The roughening transition of crystal surfaces. II. experiments on static and dynamic properties near the first roughening transition of hcp 4He

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The roughening transition of crystal surfaces. II.

experiments on static and dynamic properties near the first roughening transition of hcp 4He

F. Gallet, Sebastien Balibar, E. Rolley

To cite this version:

F. Gallet, Sebastien Balibar, E. Rolley. The roughening transition of crystal surfaces. II. experiments

on static and dynamic properties near the first roughening transition of hcp 4He. Journal de Physique,

1987, 48 (3), pp.369-377. �10.1051/jphys:01987004803036900�. �jpa-00210451�

The roughening transition of crystal ^surfaces.

II. Experiments ^on static and dynamic properties

near the first roughening transition of hcp ^4He

F. Gallet, S. Balibar and E. Rolley

Groupe ^de Physique des Solides de l’Ecole Normale Supérieure, ^{24, rue} Lhomond, 75231 Paris Cedex 05,

France

(Reçu ^{le 22} juillet 1986, accepté le 18 novembre 1986)

Résumé.

Nous récapitulons ^{dans cet article nos} connaissances expérimentales ^{sur la} transition rugueuse de l’interface (0001) des cristaux hcp d’4He, ^à _TR = 1,28 ^{K. Nous} décrivons dans le détail de nouvelles mesures précises

du taux de croissance, effectuées par ^un procédé interférométrique, ^et qui ^{montrent un} élargissement de la transition lié à l’écart à l’équilibre. ^{Ces nouveaux} résultats, ainsi que les ^mesures de l’énergie ^{de marche et} de la tension de surface, ^sont analysés dans le cadre des développements ^récents de la théorie de la transition rugueuse, exposés ^dans

l’article qui précède. ^{L’accord est} quantitatif, ^ce qui démontre que la transition rugueuse ^est effectivement une

transition de type Kosterlitz-Thouless.

Abstract.

We summarize in this paper ^our present experimental knowledge ^{about the} roughening transition of

(0001) interfaces of hcp 4He crystals, ^at TR = 1.28 K. We describe in detail new and precise measurements of their

growth rate, obtained by ^an interferometric method. They ^{show a} broadening of the transition due to non-

equilibrium conditions. These new results, together ^with step energy and surface stiffness measurements, ^are

analysed ^{in the} light ^{of the recent} developments ^{of the} theory ^of roughening, exposed ^{in the} preceeding paper. The agreement ^is quantitative, and proves that it is ^a smooth, infinite order transition, of the Kosterlitz-Thouless type.

Classification

Physics ^Abstracts

61.50C

64.00

67.80

68.00

1. Introduction.

As was first explained by Burton, ^{Cabrera and Frank}

[1], ^the roughening transition occurs when a crystalline

interface changes ^{from a smooth to a} rough ^state.

Below the roughening temperature TR, ^{a flat facet} develops in the direction undergoing the transition. In contrast, ^above TR, the surface is rough and may be curved when a difference in pressure is applied.

Moreover, ^the growth velocity ^{of a} rough ^{surface is} generally ^much larger ^{than that of a} facet when the

same departure ^from equilibrium ^is applied.

In the preceeding paper, P. Nozi6res and F. Gallet

[NG] give ^a detailed theoretical description ^{of the} roughening transition. ’In the context of the critical

theory ^of roughening, ^and by applying renormalization calculation techniques ^{to the} Sine-Gordon model, ^NG

made quantitative predictions for various physical quan- tities, ^{like the} step energy, surface stiffness and growth velocity. ^Their behaviour, both close to the roughening

transition and far from it, is also discussed in this paper.

In contrast with the mean-field theory [10], ^{the critical}

theory relates the roughening transition to the Koster- litz-Thouless transitions. In the present paper, ^we

definitely establish these theoretical results, by compar-

ing ^them quantitatively ^{to new and past} [4] experiments.

Experiments performed before the one we describe here already gave ^some support ^{to the critical} theory.

The first step ^{was the} discovery ^{of three} roughening

transitions on hcp 4 He crystals, ^at three different

roughening temperatures ( TR =1.28 K, ^{1.0 K} and

0.35 K respectively ^{for the} (0001), (1100) ^and (1101)

orientations [3-4]). ^{Later on, the} surface stiffness was

measured for the (0001) ^and (1100) directions and was

found finite at their respective roughening temperatures [4-5], ^{in fair} agreement ^{with one of the} predictions ^of

the critical theory [6-7]. ^More recently, experimental

values of the step energy 8 ^{were extracted from} growth

rate measurements performed ^{on a} (0001) ^facet [4].

Some indication was found that P smoothly ^{vanishes at} TR, again ⁱⁿ accordance with the critical theory.

As in a preliminary ^letter [8], ^we present ^{in this} paper

more accurate measurements of the growth ^{rate of} hcp 4He crystals, under very small differences in chemical

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01987004803036900

370

potential: the relaxation towards equilibrium ^{of a flat}

horizontal (0001) interface is observed by ^{an inter-}

ferometric technique, around its roughening tempera-

ture TR ^{= 1.28 K.} First, in the facetted state, the results confirm with more precision ^the expected ^be-

haviour of p, ^at T $ 1.22 K. Moreover, ^{we have}

discovered a regime ^of growth, ^{which is} intermediate between the rough and the facetted one, and which

occurs just ^below TR (1.22 ^K T ,- 1.28 K ). ^{It corres-} ponds ^{to a} broadening ^{of the} transition due to crystal growth, ^as predicted in reference [2]. ^This broadening depends ^{on the} driving force exerted on the interface.

