zotero/storage/5S6WH8X3/.zotero-ft-cache

PHYSICAL RKVIE% A

VOLUME 21, NUMBER 5

MAY 198Q

Gravity and inertia in quantum mechanics
J.-L. Staudenmann and S. A. %erner
Physics Department and Research Reactor Facility, University of Missouri-Columbia, Columbia, Missouri 65211
R. Colella and A. %". Overhauser
Physics Department, Purdue Uniuersity, West Lafayette, Indiana 47907 (Received 11 September 1979)
The experiments described in this paper probe the simultaneous effects of gravity, inertia, and quantum mechanics on the motion of the neutron. Using a neutron interferometer of the type developed by Bonse and Hart for x rays, we have observed quantum-mechanical interference phenomena induced by the gravitational field of the Earth and by the Earth's rotation relative to the fixed stars. The importance of these experiments with regard to the role of the principle of equivalence in quantum mechanics is discussed.

I. INTRODUCTION

The purpose of this paper is to provide a com-

prehensive description of a series of neutron-

interferometry experiments we have carried out

over the last few years, in which the effects of

gravity and inertia on the quantum-mechanical

phase of the neutron have been studied. Separate

preliminary work have

reports on already been

vpaurbioliushseda.spe'ctWs eotwtihllis re-

view in this paper all of our previous work and

give many experimental and theoretical details
that have not been discussed in our previous re-

ports.

In 1965 it was demonstrated by Bonse and Hart

that interference effects could be obtained between

well-separated coherent beams of x-ray photons of

about 1-A wavelength. In their experiment an

x-ray beam was split into two spatially separated

coherent beams, a few centimeters apart, and re-

combined in such a way that intensity oscillations

could be observed as the optical path along one of

the beams was changed. This remarkable achieve-

ment opened up the field of interferometry in the

angstrom region. The question was immediately

asked whether or not the same principles could

be used to get interference effects between co-

herent beams of thermal neutrons, which can be

diffracted in crystals in the same way as x rays.

The feasibility of neutron interferometry was, in

fact, mer,

fainndallyBodnesme,o'nusstrinatgedessiennt1ia9l7ly4

by Bauch, the same

Trei-

scheme, apart from dimensions, adopted for x

rays. The principles upon which x-ray and neu-

tron interferometry are based are obviously very

different from those applied in optical interfero-

metry. Two new features are at the basis of in-

terferometry in the angstrom region: (a) Bragg

diffraction, and (b) highly perfect crystals, free

of lattice defects. The usual requirements in

21

polishing and aligning optical surfaces to frac-

tions of wavelength would obviously not be satis-

fiable in x-ray and neutron interferometry. We

will see, however, in the next section how the

very same two new features (a) and (b) above en-

able us to circumvent this apparently insurmount-
able difficulty. When a neutron beam is coherent-

ly split and recombined, a new property of the

neutron becomes available for investigation. the

phase. In the language of quantum mechanics we

can say that it is the wave function, along with its

own phase, that becomes measurable, whereas

before the advent of neutron interferometry only

the probability density ~P~ could be measured.

A number of experiments have been performed

since 1974 in which the neutron phase has been

probed in one way or another. We will concen-

trate in this paper on those experiments in which
the neutron phase is affected by the Earth's gravity and by its rotation. Among the various interactions in nature, gravity is by far the weakest

one. In the hydrogen atom, for example, the

gravitational attraction between the proton and the

electron is only 10 times the electrostatic

(Coulomb) attraction. The fact that neutrons are

subject to a gravitational pull toward the center

of the Earth has been demonstrated by verifying

that tory.

a'

neutron

beam

follows

a parabolic

trajec-

The gravitational constant involved in interpre-

ting the observed parabolic paths has been veri-

fied to coincide, with reasonable accuracy, with

the accepted value for the gravitational accelera-

tions as measured with macroscopic bodies. This

result has been obtained in a recent improved

version of this experiment. The important result

here is that the gravitational mass of the neutron

is found to coincide with its intertial mass, as

required by the principle of equivalence. This

principle has been verified with great accuracy,

1419

1980The American Physical Society

1420

STAUDKNMANN, %KRNKR, COLKLLA, AND OVERHAUSKR

21

10, better than 1 part in

for macroscopic bod-

ies. There is no guarantee, however, that the

same principle holds for the quantum-mechanical

behavior of isolated elementary particles. We

want to emphasize that the parabolic fall of a

neutron is a classical experiment, in the sense

that no quantum features of the neutron are being

observed. This would merely be a requirement

imposed by the correspondence principle.

The experiment which we have carried out is

one in which a neutron interferometer is oriented

in such a way that the two coherent beams, into

which the primary beam is split, propagate in

regions of space with different average gravita-

tional potential. A gravity-induced change of

phase provides a manifestation of gravity effects

on a quantum-mechanical feature of the neutron.

If the change of phase agrees with a gravitational potential of the form m~g ' r, where m~ is the
gravitational mass of the neutron, g is the gravitational acceleration, and r is the position vector

of the neutron, then we can conclude that we have

verified the principle of equivalence for neutrons

in the quantgm Bmit. It will be shown, in fact,

that the fringe shift can be expressed by a formula

in which the gravitational constant and Planck's

constant are inseparably linked. This is, to our

knowledge, the only experiment in physics in

which the outcome depends on a simultaneous di-

rect combination of gravitational and quantum-

mechanical properties of an elementary particle.

A complete test of the principle of equivalence

in the quzntum limit would involve repeating the

experiment in an accelerated frame of reference,

traveling in a gravitation-free space. We have

not directly done this experiment. However, we

surmise that this experiment does not need to be

done, if we believe that the Schrodinger equation

holds in an accelerated frame. In such a case,

it is possible to derive the same expression for

tthioenaflrincgaeses.h'ift as that obtained for the gravita-

Since the coordinate frame in which our experi-

ments are carried out is not an intertial frame,

the Hamiltonian governing the neutron's motion

will involve a third term in addition to the kinetic

energy (relative to the Earth) and the gravitational

potential energy. Our neutron experiment design-

ed to detect this effect is the quantum-mechanical

analog of the optical interferometry experiment

of Michelson, Gale, and Pearson" carried out in

1925. An experiment which demonstrated the

principles of detection of rotation by optical inter-

fSeargonmaect.r'y

was carried The physical

out earlier in 1913 by principle involved for ms

the basis for the ring-laser Sagnac gyroscope.

In the latest version of our experiments, a geo-

metry has been chosen in which the effect of gravity is suppressed. In this way only effects associated with the relative motion of the Earth with respect to the stars is detected. If our results agree with the insertion in the neutron Hamiltonian of a term of the form cu ' L (where &u is the Earth' s angular rotational velocity, and L is the neutron's angular momentum with respect to the center of the Earth) we can conclude that the principle of equivalence for neutrons has been verified in the quantum limit, in an accelerated frame free of
gravity. In the next section we will discuss the basic
principles of neutron interferometry. In Sec. III we will give some experimental details relevant
to the neutron source, the neutron monochromator, the detection system, and the construction of the interferometer. In Sec. IV we give the theoretical background necessary to understand
the experimental results. Section V is devoted primarily to a discussion of gravitationally-induced quantum interf erence. The experiments discussed in this section were carried out with an incident beam directed horizontally, that is, parallel to the local surface of the Earth. The effect of the rotation of the Earth on the neutron phase has been accurately detected with a verti-
cally directed incident beam. These experiments
are discussed in Sec. VI. In Sec. VII we make some concluding remarks.

H. PRINCIPLES OF NEUTRON INTERFEROMETRY

Various schemes have been proposed, and in

part realized, for obtaining interference effects

between spatially separated beams having a wavelength

coherent neutron in the angstrom range.

