig. 11.

It should be mentioned once again that this 180' phase

shift only occurs when we compare the analyzed C3

beam with the overall C2 beam. The overall C2 and C3

beams are always 180 out of phase. With the analyzer,

however, we can pull out of the beam a certain range of

components whose mean phase is different from that of

the overall beam. This is only possible due to the non-

Czaussian shape of the wavelength distribution.

IV. CONCLUSIONS
In this experiment, we were able to restore coherence to an apparently incoherent beam by filtering out of that beam those neutrons whose wavelengths lie in a narrow band o.z. Indeed, with the nearly-perfect-silicon analyzer, we extracted essentially 100% relative contrast

TABLE VIII. Calculated longitudinal coherence lengths bx of the neutrons in the different analyzer configurations.

Beam

Ax (A)

Direct C3 (and C2) PR analyzer, (111) parallel PR analyzer,
(111) antiparallel NP analyzer,
(111) antiparallel

86.2 97.5
148
3450

from a "contrastless" beam, even with 20 mrn of bismuth

dephasing the beam. We were also able to show that the

mean phase of the neutrons in this wavelength band may

not be the same as that of the beam as a whole, but may instead differ by 180'.

Although the results make perfect sense from a wave-

component viewpoint, it brings into question the connec-

tion between particles and localized wave packets. When

we say there is a neutron wave packet of coherence

length Ax, we often think semiclassically that the wave
packet somehow "is" the localized neutron particle. If the two wave packets traversing paths I and II in the in-

terferometer overlap within a distance Ax in the region of

recombination, we picture the neutron overlapping with

itself, giving rise to interference.

But consider what this conceptual picture means in

light of this experiment. By reducing the wavelength spread, we increase the longitudinal coherence length of

the packet, according to the uncertainty principle:

„= hx(FWHM) = v'81n2o 1.034 A

(29)

for

A. =2.35

0
A.

The

calculated

coherence

lengths

are

given in Table VIII. The thing to keep in mind is that we

determine the coherence length after the interference has

taken place, far downstream from the interferometer; Ax

can be 86, 149, or 3450 A, or any other value for that
matter. If the wave packets "were" the neutron particle,

we could not vary their physical extent, at will, after the

fact, as we have apparently done in this experiment.

The conclusion to be drawn is a familiar one in quan-

tum mechanics: matters waves are not particles, and we

have no right to think of them as such, even in a semi-

classical way. The neutron wave-packet formalism is

merely the mathematical description of Wheeler's

quantum-mechanical "great smoky dragon" [12]. We

know the neutron is a particle when it is emitted, and

again when it is detected, but between these two times,

the physical connection between the neutron particle and

the wave packet remains hidden, no matter how diligent-

ly we try to analyze the quantum questions with our clas-

sical tools.

ACKNO%'LEDGMENTS

This work was supported by the Physics Division of the NSF (Grant Nos. PHY-8813253 and PHY-9024608}, the U.S.-Austria program at NSF (Grant No. INT-
8712122}, and Fonds zur Forderung der wissenschaftlichen Forschung (Project No. S4201) in Austria.

42

KAISER, CLOTHIER, WERNER, RAUCH, AND WOLWITSCH

*Present address: Physics Department, Bethany College,
Bethany, WV 26032.
[1] R. Clothier, H. Kaiser, S. A. Werner, H. Rauch, and H. Wolwitsch, Phys. Rev. A 44, 5357 (1991).
[2] S. A. Werner, R. Clothier, H. Kaiser, H. Rauch, and H.
Wolwitsch, Phys. Rev. Lett. 67, 683 (1991). [3] D. Petrascheck and R. Folk, Phys. Status Solidi (A) 36,
147 (1976).
[4] V. F. Sears, Neutron Optics (Oxford University, London, 1989), pp. 245-252.
[5] H. Rauch and M. Suda, Phys. Status Solidi (A) 25, 495
(1974).
[6] W. A. Hamilton, A. G. Klein, and G. I. Opat. Phys. Rev.

A 28, 3149 (1983). [7] H. Kaiser, S. A. Werner, and E. A. George, Phys. Rev.
Lett. SO, 563 (1983). [8] G. Comsa, Phys. Rev. Lett. 51, 1105 (1983).
[9] H. Bernstein and F. Low, Phys. Rev. Lett. 59, 951 (1987). [10] A. G. Klein, G. l. Opat, and W. A. Hamilton, Phys. Rev.
Lett. 50, 563 (1983).
[11]R. Clothier, Ph. D. thesis, University of Missouri-
Columbia, 1991 (unpublished).
[12] W. A. Miller and J. A. Wheeler, in Proceedings of the In
ternational Symposium on the Foundations of Quantum Mechanics, Tokyo, 1983, (Physical Society of Japan, Tok-
yo, 1983), p. 38.