Two-velocity elasticity theory and facet growth

12 Pages·0.17 MB·

by  A. F. Andreev

Two-velocity elasticity theory and facet growth
2
0
0 A.F.Andreev∗, L.A.Melnikovsky†
2

n
a
J
Kapitza Institute for Physical Problems
1
2 117334, Moscow, Russia

] Abstract
h
c Weexplainthelineargrowthofsmoothsolidheliumfacetsbythepresenceoflatticepointdefects. To
e implementthistask,theframework ofverygeneraltwo-velocityelasticity theoryequationsisdeveloped.
m Boundaryconditionsfortheseequationsforvarioussurfacetypesarederived. Wealsosuggestadditional
- experimentsto justify theconcept.
t
a PACS: 67.40.Bz,67.40.Pm, 68.35.Ct, 61.72.Ji
t
s
.
t
a 1 Introduction
m
- Existence of two distinct states of a crystal surface is well known: it may be either smooth or rough (for a
d
review, see [1]). A smooth surface is characterized by a long-range order and small fluctuations. On the
n
o contrary, a rough surface behaves differently — it does not exhibit a long-range order and its displacement
c fluctuates heavily. These equilibriumpropertiesleadto differentkinetic properties. While the roughsurface
[ is usually supposed to grow easily (as described by the growth coefficient), the smooth one (providing the

2 crystal has no dislocations) is characterized by zero growth coefficient and grows with nuclei of new atomic
v layer. In accordancewith this mechanism, one should not observethe linear growthrate if low overpressure
2 is applied. Reality is different — experiments [2] demonstrate that a smooth helium surface free of screw
5
dislocationsgrowslinearly. This workis anattempt to explainthis behaviorby the presenceoflattice point
3
defects (vacancies). TheideaissimilartothatsuggestedbyHerring[3]andbyLifshitz[4]asanexplanation
1
0 of polycrystalflow. It is quite simple: the mass flux in bulk helium is attributed to the motion of vacancies.
2 This flux is the mass transfer through the lattice. Therefore, if vacancies are allowed to be created on the
0 bottom edge of the sample (the boundary between the crystaland the wall, see Fig. 1) and to annihilate on
/
t the top of it (on the smooth crystal-liquid interface), then the crystal grows.
a
m
v
- g
d superfluid
n
o
c 4He crystal
:
v
i
X silica wall
r
a
Figure 1: Typical experimental layout
Thesuggestedcrystalgrowthmechanismmaybeexplainedasfollows. Sincethesmoothcrystalfacet(the
top one in Fig. 1) cannot move with respect to the lattice, it moves upward stuck to the lattice. Vacuities
to appear due to this on the bottom edge of the crystal transform into lattice defects (ordinary vacancies)
and go up through (and faster than) the bulk helium. They finally vanish in the liquid on the top smooth
∗E-mail: [email protected]
†E-mail: [email protected]
1
surface of the sample. In other words, the crystal grows on the boundary between helium and the wall,
rather than on the smooth solid–liquid interface (which nevertheless provides mass supply for the growth).
It is important to emphasize that this scenario may occur if and only if the vacancies are allowed to emerge
on the bottom edge of the crystal. One may say that this boundary is in some sense “atomically rough” —
it can grow new atomic layers. For this condition to be satisfied, the wall surface should have a disordered
shape or be slightly tilted with respect to basal planes of the crystal (thereby forming a vicinal interface).
This ensures that the surface can play the role of a source or a sink of vacancies. An atomically flat wall
parallel to the basal plane should, in contrast, behave like a normal smooth surface — it is fixed to the
lattice. This is due to the fact that for the surface to move new atomic layer nuclei have to be created.
This paper is organized as follows. In Secs. 2 and 3 we derive very general two-velocity elasticity theory
equations. They consist of conditions for the conventional elasticity theory variables (including lattice
velocity) and equations for a macroscopic description of the quasiparticle gas (including the quasiparticle
gas velocity).
