Instability of a crystal 4He facet in the field of gravity

0.15 MB·English

by  S. N. Burmistrov

Instability of a crystal 4He facet in the field of gravity
S. N. Burmistrov, L. B. Dubovskii, V. L. Tsymbalenko
Kurchatov Institute, 123182 Moscow, Russia
Weanalyze theanalog of theRayleigh instability in thefield of gravity for thesuperfluid-crystal
4 4
He interface provided that the heavier He crystal phase occupies the half-space over the lighter
superfluid phase. The conditions and the onset of the gravitational instability are different in kind
4
aboveandbelowtherougheningtransitiontemperaturewhenthecrystal Hesurfaceisintherough
or in the smooth faceted state, respectively. In the rough state of the surface the gravitational
instability is similar to the classical case of the fluid-fluid interface. In contrast, in the case of
1
the crystal faceted surface the onset of the gravitational instability is associated with surmounting
1
some potential barrier. The potential barrier results from nonzero magnitude of the linear facet
0
step energy. The size and the tilting angle of the crystal facet are also important parameters for
2
developing the instability. The initial stage of the instability can be described as a generation of

n crystallization waves at the superfluid-crystal interface. The experiments which may concern the
a gravitational instability of thesuperfluid-crystal4He interface are discussed.
J

1 PACSnumbers: 67.80.bf,47.20.Ma
1

] I. INTRODUCTION
r
e
h
The Rayleigh instability is a well-known instability of
t
o the interface between two liquids in the field of gravity
t. when the heavier liquid is placed above the lighter one.
a
The similar effect is difficult to observe in solids since
m
any displacement of a piece of the solid body is impeded
- due to appearing elastic stresses. However, there exists
d
an elastic stress-free possibility of changing the shape
n
o of a crystal, namely, remelting in the hydrostatic pres-
c sure gradient. A 4He crystal in contact with its super-
[ fluid phase could be one of most promising objects to FIG.1: The4Hecrystalgrownat0.92Kbetweentworough-

observe the manifestation of the Rayleigh instability for ening transitions as in experiment [3]. The diameter of the
1
v the liquid-solid interface. In fact, solid 4He density is visible margin is 12 mm.
9 larger than that of the liquid phase. The growth rate
9 of atomically rough crystal surfaces increases drastically
1 with the lowering of the temperature, and the change of of the cell. Unfortunately, the development of the insta-
2
thecrystalshapecanoccurinveryshorttimeintervalsof bility observed is described qualitatively and the results
.
1 about 1 s. Unfortunately, in most of crystal growth ex- of visual observations are illustrated by the schematic
0 periments [1] a 4He crystalappears either at the bottom figure alone. The main conclusion of the work is that
1 ofanexperimentalcellordropsonthesamebottomlater the spatial scale associated with the development of the
1
ifacrystalseednucleatesfirstatthe wall. Thus,prepar- instability is about 1 cm.
:
v ing the necessary configuration with the heavier crystal This result [2] seems to conflict with the stability ob-
i above its lighter liquid phase in order to have the initial served for the lower surface of the 4He crystals grown
X
condition for the development of the classical Rayleigh on the needle point at the center of a cell (Fig. 1). As
r
a instability comes across a difficulty. is seen from the figure, the lower crystal facet is under
Nevertheless, one experiment, in which the necessary conditions appropriate for the development of the grav-
liquid-crystalconfigurationagainstthe directionofgrav- itational Rayleigh instability. The single distinction is
ity is arranged, has been reported [2]. A single hcp 4He likely to be associated with the state of the crystal 4He
crystalis grownatthe bottomof anexperimentalcell at surface. Whereas in experiments [2] one deals with the
the temperature of about 1.1 K lying between the first roughstateofthecrystal4Hesurface,inourexperiments
and second roughening transitions. After the grown4He [3]oneobservesthesmoothfacetedstateofthesurfacein
crystal occupies the lower half of the optical cell, the theformofahcpcrystalbasalfacetabovethesuperfluid
cellisrotatedmechanicallythrough180◦ sothatthe4He phase.
crystal proves to be above the superfluid liquid. Then In this paper we consider an instability of the crystal
the crystal phase starts to melt, descending along the 4He surface above the superfluid phase in the field of
cell walls. In its turn, a single finger of the superfluid gravity. We analyze and compare two possible states of
phase moves in the upwarddirection atthe center of the the crystal 4He surface such as the rough and smooth
cell. Eventually,thecrystalagainoccupiesthelowerhalf faceted ones.
