quantum mechanics of the de broglie wave packet and a review of inelastic losses of ucn in bottles

*Corresponding author.

Nuclear Instruments and Methods in Physics Research A 440 (2000) 709}716

Quantum mechanics of the de Broglie wave packet and a reviewof inelastic losses of UCN in bottles

V.K. Ignatovich!,*, Masahiko Utsuro"

!Frank Laboratory of Neutron Physics of Joint Institute for Nuclear Research, 141980 Dubna, Moscow Region, Russia"Research Reactor Institute, Kyoto University, Kumatori-cho, Sennan-gun, 590-0494 Osaka, Japan

Abstract

Di!erent inelastic processes of ultra-cold neutrons (UCN) loss in traps are considered. A hypothesis of the deBroglie singular wave-packet description of the neutron wave function to explain anomalous losses of UCN isproposed. An experiment to check the hypothesis, and its results are discussed. ( 2000 Elsevier Science B.V. All rightsreserved.

1. Introduction

The anomalous loss coe$cient g"3]10~5 ofUCN in the Be bottle [1], which is 2 orders ofmagnitude higher than the theoretical one(+3]10~7), requires an explanation. We tried toexplain this phenomenon by properties of the neu-tron itself, i.e. by properties of its wave functionstructure. Thus, our goal is to present the hypothe-sis here and to show experimental results aimed atits veri"cation. However, before doing that, it isuseful to review brie#y all the possible inelasticscattering processes leading to UCN losses to see,whether it is necessary indeed to devise somethingextraordinary to explain the anomaly. We thinksuch a review is desirable to have a reference point,because there is a rumour in the UCN communitythat in the report by academician Belyaev all theUCN problems will be solved. Because of volume

restrictions we present here only the "nal results.The details of calculations can be found in Ref. [2].

2. Review of all the inelastic loss processes

First of all we should recall the de"nition of theloss coe$cient. Re#ection of UCN of energy(+2/2m)k2, where k is a wave number, from a wallwith the potential u"(+2/2m)u

0, where u

0"

4pN0b, b is the coherent scattering length (we call

it `amplitudea in the following) of the wall nucleiand N

0is the atomic density, is described by a re-

#ection amplitude R. For total re#ection (k2(u0)

in the absence of losses DRD"1. Because of lossesDRD(1, and a loss coe$cient is de"ned ask"1!DRD2. The coe$cient k is proportional tothe reduced loss coe$cient g"Im u/Reu, whichis equal to the ratio of the imaginary and realparts of the coherent scattering amplitude b

#:

g"Im b#/Re b

#, where Im b

#according to the op-

tical theorem is given by: Im b#"kp

-/4p, and p

-is the total loss cross-section, which includes

0168-9002/00/$ - see front matter ( 2000 Elsevier Science B.V. All rights reserved.PII: S 0 1 6 8 - 9 0 0 2 ( 9 9 ) 0 1 0 6 7 - 0 SECTION 5b.

absorption and many inelastic scattering cross-sec-tions. The particular inelastic process i gives itspartial contribution to the loss coe$cientg*"kp

*/4pRe b

#. In the following we shall write in

the denominator b#

instead of Re b#, because

Im b#;Re b

#.

In general, the cross-section for inelastic scatter-ing, and the related loss coe$cient are representa-ble in the form:

p"4pDb@D2k%&&

/k0Pg"Db@D2k

%&&/b

#,Db@/b

#D2b

#k%&&

"b#k1F (1)

where k%&&

is an e!ective wave vector of the neutronheated via the inelastic process considered, b@ de-notes either a coherent or an incoherent or a mag-netic scattering amplitude. In the last equality

k1

can be k-*.

"Ju0

or kT"J¹ where

¹"300 K is the room temperature, and F is thefactor which should be calculated to see how rel-evant is the magnitude of g to UCN anomaly. Sucha representation of g is convenient for calculationsbecause for, say, copper walls we have b

#"0.7]

10~12 cm, k-*.

+106 cm~1, so b#k-*.

+7]10~7,and b

#kT+2.5]10~4 for all the considered inelas-

tic processes. For substances with lower b#

thesemagnitudes are even smaller.

In considering all the inelastic processes it isuseful to remember two additional experimentalfacts:

1. recent experiments at ILL [3] have shown thatwith probability of the order of 10~6 there issome process of UCN stepwise small heating upto an energy which is approximately twice theprimary one,

2. the experiments [4] had shown that there wasno continuous broadening of the spectrum, i.e.there was no heating at the level 2]10~12 eVper single collision with the walls.