Equations to be derived are similar to those of the two-velocity superfluid hydrodynamics. Velocities of
lattice and excitation gas in our equations replace superfluid and normal component velocities of the two-
fluid hydrodynamics. Similarly to the usual linear phonon hydrodynamics (see [5] § 71) Umklapp-process
(which result in the nonconservation of the total quasimomentum in quasiparticle collisions) probability is
supposed to be low. In the low-temperature region considered here this assumption is quite reasonable.
We also neglect dissipation here. This means that our analysis is limited to the terms of the first order in
gradients.
Derivation procedure of exact (nonlinear) hydrodynamics equations for superfluid can be implemented
(see [6]) from phenomenological considerations, using conservation laws. A constitutive argument for this
derivation is the statement that the superfluid flow is potential. This is an intrinsic property of the order
parameterinasuperfluid. Suchconditionisunavailableforacrystal(moreover,thereisnoquasimomentum
conservation relation in the nonlinear description, see Eq. (15) below).
We deduce the two-velocity elasticity theory equations using a more general approach(see the paper by
Pushkarovandoneoftheauthors[7],aswellas[8]and[9]). Itisbasedonthekineticequationdescriptionof
the quasiparticledynamics. Therealizationofthis technique,particularlyinnonlinearsituation,isamatter
of considerable interest for its own sake not only for a solid but also for a superfluid (that this procedure
is possible is mentioned in [5] §77). With this technique, we find exact expressions for all hydrodynamic
variables and their dependence (in terms of the quasiparticle energy spectrum) on the relative velocity of
components. It is trivial to extend the equations obtained for the solid dynamics to the simpler case of
superfluid hydrodynamics.
Boundaryconditionsforourequationsdependonthesurfacetype;inSec.4wethoroughlyconsiderthree
possibilities: rough (Sec. 4.1.1) and smooth (Sec. 4.1.2) interfaces between solid and liquid helium, and the
rough boundary between solid helium and normal hard wall (Sec. 4.2.1). Finally, in Sec. 5 we calculate the
growth rate for the crystal.
2 Definitions
Following the principles in Refs.[7, 8, 9] we employ the Euler approach to the lattice description. We thus
introduce three “node numbers” Nα (α = 1,2,3). They are functions of space coordinates r and time t,
Nα =Nα(r,t). From now on, Greek indices (like α here) are used for the “lattice space” and Latin indices
(e.g., i in x for the components of r) for the real space components. Defining the reciprocal lattice vectors
i
as aα = ∂Nα/∂r, we get the elementary lattice translation vectors a as aαa = δα. Taking the time
β β β
derivative, we obtain the lattice velocity as w = −a N˙α. The elastic energy E of the lattice is a function
α l
of the deformation. Moreover, since it depends not on the spatial orientation of the infinitesimal sample
(the space is isotropic), but on the relative position of the aα vectors, we may write E = E gαβ , where
i l l
gαβ =aαaβ is a symmetric “metric tensor” of the lattice space.
(cid:0) (cid:1)
We are now ready to describe quasiparticle degrees of freedom. We do not specify the quasiparticle
nature at the moment (be it phonons, vacancies as in [9], or electrons as in [7]). Actually, all the equations
below imply the summation over all branches of excitations; we do not explicitly write the sum for brevity.
Any quasiparticle should be characterized by its mass m (zero for phonons, positive for electrons, and
negativeforvacancies),coordinate,andmomentum. Sincequasiparticlesexistinthelatticebackground,the
quasimomentumshouldbeused. Theirenergyintheframeofreferenceofthelatticeǫ=ǫ(a (p−mw),gαβ)
α
2
is aperiodic functionofthe quasimomentump(its periodsare2π¯haβ). Inlaboratoryframeofreference,we
have the quasiparticle energy (see [7])
∂ǫ w2
˜ǫ=ǫ+mw +m .
∂p 2
Wealsousethevariablesk=p−mwandk =a k. QuasiparticledynamicsisdeterminedbytheHamilton
α α
function
2
H =ǫ+pw−mw /2.
We now introduce the distribution function f(r,p) (it is also a periodic function of the quasimomentump).