2
II. INSTABILITY OF THE CRYSTAL SURFACE
IN THE ROUGH STATE
We start our considerationfrom the high temperature
region above the roughening transition when the crystal
surface is in the atomically rough state. Since in this
casethesurfacetensionαhasnosingularityattheclose-
packedfacetsandweaklydependsonthecrystallographic
direction, the tensor of surface stiffness γ is close to
ik
αδ . Thus, to simplify the analysis, we neglect any dis-
ik
tinction between surface tension and surface stiffness in
4
the formulas for the Laplace pressure. FIG.2: Theisotropicgrowthofa Heat1.28K.Thediameter
of the visible margin is 12 mm.
A. Flat shape of the crystal surface B. Spherical shape of a solid
Crystallizationwaves,representingsmalloscillationsof An interesting situation appears with the nucleation
the superfluid-solid 4He interface, are predicted by An- and growth of a crystal at the needle point at the tem-
dreev and Parshin [4]. The spectrum of crystallization peratures above the roughening transitions (Fig. 2). In
wavesinthe field of gravitycanbe found fromthe equa- the hydrostatic pressure gradient the crystal grows in
tion [5] the shape of a ball which melts in the upper part and
crystallizes in its lower part at the same time. In out-
ω2 ρ′ (∆ρ)2 ward appearance this looks like the motion of a crystal
ρef q +iK ω−αq2−∆ρg =0, ρef = ρ . (1) downward. In work [3], in which the similar situation is
studied, it is shown that the crystal with the isotropic
Here q is the wave vector, ∆ρ = ρ′ ρ where ρ′ and ρ growth coefficient conserves its spherical shape, and the
−
are the densities of the solid and liquid phases, K is the descending motion velocity of the sphere v is equal to
interface growth coefficient, and g is the acceleration of
∆ρ
gravity. The positive magnitude g > 0 corresponds to v =K gR(t), (3)
the usual situation when the 4He crystal lies under the ρ
liquid phase. In this case the frequencies of small inter-
where R(t) is a time-dependent radius of the sphere.
faceoscillationshavethenegativeimaginarypartsforall
Then, as follows from(3), the center of the crystalshifts
wave vectors and the crystal surface is always stable.
downwardata constantvelocity in the hydrostaticpres-
Providedthesolidphaseoccupiesthehalf-spaceabove
sure gradient provided the crystal volume remains un-
the liquid, the dispersionequationremains the same but
changed.
parameter g becomes negative. This means that the last
Let us consider stability of the spherical shape of a
termin (1), whichdominates for sufficiently small q, can
crystal against small shape perturbations in the field of
result in the positive imaginary part for the roots of the
gravity. For analysis, we choose the frame which origin
dispersion equation. The critical magnitude of the wave
is put at the center of a crystal. In this frame the cen-
vector q0 is given by ter of a crystal is fixed and the liquid phase circulates
around the crystal, outflowing from the upper part of
∆ρg the sphere and flowing into the sphere in its lower part.
q0 = =1/λ,
α We introduce ζ(t) R as a small perturbation of the
r ≪
radius R(t). The liquid phase is assumed to be incom-
where λ 1 mm is the capillary length. The instability pressible and the solid one is motionless. In experiment
∼
will develop faster for small wave vectors. For the har- [3] the velocities of the liquid flow are small and do not
monic q 0,the estimate for the time of developingthe exceed0.1mm/s. Forthe hydrodynamicpressureρv2/2,
→
instability is given by we haveanestimate 10−2 dyne/cm2. This magnitude is
much smaller than the typical hydrostatic pressure drop
ρ′ 1 ∆ρgR. In what follows, we neglect quadratic terms in
t . (2)
∼ ∆ρ gK velocity of the liquid phase.