Both facts have nothing to do with the UCNanomaly. Especially, the "rst one because it hasbeen observed in bottles with large losses, andbecause its magnitude is smaller than the anomaly.However, it is useful to take both facts into accountin the estimation of probabilities for di!erent in-elastic processes.

With relations (1) we can easily "nd the probabil-ity of stepwise small heating. Indeed, if b@+b

#and

k%&&

+k-*.

,Ju0, the partial loss coe$cient be-

comes g"b#k-*.

+7]10~7 in pretty good agree-ment with the observations.

2.1. Phonons

2.1.1. Coherent phonon cross-sectionThe coherent phonon cross-section is given by

p#,*/%-

"Db#D2 +

q P Cm

M

i2

2u2D d3qd(q!j!s)

]n(u2/¹)d(u!u

2)d3k

k0

where j"k0!k is the momentum transferred, s isa vector of the reciprocal lattice, u"

(k20!k2)/2"!i2/2#j ) k

0is the energy trans-

ferred, u2

is the phonon energy, which for smallq can be represented as cq with c being the soundvelocity, n(x)"1/(exp x!1) is the Bose}Einsteinfactor, ¹ is the temperature, m is the neutron mass,and M is the mass of the wall nuclei. Here and inthe following we use the units +"m"k

B"1,

where kB

is the Boltzmann constant. In the follow-ing, to facilitate calculations for readers, the "nalformulas for F will have the physical magnitudes intheir natural dimensions.

We can neglect small vectors k0

and q in the "rstd-function. Then, after integration over k, and sum-mation over q, which is approximated by an inte-gral, we obtain

p#,*/%-

"4pDb#D2

kT

k0

FPg"b#kTF

F"

p2C(7/2) f(7/2)

m

MAvTc B

3

Aa

jTB

3(2)

where C(n) and f(n) are Euler and Riemann func-

tions: C(7/2)"15Jp/8"3.32, f(7/2)"1.127,C(7/2)f(7/2)+3.75, and we used the relationskT"2p/j

T, k

T/c"v

T/c, where v

Tis the speed of

thermal neutrons. All the magnitudes in Eq. (2) canbe now used with their natural dimensionalities.

From Eq. (2) it follows, that the coherent 1-phonon heating contributes to the loss coe$cientg#,*/%-

"b#kTF. For Be and ¹"300 K we have

710 V.K. Ignatovich, M. Utsuro / Nuclear Instruments and Methods in Physics Research A 440 (2000) 709}716

vT"2200 m/s, j

T"1.8As , b

B%kT"2.7]10~4,

and F+6]10~3, where for c we used the soundvelocity of transverse vibrations: c"8.8 km/s.Thus, g

#,*/%-+2]10~6, and it quickly decreases

with temperature as ¹[email protected] smallest wavelength of the neutron after

inelastic coherent scattering is j+a, the in-teratomic distance. Thus, the coherent phonon pro-cess does not give small energy heating.

2.1.2. Incoherent phonon cross-section

p*/#,*/%-

"Db*/#

D2Pm

M

i2

2uq

n(u2/¹)

3u2qdu

qu3

D

]d(u2!u)

d3k

k0

"4pDb*/#

D2kT

k0

F

where F"3C(7/2)f(7/2)(m/M)(¹/¹D)3, ¹

Dis the

Debye temperature, and we made the approxima-tion j+k.

For Be the incoherent amplitude b*/#

is verysmall; however, for purposes of estimation we cantreat all inelastic scattering in the incoherent ap-proximation and suppose that b

*/#"b

#. Then at

room temperature, if we take for Be ¹D"1200 K,

we get F+0.02, thus g*/#,*/%-

"b#kTF+5.4]10~6.

For heating to small energies u1+k2

0/2 the

value of F is less by the factor (k0/k

T)7(10~14, i.e.

it is negligible.The two phonon process has an additional factor

F2, which for emission and absorption is equal to

F2"(m/M)(E

0/u

D)3(10~16, and for 2 absorp-

tions is equal to F2"(m/M)(¹/¹

D)3+0.002 (for

Be).