'

We will limit ourselves to consider the scheme

depicted in Fig. 1, consisting of three identical

perfect silicon crystal slabs cut perpendicular to

a set of strongly reflecting lattice planes, typi-

cally (220). The distances dq and d2 between the

slabs are usually a few centimeters and are equal

to within about 1 p m. A nominally collimated,

monochromatic beam is directed along the line SA

and is coherently split by the first silicon crystal

slab by Bragg reflection. These two coherent

beams are again split in the second crystal in the

regions near points B and C. Two of these beams

overlap and interfere near point D in the third Si Slab. We always assume that by "beam" we

mean a plane wave of limited, but not infinitesi-

mal, lateral extent (a few mm or so). If the beam

traversing the path I is phase shifted by increasing the "optical" path length (via some interaction

potential) relative to the beam traversing the path

D, the intensities in the detectors C2 and C3 will

21

GRAVITY AND INERTIA IN QUANTUM MECHANICS

1421

A
Ih
Lattice
A-=

hah

2

hah

II

II

phase
hf

which the surfaces of the optical components need to be polished to fractions of wavelength. The phase shift resulting from a step of thickness t on the surface is given by the formula

l. FIG. A schematic diagram of the LLL interfero-
meter. The incident beam is coherently split in the region of the first slab between A and A'. The two coherent beams I and II a.re again coherently split in the sec-
ond crystal slab and recombined in the third crystal in
the region of points D and D'. The interfering beams
are detected by counters C2 and C3, a noninterfering beam is detected by counter C&.

change. We will show here that the expected intensities in these detectors, as a function of the phase shift P are of the form

o.' cosp

and

I, = n(l + cosp),

(2)

The constants & and y depend upon the incident flux, the crystal structure and the neutron-nuclear
scattering length of Si. The basic principle of this interferometer seems
simple enough; however, there are certain subtleties hidden in the apparent simplicity. The first one concerns the alignment of the three crystals. Clearly, in order to preserve the Bragg reflecting condition for a given wavelength neu-
" tron, we must align the three crystals to within
the "Darwin width, which is typically 0.1" of arc
for neutrons. Bonse and Hart devised a simple and ingenious way to achieve this result. They cut out the three slabs from a large monolithic silicon single crystal of very high quality, free of lattice defects of any kind. As a consequence of great advances in crystal growth techniques, prompted by the needs of the solid-state-electronics industry, it is possible today to purchase from commercial manufacturers silicon crystals
of the required perfection with typical dimensions of order 5 to 10 cm. Spatial coherence in atomic
positions is preserved to a billionth of a centimeter over these distances.
The second point concerns the accuracy with which the surfaces of the slabs need to be polished. It would clearly be impossible to satisfy the
usual requirements of optical interferometry, in

where & is the neutron wavelength and n is the index of refraction, which differs from 1 by

— n-1=

A.
2'

Nb.

(4)

Here & is the atom density and b is the neutron-
. nuclear scattering length. For Si at &= 1. 4.4, (n —1) =0.67&&10 . Thus, a step t=2 p, m will
cause a phase shift corresponding to ~th of a

fringe.

The third consideration is the question of the

extent to which the incident beam is required to be

monochromatic. In our neutron-interferometry

experiments, the incident beam is only nominally monochromatic with &&/& = 0.01. The important

feature of this type of interferometer is that it

utilizes Bragg reflection from perfect crystals. This requires the wavelength, along a given tra-

jectory (ray line) to be defined to within about 1
. part in 10 But this definition of wavelength is

accomplished by the interferometer itself and not

through the preparation of the incident beam.

The final point we want to mention is a peculiar

feature of angstrom-wavelength interferometry.

We can understand. that the two beams BD and CD

in Fig. 1 are coherent and produce interference

fringes localized in space, with spatial separa-

0
tions of order 1 A.

Strictly speaking,

the inter-

ferometer could consist of the first two slabs

only. The problem is that no film or detector of

any. kind is able to resolve fringes so closely sep-

arated in the region of the overlap. The scheme

adopted by Bonse and Hart is to use the third
crystal as a receiver, or as a mixer. The crystal lattice potential Vg corresponding to the re-

ciprocal lattice vector G mixes the two waves

traveling along the rays BD and CD, so that the

outgoing beams depend upon the wave amplitudes

of each of the incident beams.

We will now look in detail at the diffraction

mechanism of the three crystal LLL interfero-

meter. (LLL stands for Laue-Laue-Laue trans-

mission geometry. ) The theory of diffraction of

neutron waves by the periodic potential of a per-

fect crystal lattice is similar in many ways to the

theory of electron motion in solids. A neutron

wave of wave vector Kp oriented on or near a

Brillouin-zone boundary will be Bragg reflected

forming a coherent state described by the wave

function

1422

STAUDENMANN, %KRNER, GOLELLA, AND OVERHAUSER

21

g g(r) = gp exp(iKp ' r) + exp[i(Kp+ G) ' r] . (5)
In electron band theory, there will be two such wave functions, one for states above the energy gap corresponding to the reciprocal lattice vector
6, and one below the energy gap. In the dynami-
cal theory of neutron diffraction, the energy of the neutron is fixed by the preparation of the incident beam, and the periodic potential causes a splitting of the allowed internal wave vectors Kp. Thus, the index y takes on two possible values. The wave amplitudes go~ are determined by the orientation of the external incident wave vector kp and the requirements of continuity of the neutron wave function across the entrant boundary. Continuity of the wave function across the exit boundary determines the amplitudes of the diffracted beam (yo) and the forward scattered beam (Xp), such
that the wave function of the neutron leaving the
crystal is of the form

X(r) =Xp exp(ikp r)+Xo exp(i& r).

(6)

For the symmetric I aue geometry, the solution to this problem is given in the Appendix. The results are given by Eqs. (A21) and (A22); they are
of the form

X, =D(~8)C,

(8)

where 4 is the amplitude of the incident plane
wave, and the coefficients T and D depend upon the angular deviation &~ of the incident wave vec-
tor kp from the exact Bragg condition for the re-
ciprocal lattice point G. Using these results, it is an easy matter to de-
rive expressions for the waves emerging from the third slab of the interferometer. The wave function in the region of detector C3 ls

U, (r) = 4$D(&8) exp(ikp ' ro)D(-&8) exp[ikp ' (r~ —ro)]T(&8)

+ T(48) exp(ikp ' re)D(&8) exp[iko ' (r~ —re )]D(-&8)e P}exp[ikp ' (r —r~)],

(9)

and the wave function in the region of detector Cq is

U&(r) = QD(68) exp(iko ' r, )D( &8) exp[ik—p ' (r~ —ro)]D(&8)

} + T(&8) exP(ikp ' re)D(&8) exP[i& ' (rD —rz)]T(- n8) exP[i& (r —rD)] .

(10)

The origin of coordinates is taken to be point A in
Fig. 1. The phase shift P is assumed to be intro-
duced into path I of the interferometer. The external diffracted, wave vector Q is given by Eq. (A29). After some algebra, we find for squares
of the wave amplitudes

I Up I

= c'

I

I

(1 + cosP) 2A (x, &)

IU21'= I4 I'[&(~, ~) »(~, ~)—cosP]

(12)

The functions A and B are given by

=, A. (x, ~) (x, +1), sin'$(x'+ cos'&)

(is)

and

=, ]]. a(x, E) x +1 , sin'&[(x'+ cos'&)'+ sin

(i4)

The definition of the symbol $ is given by Eq. (A23). These functions are dependent upon the
two dimensionless parameters

kp sin20~ V„»

P

4

and

pa

2m~le )a

2 cos0g Vcel 1~p coseB

(i6)

Here ~& is the Bragg angle, V„» is the volume of a unit cell, &G is the structure factor, and a is
the slab thickness.
Since the beam incident on the interferometer
is divergent, we must integrate these expressions over the angle &~. This is equivalent to integrating over the scaled angular variable x. We,
therefore, see that the expressions for the param-
eters o.' and r in Eqs. (1) and (2) are

~=IP

am)Eg)

P p

Sl.n 2

g

ceps

and

A(x &)dx

Y

=IP

~P
p

s4l.nm22/E8~B

/
yVce»

B(xq 'E)dà q

(18)

where Ip is the incident beam intensity. The integrals appearing in Eqs. (17) and (18) are shown in
Fig. 2. For our experiments a=0.246 cm, at
X=1.4 A, the value of p:=40, which gives y/&=2, 6. Thus the predicted contrast in the "0-beam" is
considerably higher than in the "G-beam", as
observed experimentally.
The analysis presented here assumes that each plane wave Fourier component in the incident di-

1424

STAUDENMANN, WKRNER, COLE LLA, AND OVERHAUSER

FIG. 4. Schematic diagram of the neutron interfero-
meter and the He detectors used in these experiments.
is about 2X10 neutrons/cm /sec. Three small
He detectors are mounted behind the interferometer, inside an aluminum box (25 X 25&&25 cm') which is attached to a rotator assembly. The He detectors have a diameter of & in. and are filled to a pressure of 40 atm, yielding a counting efficiency of order 90/0 for thermal neutrons. Figure 4 shows the arrangements of the three counters, with Cg mounted on a noninterfering beam for the purpose of aiding in orienting the interferometer. Occasionally the three He counters are replaced by a single Ar-filled proportional counter when x rays are used instead of neutrons. The rotator assembly, consisting of a large steel tube supported on ball bearings, is rigidly attached to the inside of a large dense masonite neutron shield which is mounted on a vibration isolation
pad.