Next, in accordancewith the experimental conditions,
ForT 1.1K,theinversegrowthcoefficient1/K 2m/s we take into account that the experimental cell is closed
∼ ∼
and t 2 s. This is in a qualitative agreement with the and no matter comes from outside. In other words, the
∼
observations of work [2]. Thus, the instability similar to total mass of the liquid and solid phases remains un-
the Rayleigh one develops at the lower surface of a 4He changed. In the frame comoving with the crystal the
crystal in the atomically rough state. center of the crystal is fixed. As a result, we can omit
3
the spherical harmonic with l = 0 from consideration
since this harmonic is associated with the change of the
volume. The next harmonic l = 1 is responsible for the
displacement of the sphere as a whole and corresponds
to the circulationof the liquid aroundof the sphere. We
also omit this harmonic from our consideration.
The determination of the oscillation spectrum is not
difficult but cumbersome. We give a scheme of solution
and then the final result. Let axis z run in the vertical
direction,andweseekforthegeneralsolution,expanding
perturbationζ insphericalharmonics. Sincetheliquidis
assumedtobeincompressible,itisconvenienttodescribe
its motion in terms of velocity potential φ according to
FIG. 3: The initial stage for the development of instability
v = φ. The mass flow across the interface is propor-
∇ for a spherical crystal with the size larger than the critical
tional to the chemical potential difference ∆µ
radius. The dashed circle is theinitial shape.
J =ρ′K∆µ.
Within this temperature range the kinetic growthcoeffi-
Thecontinuityofthemassflowacrosstheinterfaceallows cientissmallandvariesinsignificantlywiththetempera-
ustorelatethevelocityoftheliquidattheinterfacewith ture from 1/K 11 m/s at the bcc transition to 3 m/s
the interface growth rate ζ˙. After some calculations and near the rough∼ening transition. Inserting these∼values
involving the axial symmetry with respect to axis z, we intoEqs.(4,5)andsolvingnumericallythemasterequa-
arrive eventually at the dispersion relation tion for proper frequencies, we determine the critical ra-
dius R =0.61 cm. The radius of crystals studied in [3]
c
ω2 Γ doesnotexceedthismagnitude. Thismayserveasanex-
+iω (l+1) (l 1)(l+1)(l+2) a
ω2 ω2 − − l planation that all the crystals grown in that experiment
(cid:20) 0 0 (cid:21)
2R2 l2 (l+1)(l+2) display stability of their spherical shape.
− λ2 4l2 1al−1+ (2l+1)(2l+3)al+1 =0, (4) Thepropervectorsin(4)aredeterminedwithaccuracy
(cid:20) − (cid:21) to their sign. For R > Rc, this gives two possible cases
for developing the instability shown in Fig. 3. In the
where
first case at the bottom there appears a hogging in the
α ρ ρρ′ 1 upward direction. In the second case we see a formation
ω02 = R3 (∆ρ)2 , Γ= (∆ρ)2 KR, (5) of constriction. Note that the result refers only to the
initial stage of developing instability.
and a is the perturbation amplitude corresponding to
l
l-th spherical harmonic from a sum ζ(t) = a (t)Y .