2.2. Surface waves

The surface waves are important for liquids likefomblin oil and for solids too. To "nd the heatingby surface waves we solve the SchroK dinger equation

[iLt#*!u(r, t)]t(r, t)"0 (3)

with the potential u"u0h (z'f

0cos (q ) r

@@!Xt)),

where we introduced the Heaviside function h(x)which is equal to 1 or 0 when the inequality in its

argument is satis"ed or not. For small amplitudef0

of vibrations we can use the linear expansion:u (r, t)+u

0h (z'0)#u

0f0

cos (q ) r@@!Xt)d(z), and

treat the second term as a perturbation for theunperturbed potential u

0h (z'0).

The scheme for solving the equation is the fol-lowing: we go to the reference frame moving alongthe surface wave with the speed c

R"X/q, which is

the Raleigh speed. Then we "nd the di!ractionfrom a frozen wavy relief. After that we transformthe result back to laboratory frame and average itover the spectrum.

As a result the probability of inelastic scattering=, which can be identi"ed with the partial losscoe$cient g

463&.8, is

="g463&.8

+

m

M A¹

¹4DB

2 u0

k2T

(10~6. (4)

The number in Eq. (4) is calculated for Be at roomtemperature, if the surface Debye temperature is¹

4D"0.8¹

D,B%+900 K.

2.3. Liquids

In liquids there is no pure elastic scattering. If wedo not take into account the optical potential of theliquid and treat the quasielastic scattering in thesame way as for thermal neutrons, then the cross-section is

p2%"

2

k0

Db#D2Pd3k

i2D

(k2!k20)2#(i2D)2

.

The integration over the quasielastic peak givesa magnitude of order 4pDb

#D2. We can suppose

that all this cross section means losses. Then itscontribution to the loss coe$cient is g

2%+b

#k0(

b#k-*.

(10~6.The formula is correct not only for liquids, but

also for surfaces contaminated with hydrogen [5],if the hydrogen atoms are freely di!using along thesurface. In that case the magnitude of the e!ect isgH,$*&

"C(pH/p

#)b

#k0, where C and p

Hare the hy-

drogen concentration and cross-section, respective-ly. Thus, there can be an enhancement factorF"C(p

H/p

#), which is important for su$ciently

high C. However, di!usion processes should

V.K. Ignatovich, M. Utsuro / Nuclear Instruments and Methods in Physics Research A 440 (2000) 709}716 711

SECTION 5b.

broaden the UCN spectrum in the bottles, and inexperiment the heating was seen to be a stepwiseprocess.

2.4. Spin waves

Spin waves can be important for ferromagneticwalls. The spin wave cross-section in the Heisen-berg model is

p48"

2

k0

r20+q Pd

3k d3q DF (i)D2Sn(u2/k2

T)

]d(k!q!s!k0) d (k2!k2

0!ui)

where r0"ce2/m

%c2, c"1.91 is the neutron mag-

netic moment in nuclear magnetons, s is the vectorof the reciprocal lattice, S is the spin of the atom,and F(i) is the magnetic form factor, which in ourcase can be approximated by 1. Spin waves in zeroexternal magnetic "eld are characterized by energyui"Di2 with the constant D, which can be repre-sented as m

%/m

%&&, with m

%&&+0.01m

%.

The result of the calculation is

p48"4pr2

0

kT

k0

FPg48+b

#kTF

F"2pp2

6 Am

%&&m

%B

3@2

Aa

jTB

3,

where jT

is the mean wavelength of thermal neu-trons, a is the interatomic distance. The magnitudeof F is near 10~2 at room temperature; thereforeg48

is nearly the same as for phonons.

2.5. Gas scattering

There are some losses of UCN due to scatteringon residual gas. The loss coe$cient due to theselosses can be estimated as g

'!4"n

0p'!4

¸, wheren0

is the gas density, ¸ is the linear extension of thebottle. The cross section is given by

p'!4

"4pDbD2

(1#m/M)2Sm

MS4

p

kT

k0

Pg'!4

"b#kTF

F+2b#n0¸j

0Sm

M (5)

where we supposed that b'!4

"b#of the walls. If we

suppose that F"10~2, then for b#"10~12 cm,

M"25 m, and ¸"10 cm the gas density should

be n0"FJM/m(F/2b

#¸j

0)"5]1014 cm~3,

which is equivalent to the gas pressure at ambienttemperature +10~2 Torr. Usually, residual gaspressure in UCN traps is considerably lower.

2.5.1. Gas model of the wallsWe can consider a very speculative model of the

wall to be composed of gas molecules with highdensity. Then from Eq. (5) it follows that

g'!4,8

+b#kTJm/M. Since b

#kT+3.5]10~4, then

scattering by gas with mass M"25 m is near7]10~5. Of course this model is unrealisticbecause the scattering law of solids di!ers from thatof a gas.