Its kinetics is governedby the Boltzmann equation
∂f ∂f ∂H ∂f ∂H
+ − =Stf. (1)
∂t ∂r ∂p ∂p ∂r
Using this distribution function, we can obtain macroscopic quantities like the mass density as
ρ=ρ +hmfi=Mdet gαβ 1/2+mn,
l
where the angle brackets denote the integration over quas(cid:0)imom(cid:1) entum space, hi = d3p/(2π¯h)3, ρ is the
l
lattice density, M is the mass of an elementary cell, and n=hfi.
R
Weconsideraquasi-equilibriumdistributionfunction. Thecompletesetofquantitiesconservedinquasi-
particle collisions consists of their mass (proportional to their quantity for “real” particles like electrons
andvacanciesandzeroforphonons),energy,andquasimomentum(inthe low-temperatureregion,Umklapp
processes may be neglected). Consequently, the most general quasi-equilibrium distribution is a function of
ǫ−kv−mµ0 ǫ−(p−mw)v−mµ0 ǫ−pv−mµ0+mwv ǫ−pv−mφ
z = = = = ,
T T T T
where µ0−wv=φ. The Lagrangecoefficients T, v, and µ0 denote temperature, the velocity relative to the
lattice, and chemical potential of the quasiparticle gas, accordingly. For definiteness, we assume that the
excitations are Bose particles. The distribution function is then given by
−1
1 ǫ−pv−mφ
f = = exp −1 , (2)
ez −1 T
(cid:18) (cid:19)
and ln((f +1)/f)=z.1
We can now calculate other macroscopic parameters with this distribution function. For the mass flux,
we have
J=ρ w+wmn+j=ρw+j=ρw+mnv, (3)
l
where the mass flux with reference to the lattice is
∂ǫ
j=m f =mnv.
∂p
(cid:28) (cid:29)
Using J, we can write the mass conservation as
ρ˙+J =0.
i,i
The number of real (massive) particles is also conserved in the bulk, and we therefore have one additional
conservation law
ρ˙ +(ρ w ) =0.
l l i ,i
1Forfurtherconvenience, wealsoprovideheretheresultofthedistributionfunctionintegration:
ez−1
fdz=ln =−ln(f+1).
ez
Z
3
Similarly, the energy density is given by
w2 w2
E =ρ +E gαβ +hǫ˜fi=ρ +E gαβ +wj+hǫfi. (4)
l l l
2 2
This equation allows us to prove (and(cid:0)find(cid:1)) exact macroscopic(cid:0)equi(cid:1)valents of the microscopic quantities
introduced above. The total energy density of the crystal can be obtained via a Galilean transformation,
ρw2
E =E0+ +jw, (5)
2
whereE0 =E0(aα,S,ρ,K)istheenergyintheframeofreferenceofthelattice,withK=hkficharacterizing
the quasimomentum density. A reasonable expression for the E0 differential
dE0 =λijaνjdaνi +T dS+µdρ+vdK (6)
we obtain with the conventionaldefinition of the entropy density for the Bose gas,
S =h(f +1)ln(f +1)−flnfi=hfx+ln(f +1)i.
Its differential being
dS =h(ln(f +1)+1−lnf −1)dfi=hzdfi.
Subtracting the differentials of (4) and (5), we obtain
0=λ a daν +T dS+µdρ+vdK− dE gαβ − dhǫfi
ij νj i l
=λijaνjdaνi +T hzdfi+µ(cid:0)dρ+(cid:1) vdK− dEl gαβ −hǫdfi−hfdǫi
=λijaνjdaνi +h(Tz−ǫ+vk)dfi+(cid:0)µdρ(cid:1)− dEl gαβ +vhfdki−hfdǫi (7)
We now transform the part of this equation related to the lattice deformation(cid:0) (cid:1)
µ µ ∂E
ρ g dgαβ − dE gαβ +vhfdki−hfdǫi= ρ g dgαβ − l dgαβ −mvhfdwi
2 l αβ l 2 l αβ ∂gαβ
(cid:0) (cid:1)
∂ǫ ∂ǫ
− f ((p−mw) da −ma dw)+ dgαβ
∂k α α ∂gαβ
* α (cid:18) (cid:19)kα !+
∂E ∂ǫ ∂(Tx+pv)
=− l + f dgαβ + µρ δ − f (p −mw ) aα da
∂gαβ ∂gαβ l ij ∂p i i j αi
* (cid:18) (cid:19)kα+! (cid:18) (cid:28) j (cid:29) (cid:19)
∂E ∂ǫ
=− l + f 2aαaβa daν
∂gαβ ∂gαβ i j νj i
* (cid:18) (cid:19)kα+!