A. The neutron wavelength ) o
The interpretation of all our experiments re-
quires an accurate knowledge of the incident neutron wavelength &0 as measured in the laboratory frame of reference. %e have employed a techni-
que to measure &0 which is schematically illustrated in Fig. 5. A pyrolytic graphite crystal is placed in the beam which passes directly through
the interferometer. The beam transmitted through
this crystal is counted with a fission chamber. By rotating the pyrolytic graphite crystal through
the same (004) reflection used for monochromation of the incident neutrons; a dip in the transmission is observed. A similar dip is observed by reflecting the beam to the right instead of to the left. The difference in the crystal rotation
angles for minimum transmission, i. e. , 02 —eq,
determines the neutron wavelength in terms of the lattice parameter of pyrolytic graphite (c
=8.V08 A). The analysis of the data presented
here is based on this method of determining &0.
B. The interferometer
The interferometer used in the experiments
described in this paper is a considerably improved
version of the same kind of device used in our early work. ' It has been cut with a 600-grit, 4-in. -diameter, diamond blade from a 5-cm-diameter high-purity silicon perf ect- single-crystal ingot. The room-temperature resistivity is p

20000counts
54 sec
l5000-
IOOOO-

76:= 8= l9.

Xo- I.I54 A

Ilelation

pa configu

5000-
0-

Interfero box
double monochromator

M counter
rotating graphite crystal

l8

l9

20

I

I

I 59

l60

IGI

8 (deg )

FIG. 5. An example of the method used to determine the neutron wavelength by measuring the angular spacing between the transmission dips observed by rotating the pyrolitic graphite crystal in the beam transmitted through the interferometer.

IT Y AND INKR TIA IN QUANTUM MECHANICS

1425

C2
c3
monitor
FIG. 6. Geometry of thee Si-slab experiment shown in Fig. 7. Th e same geometr y wass used for the x-ray experimimeennts s designed to measure qb, ~d (Sec. V}, in which
case the Si slaab is replaced by a plastic slab.
. = 3600 ohm cm. The slaab th1ckness is a=2.464
mm, and the distance between the slabs is d==34.518+ 0.002 mm. It has been f
n s or silicon do n n most cases, a surface originally flat corn he departure from flatness du e to th is effect is enough in some cases to produce an appreciabl p ase shift acrosss the beam dimensions. Alth ough no silicon etchant 1s immune from this undesirab
in a mixture of 3 parts HÃQ (70%), 2 arts l re or min. The quality of our interfero-

meter can be seen from a measurement of the

y, phase shift induced by th e mean neutron-nucleaear

1nteraction potential. Placin g a

p

a o homogeneous materiail in thee inntterfero-

meter as shownn in Fi' g. 6 and rotatinggiit aboouuttan

axis perpendicularar to th e interferometer scatter-

1ng plane results in an osc'liat'

t'

. n

We show data obtained in

zs way for a Si slab (thickness 7' —0.2

ig. 7. The slit size on the neutron beam was
. . 6 mm It i' s eas y to show that the phase

corresponding to a given rotat 1on angle ~ is

sin5 singe
cos'8& —sin'6 '

where N is the atom density, b the nuclea e interf erometer Brraagg angle
. (220 planes), annd A. th e neutron wavelen gth. The
period of these oscciillaat 1ons therefore provides a precision measuremenetn of th e scattering length b.
The experiments described in this pa er presented unusuaal deaf 1culties with regard to mounting the interferometer. The ex e
ol ot t'ing th e interferometer about a h or1zontal axis and alssoo ab out a vertical axis. W have therefore beeenn forced to find a method for
. holding the interferommetee er without strain' ing it.
We have tried a lar g e sele ct1on of waxes, gl an epoxies. In all cases th e 1' nduced strain spo1led the performance e of th e interferome eter. After considerable e eff ort, we have discovered that

4000-
counts 239 sec
5000-

2000-

IOOO-

0-

I

I

-0.26I8 -0.)745 -0.0873

X.= I. 229 A

neutrons

I

I

0

0.0875

I'
O.J745

0.26I8

sin 8 sin 8 (radians)

FIG. 7. Example of data obtaineed by rotating an Si slab in thee innterferometer (Fig. 6}. Th e counts in detector C3 are
shown.

1426

STAUDKN MAN N, WKRNKR, COLE LLA, AND OVKRHAUSKR

21

"double-sticky-back" plastic tape works. It provides the necessary adhesive character, yet it is sufficiently pliable and resilient so as to minimize the transmitted strain.

IV. THEORY GF THE EXPERIMENTS

Classically, the Hamiltonian governing a neutron's motion in the gravitational field of the ro-
tating Earth is

P2 Q

g ~~J

(2o)

2m;

Here the angular momentum of the neutron's mo-
tion about the center of the Earth (r =0),

L=rxp

(21)

p is the canonical momentum of the neutron, co is
the angular rotation velocity of the Earth, M the
mass of the Earth, m& the inertial mass of the
neutron, and m~ the gravitational mass of the neutron. From an epistomological point of view,
it is not possible to be confident that this Hamiltonian correctly describes quantum- mechanical phenomena, especially. those involving interference. However, for lack of evidence to the contrary, we will assume it is also the correct quantum-mechanical Hamiltonian, and then see if the predictions based on it agree with the experiment.
The principle of equivalence would require that the inertial mass m& and the gravitation mass m~ in Eq. (20) are equal.
Since the distances involved within the neutron interferometer are very small compared to the radius R of the Earth, we can write (20) as

=p 2
2m) +mego' r —v L+ Vo,

(22)

where Vo is the gravitational potential energy at

some reference height above the Earth (say, the

center of the interferometer), and gp is the acceleration due to gravity. The classical equa-

tions of motion are Hamilton's equations

=, r=

ea '
Bp

and

.
p=-

. a-- a—
Br

Here, the dot implies a time derivative, so that r is the neutron velocity in the coordinate frame of the rotating Earth. The first of these equa-
tion gives the canonical momentum

p = m~r + m~Q) X r ~

(24)

and the second gives its time derivative

p =mego —co Xp

(25)

Combining Eqs. (24) and (25} we obtain the wellknown equation of motion for a classical particle

in a rotating frame,

r. ~~
m;r =m — gp m&&u& ((exr) —am, (ox

(26)

-~ Thus, the term ' L in the Hamiltonian gives

rise to both the centrifugal acceleration and the

Coriolis acceleration. Since we will only be in-

terested in the neutron's motion over distances

corresponding to the dimensions of the interfero-

meter, which are very small compared to the

Earth's radius, we can define an effective gravi-

tational acceleration in the usual way

g =gp+ (m, /m, )(o && ((u &&%),

(2&)

which we take to be independent of position r. Un-
der this assumption, we can solve (26} for the local motion of the neutron in the frame of the ro-
tating Earth. To leading order in (o, the solution
ls

. r = rp+ vpt+ —'(,gt') + p((ut'~&& g)

(26)

The transit times for thermal neutrons through

the interferometer in our experiments are of or-

. der 5x10' sec.
56 min, one has

.Based on a co=V.29x10

s'idseerceal

day of 23 h, Thus, we

see that the term in (28) involving a& is smaller
, . than 'gt by a fa—ctor of about 10 Therefore,

the Coriolis force has a negligible effect on the

trajectory over these small distances; however,

its effect on the neutron phase is not negligible

as we shall see.