2.6. Cluster model

2.6.1. The wall of clustersIf we consider the substance to be a collection of

gas clusters, each one containing n atoms, then theelastic scattering cross-section of a cluster is pro-portional to (b

#n)2, their density is N

0/n, and

inelastic scattering for a gas cluster contains the

factor Jm/nM as follows from Eq. (5). In

that model the e!ective g#-

is g#-"Jng

'!4. The

clusters have thermal velocity vT,/

"vT,1

/Jn.Thus, for neutrons to acquire the velocity 5 m/s it isnecessary to have clusters of n5104 nucleons.They have dimension less than the neutronwavelength, so the scattering on them can be con-sidered in s-wave approximation. The value ofg#-

can be su$ciently large to explain any magni-tude of observed g. However, this model seems notto be plausible.

2.6.2. Gas of clustersA more reasonable suggestion is that the storage

volume contains dust consisting of such clusters.The losses of neutrons in such a gas of clusterscan be represented by g

#-"Cp

#-¸, where C is the

cluster concentration. If clusters are made ofn atoms with the same amplitude b as the walls,

then g#-"4pb2Cn3@2¸(k

T/k

0)Jm/M"Fbk

T, where


F"2Cn3@2¸bj0Jm/M. Thus, for losses with given

F the clusters in the volume should have the con-centration

C"Fk0JM/m/4pb¸n3@2.

For F"10~2, M/m+25, ¸"10 cm, b"10~12cm,and n"104 we obtain C+109 cm~3. The pressureof such a gas at room temperatures is nearly 10~7

Torr, which is below the sensitivity of vacuumetersused in UCN physics.

2.6.3. Gas of large clustersIn the case of large clusters with dimensions

d"100 As , UCN are totally re#ected from themand on average increase their energy by w2 afterevery collision, where w is the thermal cluster velo-city. We can again use the relation g

#-"Cp

#-¸ but

with p#-"pd2 determined only by geometrical di-

mensions. Thus, the density of such a gas should beC"g

#-/p

#-¸. For d"100 As , g

#-"10~5 and

¸"10 cm we have C"106/cm3. Such clusterscontain 105 atoms and their velocity is near 1 m/s.

2.6.4. Large clusters on wall surfaceIt is also possible to imagine the dust with dimen-

sions d+100 As on the wall surfaces. In that caseneutrons can be totally re#ected from a dust par-ticle, and the probability of inelastic scattering ateach collision with the wall can be estimated asa probability of interaction with the dust particle.The probability of inelastic scattering is g

$645"

C4d2, where C

4is two-dimensional density of the

dust particles. g$645

"10~5 requires C"107 par-ticles/cm2. The thermal velocity of suchparticles is of the order 1 m/s, thus almost everycollision with a dust particle leads to a step-wiseheating.

2.7. Acoustics

Surface vibrations with frequencies lower than109 Hz cannot be treated like phonons, because theneutron interaction time with the wall becomessmaller than the vibration period, and we must takeinto account the velocity of the wall with respect tothe neutron at the moment of collision. Thus, the

neutron becomes re#ected from the moving wall,and it changes its energy according to classical laws.

2.7.1. UltrasoundThe classical considerations show that the prob-

ability of heating at every collision with the wall isclose to unity. If the energy gain at a single collisionis *E then the number of collisions that the neutroncan withstand during the storage is N+u/*E,where u is the wall potential. N can be large, andthe storage time long, if *E is small. However, if*E+u, as was observed in Ref. [3], then the neu-tron becomes lost at every collision, and the storagetime should be very short. We can obtain the stor-age time to be long and the one step heating to belarge, if we suppose that the spectrum of vibrationshas frequencies higher than 109 Hz. In that case theclassical interaction takes place only with someprobability, which can be estimated as the fractionof the spectrum of frequencies below 109 Hz in thetotal spectrum.

Let us suppose that the wall is trembling with anamplitude A and frequency u. If the period¹u"2p/u'1/Juv+2]10~9 s, where v is theneutron velocity, then the interaction with the wallcan be treated as classical, and the neutron energyafter collision is on average equal toE"S(v#2w cos(ut))2T+E

0#2w2, where

w"Au. If we suppose that w"1 m/s atu,u

0+109 rad/s, then A+10 As . At smaller fre-

quencies the amplitude A, to get the samew"1 m/s, should be larger: A(u)"A

0u

0/u. If the

total spectrum of the sound is limited by the fre-quency u

0"109 rad/s, then neutron will acquire

energy Jw2 with probability 1. If we want to getthe probability smaller, we should extend the spec-trum to higher frequency u

1. Then probability of

classical heating will be unity for frequencies insidethe range 0(u(u

0and 0 outside of it. Thus, the

total probability for the whole spectrum will be lessthan 1. Suppose the spectrum to be g(u)"3u2/u3

1up to the highest frequency u

1, and zero above it.