In this _sense, our new experimental ^results clarify ^the

influence of dynamics ^on the transition.

Present and past experiments concerning growth

rates, step energy and surface stiffnesses constitute a set of data, ^{which can be} quantitatively compared ^with theory. ^The agreement ^{with NG’s} predictions ^is good,

which confirms that the roughening transition is well described by the critical theory : ^{it is an infinite order}

transition of Kosterlitz-Thouless type.

This paper is organized ^{as follows.} First, ^we report the main experimental ^results recently ^{obtained on} 4He crystals. ^{We also} present ^{a detailed} description ^of

our experimental ^set up and procedure, ^{and of the}

methods used in analysing the data. Our results are

presented together ^with previous ones : we make a

comparison ^{between them,} and with the theory. ^Start- ing ^{with NG’s} equations ^we explain ^the procedure ^to

calculate the fits, and deduce the main parameters ^of the theory, namely ^the roughening temperature ^itself

( TR = 1.28 - 0.01 K) ^{and the} coupling parameter

(tc

^{0.63 ±} 0.13).

2. Previous results on 4He _crystals.

4He is not the only system in which the roughening

transition has been observed [9]. ^However, quantum properties ^of superfluid ^helium, associated with the

high purity ^{of the} samples, ^{make it a} very good

candidate for a quantitative study ^{of this} phenomenon.

Since the discovery ^{of three} roughening transitions on

hcp 4He crystals [3], systematic measurements have been performed ^on equilibrium ^and growth shapes, ^to

understand static and dynamic properties ^{of these} crystals. ^The transparency ^of liquid and solid helium makes a direct optical observation possible. ^{One of}

such experiments ^concerns the surface stiffness y in the direction undergoing ^the transition. According ^{to Fisher}

and Weeks [6], ^{there is a} simple ^relation between the

roughening temperature TR ^{of a} crystal ^{surface and its}

two principal surface stiffnesses _{yi and} y2, at TR :

This relation is universal in the sense that it does not

explicitely depend ^{on the nature of} crystalline ^interac-

tions. The only parameter characterizing ^the crystal ^is

the distance a between equivalent ^lattice planes parallel

to the surface (for (0001) planes ^of hcp 4He, ^{a is}

2.99 A, i.e. half the true crystalline periodicity). ^{As a}

consequence, ^{when a} difference in pressure is applied

between the crystal ^{and the} liquid, ^the equilibrium

curvature of the interface is finite above and at

TR, ^{and zero below} [7]. ^First attempts ^to verify ^this prediction ^were performed by ^{Wolf et al.} [4] ^and

Babkin et al. [5]. ^{Both these} papers describe ^curvature measurements on a hcp meniscus around its (0001)

orientation (also ^called c-orientation), ^{near its} roughen- ing temperature TR = 1.28 K. This curvature, _pro-

portional ^to 1/y, remains finite at TR, and is of the order of the expected value. More precisely, ^{Wolf et al.}

measured _y

0.245 ± 0.03 erg/cm2, ^{and Babkin et al.}

y

0.21 ± 0.015 erg/cm2 (the c-direction is a six fold

symmetry axis, ^{so that y}

y,

y2). The first value is 21 % smaller than the expected ^{one, the second 32 %}

smaller (1). ^As explained ⁱⁿ [4], ^{this may be due to the} anisotropy ^{of y near the} c-direction : the measurements

give ^an average value ^{over a} finite angular ^domain,

which does not reveal the variations of y with the

crystallographic angle.

Part of this anisotropy ^was taken into account in reference [4], ^which probably explains ^{the difference}

between the two measurements. This will be discussed with more detail in section 5. At least the mean field

theory ^{was ruled out} by experiments, ^{since it} predicted

a continuous vanishing ^{of the curvature at} TR. ^{Let us}

also mention that the relation (1) ^{has been} approxi- mately verified for the roughening transition of the

(1100) orientation. Finally, the very ^recent discovery ^of

a first roughening transition on bcc 3He crystals, ^at

T ⁼ 0.1 K [11], ^is consistent with the universal charac- ter of equation (1).

Growth experiments [4] performed ^{on a} (0001) ^facet

below its roughening temperature gave ^some informa- tion about the variation of the step energy j6 ^near TR. ^{In such} experiments, ^the crystal ^{was forced to} grow

through ^{a little} hole into a box suspended ^{in the} experimental cell. This was a way ^to obtain a crystalline

surface relaxing ^towards equilibrium ^{under the effect of}

a known departure ^from equilibrium. ^{The interface out}

of the box remained fixed and its level was used as a

reference. The difference in chemical potential AA (per

unit mass) ^{across the interface} relaxing ^{inside the box}

was proportional ^{to the} hydrostatic overpressure measured from this reference level. I1JL ^{and the} growth velocity ^{v were measured} simultaneously by watching

the position ^{of the interface as a function} of time. For the (0001) ^{facet, the} growth characteristics v(åJL)

indicated that, ^near TR, ^{the dominant} growth ^mechan-

ism is the nucleation of two dimensional terraces of

(1) ^{See note added in} proof.

atomic height ^a. According ^to the relation [NG - 36],

v should be given by

(The driving force F is equal ^to p, ApL, ^where Ps is ^the density ^{of the} solid). ^{In this formula,} f is ^a

function of AA ^and /3 which varies slowly compared ^to exponential, ^{and is not} easily calculable a priori. By using ^a slightly ^different approach, ^other authors find that f ^{should not} depend ^on j3, ^{and should} be pro-

portional ^to (ð.JL )5/6 [12]. Experimentally, ^{Wolf et al.}

found good agreement ^with (2) ^when choosing f (ð.JL ) - Atk. Indeed, they plotted Log (v/Att ) ^versus

AA, ^{and found} that the relation was linear. They

inferred values of /3 ^{from the} slope ^{of the obtained} straight lines, ^{at various} temperatures. These first results indicated that j6 ^{vanislies at} TR, ^{in a} way which is

compatible ^{with the} predictions of the critical theory.