The discussion so far has been based on classi-

cal mechanics. In order to calculate the phase

shift P in a neutron interferometer experiment,

we assume that we can associate a de Broglie

wave of wave vector k with the neutron having

canonical momentum p:

p=@k q

(29)

where @ is Planck's constant divided by 2r. The
phase difference for the neutron wave traversing the path ACD in Fig. 4 relative to the neutron
wave traversing the path ABD is then given by

P=

—t
@ ~ACD

p

dr-

— t p'dr.
@ ~ABD

1 p dr,

(so)

The momentum appearing in this line integral on

the path ACDBA around the interferometer is

™ ™ given by Eq. (24}. The phase shift thus involves
two terms,

p=

tp r ' d r +

. (&u x r) ' d r

(sl)

The velocity r is obtained by differentiating Eq. (28). To a very high order of approximation we

GRAVITY AND INERTIA IN QUANTUM MECHANICS

1427

can regard the trajectories between the interfero-
meter slabs as straight rather than parabolic curves. The angular deviation, from a straight
line over these distances (for &=1.4 A) is of order 0.01" of arc. It is fortunate that this angular deviation is about 10 times smaller than the
"Darwin acceptance width" for Bragg reflection in the silicon crystals. If this were not so, a neutron on the trajectory AB, say, would not be Bragg reflected by the middle crystal slab of the interferometer. The first term in (31), which we
will call P„„, is the phase shift due to the gravi-
tational field of the Earth. To work out the integral for P~„requires us to specify the direction of the incident beam and the orientation angle P of the interferometer with respect to this direction (see Fig. 4). For a horizontally directed incident beam the result is
P~„=-2mm, m~(g/h )&OA' sing -=-q~ sing .

(32)

The area A' is given by

A' = (2d + 2ad cos8e) tanee .

(33)

The angle P is defined to be zero when the plane ABDC of the interferometer is horizontal. The laboratory neutron wavelength i.s related to its
velocity by

~, =h/m, . )rf,

(34)

and 8~ is the Bragg angle. The result for a verti-
cally directed beam is given in Sec. VI.
The second term in Eq. (31), which we call
P~~, is due to the rotation of the Earth. Using
vector calculus, this integral is easy to evaluate,

(35)

The normal area A enclosed by the beam paths is

. A = (2d + 2 ad) tan8e

(36)

The formula (35) was obtained by Page using wave-optical arguments, and by Anandan 20 and Stodolsky within the framework of general rela-
tivity. Recently, an interesting derivation has
been given by Dresden and Yang in which the phase shift for either a rotating neutron or optical
interferometer is derived from the point of view of a Doppler shift due to a moving source and moving reflecting crystals.
The actual path of any given neutron within a crystal slab is more complicated than the line drawn straight across the crystal as shown in
Fig. 1. There is a current j carried by the -
branch part of the wave function and a current j~ carried by the P-branch part of the wave func-

tion. In addition, there is a current j ~ due to the interference of the o.'- and P-branch wave func-
tions which leads to a sinusoidal trajectory for the neutron. This is the phenomenon which leads
to Pendellosung interf er ence fringes. Consideration of these effects has recently been treated in extensive detail by one of us. The conclusion reached is that to very high order of approximation the trajectory can be regarded as a straight line across the crystal, and that the microscopic details of the trajectory do not play a role in calculating the nest phase shift due to gravity within
the crystal medium.

V. GRAVITATIONALLY INDUCED QUANTUM INTERFERENCE

In this section we will confine our attention to experiments in which a horizontally directed inci-
dent beam is utilized as shown in Fig. 4. The
experimental procedure involves turning the interferometer, including the entrance slit and the
three detectors Cq, C2, and C3, about the incident beam line AB. At each angular setting P, neutrons are counted for a preset length of time
(actually based on the incident beam monitor).
This procedure allows the neutron on the beam
trajectory CD to be somewhat higher above the surface of the Earth than for the beam path AB. The difference in the Earth's gravitational poten-
tial. between these two levels causes a quantummechanical phase shift of the neutron on the tra-
jectory ACD relative to the trajectory ABD. The phase shift on the rising path AC is exactly equal
to the phase shift on the opposite rising path BD, as can be shavn by applying Huygen's principle.
The phase shift P~„depends on the product of the inertial and gravitational masses m, m~ of the neutron. Thus, measuring this phase shift induced by the Earth's gravity can be r egarded as a test of the princple of equivalence in the quan-
tum limit if we compare the mass

m„=(m, m, )'"

(37)

with the neutron mass obtained from mass spectroscopy results on the proton and the deuteron according to the formula

. m„= m~ —m~ + E„/c

= 1.6747 x 10

4
g

(36)

The deuteron binding energy is obtained from the
radiative capture 'gamma-ray energy &„(=2.23
MeV). The fact that this experiment is a test in the quantum limit is apparent since quantummechanical interference is involved, and Planck's
constant appears explicitly in the formula (32).

STAIJOENMANN, %KRNKR, COLKLLA, AND OVKRHAUSF R

A. The total phase shift
Unfortunately, as we have pointed out previously, there is an additional effect on the measured phase shift P resulting from bending (or warping) of the interferometer under its own weight. This effect is dependent upon the rotation angle P, since the experiment involves turning the interferometer about an axis which is not an axis of elastic symmetry. We eall this effect
Pbeag =-qbeng Sm4 ~
In Subsec. B below we will justify writing P„„„in
this form. For a horizontally directed incident beam we can
easily work out the dot product involved in evalua-
ting Ps~ in Eq. (35); the result is
}, P8sa~ge ac 4m—m;' ~&( cos@ cos&~ + sing sini' sin8~

where ~L, is the colatitude angle, I" is the angle of
the incident neutron beam west of due south.
Beam port B is oriented nearly exactly along a
north-south line, such that the monochromatic
beam incident on the interferometer is directed due south. Thus, the angle I' in Eq. (40) is zero for our experiments, and we can write (40) in the
form

PS agnac qgagnae COSg

(41)

where

qs~ —(4~~,.(uAlh) cos8~ .

(42)

Consequently, the total phase shift in these experiments involves three contributions

P =Pgeae+Psagnae+ Pbeng
~ =-q~„sing + qs cosQ —qb„g sing . (43)
We see that the Sagnac effect is maximum for P

=0, while the gravity and bending effects are
maximum for g =90'. We can rewrite Eq. (43)
as

P = q sin(p —p, ),

(44)

where

q' 2 =(q

+qb

2
g)

+qg2

(45)

qg aan ae
+ ~grav ~bend

(48)

The fact that the phase shift due to the Earth's ro-

tation (Sagnac effect) depends upon cosg and not

sing comes about as a result of our selection of
due south as the incident beam direction [I'=0

in Eq. (40)]. This leads to an important experi-
mental circumstance. Although qs~~ is of or. der 2.5% of q~„, its contribution to the total fre-

quency of oscillation q of the interference pattern
is very small (of order 3 parts in 10 ). However,

it leads directly to a shift P, in the center of the

interferenee pattern. Table I gives the calculated

qs~, wavelength dependence of q~„,
We have used the colatitude angle

81, —5a1nd.37Pga.nd

the acceleration due to gravity q = 980.0 cm/sec

at Columbia, Missouri, along with the dimen-

sions of our interferometer given in Sec. III to

compile this table.

B. Bending
The values for qb„d given in Table I have been obtained from a series of experiments using x rays. The procedure involves using molybdenum Kn x rays (&=0.71 A). We direct a beam of x rays along the same incident line AB (Fig. 4) and observe the interfering x-ray beams with an xray sensitive porportional gas detector as a func-
tion of rotation angle P. The effect of gravity
(gravitational red shift) on the x rays over the

TABLE I. Calculated frequencies of oscillation, q.