l l l C. Crystals of the limited sizes
Thus, we havea determinant of the infinite orderfor de-
P
terminingthepropervalues. Inthelackofgravityλ=
the off-diagonal terms vanish. The roots of the mast∞er Another factor which stabilizes the flat surface is a
equationgivetheoscillationspectrumofasphericalsolid finiteness of the crystal size. In this case the spectrum
with damping. All the frequencies havea negativeimag- ofcrystallizationwavesbecomesdiscrete. Theinstability
inary part, resulting in the conclusion that in this case willappearattheminimumwavevector. Toestimate,we
the spherically shaped solid is stable against small per- replace the hexagonal crystal shape with the equivalent
turbations of its equilibrium shape. cylindrical one of radius R. Then we can show that dis-
In the presence of gravity we must involve the off- persion equation (1) holds its form but the perturbation
diagonal terms in (4). Note that the diagonal terms in- amplitude ζ of the flat crystal surface is given by
creaseproportionaltol3asl ,whiletheoff-diagonal
terms remain finite and have→th∞e order of (R/λ)2. This ζ(r, t)=ζ0Jm(qr)eimφe−iωt,
means that, as the degree of harmonic l increases, the
where φ is the azimuth angle, J is the Bessel function,
relative effect of the pressure gradient reduces, agreeing m
andristhedistancefromthecenterofacylinder. Forthe
with the stability of the flat interface at large wave vec-
circular symmetry at the fixed volume of a crystal, the
tors. As in the case of the flat interface, the instability
minimum wavevectorq0 is determined fromthe relation
developsinthe firstturnfor the minimumdegreeofhar-
monics l =2. J1(q0R)=0, q0 3.83/R. (6)
Thesphericalshapeofheliumcrystalsinthehcpphase ≈
isobservedabovethe firstrougheningtransitiontemper- From expressions (1–2) and (6) one can obtain that the
atureatT >1.25K (Fig.2). Onthe other hand,the hcp instability should appear for a crystal with the diameter
phase is limited by the hcp-bcc transitionat T =1.44 K. larger than 6 mm.
4
III. INSTABILITY OF A CRYSTAL FACET Two types of functions can be solutions of the equation
1
Below the roughening transition temperature a singu- η(x′)=η0 sin(x′ x′)/γ ,
larityappearsasanonanalyticangulardependenceinthe (cid:26) − 0
behaviorofthesurfacetensionversusangleθbetweenthe
i.e., either a constant or a sine function.
direction of crystallographic axis and the normal to the
Theheightofapotentialbarrierfordestructingtheflat
crystal surface [6]
crystal facet as well as the type of a critical fluctuation
dependonthemagnitudeγ. Forsufficientlysmalllength
α(θ)=α0+β(T) θ + , (θ 1).
| | ··· | |≪ of the facet L < Lc1 = πλ, there exists a single trivial
solution η(x′) = 0, and the flat crystal facet remains
Here β is the quantity proportional to the energy of a
crystal facet step and is positive below the roughening stable. For L > Lc1, there appears a nontrivial solution
which consists of two half-sinusoids and flat segment
transitiontemperature. Thisentailsthatthesurfaceten-
sion and surface stiffness represent the drastically differ-
1, x′ 6x′ = 1−πγ
ent quantities. The surface stiffness becomes infinite for | | 0 2
tzhereosahnagpleeoθf=th0e.cTryhsetaslinfagcuelatraitnydpqruevaelinttastifvreolmy cvhaarnyginegs η(x′)=η0 sin1/2−γ|x′|, x′0 <|x′ |61/2 .
the conditions for stability. Let us consider the distinc-
Here the function η(x′) is continuous together with its
tionsinappearingtheRayleighinstabilityincomparison
derivative at points x′ = x′ and vanishes at x′ =
with the case of the rough crystal surface. 0
±
1/2. The critical perturbation amplitude of fluctuation
is equal to
A. Rayleigh instability of the horizontal crystal
2 2β 1
facet ηc1 = or ζc1 = .