There again, ^{a mean field} theory ^would predict ^a

square ^root cusp ^at TR ^{and is} clearly ^{not valid.}

However, the critical behaviour of j8 very close ^to

TR ^{was not} observable, mainly ^because the accessible values of Att ^{still remained too} large. Indeed, ^when j6 becomes smaller, ^{one also needs} smaller values of AA

to measure it, ^{otherwise the} argument ^{of the} exponen- tial in (2) ^{does not} appreciably ^{differ from zero. In the} experiment performed by ^{Wolf et al., the minimum}

value of AtL ^{was limited} by ^the capillary depression ⁱⁿ

the box where the interface relaxes, i.e. =.-,: 0.2 mm.

Moreover, ^{the end of the} relaxation was masked by ^the

thickness of the meniscus of the reference interface,

which is of the order of 0.5 mm. We present ^{now a} growth experiment ^{which is based on the same} prin- ciple, ^{but with two} important improvements : ^{first, the}

box has been made larger ^{in order to} reduce the

capillary depression, second, ^the relaxation is observed from the top, by ^an interferometric technique. Thus,

one can follow the level of the relaxing ^{interface closer}

to equilibrium ^{and with a} much better precision.

3. Experimental ^set up and procedure.

We used the same pure 4He _cryostat as in reference [4].

In its high pressure cell which has four sets of windows at right angle, ^an interferometric cavity ^{is set} up (see Fig. 1) : ^{a mirror and a 50 %} reflecting plate (30 ^{mm in} diameter) respectively form the bottom and the top ^of this cavity. They ^are separated by ^a glass cylinder (13 ^{mm in inner diameter, 8 mm in} height), ^which

maintains them as parallel ^as possible, ^and glued together. ^An enlarged ^and parallel ^{He-Ne laser beam}

goes horizontally into the cell through ^the windows, ^is

reflected on a 45° mirror, ^and illuminates this inter- ferometric cavity ^{from the} top.

The angle between the top and bottom plates ^is roughly ^one milliradian, ^{so that one observes} parallel wedge fringes ^{in the} cavity. Any displacement ^{of a flat}

horizontal interface between the two plates ^{induces a} proportional ^{shift of these} fringes. From the known

Fig. ^1.

Scheme of the experimental apparatus. On the left : top view of the whole set up. The pressure cell is illuminated

by ^a parallel laser beam. Two photomultipliers ^{are located on}

an enlarged image of the cell. They record the passage of

fringes ^{due to} the motion of the crystal respectively ^{inside and}

outside the interferometric cavity. ^{On the} right : side view of the cell showing the interferometric cavity. The motion of the

growing crystal produces ^a proportional ^{motion of the} wedge fringes.

densities of liquid ^{and solid 4He} [13] ^{at T =1.3} K, ^one

calculates their difference in refractive index An

3.43 x 10- 3. A shift of one fringe ^{thus corres-} ponds ^{to a} 92 R ^vertical displacement. ^{A little} hole, 0.8 mm in diameter, ^is drilled through ^a thin copper shield fixed on the side of the cavity, ^near the bottom.

Four other holes, ^{about 1 mm in} diameter, ^{have been} drilled similarly ^{near the} top ^{to allow} liquid ^helium

flow. As described in [4], ^we grow ^a hcp crystal by adding helium in the cell at constant temperature. ^The

first seed can be oriented with the (0001) ^surface roughly horizontal [5]. To make it as horizontal as

possible, which is crucial for the success of this exper-

iment, ^one proceeds ^as follows : the first seed is grown

at T- 1 K, ^{so that a} large (0001) ^facet develops. ^At

this temperature ^{the facet moves} very slowly ^{and its}

intersections with any wall ^are thin, easily visible, straight ^{lines of contact.} By tilting ^{the whole} cryostat, which is supported by springs, it is easy ^to superimpose

the two lines of contact on front and back windows in a

horizontal cathetometer. The final precision ^{on the} horizontality ^{of the} (0001) orientation is about 1 milliradian. Once this is done, ^one may ^warm the cell up ^to the desired temperature, ^while keeping ^a properly

oriented seed. The cell is then slowly filled with crystal, step by step, ^{and a stable} crystalline meniscus forms on

the bottom hole of the cavity. ^As already explained [4],

this meniscus becomes unstable only when the level of

crystal ^{above the} hole, outside the cavity, ^{reaches some}

critical value.

At this point, ^one stops adding ^{helium in the cell, the}

meniscus _grows, invades the inner part ^{of the} cavity,

and relaxes toward equilibrium ^{with a flat horizontal}

interface (the equilibrium ^level nearly ^{coincides with}

the outer level). ^{In the same} time, ^{the outer level melts}

372

Fig. ^2.