0

Og

(deg)

A.
(cm2)

(cm2)

&gray
(radians)

~Sggnac
(radians)

&bend
(radians)

0.6 8.9885 4.038 56 4.035 25

0.8 12.0236 5.437 91 5.429 97

1.0 15.0934 6.885 82 6.869 99

1.2 18.2083 8.398 46 8.370 45

1.4 21.3800 9.995 43 9.949 61

1.6 24.6220 11.701 1 1.8 27.9505 13.547 1

11.630 3 13.441 8

2.0 31.3851 15.575 4 15.423 6

2.2 34.9503 17.844 5 17.630 1

2.4 38.6781 20.438 7 20.140 0

9.5255 17.0904
27.0285 39.5181
54.8024 73.2109
95.1913 121.362 152.596 190.168

0.582 396 0.784 196
0.992 997
1.211 13 1.441 43 1.687 41 1.953 62
2.24612 2.573 33 2.947 44

1,406 41
1.875 21
2,344 01 2.812 82
3.281 62 3.750 42 4.219 23 4.688 03
5.156 83 5.625 63

6}0
(deg)
3.297 62 2.512 07 2.029 54 1.703 26 1.468 07
1.290 62 1.152 08 1.041 03
0.950109 0.874 415

21

IOK
cooqts
68 sec, IOK-
5K-
IOK-
5K-
IOK 5K IOK5K
IOK-
5+
IOK5KIOK5KIOK5K-

~

~

I i~
-0.)

~I

I ~ ~ ~

~

X RAyS
(220)

I

~

e

~

~

0p5 0

20
l5 Ip
0
-lp
-20
pp5

sin 5 sin 8

ction of the

sett ings

4,

ofothe'eni'tnatieornferoanmgeleterQr.(seTehisFigda.~ ta6)

for
'

varloQs

0 reflection in S

p

o this data

e rmine q bend

distancess i'nvolved in the int

bl 8 c se th e x-ray wave field may not remm
y uniform (coh erentp over the cr

th ee 'Ulterferomet

o warping of th ensity

P will not necessarily give abend o e reme car

y~ n

us

and we do not observe

e circumvent theeese problems b

suring the shift in the phassee of an interf

oscillation patt

s lt

Fi. fit lb f p1astic in the ' te

see

6 fo

me ry) as a function of thee angle

P. Data obtained i

il tof 1 t fo th

p erlI.n ig. 10 wheere we plot th

shift

rn& vs amer W e observe a lmear relation

ch justifies thee form of Eq. (89

se plots

s Q&end' It is reason ably cer-

coUnts
&42 sec.
. 4Qpp.

X RAYS (44p)

2po = &g&

40OO45oo. -
5000-
55oo.
«pp—
50oo550p-

@= l5O loo 5o ~
0' .
5o
@--IP---

550p
«oo-
500p.

C'=- I5
@*-20..

4ooo

I
-Q.I745

-0087'

sins sin g cos g

(radians)

FIG. 9. X-ray data for the (440) reflection in Si.

tain that the functional form o

p

Pb„~ —krM= (2m/X 0)6d

where &d is th e di'f ference in
relative to ACD d

g or the ng o

the in

the fact that th

a xs very nearl y four times the

I

I

I' I

I

I

I

300-
PHASE
(deg}
200-

IOO-

0'I -0.4 -0.2

I

I

I

I

0

0.2 0.4

sin 4'

FIG. 10. A plot of the ph i ions re '

ro ating a plastic

erring x ra b

"l. "- l

i- 8-. Th

~ ese plots gives q

STAUDENMANN, %ERNER, COLKLLA, AND OVKRHAUSER

21

4000
3000
C3 2000,

+1.060 A

100
+ FOURIER TRANSFORM 1.060 A
80

1000

60
0-40 -32 -24 -16 -8 0 8 16 24 32 40 g [deg ]

FIG. 11. Gravitationally induced quantum interference

experiment at X 0=1.060 A. The counting time was

40

about 5 min per point.

slope of the (220) data, we conclude that

. ~do- sin'8&

(48)

Therefore putting the arguments together that have led to (39), (47), and (48) we find

Pb„d =—q„„~sin@ =-(C/%0) sin es sing;

(49)

the numerical value of the constant from the
data is

C =34.5V rad A.

(50)

The reason why the bending effect seems to depend quadratically on sineB is not yet understood.

C. Experimental results

We show in Figs. 11 and 12 representative data
obtained at two wavelengths: ~0 —1.060 and 1.419
A, respectively. The neutron counting rate in
detector C& is plotted versus the interferometer rotation angle P. The contrast (maximum/mini-
l. mum) of these data is seen to be about 3 to
Contrasts as high as 8 to 1 have been observed in some runs. To obtain the frequency of oscilla-
tion q we Fourier transform the data numerically according to

N
sin eQQ si5
g=1
where I is the oscillatory

part of the neutron

(8i) in-

8000
6000-
C3
4000

g 1&19 A

2000

0-40

I
-32

I
-24

I
-16

I
8-.

0

I

I

8 16 24 32 40

|t)[deg ]

FIG. 12. Gravitationally induced quantum interfer-
ence experiment at X 0= 1.419 A. The counting time was

about 7 min per point.

20

00

l

I

i

i

10 20 30 40 50 60

FIG. 13. Fourier transform of the data of Fig. 11.

j tensity, and the index runs over all & datum
points. The Fourier transforms of the data of Figs. 11 and 12 are shown in Figs. 13 and 14.
There is loss of contrast at larger rotation
angles g, which we believe to be due to warping of the interferometer under its own weight as the interferometer is rotated. This explanation of the effect is in accord with various experiments we have carried out with reduced slit sizes in
which the loss of contrast is reduced. We have found that as the neutron wavelength becomes
larger the loss of the contrast occurs at smaller
rotation angles P. This observation is also in
agreement with the above explanation, since the bending effect measured with x rays is porportion-
al to sin'es(= &OG'/4).
In any interferometry experiment, the long-term

120100 Ill 80

FOURI FR TRANSFORM
X; 1.419 A

60 40

20

I
0 1 0 20 30 40 50 60 70 80 90

FIG. 14. Fourier transform of the data of Fig. 12.

GRAVITY AND INERTIA IN QUANTUM MECHANICS

o=

l. 05

0
4

5000-
counts
360 sec.

data taken from

58.4 ~M
26

t"o "

-4I.6 deg -58 4 -52.4

a 26

' -26 4

~ 20

-20.4

s mean value

4
~ 0+

I

2000-

~%'s'
00

L

0*

)000-

s0
~~

'N 4e 0
0

~g pO
*
0
s
*
D

30 20

IO

0

-CO

-20

-30 -40

4 {deg)

FIG. 15. Gravitationally induced quantum interference experiment at & p=1.050 A. These are data from 6 runs taken
over a period of about 75 h. These data show the long-term phase stability of the interferometer.

phase stability is an important consideration. We
show in Fig. 15 data obtained at &p —1.050 A in
which we have tested the phase stability of our interferometer. This figure shows the gravita-

Frequency
of
oscillation
q
loo-

Gravitationally induced quantum interference

/ / /
l4~+/2
/
l5$
/

50-

/
/
~I /3l. l O.I / l2.2
/g2 4/26 17 8/30
25'
7/34/34. I
/

qbbendd( experimentao

I

0 0.5

I

l5

2

&. (A)
FIG. 16. A plot of the frequency of oscillation of q of a large series of scans of the type shown in Figs. 11
and 12 as a function of wavelength Ap. The dashed. curve is the least-squares fit to the data using Eqs. (32} and (49). The labels next to the data points are run numbers.

tionally induced quantum interference fringes obtained on 5 separate runs. The total data collection time was about 75 h. We see that the phase stability over this period of time is extremely
good. The fact that the interferometer is mounted inside a heavy aluminum box, which in turn is mounted inside a large heavy masonite neutron shield, creates isothermal conditions sufficient
to obtain data of this quality.
We have now taken data of the type shown in
Figs. 11 and 12 at a wide selection of neutron wavelengths. The frequency of oscillation q for each run has been obtained both by the Fouriertransform method discussed above and by a leastsquares-fitting procedure (which is discussed in
the next section). These two data analysis techniques agree to within the statistical uncertainty of the raw data. The results of this extensive set of
measurements is summarized in Fig. 16 in which we have plotted the observed frequency gf oscilla-
tion q versus the incident neutron wavelength &p.
The dashed line is the prediction of theory, based on Eq. (32) and the measured frequency of oscil-
lation due to bending. We have fitted the theoretical curve to the data leaving (m, m~)" as an adjustable parameter. We find

(m, m~)' = (1.675 + 0.003) && 10 g

(52)

which agrees with the rest mass of the neutron
obtained from mass spectroscopy to within the
limits of error.