1 πγ ∆ρg L Lc1
− −
Belowtherougheningtransitiontheenergyvariationof The height of the potential barrier, separating the tran-
the surface for sufficiently small perturbation amplitude sitionoftheflatcrystalfacettoadistortedstate,isgiven
ζ(x, y) from the horizontal flat facet is equal to by
∆E = α0( ζ)2+β ζ ∆ρgζ2 dxdy. ∆E = 2 = 2β2 1 . (8)
Z (cid:18) 2 ∇ |∇ |− 2 (cid:19) D 1−πγ ∆ρg L−Lc1
The smallnessofζ and ζ forthe correctandconsistent As the size of a crystal facet increases, there appear ad-
∇
use of the expansion of energy in ζ is provided by in- ditional possibilities for other fluctuations consisting of
equality β α0. Let us start from the one-dimensional a combination of flat segments and half-sinusoids. The
≪
case, namely, flat bar of length L. The energy per unit correspondingsolutions, composed with n flat segments,
length ∆E/D reads appearasL>Lcn whereLcn =nLc1. Asthenumberof
the flat segments increases, the perturbation amplitude
L/2 2 of critical fluctuations does as well
∆E = α0 ∂ζ +β ∂ζ ∆ρg ζ2 dx,
D −LZ/2 2 (cid:18)∂x(cid:19) (cid:12)(cid:12)∂x(cid:12)(cid:12)− 2 ! ηcn = 1 2πnγn or ζcn = ∆2ρβg L nLcn .
(cid:12) (cid:12) − −
(cid:12) (cid:12)
where D is the width ofthe crystalbar in the transverse The potential barrier height grows as the number n of
direction. The boundary conditions at the ends of the possible flat segments increases
bar
∆E 2n2 2β2 n2
n
= = .
ζ( L/2)=ζ(L/2)=0 (7) D 1 πγn ∆ρg L L
− − − cn
correspond to the case when the crystal surface is im- Provided the experimental conditions correspond to the
mobile at these points. It is convenient to introduce conservation of the total mass including the both liquid
the dimensionless quantities according to x′ = x/L, and solid phases, the solutions should satisfy an addi-
η =ζ(∆ρgL)/β, and γ =λ/L. Then, tional requirement
∆E[η(x′)] β2 1/2 γ η2 L/2
= η′2+ η′ dx′ ζ(x)dx=0.
D ∆ρgLZ−1/2(cid:18)2 | |− 2 (cid:19) Z−L/2
The extremum satisfies the equation In this case the nontrivial solutions for critical fluctua-
tions can be realized for the even numbers n and, corre-
γ2η′′+2δ(η′)η′′+η =0. spondingly, first critical length becomes equal to Lc2.
5
For the temperatures well below the roughening tran- and the following boundary conditions are fulfilled
sition, the facet step energy coefficient β is measured
ζ( L/2)=0. (10)
[1], and the numerical estimate of the coefficient in
±
Eq.(8)givesthemagnitudeofthepotentialbarrierabout
Here L is implied as a size of the crystal shape fluctua-
10−5 ergor1011 Kforthebasalfacetofsize 1cm. The
∼ tion. The surface fluctuations satisfy the conservationof
overcoming of such barrier is practically impossible dur-
the total mass of the liquid and solid phases
ing the time of experiment. Thus, the appearance of the
singular angle dependence in the function α(θ) results L/2
in a drastic change for stability conditions of the crys- ζ(x)dx=0. (11)
tal surface. With very small cooling below the roughen- Z−L/2
ing transitiontemperaturethe potentialbarrierbecomes For ϕ = π/2, expression (9) goes over to the expression
sufficiently large and the probability of its overcoming for a horizontalcrystal facet with the small vertical per-
due to thermal or quantum fluctuations is vanishingly turbations. For ϕ = 0, we have the expression for the
small. This conclusion is confirmed by the experimen- energy of a vertical crystal facet with the small pertur-
tal evidence for stability of the crystal shape below the bations in the direction normal to the facet.