Photograph showing ^a partial top ^{view of the} cavity,

while a hcp He 4 crystal fills it from its side. The (0001)

orientation is parallel ^{to the} plane ^{of the} figure. Although

T= 1.30 K is above all the roughening temperatures, ^the growth shape ^remains anisotropic. Note the substructure of

wedge fringes.

down a little to ensure mass conservation : this displace-

ment is about 0.04 times that of the inner interface,

which corresponds ^{to the} ratio of the inner to the outer area. The photograph ⁱⁿ figure ^{2 shows a} top ^{view of} the interferometric cavity, ^{with its} fringe pattern, ^while

a little crystal ^is growing ^{inside. Two} photomultipliers,

located on an enlarged image ^{of the} cell, ^{record the}

passage of fringes respectively ^{out of the} cavity ^{and in}

its centre (Fig. 1). ^Moreover, during ^the recording, ^two diaphragms ^{limit the} illuminated field on the two small

interesting ^{areas, so} that the total incident power from the laser in the cell is less than 100 RW (most ^{of it is} reflected). Figure ³ represents ^a typical recording ^ob-

tained at 1.253 K, during the relaxation of a (0001)

orientation. In addition to the main structure, ^one observes secondary maxima and minima which are due to multiple reflections of light between the plates. ^This interfringe pattern identically reproduces ^{several times}

as time elapses. ^{Note that it is} scanned in opposite

senses in the two presented recordings : ^{one interface}

grows (in ^the cavity) while the other one melts (out ^of it). By counting ^the fringes, ^{one measures the level of}

the two surfaces at any time. An interpolation ^{is made} by comparing ^the recording ^{to a reference} one, ob- tained while adding helium in the cell at constant rate :

Fig. ^3.

^A typical recording of the passage of the fringes

versus time, inside and outside the cavity. ^It corresponds ^to

the relaxation of the (0001) orientation at T = 1.253 K.

the surface then moves at constant velocity, ^{and the}

recorded pattern ^is periodic ^{in time. The} definition is such that one appreciates ^{a variation of one hundredth}

of a fringe, corresponding ^{to a 1 J,L} displacement ^{of the}

interface. The heights hi ^and ho ^of the surface respect- ively ^{in and out of the} cavity ^{are measured from their} respective equilibrium ^{levels at} the end of the relaxation

(when ^the fringe pattern ^{does not move} anymore). ^The

difference in chemical potential ^{across a} flat interface is

AR = [(lips) - (llpl)] ^{Ap, where} Ap ^{is the deviation}

from the liquid-solid equilibrium pressure, and ps, p I are the respective ^{densities of solid and} liquid

helium. Now, ^{an accurate treatment also must consider}

the capillary depression he ^{inside the 13 mm} cylinder ^of

the cavity. ^{From the} known value of the surface stiffness around the c-axis [4], ^we calculated hc = 13.5 --t 2 u. ^We finally ^{assumed that} dp =

p I g (hi ⁺ hc), ^so that, ^for the inner surface :

with H

hi ⁺ hc.

Let us comment on the validity ^of (3). ^{If the relation}

v

dH/dt

v(âJL) may be considered as the same

for the growing ^{and the} melting interface, ^{and in the} particular ^{case where v is a} linear function of Au, ^the

equilibrium ^level Ap

0 coincides at any time with the final reference level H

0, ^{and the relation} (3) ^is

exact. This is the only ^case where the motion of the two interfaces are strictly ^not coupled. ^{In all other cases,}

one cannot write a simple ^{and exact} expression ^for Ap

without knowing ^a priori V(d/L). However, ^the slight melting ^{of the outer} interface is supposed ^{to disturb}

very little the growth of the inner one. The relation (3)

is then correct within a few percent ^{maximum error,} and we shall use it in the following.

Second, ^{curvature effects of} the interface have been

neglected. ^This is reasonable as long ^{as the} growth shape presents ^a large ^flat interface. Experimentally,

this is true in a steady ^state regime ^of growth, ^when

H;-.> hc. ^{In contrast, at the very end of the} relaxation,

the curvature increases, ^{so that} AtL ^{is zero when}

H

hc. ^{Out of our} results, ^{we shall use} only ^data

obtained for H > hc.

Finally, ^{let us mention that the formula} (3) ^differs slightly ^{from the one used in reference} [8]. ^{The above}

detailed analysis explains ^{this new} choice. However, the results shown in section 5 show that the difference is

hardly perceptible.

4. Results.

We performed ^{the same} experiment ^{on different crys-} tals, suitably ^{oriented, at different} temperatures ^from 1.16 K to 1.34 K, around the roughening temperature

of the c-orientation. The growth characteristics v (A tt )

(or ^v (H)) ^is calculated numerically : ^each recording ^is

digitized, ^{and the} variation of H versus time t is

Fig. ^4.

Some of the typical growth characteristics

v(H) ^{obtained at various} temperatures ^{for the same} (0001)

orientation. The height ^{H is} proportional ^to the difference in chemical potential ^{across the} interface. The growth ^rate continuously changes ^{from a low} temperature, ^{non linear} regime (2D nucleation on a facet) ^{to a} high temperature ^linear

one (growth ^{of a} rough surface).

computed point by point. ^{Then v}

(dH/dt ) ^{is calcu-}

lated as the local derivative of the least square parabola interpolating ^{a set of five} H(t ) points.