+(δ (T hln(f +1)i+µρ )+v P −mnv w )a daν
ij l i j i j νj i
w2
= δij TS+ρ 2 +El−E+Pv+µρl+mnµ0 −Λij +viPj −mnviwj aνjdaνi.
(cid:18) (cid:18) (cid:19) (cid:19)
Finally, from (7) we get:
0=(µ−µ0)mdn
w2
+ λ −Λ +δ TS+E −E+µρ+Pv+ρ +v P −mnv w a daν, (8)
ij ij ij l 2 i j i j νj i
(cid:26) (cid:18) (cid:19) (cid:27)
where we introduced P =hp fi and
i i
∂E ∂ǫ
Λ =2aαaβ l + f . (9)
ij i j ∂gαβ ∂gαβ
* (cid:18) (cid:19)kα+!
The terms in (8) are independent, and each of them must therefore be equal to zero. That is,
µ=µ0,
w2
λ =Λ −δ TS+E −E+µρ+Pv+ρ −v P +mnv w . (10)
ij ij ij l i j i j
2
(cid:18) (cid:19)
4
3 Equations and Fluxes
Here,wederivedynamicsequationsandthermodynamicfluxesforthesystem. Neglectingdissipationatthis
point, we assume that the entropy conservation law is valid,
S˙ +F =0,
i,i
where the entropy flux F is determined by
i
F=S(v+w).
We continue with the equation for the momentum flux found in [7],
J˙ +Π =0, (11)
i ik,k
where
∂E ∂ǫ
Π =ρw w −E δ +2aαaβ l + f +w j +w j
ik i k l ik i k ∂gαβ ∂gαβ i k k i
(cid:18) (cid:28) (cid:18) (cid:19)kα(cid:29)(cid:19)
∂E ∂ǫ
=ρw w −E δ +mn(w v +w v )+2aαaβ l + f
i k l ik i k k i i k ∂gαβ ∂gαβ
(cid:18) (cid:28) (cid:18) (cid:19)kα(cid:29)(cid:19)
=ρw w −E δ +mn(w v +w v )+Λ , (12)
i k l ik i k k i ik
where we employed the definition (9) for Λ .
ik
Taking the appropriate equation for the energy flux from [7], we have
E˙ +Q =0, (13)
i,i
where
∂H w2
Q =w E + ǫ f − J +w Π .
i i l i k ik
∂p 2
(cid:28) i (cid:29)
To find the second term, we again use the distribution function from (2),
∂H ∂ǫ
ǫ f = ǫ +w f
∂p ∂p
(cid:28) i (cid:29) (cid:28) (cid:18) i (cid:19) (cid:29)
∂z ∂z
= ǫ T +v +w f =(v +w )hǫfi+T (Tx+pv+φ) f
i i i i
∂p ∂p
(cid:28) (cid:18) i (cid:19) (cid:29) (cid:28) i (cid:29)
∂ln(1+f)
=(v +w )hǫfi−T pv =(v +w )hǫfi+Tv hln(1+f)i
i i i i i
∂p
(cid:28) i (cid:29)
f +1 ǫ−pv−mφ
=(v +w )hǫfi+Tv ln(1+f)+fln −Tv f
i i i i
f T
(cid:28) (cid:18) (cid:19)(cid:29) (cid:28) (cid:29)
=(v +w )hǫfi+Tv S−v hfǫi+v vP+mnφ=w hǫfi+v (TS+vP+mnφ).