roughening transition. To find the minimum magnitude of the potential bar-
The one-dimensional problem and boundary condi- rier which prevents from the development of instability,
tions (7) are chosen as simplest ones in order to illus- we must consider extremum of functional (9). The ex-
trate the method of solution and to obtain an analyti- tremum of functional (9) satisfies the equation
cal estimate of the potential barrier height for develop-
ing gravitaional Rayleigh-like instability. The full prob- α0ζ′+βsgnζ′ ′+∆ρg xcosϕ+ζsinϕ =0. (12)
lem should be solved employing the real crystal shape
with the surfaces connecting the facets. We have ana- Pr(cid:0)ovided the deri(cid:1)vative ζ′((cid:0)x) does not van(cid:1)ish for all x
lyzed an onset of instability at the circular facet on the within L/2<x<L/2, i.e.,
−
analogy with Sec. IIC. The mathematical treatment be-
ζ′(x)=0, (13)
comes more complicated but the final result for the bar- 6
rier height differs from Eq. (8) with a numerical coeffi- equation (12) takes the simple form
cient of about unity.
α0ζ′′(x)+∆ρg xcosϕ+ζ(x)sinϕ =0. (14)
If the condition (13)(cid:0)is satisfied, the gen(cid:1)eral solution of
B. Instability of the tilted crystal facet
Eq. (14) reads
Here we analyze the stability of a crystal facet tilted ζ(x)= xcotϕ+Asin(x/λ )+Bcos(x/λ ). (15)
ϕ ϕ
withangle(π/2 ϕ)againstitssmallperturbations. The −
−
variation of energy ∆E per width of the crystal facet Unknown coefficients A and B are determined by condi-
reads tions (10) and (11). Finally, we arrive at
∆E = L/2 dx α0 ζ′2(x)+β ζ′(x) ζ(x)= ssinin((Lx//2λλϕ)) − 2Lx L2 cotϕ. (16)
D Z−L/2 (cid:18) 2 | | (cid:18) ϕ (cid:19)
ζ2(x) Here λϕ plays role of an effective capillary length and
∆ρg xζ(x)cosϕ + sinϕ , (9)
determinesthetypicalscaleoflengthatwhichζ(x)varies
− 2
(cid:19)
(cid:0) (cid:1)
where ζ(x) is a perturbation amplitude taken from the λ = α0 = λ .
ϕ
flatsurface. The angleϕ=0means the verticalposition ∆ρgsinϕ √sinϕ
r
of the crystaland ϕ=π/2 correspondsto the horizontal
Note that, as ϕ 0 for the vertical position of a crystal
position. →
facet, the typical length λ diverges.
Let axis Oz run along the normal to the crystal facet. ϕ
Substituting (16) into (9), we have
The axis Oy is perpendicular to the axis Oz and the
acceleration of gravity as well. The third axis Ox, lying
inthe plane ofthe facet,is perpendicular to the axesOy ∆E = α0 λ3ϕ cos2ϕ L˜/2 dx a2cos2x 2acosx
andOz. TheaxisOxrunsattheangleϕtothedirection D 2 λ2 sinϕ Z−L˜/2 (cid:18) −
of the accelerationof gravity.
Let perturbation amplitude of the crystal facet ζ(x) +1+x2+2(β/α0) acosx 1 tanϕ (17)
| − |
from its initial position ζ(x)=0 take place in the direc- (cid:19)
tionnormaltothefacetsurfaceandbeindependentofthe Here we have introduced the dimensionless quantities
coordinateydirectedhorizontallyalongthecrystalfacet.
In addition, we assume that the small perturbation am- L˜ = L and a= L˜/2 .
plitude ζ(x) is finite within the region L/2< x < L/2 λϕ sin(L˜/2)
−
6
The magnitude of energy (17) can be written as which the facet step energy vanishes (Fig. 4). In Fig. 4
the image of a 4He crystal at 0.901 K is given. Note
∆E = α0L λ2ϕ cos2ϕ L˜2 2+L˜cotL˜ tshtraattitnhgetlhaeterroaulgfahceentiinnggtdraisnaspitpieoanrsfoartt0h.e91a0-faKc,etdse.mTohne-
D 4 λ2 sinϕ 6 − 2
(cid:18) (cid:19) crystalhas a clear lateralfacet which remained stable in
λ3 L˜/2 the course of experiment during, at least, 10 min.