We present ⁱⁿ figure (4) ^some typical growth ^charac-

teristics v (H) ^{obtained at various} temperatures. ^Values of H range from about 1 ^{mm to} nearly hc. ^{For a} typical

value H = 0.1 _mm, the corresponding Au ^{is about}

3 x 10 - 8 k8 T/m4, ^{so that the} departure ^from equilib-

rium always remains very small. The measured vel- ocities vary from about 100 JL/S ^to less than 1 JL/s. ^The shape ^{of the} characteristics v (H) changes ^with tempera-

ture. For T =::: 1.28 K, the relation between v and H appears linear. On the contrary, below T = 1.21 K, ^this

relation is non linear, ^{v seems to vanish} smoothly ^when

H goes ^{to zero.} The change ^{from linear to non linear is}

continuous when T decreases. Our results are also

presented ⁱⁿ figure 5, ^{in a different} _{way :} ^we _compare

the measured velocity ^{v of the} c-interface to the

velocity v, of ^a typical ^{surface at the same} temperature, which is completely rough. Figure ⁵ represents ^the variations of vlv, ^{versus T, for} three different and fixed values of H (H

^{0.7 mm, 0.22 mm, and 0.07} mm) ;

vr is taken as 2.73 x 10-4 H exp(7.8/T) (T ⁱⁿ K, ^{H in}

cm, vr in cmls), ^which properly ^{describes the} tempera-

ture evolution of the velocity ^{of a} typical rough ^surface [14]. The maximum value H

0.7 mm is smaller than the one used in [8] : ^{we have now realized that, at the} beginning ^{of the} relaxation and till H s 1 _mm, the meniscus does not fill completely the whole bottom of the cavity, because the bottom hole is located on its side. In this situation, ^{curvature effects are not} negli- gible ^{and one cannot use} (3) ^{to calculate Atk. In the}

other limit, ^at the end of the relaxation, hc ^{is no} longer negligible compared ^{to H} when H s 50 _JL. This explains why ^we only present data for 0.7 -- H , 0.07 mm.

Fig. ^5.

^Reduced velocity vlv, ^{of a} (0001) orientation

plotted ^versus temperature, for three different values of the

height ^H. vlv, ^vanishes continuously ^{to zero from} high ^{to low} temperatures, ^{which is a} signature ^{of the} dynamic broadening

of the transition. The theoretical fit (full lines) is drawn for the same values of H, the adjustable parameters being

= 0.63, TR ^{= 1.28} K, Ao = 1.161a (see text).

The data in figure ^{5 can} be divided in three tempera-

ture regions. First, for T > 1.28 K, ^{the ratio} v IVr ^is comparable ^{to one and does not} depend ^{on the} departure ^from equilibrium ^{H. On the} contrary ^below T = 1.21 K, vlvr is small and its dependence ^{in H is} important: v/vr ^increases by ^{more than one decade}

from H

0.07 mm to H

0.7 mm. Finally, ^between

1.21 and 1.28 K, v/vr changes ^from nearly ^{zero to its}

maximum high temperature value. The width AT of this intermediate regime ^is relatively small, and increases

slowly ^{with H} (Ar/Fp - 0.05 for H

0.07 mm and

AT/TR - 0.07 for H

0.7 mm). ^{We show now that}

the interpretation of the results is different in each of these three temperature ^domains.

5. Interpretation. Comparison ^with theory.

Let us first consider the domain T 2: 1.28 K. The ratio

v/vr ^{does not} depend ^on H, ^{which means that}

v/H ^{does not} depend ^on H, ^but only ^{on T. This}

behaviour is characteristic of a rough surface for which the ratio v/Au, also called mobility, ^{is a} function of temperature only. Consequently, ^{at T 2: 1.28 K, the} (0001) ^{interface is} completely rough. The fact that

v/vr remains smaller than one simply ^{means that the} mobility ^of rough ^{surfaces is} anisotropic, ^as already

observed by ^{Leiderer et. al.} [15].

Let us now analyse ^{the data obtained at T s}

1.21 K. In this temperature range, Wolf et al. showed that 2D nucleation of terraces is the dominant growth

mechanism. Following ^their procedure, ^we plotted Log (v/H) ^versus 11H ^{for each} relaxation : we noticed that the experimental points ^{are located on} straight

lines with negative slopes. ^This slope only depend ^on

temperature and vanishes when T increases. This

374

behaviour is characteristic of 2D nucleation. Thus, ^we inferred from the slope ^{of these} straight ^{lines new}

values of the step energy a, ^and plotted ^{them in} figure ^6, together ^{with some of those} given ^{in reference} [4]. Note first that our measurements extend towards

higher temperatures. Moreover, ^{at the same} tempera- ture, ^{our values of} a ^are compatible ^{to those} given by

Wolf et al., although slightly smaller. The new results confirm the previous ^{ones, and show more} clearly ^that /3 ^{vanishes at} TR ^{with a zero} derivative (dB/dT ^also

vanishes when T approaches TR). ^{This is a further}

argument against ^{mean field} theory, ^which predicts

,8 - (TR - T)n2, ^{and in} favour of the critical theory

which predicts 8 - exp - c (TR - T)- ^u2.

Fig. ^6.

Step energy f3 / a ^{of the} (0001) orientation plotted

versus temperature. ^Different symbols ^{refer to different} crystals. Pla ^vanishes smoothly ^{when T} approaches TR

1.28 K, which is consistent with the critical theory. ^{The fit} (full line) is drawn for the same values of the adjustable parameters ^{as in} figure ⁵ (tc

0.63, TR ^{= 1.28} K, Ao = 1.16/a).

One can a posteriori verify that 2D nucleation is

really ^dominant compared ^{to other} possible processes of growth, especially spiral growth ^{from screw disloca-}

tions emerging ^at the interface : the growth velocity

Vd due to this process would be such that vd - vr a 2p, A,4 /20 ^{P, in a crude} approximation ^which neglects co-operative effects of several dislocations [1].