i i i i i i i
For the energy flux, we finally have
w2
Q =w hǫfi+v (TS+vP+mnφ)− (ρw +mnv )+w (ρw w +mn(w v +w v )+Λ )
i i i i i k i k i k k i ik
2
w2
=w hǫfi+v (TS+vP+mnφ)+ (ρw +mnv )+w (mnw v +Λ )
i i i i k i k ik
2
w2 w2
=w hǫfi+wj+ρ +v TS+vP+mn φ+ +w Λ
i i k ik
2 2
(cid:18) (cid:19) (cid:18) (cid:18) (cid:19)(cid:19)
w2
=v TS+vP+mn φ+ +w (E−E )+w Λ . (14)
i i l k ik
2
(cid:18) (cid:18) (cid:19)(cid:19)
This formula completes the list of the conventional elasticity theory equations. An additional equation is
requiredto governthe quasiparticledegreesoffreedom. Wenow find the time derivativeof P . We multiply
i
5
Boltzmann equation (1) by p and integrate over the momentum space. We temporarily neglect Umklapp
i
processes,whicharesupposedlyrare. Ifneeded,dissipationcanbeexplicitly introducedintothefinalresult.
Inotherwords,thequasimomentumpisconservedin(normal)collisions,andtheterminvolvingthecollision
integral is therefore zero. The left-hand side of the Boltzmann equation gives
∂f ∂H ∂f ∂H
P˙ = p −
i i ∂p ∂r ∂r ∂p
(cid:28) (cid:18) (cid:19)(cid:29)
∂f ∂ǫ ∂w ∂f ∂ǫ
= p +(p −mw ) j − +w
i ∂p ∂r j j ∂r ∂r ∂p
(cid:28) (cid:18) (cid:18) (cid:19) (cid:18) (cid:19)(cid:19)(cid:29)
∂ǫ ∂w ∂2ǫ ∂w ∂f ∂ǫ ∂f
= −f +(p−mw) −fp + −p −p w
∂x ∂x i ∂r∂p ∂r i∂r∂p i∂r
(cid:28) (cid:18) i i(cid:19) (cid:18) (cid:19) (cid:29)
∂ǫ ∂2ǫ ∂f ∂ǫ ∂w ∂w ∂w ∂f
=− f +fp +p +hfimw − p f +p f +p w .
∂x i∂r∂p i∂r∂p ∂x ∂x i ∂r i∂r
(cid:28) i (cid:29) i (cid:28) i (cid:29)
The first term can be transformed as
∂ǫ ∂2ǫ ∂f ∂ǫ ∂2ǫ ∂ǫ ∂f ∂z
f +fp +p = fp +f +p T +v
∂x i∂r∂p i∂r ∂p i∂r∂p ∂x i∂r ∂p
(cid:28) i (cid:29) (cid:28) i (cid:18) (cid:19)(cid:29)
∂2ǫ ∂ǫ ∂f ∂f ∂z ∂ǫ ∂v ∂φ ∂T
= fp +f +p v+p −p k −m −z
i∂r∂p ∂x i∂r i∂z ∂p ∂r k ∂r ∂r ∂r
(cid:28) i (cid:18) (cid:19)(cid:29)
∂f ∂φ ∂v ∂T ∂v ∂z ∂T
= p v+mf +fp +fz +p f +p f
i∂r ∂x ∂x ∂x i ∂r i ∂p ∂r
(cid:28) i i i (cid:29)
∂φ ∂v ∂ ∂z ∂T
=nm +P + (P v)+ δ fz+p f
∂x ∂x ∂r i ik i ∂p ∂x
i i (cid:28) k(cid:29) k
∂φ ∂v ∂ 1+f ∂z ∂ ∂T
=nm +P + (P v)+ δ fln −p ln(f +1)
∂x ∂x ∂r i ik f i∂p ∂z ∂x
i i (cid:28) k (cid:29) k
∂φ ∂v ∂ 1+f ∂T
=nm +P + (P v)+ fln +ln(f +1)
∂x ∂x ∂r i f ∂x
i i (cid:28) (cid:29) i
∂φ ∂v ∂ ∂T
=nm +P + (P v)+S .