STAUDKXMA WX, WKRNER, COLKLLA, AND OVKRHAUSKR

VI. NEUTRON SAGNAC EFFECT
We pointed out in the previous section that the
contribution of qs ~ to the total frequency of oscillation of the interference pattern is very small as a result of our selection of the direction of the
incident neutron beam. However, the Sagnac effect leads to an angular shift $0 in the center of the interference pattern given by Eq. (46). ln Subsec. A we will discuss our results on the measurement of $0. We have also pursued an alternative approach, utilizing a vertically directed incident beam, to observe the neutron Sagnac effect.
These experiments are described in Subsec. B.

A. Horizontal-incident-beam experiments

According to Eqs. (2) and (44) the counting rate

in detector C3 should vary with the rotation angle Q according to the formula

I,(p) = o'f&+ cos[q sin(g —gp) +tIO]],

(53)

where P, is the "zero-phase" of the interferometer resulting from the fact that the two legs ABD and ACD are not precisely equal. It is ap-
parent that in order to separately measure $0 and Pp data must be accumulated over an angular range of (@—$0} where the sine function departs from linearity.
This presents special difficulties since the warp-
ing of the interferometer at large P tends to wash out the contrast. Because of this loss of contrast
the analytical form of the intensity I,(g) is more complicated than Eq. (53). We have pursued this
problem by a least-squares fitting procedure in which we use a parametrized form of the inten-
sity profile. Since the experimental results of .the type shown in Figs, 11 and 12 exhibit an envelope and a weakly sloping mean level, we have
chosen the form

I~(g) =A + BP + CP

+ D cos[@ sing + &q] cos[q sin(g —g p) + P p]

(54)

as a phenomenological generalization of Eq. (53).

There are nine parameters in this equation. '
. B C D Q &g Pp q, and P, We are, of course,
only interested in q and Po. The results of an ex-

tensive series of measurements e,nd data analyses

are shown in Fig. 17. The dashed line is the re-

sult of a numerical calculation based on Eq. (46)

qa, „. relating the angle $0 to qs,

Since

is in-

dependent of wavelength (aside from its depen-

dence on the area A), and q~,„ is proportional to

the wavelength (aside from its dependence on the

area A'), we see that except for the small correc-

(deg)

Shift C, of the center of
the interference pattern
gIe

theory

g l3

34 ~2e

~12.2 ~ I I X g24

le g

Q I6

3OQ

+25

3I.I

~32

~ Sm ~+

+i5

0

0 05

l. 5

).(A)

FIG. 17. Experimental results for the shift ft)0 of the
center of the interference patterns as a function of
wavelength Xo. These results were obtained by leastsquares fitting of the functional form of Eq. (54) to a large series of data of the type shavn in Figs. 11 and 12. The angle f3t 0 is related to the frequency of oscillation due to the neutron Sagnac effect as given by Eq. (46). The dashed curve libeled "theory" is the result
of using Eq. (46) and the geometric parameters of the
interferometer given in the text. The labels next to the
data points are run numbers.

tion due to q„.„d in Eq. (46}, Qp should be inversely proportional to ~0. From this data it is clear that we are observing the neutron Sagnac effect. However, the scatter in the data is rather severe.

B. Vertical-incident-beam experiments

We show in Fig. 18 a schematic diagram of our

most recent experiments designed to detect the

neutron Sagnac effect. The initially horizontal

beam from the monochromator is reflected by a
beryllium crystal through 90, such that the beam

incident on the interferometer is vertical (along a
. 'plumb line). The experimental procedure involves

turning the interferometer about the vertical line

AB through various angular settings Q. For a

P, beam which is precisely vertical, the phase shift

due to gravity,

„, is independent of the angle

@, as can be seen from symmetry. However, the

angle between the rotation axis ~ of Earth and the

interferometer normal area vector A is P depen-

dent. Thus, this incident beam orientation allows

GRAVITY AND INERTIA IN QUANTUM MECHANICS

1433

vert ical phas shifter

coll ima to r&

. beam from & ..... ...... ......,l

monochromato r

P/Ff PFf F I

pp

li
p Se crystai

FIG. 18. Schematic diagram of the configuration of the apparatus for the vertical-beam experiments designed to measure the neutron Sagnac effect. The drawing is not to scale. The collimator is approximately 1 m in length, and the interferometer is approximately
8 cm long from point A to point D. The angle 6 of the phase shifter is defined to be zero when it is parallel to the three interferometer slabs.

us to experimentally suppress the effects of gra-
vity, leaving only the effect of the rotation of the Earth on the phase shift. The bending effect is
also independent of P since the orientation of the interferometer with respect to g is fixed. The suggestion that the Sagnac effect could be obser-
" ved with this geometry was first made by Anan-
dan.
Using Eq. (35) it is an easy matter to work out
the formula for P8. „as a function of @; we get

Pa = (4vm, /h)&uA sing~ sing

= g@,~„sing .

(55)

The superscript "v" indicates that this expression

is for a vertically directed incident beam. For

these experiments the incident wavelength is fixed
at &o —1.262 A, which in turn determines the value
. of the normal area, whicI '.s&=8.864 cm The

angle P is defined to be zero, when the normal

area vector A is directed due west. The predicted

frequency of oscillation is

„=91.92 deg = 1.604 rad. (theory). (56)

As we turn the interferometer through various
angles P, the counting rate is expected to vary according to

f(g) =A+Bcos(qe, „sing+ po).

By allowing A and B to be numerically different, we have taken into account the facf that perfect contrast is never actually realized in practice. The results of such an experiment are shown in

l000-
Cp COUA)S
500 sec. 500
tOO"
50" 800"
500--

A) I(C) )
l
nterferometer at Bragg pos.

r,. (B)

~

~

I

i

I(Cgb«
}

lriterferometer off Bragg pos.

(C) l(C)) = l(4))-l(4)) b

400
200-loo-
0" 400
200

4 ~~a l

0 Ct

0

0

(P') g (61

0

s~ 'r

~4 —~

41

(H) ~(C)

(i) f(

l
N

E

S

W

N

. . l

l .l .l

l

90 I80 270 0 90

4(deg )

FIG.. 19. Data taken in a "direct" measurement of the
effect of the Earth's rotation on the neutron phase snift.
The various parts of this figure are explained in detail in Sec. VIB.

Fig. 19. There is a difficulty in directly interpreting the counting rate in detector C, to be f(P). There is a natural variation of the "effective" incident beam intensity which is the angular ac-
ceptance range for Bragg scattering by the silicon interferometer. This is due to the energy-angle
correlations in the incident beam resujting from
the monochromation process using single crystals. However, one can measure this variation sepa-
rately by blocking off the beam in one leg of the interferometer (and then the other) and measuring
the effective beam strengths under noninterfering
conditions. The series of curves in Fig. 19 are the results of utilizing this idea. Part (A) of this figure is the raw data for the counting rate in de-

1434

STAUDENMANN, WKRNKR, COLKLLA, AND OVKRHAUSER

21

tector Cz versus g. Part (B) is the background
counting rate with the interferometer rotated off
the Bragg reflecting condition. Part (C) is the
raw data minus the background, which we call
i(P}. The sets of data (D) and (E) are the counting rates in detector C3 when beam I is blocked off and then beam II is blocked off with a cadmium
absorber. Parts (F) and (G} are the background counting rates under these conditions. Part (H)
gives the average of the data in scans (D) and (E)
minus the average background of scans (E) and
(G). The data of part (H), called o,'(@), is directly
proportional to the effective incident beam intensity. The final graph, part (I), is obtained by dividing f(g) in part (C) by o.'(g) in part (H), that

lsq

f(p) =f(p)/o'(&) .