ϕ
+β cosϕ dx acosx 1 . (18)
λ2 Z−L˜/2 | − |
For L˜ =L/λ 1, equation (18) gives IV. CONCLUSION
ϕ
≪
∆E = α0L cot2ϕ L 4+ βLcotϕ L 2. 4HTehseurgfaracve,itwathiiocnhadleivnesltoapbsiluitnydaerttthheelaactokmoficaanllyyprootuegnh-
D − 80 2λ 12 2λ
(cid:18) ϕ(cid:19) (cid:18) ϕ(cid:19) tialbarrier,issimilartotheclassicalRayleighinstability
As a result, for sufficiently small-sized fluctuations with
L L , the energy of fluctuations is positive ∆E > 0
cr
≪
and such fluctuations are of low probability. The tilted
facet is practically stable for such perturbations. The
critical length is equal to
5β 4 5β/(3∆ρg)
L =4λ tanϕ= .
cr ϕ
r3α0 p√cosϕ
This means that the crystal facet is practically stable if
its size does not exceed the critical length. In fact, the
magnitude of the energy barrier ∆E β2/∆ρg proves
to be about 10−5 erg or 1011 K. As th∼e size of a crystal
becomeslargerthanthecriticallengthL>L ,thetilted
cr
crystal facet becomes unstable against distortion of its
shape.
Theanalysisofthe crystalfacetstabilityforthelarger 4
FIG.4: Thelateral facet ofa Hecrystalisstable. Thetem-
lengths L λ is simple. Let us represent Eq. (18) in
ϕ perature is below the second roughening transition by about
∼
the form of the following inequality 20 mK.
∆E 6 α0L λ2ϕ cos2ϕ L˜2 2+L˜cotL˜ whenthe heavierliquidliesabovethelighterliquid. The
D 4 λ2 sinϕ 6 − 2
(cid:18) (cid:19) distinction is that the time necessary for the develop-
λ3ϕ L˜/2 ment of instability is determined by the kinetic growth
+β cosϕ dx( acosx +1) .
λ2 (cid:18)Z−L˜/2 | | (cid:19) coefficient of a crystal surface.
As for the smooth faceted crystal surface, having a
The right-hand side of the inequality is obviously nega- singularity in the surface stiffness, the development of
tive for L˜ & π, i.e. ∆E < 0. This means that a crystal the surface instability becomes possible due to thermal
facet distortion with such lengths is energetically favor- or quantum fluctuations if the size of the facet surface
able. Thetiltedcrystalfacetbecomesabsolutelyunstable exceeds the critical one. However, the large height of a
if the facet size L&λ . potential barrier makes its overcoming impossible for a
ϕ
In conclusion, such high potential barrierscan explain experimentally reasonable time. This explains the sta-
a gravitationalstability of a crystal facet in the immedi- bility of the lower facet for a free-growing 4He crystal
ate vicinity of the roughening transition temperature at observed in the experiment during a few hours.
[1] S.Balibar,H.Alles,andA.Ya.Parshin,Rev.Mod.Phys. 75, 1511 (1978) [Sov.Phys. JETP 48, 763 (1978)].
77, 317 (2005). [5] A. O. Keshishev, A. Ya. Parshin, and A. V. Babkin, Zh.
[2] C. D. Demaria, J. W. Lewellen, and A. J. Dahm, J. Low Eksp.Teor.Fiz.80,716(1981)[Sov.Phys.JETP53,362
Temp. Phys. 89, 385 (1992). (1981)]. 114, 1313 (1998) [JETP 87, 714 (1998)].
[3] V.L.Tsymbalenko,Fiz.Nizk.Temp.21,162(1995)[Low [6] L. D. Landau, The Equilibrium Form of Crystals, in Col-
Temp. Phys. 21, 120 (1995)]. lected Papers (Pergamon, Oxford, 1965).
[4] A. F. Andreev and A. Ya. Parshin, Zh. Eksp. Teor. Fiz.