∂x ∂x ∂r i ∂x
i i i
Consequently,
∂w ∂w ∂ ∂φ ∂v ∂ ∂T
P˙ =nmw −P − P w−nm −P − (P v)−S
i ∂x ∂x ∂r i ∂x ∂x ∂r i ∂x
i i i i i
2
∂ w ∂T ∂ ∂
=nm −φ −S −P (w+v)− (P (w+v)). (15)
∂x 2 ∂x ∂x ∂r i
i (cid:18) (cid:19) i i
The desirable complete set of the two-velocityelasticity theory equations consists ofEqs.(15), (13) (with Q
defined by (14)), and (11) (with Π defined by (12)).
ij
4 Boundary Conditions
Wenowturntoboundaryconditions. Theyimmediatelyfollowfromtheconservationrelationstobesatisfied
at the interface. It is much easier to perform all transformations in the frame of reference of the interface
itself. All the velocities are therefore taken relative to the boundary. Moreover, we simplify the problem
by restricting it to the one-dimensional case: all fluxes are supposed to be perpendicular to the flat surface;
we let the z axis run along this direction. Since no curvature is ascribed to the surface, we ignore capillary
effects. Allcalculationsdoneherearevalidwithinthelinearapproximation. Naturally,boundaryconditions
should depend on the type of the boundary and on the type of the media on the other side of the interface.
We begin with the situation extensively discussed in literature, the solid–liquid interface [1]. Because the
possibility of the mass flux through the lattice is taken into account, the results are different, however.
6
z L vL vsL superfluid
n
crystal
wS vS F Q
S ai
α
Figure 2: Solid–liquid boundary: fluxes in one dimension
4.1 Solid–Liquid Interface
Theliquidonthe othersideofthe interface(being superfluid)ischaracterizedbythe chemicalpotentialµL,
normal and superfluid densities ρL and ρL, normal and superfluid velocities vL and vL, temperature TL,
n s n s
pressure pL, and the entropy density SL (see Fig. 2).
R+SS(vS +wS)=SLvL,
n
wS(ES −ElS +ΛSzz)+vS(TSSS +mSnSφS)=µL(ρLsvsL+ρLnvnL)+SLTLvnL,
(16)
 vSmSnS +ρSwS =ρLsvsL+ρLnvnL,
ΛS −ES =pL.
zz l
The superscript S indicates that the appropriate quantities refer to the solid. The first equation is the
entropy growth condition, where R is the surface dissipative function. The last three equations in (16) are
simplytherequirementsfortheenergy,mass,andmomentumconservationforthesurface,respectively. The
surface dissipative function must be a positive square form. Using (16) and (10), it can be expressed as
RTL =vS SS(TS −TL)+mSnS(φS −µL) +wS ES −ES +ΛS −ρSµL−TLSS
l zz
(cid:0) =vS mSnS(φS −µL)+(cid:1)SS(TS(cid:0)−TL) +wS λSzz +ρS(φS −µL)(cid:1)+SS(TS −TL) .
We now recall that th(cid:0)e solid–liquid boundary can be (cid:1)either (cid:0)atomically-rough or atomically-smo(cid:1)oth,
depending on the temperature. The nature of the surface may (or may not) impose certain restrictions on
the dynamics. For both types of the surface, the equation
ΛS −ES =pL (17)
zz l
is satisfied.
4.1.1 Rough Surface
Employing the Onsager principle, we obtain
vS =α mSnS(φS −µL)+SS(TS −TL) +η λS +ρS(φS −µL)+SS(TS −TL)
zz
(18)
wS =η (cid:0)mSnS(φS −µL)+SS(TS −TL)(cid:1)+ν (cid:0)λSzz +ρS(φS −µL)+SS(TS −TL)(cid:1).
(cid:0) α η (cid:1) (cid:0) (cid:1)
The kinetic matrix is positively definite.
η ν
(cid:18) (cid:19)
4.1.2 Smooth Surface
A smooth surface implies immobility of the interface relative to the lattice. That is,
wS =0.
7
For the quasiparticle gas velocity, we then obtain a restricted version of (18),
vS =α mSnS(φS −µL)+SS(TS −TL) , (19)
(cid:0) (cid:1)
with the kinetic coefficient α>0.