Let us suggest that the fraction of the spectrumbelow u

0is 10~5. It gives the equation for u

1:

(u0/u

1)3"10~5Pu

1+50u

0.

It is important to estimate the total energy densityof vibrations with the considered spectrum. It is


SECTION 5b.

equivalent to the sound pressure:

p!#"oP

u1

0

A2(u)u2g(u) du"oA20u2

0"1 N/cm2

where o, the density of the substance, was taken as10g/cm3. We can compare this with the density ofthermal energy: P

T"3N

0kB¹, where k

Bis the

Boltzmann constant and N0

is atomic density,which we can take to be equal to 0.1 of theAvogadro number per cm3. It is easy to "nd, thatp!#+10~4P

T. Thus, p

!#is small. However, if we

compare it with the fraction (u1/u

D)3 of thermal

energy density for frequencies 0(u(u1, cal-

culated in the Debye approximation (uD

is theDebye frequency of the order 1013), we "nd thatp!#

is 104 larger than this fraction.The intensity of the sound is measured in deci-

bels: I"10 log(P/P0), where P

0is the reference

pressure: P0"10~16 N/cm2. Thus, the intensity of

the vibrations is 160 dB. However, the frequencies109 are too high to be considered as acoustical. Soto estimate how much of acoustical energy is con-tained in pure acoustical vibrations we should limitourselves to frequencies u

2+105. This range con-

tains only (u2/u

1)3+10~16 of the full energy, thus

the fraction of total supersound energy in acousti-cal range is of the order 0 dB.

2.7.2. Acoustical soundThe real acoustical sound is limited to the range

up to 104 Hz, and every collision with the wall isinelastic with probability 1. If the amplitude of themotion is A, the velocity of the interface is w"Au,then the neutron acquires an energy Jw2 at everycollision with the walls. If the neutron should sur-vive 105 collisions, the acquired energy should bew2"10~5u which is equivalent to +10~12 eV. Inthat case the UCN spectrum in the trap broadenscontinuously with time, which was not observed inthe experiment [4].

3. Quantum mechanics of the de Broglie wavepacket

In the previous section we considered all possiblechannels for UCN heating. The presence of ultra-

sound seems unreasonable, because its total energyis so high compared to thermal one, and thesource of such a sound is unknown. The hypothesisof "ne dust or cluster gas is not yet checked, how-ever, it may be rejected too, so we should beready to seek for another explanation of the UCNanomaly.

One hypothesis was formulated in Ref. [6]. Wesuggested there that the explanation is related tothe structure of the wave function of the free neu-tron, which is described by the de Broglie wavepacket [7]:

t(r, t)"Ss

2pexp(!sDr!ttD)

Dr!ttDexp(iv ) r!iut)

(6)

where s is a parameter which determines the widthof the packet in momentum space and the inversewidth in coordinate space, * is the central wavevector, and u"(v2!s2)/2. The wave packet (6) isfundamentally di!erent from the superposition ofplane waves used in conventional descriptions. Thewave packet (6) is normalizable, non-spreading,and satis"es the SchroK dinger equation everywhereexcept at one point:

(iL/Lt#*/2)t(r, t)

"!J2pse*(v2`s2)t@2d(r!r(t)) (7)

where r(t) is the coordinate of the neutron itself.This point can be considered as a source of thewave function. At present we shall consider onlythe free motion with r(t)"tt.

The wave number spectrum of the packet (6)

t(r, t)"Ss

2pPd3p/2p2

(p!*)2#s2exp(ip ) r!iX(p)t)

(8)

where X(p)"[p2!(p!t)2!s2]/2, has a long tailextending far away from the central wave vector t,and the anomalous losses of UCN can be at-tributed to nontunnelling transmission of neutronsover the potential barrier. This transmission alwaystakes place even if the height u of the barrier isconsiderably higher than the kinetic energy v2/2 ofthe neutron.