One finds that Vd ^is smaller than the measured vel- ocities.

However, ^our analysis ^{in terms of 2D} nucleation fails for Tz- 1.22 K for a different reason. Indeed, ^as discussed in section 4.1 of reference [2], the 2D nuclea- tion model assumes that the steps limiting ^{the terraces}

are sharp-edged objects. ^In fact, ^{when T} approaches TR, j3 goes ^{to zero} and the mean amplitude ^of step fluctuations diverges ^like (ya2/i ) ^{at TR [NG-40]. In}

particular, nucleation is no longer ^an activated process

as soon as ) is greater than rc

8 lap, ^{âJ.L, the critical}

nucleation radius of round terraces. In this _case,

thermal excitations produce ^a large density ^of terraces,

at a high ^{rate :} the interface should no longer ^behave

like a smooth facet, but rather like a rough ^{surface, in a}

sense which ^we now make more precise. ^{The condition}

rc > § ^is equivalent ^{to H} Hmax

^{2.5 mm at T}

1.18 K or H Hmax = 25 f.L ^at T =1.22 K. For T 1.22 K, ^we calculated 13 only from the data points ^such

that H Hmax : ^our result is meaningful. ^In contrast, for T> 1.22 K, the data do not satisfy the criterion H Hmax, ^{so that one should not be confident in the}

values of /3 ^{obtained at such} temperatures.

We have not plotted ⁱⁿ figure 6 values of p ^obtained by Wolf et al. at T > 1.20 K. Indeed, the criterion H Hmax ^{was no} longer satisfied in their experiment ^at

such temperatures. ^This argument may also explain ^the

small systematic difference between Wolf et al. and our

results at T 1.20 K.

Together ^{with the} experimental ^{data, we} present ⁱⁿ figure ^{6 a} theoretical fit for /3, ^{which was} numerically

calculated from Q

(4 al’IT)( ’Y V)112 ^{[NG-40]. Here, y}

and V are respectively the renormalized values of the surface stiffness and of the pinning potential ^{of the}

interface on a macroscopic ^{scale. These values are}

obtained by integrating ^formula [NG-21], ^{yo and} Vo being the values of _y and V on the smallest scale

1/Ao ^{where the} renormalization starts. Of _course, yo, Vo ^and Ao ^{are not} accessible to experiment ^{and thus}

are the three parameters ^{of the fit. In} fact, ^{it is} convenient ^to replace ^{two of them} (yo ^and Vo) by ^two

other quantities which have a more accessible physical meaning. ^{The first one is} simply ^the roughening ^tem-

perature TR, the second is a dimensionless parameter which characterizes the strengh of the lattice potential, namely tc

2(A (2) )lIZ (lTZ Vol AJ ^{yo a2). In this ex-}

pression, ^and according ^to [NG-13b], 2 (A (2))1/2 --

1.26. Indeed, ^below TR, y and ^{V should} diverge ^when

the renormalization scale 1/A ^{goes to} macroscopic

values. In fact, ^{one has to} stop the renormalization when the development in powers of V is no longer valid, i.e. when V/A 2 ^is comparable ^to kB T. The fit

presented ⁱⁿ figure ^{6 shows values of} jS calculated on

the scale 1 I A * ^{such that} V / (A *)2 = kB ^{T. We checked}

that the result depends slowly ^{on the cut off} 1/A*. ^The

parameters of the fit are 1/Ao = all.16 = 2.6 A,

TR ^{= 1.28} K, tc

0.63. The choice of these adjustable values, ^which give ^{the best} fit, ^is justified ^{below. Note}

that agreement ^with experiment ^is good ^{in the} tempera-

ture domain where the measurements of /3 ^are meaning-

ful. The scale is such that only the first few percentage variation of 8 ^near TR ^are represented. By extrapolating

the fit to lower temperatures, ^one would find

,BlalT=O - ^{2 x 10-2} erg/cm2. This is about one tenth of the mean value of _y, which is quite reasonable.

We now turn to the intermediate regime ^of growth,

at 1.22 T:5 1.28 K. We already ^mentioned that, ⁱⁿ this _{range of} temperature, ^the growth ^{of the interface}

does not involve the same mechanism as a classical

facet, ^at least for the typical driving ^{forces that we}

apply. Fluctuations are large enough ^{to make the}

terraces rough. However, ^the whole interface is not

completely rough, ^{since its} velocity ^{has not} yet ^reached that of arbitrarily ^oriented rough surfaces, ^{and since} v/H depends slightly ^{on H. In} fact, ^{such a situation has}

been theoretically ^discussed in NG-section 4.2. The main predictions ^{are the} following : in the limit where the driving force tends to zero, the mobility ^{of the}

interface K

vlAA

p,lq (TJ ^{is a friction} coefficient) jumps ^{from a} finite value above TR ^{to zero below and}

has a square ^root cusp ^at TR (the ^{fact that,} experimen- tally, v I vr still increases with T above TR ^{is a} signature

of this cusp). ^{When a finite} difference in chemical

potential AA ^is applied ^{across the} interface, ^the jump ⁱⁿ mobility ^{is broadened and K} continuously ^{vanishes on}

some temperature domain. This domain extends below

TR, and its width is an increasing function of AtL. ^{This is} exactly ^{what is} experimentally observed : figure ⁵

shows the first evidence for a broadening ^{of the}

transition due to the finite growth velocity ^{of the}

interface.