4.2 Solid–Wall Boundary
z S wS vS F Q crystal
ai
α
W Q concrete
Figure 3: Solid–wall boundary: fluxes in one dimension
By a wall, we imply a macroscopicallyflat structureless medium, in short, “concrete”. The “Solid–wall”
boundary occurs between solid helium and some normal rigid solid (silica in experiment [2]). A “concrete”
wall is characterized by no mass flux in it (i.e., through the interface). The wall can supply an arbitrary
energy flux; we let Q denote the flux and TW the wall temperature (see Fig. 3). Concrete is characterized
by fewer variables than liquid, and the appropriate equations are therefore somewhat simpler.
Just like for the solid–liquid interface, the actual boundary conditions must depend on the microscopic
patternofthe surface. One caneasilyimagine asmoothbasalplane of the crystaladjacentto anatomically
flat concrete wall. This plane must stay at rest with respect to the wall since its motion presupposes the
creation of new atomic layer nuclei. The plane is similar to the smooth solid–liquid interface, and we can
naturally say that such an interface is smooth. The boundary condition is then given by wS =vS =0.
Another, much more interesting scenario is realized if the interface is slightly tilted with respect to the
basalplane. Suchplanesmaymovebygrowingadditionalnodesattheedge. Thismeansthatnorestrictions
are imposedonthe lattice velocity nearthe interface. Inother words,vacanciesareallowedto freely appear
and vanish on the surface (in this sense, the surface is similar to a grid of dislocations arranged at the
boundary of the crystalthat serve as sourcesorsinks for vacancies;similar speculations may be found in [4]
in explaining polycrystalplasticity). We call this type of the interface “rough”. In this sense, the solid–wall
boundarycanbeeithersmoothorrough. Thesuggestedgrowthmechanismcanbeappliedonlytotherough
boundary.
4.2.1 Rough Boundary
Assumingtheboundarytoberoughandusingthesameapproachasfortheliquid,wewritetheconservation
laws
SS(vS +wS)=R+Q/TW,
wS(ES −ES +ΛS )+vS(TSSS +mSnSφS)=Q,
 l zz
 vSmSnS +ρSwS =0.

8
Again, R is the surface dissipative function,
mSnS
RTW =vS (ES −ES +ΛS −ρSφS −TWSS)+SS(TW −TS)
ρS l zz
(cid:18) (cid:19)
mSnS mSnS
=vS λS + 1− SS(TW −TS) .
ρS zz ρS
(cid:18) (cid:18) (cid:19) (cid:19)
It must be positive, and for the quasiparticle velocity on the surface we therefore have
mSnS mSnS
vS =β λS + 1− SS(TW −TS) , (20)
ρS zz ρS
(cid:18) (cid:18) (cid:19) (cid:19)
where β >0 is the surface kinetic coefficient.
5 The Growth Rate
We now use the equations and boundary conditions obtained above. The physical system discussedin what
followsissolidheliumwithelementaryexcitationsrepresentedbyphononsandvacancies. Wefirstintroduce
a certain amount of friction between the quasiparticle gas and the lattice. To obtain a physically sound
result, we again restrict our analysis to one dimension. Furthermore, for simplicity, all our calculations are
performedwithinthelinearapproximation. We canwritethequasimomentumdensity(withthesuperscript
S omitted) as K =ρ v, where
K
¯h T4
ρ ∼ . (21)
K a4Θ4 c
D
Here, Θ is the Debye temperature, c is the velocity of sound in the crystal, and a is the lattice period.
D
The last equation is quite obvious. It follows from the fact that in the low-temperature region, the quasi-
momentum is mainly associated with phonons (the number of vacancies is exponentially small). The result
thereforecoincideswiththeoneforthe massdensity(andthemomentumdensity)ofthe normalcomponent
of the superfluid, ρ ∼ρ ∼T4/¯h3c5 (see [10]).