(58)

This procedure allows us to divide out the angle-
energy correlation effects, leaving orily variations with P due to interference. The solid line in part (H) is a least-stiuares fit of the data to the func-
tional form (57). We find

qs„„——104.4+ 0.4 deg

= 1.822 + 0.007 rad (expt. ).

(59)

This value is to be compared with 91.92 deg in Eq.

(56). There are several difficulties with this tech-

nique. The most serious one is that it involves

a number of independent steps in the sequence in

arriving at the corrected plot in part (I). Thus,

the experimental systematic and statistical errors

accumulate.

In order to circumvent this difficulty we have

developed a technique in which we directly mea-

sure the phase shift. We insert a slab-shaped

phase shifter into the interferometer as shown in

Fig. 18. This slab is, in fact, another Si single
crystal of thickness T =0.2931 cm, although it

could be made of any material. Rotating this slab

through an angle & about an axis normal to the

parallelogram ABDC results in a phase shift aris-

ing from the mean neutron-nuclear potential. The

formula for this phase shift was given in Sec. III,

,Eq. (19). As we rotate this phase shifter through various angles &, the counting ratios in detectors

C2 and C3 are observed to oscillate. Repeating

this procedure at another setting Q of the inter-

ferometer results in another oscillating pattern,

of the same period, but shifted in phase with re-

spect to the first pattern. We show in Fig. 20(a)

data taken at
at p =-90'.

@=0 and
The phase

in Fig. 20(b)
shift between

data taken these two

patterns is due to the rotation of the Earth. The

results of an extensive series of measurements

are shown in Fig. 21. Each datum point in this

200

I

I

I

. 1000

counts

..2pp (e) /=0

I

I

I

-8 -6 -4 -2

0

I
2

I
4

I
68

S fdeg)

FIG. 20. Typical oscillating counting rates observed
in detector C3 at two orientation settings fII} of the interferometer. The counting time for each datum point was
approximately 600 sec.

figure was obtained by least-squares fitting of a sine wave of unknown phase to data of the type shown in Fig. 20. Because of long-term drifts of the interferometer phase, measurements at a reference angle (usually A pointing east or west}
were repeated after each new setting P.
The labeling of north, south, east, and west on this diagram was achieved through an astronomical sighting of the star Polaris. This line of sight was carried inside the reactor hall (which is below ground level) by precision surveying techniques and transferred onto the interferometer with a laser mounted on a rotary table.
The solid curve in Fig. 21 is the result of a least-squares fitting of the data to a sine wave.

P (deg)

FIG. 21. A plot of the phase shift P due to the Earth' s rotation as a function of orientation fII} of the normal
area A of the interferometer about a vertical axis. The symbols N, W, S, and E indicate north, west, south, and east. These data were taken in six sections as dis-
cussed in the text. The different symbols are for the interferometer box facing various directions.

GRA VITY AND INERTIA IN QUANTUM MECHANICS

It gives

q~, —96.8+0.2 deg

=1.689+ 0.003 rad (expt).

(60)

This result is in closer agreement with theory than the result (59). However, it is obvious from the data that there is still a substantial problem. These data were taken in six steps. Because of geometrical limitations of the rotator inside the heavy masonite shield, we were only able to accu-
mulate data over an angular range of Q of about
130'. To obtain data through other ranges of P it
was-necessary to turn the entire masonite shield
box through large angles (typically 90 ). This
required releveling and orienting of the rotator
assembly. One notes that the data from each of the six sequences do not fit perfectly together. This fact points up the need for extreme care and
precision in this experiment. Since the magni-
tude of the frequency of oscillation due to gravity is about 50 times the frequency due to the Earth' s rotation, a small misalignment of the beam axis
off verticality results in a contribution from P~„
If the incident beam is off the axis of a plumb line
by an angle z, the P-dependent part of the phase shift due to gravity is

P~, =q~„sing siny,

(61)

q„„ where

is given in Sec. IV. To give an idea

. of the size of the effect, suppose y'=0. 1 At ~()

=1.262 A, q „=44 deg rad =2521 deg, giving

P~„=4.4 deg. Thus,
of the beam (and also

a misalignment of the axis of rotation)

the of

a0x.i1s'

will result in an error of about 5% in the mea-

surement of the neutron Sagnac effect. On the

basis of these considerations, we now feel that

the unusual agreement of experiment with theory

reported in our preliminary paper was somewhat

fortuitous.

We have now modified the heavy masonite shield

so that the rotator assembly can be turned through

a full 360 deg without realigning the beam axis.

We have exercised extreme care in aligning the beam axis using precision levels. Our most re-

cent data taken under these conditions are shown

in Fig. 22. The solid curve is again a least-

squares fitting of this data to a sine curve. We

find

q~„=94.6 + 0.3 deg

=1.651+ 0.005 rad (expt).

(63)

This result is within 3/z of the theoretical prediction (56). We believe that to improve upon this result would require new techniques, of which we are not aware.

P (deg)

0 50 too l50

t
N

-50--E

-l00--

4 (deg)

FIG. 22. A plot of the phase shift' P due to the Earth' s
f rotation as a function of orientation as in Fig. 21.
These data were taken after modification of the rotator assembly to allow a complete 360' sequence of scans
to be performed without realignment of the verticality of the incident beam.

VII. CONCLUSIONS

Our observation of quantum- mechanical inter-

ference phenomena in these experiments confirms

that the Newtonian potential mg ' r must be included

in Schrodinger's equation, and that this potential

influences the Phase of the neutron wave function

in a manner expected for any other potential. We

believe that this result has a fairly deep signifi-

cance which concerns the principle of equivalence.

The equality of inertial and gravitational mass

eisxpoenreimsetnattesme'ntof

of this Eotvos

principle. and Dicke

The have

classical verified

this equality to very high precision. An alterna-

tive and stronger statement of this princple re-

quires that the results of an experiment carried

out in a uniform gravitational field cannot be dis-

tinguished from the results of an experiment car-

ried out in a gravity-free laboratory experiencing

a constant acceleration.

All verifications of the equivalence principle,

prior to our experiment, have been in the classi-

cal domain. The experimental results did not

depend on Planck's constant. For the experiments

described in this paper, the number of interfer-

ence fringes observed for a given rotation of the

interferometer depends on the numerical value of

Planck's constant, and therefore represent a test

of the principle of equivalence in the quantum

limit.

Since the phase shifts observed in our gravita-

tionally induced quantum-interference experiment

depend upon the product m;m~, and the phase shifts

in the Sagnac experiment depend only on m&, we

can certainly claim that the combination of these

experiments demonstrates the equivalence of

inertial and gravitational mass in a quantum-

1486

STAUDKNMANN, %KRNKR, COLKLLA, AND OVERHAUSKR

mechanical phenomenon. We would like to propose that a stronger and deeper conclusion can be reached. In order to truly verify the principle of equiva-
lence one must carry out tmo experiments —one on
the surface of the Earth in a laboratory at rest, and another in a laboratory far out in space having
an acceleration g. We have not done this experi-
ment. However, we suggest that the second experiment need not be done if one accepts the validity of Schrodinger's equation in a gravityfree, inertial frame. If this is granted, then the outcome of an interferometer experiment in an accelerated laboratory can be calculated with
certainty. This has been done, and we find that the observed phase shift in our Earth-bound ex-
periment agrees with this prediction if we replace
the laboratory acceleration a with g in the final
formula.
To summarize then, one must either question the validity of Schrodinger's equation under zerogravity conditions or assert that we have verified the stronger statement of the equivalence principle
in the quantum limit. The first alternative seems
unacceptable. It would be very exciting to carry out these ex-
periments at both very low neutron energies and at very high neutron energies. In the ultra-cold neutron energy region where the change in gravitation potential energy is comparable to the neutron kinetic energy, the WEB approximation for calculating phase shifts fails and the neutron trajectory is not well defined. In the very high neutron energy region where terms of order (v/c) cannot be neglected one will begin to see general relativistic effects as discussed by Anandan and Stodolsky. We believe that experiments in both regions of neutron energy are possible, and we are currently pur. suing certain new ideas along these lines.
ACKNOWLEDGMENTS
The role played by C. Holmes in the successful
completion of this experiment was extraordinarily significant. We would also bke to acknowledge the support of the staff of the University of Mis-
souri Research Reactor, in particular, %. B. Yelon, R. Berliner, C. Edwards, and R. Brugger.
We are grateful for the excellent single crystals
J. of silicon provided us by B. Stone and . Burd of
the. Monsanto Corporation. We thank our col-
leagues G. W. Ford, B. DeFacio, and D. Green-
berger for their interest and many helpful discussions over several years. And finally, we are
very indebted to S. Paiva and K. I eu for the as-
tronomical sighting and surveying. This work was supported by the National Science Foundation

Atomic, Molecular, and Plasma Physics Program through Grant No. '?6 08960.