We are able to make a quantitative comparison

between theory ^and experiment, ^{in the} following ^man-

ner : as described in NG-section 4.2, ^{a finite} velocity ^v

introduces a characteristic time scale ^T

a/v. ^{At a} given ^scale 1/A

^{L of the} renormalization, ^{this time T}

must be compared ^to the time T’

L2 ps/Ky, ^charac-

terizing ^{the diffusion of a} fluctuation on the scale L

along the interface. If T’ > T (i.e. ^if L > L1=

(i’al Ps åJL )112), ^the crystalline potential, ^{as seen} by

the moving interface, ^is averaged ^to zero, and the renormalization stops. ^More precisely, ^{one can model}

the effect of a finite velocity ^v by adding ^{an extra term}

cos (2 lTt IT) ^{in the} expression of the coefficient

B (n ) ^which pilots ^the renormalization of _q

p,IK (see [NG-21]). ^{For L .e,} Ll, B (n ) ^is unchanged ^{and the}

renormalization acts as in the static case. On the contrary, for L > Ll, B (n ) ^{is zero and the} renormaliza- tion of K stops. ^{A numerical} calculation indicates that,

for the typical ^{value n}

2, B(n) ^is unchanged ^com- pared ^{to the static case until L2} L,2/10. ^{Then, it drops}

rapidly ^{to zero, since} B (n) 0 ^{when L 2-} Lr/2.

B(n) ^{is divided} by ^{two when LZ =} Lf/5. ^{Instead of}

taking ^{into account the exact variation of} B(n), ^we approximated ^it by sharpl stopping ^the renormaliza- tion of K when L

L1/ 5. ^{As far as the} renormaliza- tion is piloted by Log ^{L, this} approximation ^seems

reasonable (the typical range of L spreads ^from 10

^{10- 8 cm to} L1 - ^{10- 4 cm, much} larger ^{than the}

width åL1where the variations of B (n ) ^are important).

Together ^with experimental points, ^{we have drawn in} figure 5 ^{the three} theoretical curves representing (K/Ko ) ^{for the same} ApL ^{values as in the} experiment.

Ko ^{is the non} renormalized mobility of the interface at the scale 1/Ao, and contains the same temperature dependence ^{as the} mobility Kr ^{of a} completely rough

surface. Thus, ^the comparison ^between (K/Ko ) ^and (vlv,) = (K/ Kr) ^is meaningful. Ko ^{is a} priori ^different

from Kp ^since it reflects the intrinsic anisotropy ^{of the} crystal. ^We adjusted ^the proportionality ^{constant in}

front of (K/Ko ) ^{in order to} fit the data at T -- 1.28 K. The other adjustable parameters ^are again TR, tc, Ao, ^{as for} the fit of the step energy Q. ^{Of course,} both the fits of /3 ^and (KIKo) ^{are drawn with the same}

values of these parameters : TR ^{= 1.28} K, tc

0.63,

11 Ao

^2.6 A. Agreement ^with experiment ^is good ^at

least for T:> 1.22 K. Below this temperature, ^{we know}

that 2D nucleation takes place, and that the above renormalization calculation is no longer ^{valid. The}

lowest temperature for which the fit is calculated

precisely corresponds ^{to = rc.}

We would like now to synthesize past ^and present experimental results. The theory developed ⁱⁿ [NG]

also includes predictions ^{which concern static} properties

of the interface, like its stiffness _y. We recall that the curvature of the interface, being inversely proportional

to y, jumps ^{from a} finite value above TR ^{to zero below.}

This is true only ^{in the exact direction} undergoing ^the

transition. On the contrary, ^{in the case of a vicinal} surface, ^an angle 0 away from the previous ^{one, finite}

size effects are important (see NG-section 3). ^Below TR, when -> al 0, the surface stiffness y ^{must be} renormalized only up ^{to a} scale 1/A of the order of the

mean distance L2

alO ^between steps. In the other

limit alO, ^the steps ^{are well defined} individual objects, ^{and the} renormalization calculation fails. How- ever, in our experimental conditions, this limit concerns a microscopic ^{area of the} crystals (at T - 1. 20 K, 6 - 1 f..L is of order of a/ e when 0 ooo-o 3 x 10- 3 rad), ^and

we shall not discuss this case here. Thus, ^as long ^{as (J is}

not too small, the surface stiffness y ^{of a} vicinal surface remains finite even below TR : ^{in the} same manner as

the jump ^of mobility ^{is broadened in a non-static} situation, ^the jump ^of 1/y ^{is also broadened by these}

finite size effects.

This has been experimentally ^observed [4] : ^we

report ⁱⁿ figure ⁷ measurements of reduced curvature on a hcp meniscus, ^{near the} c-axis, ^{close to its} roughening transition. This reduced curvature C is defined as (17 12 )(kB TR/ya2), ^is proportional ^to 1/y, ^{and is} expected ^{to be} equal ^{to one at T}

TR, ^and

zero below. Wolf et al. explained why they ^{do not}

observe the jump ^of curvature, ^and why ^{the value of C}

at TR ^is greater ^{than the} expected ^{one :} they ^{measure in}

fact an averaged value of C over some angular ^window

0 which is of order of ± 0.15 rad. Comparison ^{of data}

with theory ^needs again ^a procedure ^of stopped ^renor-

malization : when the scale 1 A = ^{L is} larger ^{than the}

distance L2

alO ^between steps, ^the crystalline poten- tial is zero and the renormalization stops. ^{The exact} variation of A (n ), piloting the renormalization of _y, is

given by [NG-28] ^{when 0 0 0.} Numerically, ^{one ob-}

serves that, ^{for the} typical ^{value n}

2, A (n ) ^is unchanged compared ^to the static case until 0Lla ooo-o

1/20. It is nearly ^{zero when} OLIA - 1/4, and it is

divided by ^{two when} 0 L la ooo-o 1/6. As for the calculation

376

of K, ^we adopted ^a sharp ^cutoff procedure ^{at L}

al6 0 ^{for the} calculation of y.