The losses described by the over barrier penetra-tion are given by

="

(4pc)2

(2p)3 P=

~=

h(DpMD'v

-*.)pM

DpMD

][1!DR(p

M)D2]DJp2

M!uD

[(p!t)2#s2]2d3p (9)

where h is the Heaviside function de"ned in Section2.2, and

R(pM)"

DpMD!DJp2

M!uD

DpMD#DJp2

M!uD

is the re#ection amplitude from the potential stepu of the plane wave with the normal component ofthe wave vector equal to p

M. The calculation gives

=+

s

Ju. (10)

If we identify this value with the averaged losscoe$cient

k6 (v)"2gz2

(arcsin z!zJ1!z2)

+43g

v

Jufor small z (11)

where z"v/Ju, and g+3]10~5 [1] then we getthat

s"4gv/3+4v]10~5. (12)

In principle, it is not necessary to get s propor-tional to the velocity of particle. From a compari-son of Eqs. (10) and (11) it is su$cient to estimates+4v

#]10~5 with some constant v

#of the order

of the Be critical wave nunber, however, at thebeginning proportionality of s to v seemed to bevery attractive, because it shows why at low ener-gies particles are more wavelike and at higher ener-gies have more corpuscular behavior. However, theexperiments show that the parameter s does notchange with velocity.

The wave packet description leads to conse-quences that are important not only for UCN. For

instance, the total re#ection of thermal neutronsfrom plane mirrors should always be accompaniedby a small fraction of refracted neutrons. This pre-diction can be and was veri"ed experimentally.

We performed the experiment [8], the scheme ofwhich is shown in Fig. 1. The well mono-chromatized and collimated neutrons ofwavelength j"12 As and *j/j+0.01 were re#ec-ted from a thick Si mirror at glancing angles h be-low the limiting one, h

-"0.5773, and the intensity,

transmitted through side surface and registered inchannels 550}670 of a position sensitive detectorP (see Fig. 2) was measured. The spectrum of theseneutrons was analyzed by transmission through Infoils. Our results at present can be formulated asfollows: We see the transmitted neutrons, that havethe same energy as in the incident beam. The frac-tion of these is near 10~4, which is in good agree-ment with the predicted magnitude if the parameters in Eq. (6) does not depend on velocity v of theneutron, as shown in Eq. (12), but is supposed to bea constant of the order 10~4v

#with some wave

number v#"Ju where u is close to the Be poten-

tial uB%

. Some new experiments are planned at theILL reactor in Grenoble with di!erent glancingangles, and with the incident neutron beam lesscontaminated by higher-energy neutrons. We arealso going to check very carefully the possible e!ectof surface roughnesses, which can give small angleand wide angle scattering. If the observed e!ectwith improved experimental conditions decreasesbelow 10~5, we shall decide that the de Brogliewave packet cannot explain the UCN anomaly.

4. Conclusion

The most important feature of recent experi-ments [3,9] in Grenoble is the discovery of a step-wise heating of stored UCN slightly above thelimiting one. However, this result was obtained forvessels with low storage time. The observed e!ectcan have no relation to the anomaly observed inclean and cold Be bottles. It can be explained, inparticular, by a dust of little clusters with dimen-sions up to several hundreds of angstroms. How-ever, the small heating can also be explained by


SECTION 5b.

Fig. 1. The experimental layout: M } monochromator; C } col-limator; n } neutron beam; Cd1 } cadmium shield at the en-trance surface of the sample to exclude the part of the directbeam splitted o! at the edge a; Cd2 } second cadmium shield toeliminate the part of the direct beam, which propagates withoutinteraction with the sample surface; Si } silicon mirror sample;P } position sensitive detector (PSD). The numbers on theright-hand side are related to the channels on the PSD at theglancing angle of 0.4003.

Fig. 2. Neutron counts by PSD when the silicon mirror is ata subcritical angle of 0.3813.

properties of the de Broglie wave packet. Indeed,our theory permits us to calculate the nontunnell-ing overbarrier transmission, but it does not tell ushow the neutron behaves inside the medium. Be-cause its kinetic energy is lower than the potential,the neutron may be at rest at every point in themedium, and it can be kicked out by the motion ofsurrounding matter, resulting in stepwise heating.

Acknowledgements

One of the authors V.K.I. is grateful to V. Fur-man, P. Geltenbort, V. Nesvizhevskii, V. Novitskiiand O. Zimmer, because without their help hewould not have been able to present this report.

References

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quantum mechanics of the de broglie wave packet and a review of inelastic losses of ucn in bottles

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