APPENDIX A: DYNAMICAL THEORY OF NEUTRON DIFFRACTION

We review here the essential aspects of the dy-

namical theory of diffraction for the symmetrical

Laue-transmission geometry (Fig. 23). For fur-

ther details
articles.

'weWereafessrumthee

reader to various review that the absorption is

zero, which is a very good approximation for

silicon.

Let the incident-neutron wave function be given

by the plane wave

y(r)=C exp(ik, r).

(Ai)

To find the wave function g(r) inside the crystal, we must solve the Schrodinger equation

[ (k '/2-~) v'+ V(r)]y = ~,y,

(A2)

where V(r) is the periodic interaction potential of the neutron with the lattice. Defining

v(r) =- (2m/k') V(r)

(AS)

k20=- (2m/0') &„

(A4)

Eq. (A2) can be written as

(V +ko)g=v(.

(A5)

We now write P(r) as a Bloch function

=g g(r) go exp(iko'r+iG r),

(A6)

and expand v(r) in the Fourier series

v(r) =Q v-e@' .

(Av)

The vectors G are the reciprocal lattice vectors
', of the crystal. Putting (A6) and (A7) into the wave
equation (A5) and equating coefficients of e'o we
find

y-,[k, -(K, +G') ]=~ v; --q

(Aa)

+(P):- X
FIG. 23. Schematic diagram of the symmetric Laue geometry. p is the incident plane wave. P is the wave function inside the crystal and x is the wave function emerging from the back face of the crystal.

21

GRAVITY AND INERTIA IN QUANTUM MECHANICS

We now make an approximation. We assume that
ko is oriented very close to the Bragg condition for a particular reciprocal lattice vector G, and
assume that the internal incident wave vector +p
=kp, as we can easily verify. Thus, the "resonance factor" [ko —(K, +G') ] is small only for
G' = G and for G' =0. That is, only $0 and go will
be large. Under these conditions, the above infinite set of equations reduces to two equations; in matrix form they are given by

(K -Ko) -u 6

=0,

-VG (K'- K', ).

where we have defined
&2==&2p-vp

(A10)

and

K~ =K, +G.

(A 11)

For a nontrivial solution of (A9) to exist, the determinant of the matrix of coefficient must be zero, thus

(K- Ko)(K —Ko) = goy g/4ko.

(A 12)

In this equation we have made the approximations
K+K0=2ko and K+Ko =2ko. Equation (A12) de-
fines the dispersion surface, that is, the locus of allowed internal incident wave vectors Kp in k space. The dispersion surface has two sheets, which are hyperbolas as shown in Fig. 24. We call the one branch of this surface + and the other
branch P
The neutron wave function must be continuous
across the entrant surface. For perfect phase

G

'0
FIG. 24. Diagram showing the two hyperbolas of the dispersion surface giving the locus of the allowed internal wave vectors for the symmetric Laue case. The
asymptotes (dashed lines) are circles of radii E [Pq. (A10)] drawn about the point 0 and the point G. The
reciprocal lattice vector is 0 and ko is the external in-
cident wave vector.

matching, this requires that the internal wave vector Kp can differ from ko by only a component normal to the surface. There are two possible values for Kp for each incident wave vector kp, one on the o.' branch and one on the P branch. Thus

Kp —kp —N

(A13)

and
= K~o ko —N

(A 14)

The relation between these vectors is shown in Fig. 24. Thus, the internal wave function is com-
posed of four plane waves:

q(r) =rP() exp(iKO r)+P() exp(iKO r}
+ P~ exp(i'~ r}+P~ exp(iK~ r) . (A15)

The diffracted part of this wave function must be zero along the entrant boundary, and the incident part must match the external incident wave (Al).

Thus'

g-+/=0

(A 16)

4. 4o+ Wo —

The ratio of the diffracted wave amplitude Po to the incident wave amplitude g, for each branch is determined by (A9). If we define C" to be go/P~
(where 'Y= o' or P), then

2k 0(K —K"0)

v6

vo

2ko(K-IPg) '

(A 18)

which are known for each incident wave vector ko. Thus, we can use (A16) and (A 17) to express all of the internal wave amplitudes in terms of the
amplitude of the incident wave 4. The results
are

g =[c'/(c' c )]c,

y,'=-[c /(c~- c )Jc,

(A19)

g&=[c c'/(c~-c )Jc,

y~= [c c~/(c~-c )Jc.

The wave function X(r) emerging from the back
face of the crystal is the sum of two coherent
plane waves

. . y(r) =y., exp(ik, r) + yo exp(ik . r)

(A20)

The boundary condition for continuity of the neutron wave function across the back face of the
crystal requires us to match the incident part of X(r) with the incident part of g(r), and also the diffracted part of y(r) with the diffracted part of g(r). The algebra is rather tedious, but straightforward. The results are

1488

STAUDKNMANN, WERNER, COLELLA, AND OVERHAUSER

21

Xp=

t'. Iz

~

q sin) z +&z zgz

+ cosh )I~

and

Xo=

. k()
z

vo (

P2
+Pz') za

sing

8

4'q

where

(A21) (A22)

g = a(qz+ pz)'~/2 cose, , = '5p a[(vp/kp) + z)]/2 cos&z,
. 5G = 50+ 2koa&e sining

(A23) (A24) (A26)

The crystal thickness is a, ~& is the nominal Bragg angle, and &~ is the angular deviation of the

incident wave vector ko from the exact Bragg condition. The quantities P and g are given by

= p I va I/&p

(A26)

and
. g = ko&e sin20~

(A27)

The Fourier components of the neutron-crystal interaction potential are related to the structure factor ++ by

vo =4mFo/1 „»

(A26)

where the volume of a unit cell is V„». The wave
vector of the diffracted wave can be seen from the
geometry of Fig. 24 to be

ko = G+ k(, +2k,«sine, n,

(A29)

where n is a unit vector normal to the surface.
Note that

I xo I'+ I x, I' = I c I',

(A3O)

as it must, for the zero-absorption case we are
considering;

'A. W. Overhauser and B. Colella, Phys. Rev. Lett.
33, 1237 (1974). 2R. Colella, A. W. Overhauser, and S. A. Werner,
Phys. Rev. Lett. 34, 1472 (1975).
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tron Scattering, Gatlinburg, Tennessee, 1976, pp. 1060-1072, OBNL Report No. OBNL-USEBDA CONF
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For recent reviews of neutron and x-ray interferometry
see U. Bonse and W. Graeff in X-ray Optics, Topics
J. in Applied Physics, edited by H- Queisser (Sprin-
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J. (Springer, Berlin, 1978), pp. 303-351.
9J. W. T. Dobbs, A. Harvey, D. Daya, and H. Horstmann, Phys. Bev. 139, 756 (1965). L. Koester, in Neutron Physics (Springer, Berlin,
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~~R. Eotvos, Math. Nat. Ber. Ungarn. 8, 65 (1890);
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Dicke, Ann. Phys. (N.Y.) 26, 442 (1967). 2For a discussion of this point see D. M. Greenberger
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W. Bauspiess, U. Bonse, and W. Graeff, J. Appl.
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