The theoretical fits in figure ⁷ represent the reduced curvature C calculated for an orientation such that (J

0.15 rad. As pointed ^out by ^NG (Sect. 3), ^{and in}

reference [4], ^{we are in the case where y varies} slowly

with 0. Indeed, ^{when 0} > al , y (0, T) ^{is not} signific- antly ^{different from} y (0, TR), ^{and is} approximately given by [NG-31]. ^Thus, y ( 8, T ) depends only ^on Log (J, and it is reasonable to compare the calculated curvature for 0 = 0.15 rad with the measured _one, which is averaged ^{over some} angular range of the order of 0. This comparison ^{is shown in} figure ^{7 : curve} (b)

represents ^the fit obtained for the same values of the

adjustable parameters ^{as in} figures 5 and 6. The agreement ^{is as} satisfactory ^{as for the} step energy and the reduced velocity. We also checked that this agree- ment remains fair when calculating ^{C for (J}

0.20, 0.10 ^{and 0.05 rad.}

Fig. ^7.

^{Reduced curvature} C = 2 kB TR/ 7T ya 2 ^{of the}

(0001) orientation versus temperature (data ^{are taken from}

Ref. [4]). ^{C should be} equal ^{to 1 at} TR, but is in fact larger

because its measured value is an averaged one over a finite

angular range. The three fits (a), (b), (c) ^are drawn for the

respective ^{sets of} (TR, t,,, AO) equal ^to (1.275 ^K, 0.75,

0.87/a), (1.28 ^K, 0.63, 1.16/a) ^{and (1.285 K, 0.50,} 1.551a).

(b) ^is obviously ^{the best one.}

From these comparisons, ^we may already ^conclude

that the theory developed ⁱⁿ [NG] gives ^a qualitative

and quantitative explanation for all the experiments performed up ^{to now on} the roughening transition of helium crystals.

We would like now to reconsider the growth exper- iment described in section 3, ^to understand why ^the alignment ^{of the} (0001) ^{interface with the} horizontal is

so crucial. The growth ^{of a} vicinal interface is discussed

theoretically ^{in section 4.3 of NG :} according ^{to the}

value of 0 and A tt, ^{the finite size} effects due to the tilt may dominate the effect of the finite velocity ^{of the}

surface. One can reasonably argue that, ^if L2

alO ^is

smaller than L, = (yalp, Au )112, ^the renormalization of y and K ^{has to be} stopped ^{at the scale} L2, ^{so that the} broadening ^{of the} transition will essentially ^{come from}

the tilt of the interface. In our experimental conditions, A/i - ^{0.5 mm and 0 _} 10- 3 rad gives L1 - L2. ^The

resolution on the orientation of the interface is precisely

0 _ 10- 3 rad. The fact that the broadening ^{of the}

transition depends ^on A/i confirms that this broadening

is essentially ^{due to} dynamic ^effects. Moreover, ^the anisotropy ^{of the} growth ^rate, which tends to enlarge

the (0001) orientation on the growth shape, may also

help, by ejecting tilted orientations to the sides of the interferometric cavity.

Let us finally ^{comment on the choice} of the par- ameters ( TR ^{= 1.28} K, tc

^0.63, 1 / Ao

^2.6 A) ^for

the fits of surface stiffness, step energy and reduced

velocity. It appears that the quality ^{of the fit is} quite

sensitive to a small change ^{of one of the three} par- ameters (the ^{two others} remaining constant). ^However,

in the space (TR, t,, A)) ^{there is a set} of values for which the quality of the fit is good, ^at least for the step energy and the reduced velocity. ^{This is} illustrated in figure 8,

Fig. ^8.

Experimental values of the step energy (from growth measurements in the 2D nucleation regime) ^{and of the}

reduced velocity (from ^the quasi ^linear regime) plotted together, showing ^{a crossover at T ooooo} 1.21 K. Three fits are

drawn, corresponding ^to (TR, tc, AO) = (1.275 K, 0.75,

0.87/a), (1.28 K, 0.63, 1.16/a) ^and (1.285 K, 0.50, 1.551a),

as in figure ^7. They ^are nearly superimposable, ^{so that one}

can decide between them only by looking ^at figure ^7.

where we plotted together ^the step energy ^{and the} reduced velocity (for ^H

^0.07 mm). Three fits are also

drawn, corresponding ^{to three sets of} parameters (TR, tc, 1/Ao), namely (1.285 K, 0.50, 1.9 A), (1.28 ^K, 0.63, ^2.6 A) ^and (1.275 ^{K, 0.75, 3.4} A). These three theoretical curves are nearly superimposed. ^For- tunately, ^{the fit of the curvature allows to determine}

the best values of TR, tc, Ao, ^{as shown in} figure ^{7. The}

three represented ^curves correspond ^{to the same three}

sets ( TR, tc, AO) ^{as above, and} they surround the

experimental points. Thus, ^{we are able to} give ^{an order}

of magnitude ^{for the} uncertainty ^{attached to the}