K n
z
L vL vsL
n
2
wS vS F Q h
S
ai
α
1
W Q
Figure 4: Crystal growth in one dimension
To describe (rare) Umklapp events, we introduce the appropriate relaxation time parameter τ . It is a
U
“between-Umklapp-collision time”. From (15), we then have
K v
0=K˙ =−mn∇φ−S∇T − =−mn∇φ−S∇T −ρ .
K
τ τ
U U
9
It is worth mentioning that τ may well depend on both phonons and vacancies, despite the fact that the
U
population of vacancies is far lower than that of phonons. For instance, if the vacancy energy band is
sufficiently narrow, then the probability of Umklapp processes is significantly higher for vacancies than for
phonons. This might overcome the low concentration of vacancies.
Interestingly enough, these formulas allow us to obtain the growth rate for a smooth surface. The
quasiparticles playing the crucial role here (that of mass carriers) are vacancies, with their mass given by
m=−mHe.
Toestimatetherate,wewritethetemperaturegradientas∇T =(T2−T1)/h,wherethesubscripts1and
2 standfor the solid–wallandthe solid–liquidinterfaces respectively (see Fig. 4). Likewise, for the chemical
potential we write ∇φ=(φ2−φ1)/h. We now use the boundary conditions (19) and (20),
mn mn
(α+β)v =αβ mn(φ2−µL)+S(T2−TL)+ λzz1+ 1− S(TW −T1)
ρ ρ
(cid:18) (cid:18) (cid:19) (cid:19)
1
=αβ mn(φ2−φ1)+S(T2−T1)+S(TW −TL)+mn φ1−µL+ λzz1−S(TW −T1)
ρ
(cid:26) (cid:18) (cid:19)(cid:27)
v 1 (cid:0) (cid:1)
=αβ −ρKh +S(TW −TL)+mn λzz1−S(TW −T1) +φ1−µL .
τ ρ
(cid:26) U (cid:18) (cid:19)(cid:27)
(cid:0) (cid:1)
In other words,
1 1 ρ h 1
v + + K =S(TW −TL)+mn λzz1−S(TW −T1) +φ1−µL ,
α β τ ρ
(cid:18) U (cid:19) (cid:18) (cid:19)
(cid:0) (cid:1)
where (using Eqs. (10) and (17))
λzz1 =Λzz1−T1S1−El1+E1−φ1ρ1 =pL−T1S1+E1−φ1ρ1.
In equilibrium, λzz1 = 0, φ1 = µL, and TW = TL = T1. If the liquid temperature and pressure change by
∆T and ∆p, we can write an equation for the growth rate v ,
g
ρ 1 1 ρ h
K
−v + + =−S∆T
g
mn α β τ
(cid:18) U (cid:19)
1 SL∆T −∆p
+mn (∆p−S∆T1−T∆S1+∆E1−ρ∆φ1−φ∆ρ1+S∆T1)+∆φ1+
ρ ρL
(cid:18) (cid:19)
∆p SL∆T −∆p mn 1 1
=−S∆T +mn + =− S−SL ∆T +mn − ∆p, (22)
ρ ρL ρL ρ ρL
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) (cid:19)
where we used Eq. (6), the thermodynamic equality dµ = (dp−SdT)/ρ for the liquid, and the obvious
relation v =−vmn/ρ.
g
We now considerthis equationwith the secondterm inthe right-handside equalto zero. This is a usual
scenario for heat conductivity measurements. The heat flux Q=vTS can then be expressed as
∆T
Q=− ,
RK1+RK2+h/κ
where RK1 and RK2 are the Kapitza thermal resistances on the solid–wall and solid–liquid boundaries and
κ is the heatconductivity ofthe crystal. Takingthe inequalities RK2 ≪RK1,mn≪ρ, andρ−ρL ≪ρ into
account, we immediately obtain
1 τ TS2
U
RK1 = βTS2, κ= ρ .
K
As a result, the growth rate is given by
mn ρ−ρL
v = S∆T +mn ∆p .
g ρ(TS2R +TS2h/κ) ρ2
K (cid:18) (cid:19)
Strictly speaking, the last equality implies that thermodynamic properties of the crystal mainly depend on
phonons,whilethecontributionofvacanciestotheeffectunderconsiderationislimitedtothemasstransfer.
10