# Observation of a jump-like transition from fast to normal kinetics of facets
for free-growing helium crystals.

V. L. Tsymbalenko<EMAIL_ADDRESS>Kurchatov Institute, 123182 sq.Kurchatov 1
Kapitza Institute for Physical Problems RAN, 119334 Kosigin st.2
Moscow, Russia

###### Abstract

The growth kinetics is studied of free-growing helium crystals within the
temperature range 0.5-0.83K and supersaturations to $\sim$10mbar. For the
pressure supersaturations above $\sim$0.5mbar, the numerous drastic changes
are observed in the growth kinetics for all crystalline facets from the normal
and slow kinetics to anomalously high or burst-like one. It is found for the
first time that the anomalous-to-normal transition in the facet growth
kinetics occurs in a jump-like manner within the time 80$\mu$s interval of the
measurement record.

###### pacs:

67.80. -s, 68.45. -v

##  I Introduction

The theoretical paper by Andreev and Parshin in 1978 AP  has aroused
experimental interest in the growth kinetics of 4He crystals. They have
suggested that the quantum nature of 4He crystals should result in
delocalizing the elementary defects as crystal facet steps in the surface
structure. Above the roughening transition temperature the free energy of
crystal steps vanishes and thus their equilibrium concentration is high. The
deviation from the phase equilibrium measured in units of pressure $\delta p$
sets the crystal steps in motion limited by the small viscosity of phonon gas.
Accordingly, the large mobility of the crystal steps results in a high
mobility of the crystalline surface. The mobility of crystalline surface
increases as the temperature lowers. This theoretical prediction has received
the experimental confirmation resulting in a wide theoretical and experimental
study of the effect, see review BPH .

Below the roughening transition temperature the free energy of crystal steps
is positive, leading to appearance of flat segments on the equilibrium
crystal, namely, facets. The steps on the crystal facets are delocalized and
have high mobility. The thermodynamically equilibrium step concentration
decreases exponentially with decreasing the temperature. This should lead to
the complete loss of the facet mobility. The facet growth in this temperature
range is provided by the topological sources of the steps. Termination of a
single screw dislocation to the surface generates a spiral facet growth. A
pair of screw dislocations with opposite sign represents the Frank-Reed
source. The concentration of such sources depends on the crystal defects
different at the equivalent facets. As a result, the free-growing crystals
have an irregular hexagonal shape VLT1995 . The facet growth rates measured
experimentally have demonstrated the specific features inherent in the
indicated mechanisms typical also for classical crystals. The growth of
crystal facets starts above the threshold supersaturation $p_{FR}$ and the
growth rate depends nonlinearly on the magnitude of the deviation from phase
equilibrium $\delta p$. For small deviations from phase equilibrium, the facet
growth kinetics of quantum crystal does not differ in kind from that of
classical crystal facets. For this reason, the interest in such studies was
low.

The situation changes drastically as a result of studying the crystal facet
growth by the Finnish-Russian group in the 2-250 mK range RF-1 . In a number
of experiments it was possible to grow the defect-less crystals. For the
above-mentioned temperatures, the facet is free of thermodynamically
equilibrium steps and, as is discussed previously, the facet should not grow.
However, in the experiments under constant helium flow into the container it
is found that the pressure increases monotonically until the critical
supersaturation $\delta p_{c}$ is reached after which a drastic drop has been
observed. This means that the ideal facet does not grow until qualitatively
new growth mechanism of unknown nature starts in operation. This provides the
facet growth for some time sufficient for crystallizing an excess mass of the
supersaturated liquid and decreasing the pressure close to the phase
equilibrium one. Then this mechanism disappears, the facet looses its mobility
and the process repeats.

The similar phenomenon is discovered in the same year in our experiments on
the crystal growth from the metastable liquid at temperature 0.44 K VLT1996 .
The crystal growth time at initial supersaturation $\delta p_{0}\sim 14$mbar
is shorter than 0.7ms, corresponding to the growth rate higher 1 m/s. This
magnitude is much higher than the growth rate extrapolated to 14mbar from the
measurements carried out at $\delta p\sim 1$ mbar VLT1995 . Formally
introducing the kinetic growth coefficient $K$ as the coefficient of
proportionality between the average growth rate of the facet and the
difference in the chemical potentials of the crystal and the liquid, it is
possible to quantify the change in kinetics in this process. The rapid growth
results from the growth $K$ coefficient enhancement by two or three orders of
magnitude as compared with those measured for the crystals free from such
transition VLT1995 . The further studies show that this effect has many common
features with the phenomenon observed in Helsinki. The temperature dependences
$\delta p_{c}(T)$ are consistent. The nature of small 3He impurity effect on
the critical supersaturation is similar in kind. The review of studies on
abnormal growth (burst-like growth) is given in paper UFN .

The conditions are studied in detail for the transition of crystal facets to
the anomalously fast growth. The reverse process of restoring their normal
kinetics is studied, in fact, only in a single work on the free-growing helium
crystals VLT2004 . After $0.02-0.1$ s the growth rate of the facets returns to
the magnitudes typical for the normal crystals. According to RF-1 , one can
only estimate the time of returning to the growth-free state after the
pressure drop. This time is determined by the pressure measurement time
constant $\sim 0.5$ s. After this time, the facet becomes motionless again.

The change in the growth kinetics in the 0 – 0.02 s range, where the main
relaxation of the facet kinetics occurs, remains unknown. To solve this
problem, the crystal growth technique is modified VLT2021 . After nucleation
in a metastable liquid, the crystal continues to grow under constant flow of
helium into the container. In these experiments the liquid helium flow exceeds
significantly that in our previous experiments. It turns out that the pressure
records show the numerous jumps similar to those observed in RF-1 . In other
words, new technique makes it possible to observe numerous transitions of
crystal facets from the normal slow growth to anomalously fast growth with the
high time resolution. In this work we continue to study the return of the
crystal facet kinetics to the normal one after the anomalously fast growth
stage.

##  II Experimental Method

The crystals are grown in the opaque container as described earlier, see UFN .
The inner volume of the container is 330 mm3. The maximum initial pressure
supersaturation is observed during the crystal nucleation on the container
wall and achieves 23 mbar. The reduction of the starting pressure to 0.5–15
mbar is obtained by applying the high voltage to the tungsten needle located
at the center of the container. The distance from the needle tip to the inner
container walls exceeds 1.1 mm. Accordingly, the crystals grow freely without
touching the walls. One of the container walls operates as a membrane of
capacitive sensor with the resonance frequency 13.6 kHz. The pressure in the
container $p_{1}(t)$ is recorded with step 80 $\mu$s.

At the end of crystal growth when the facet kinetic growth coefficient becomes
sufficiently large, the pressure oscillations are observed on the wall of the
capacitive sensor. Their period, estimated from the container volume for the
starting supersaturations 2–23 mbar, lies in the range 120–190 $\mu$s VLT2000
. The registration of oscillations performed in increments 80 $\mu$s does not
allow us to determine accurately the shape of dependence $p_{1}(t)$ and to
calculate the magnitude of the average growth coefficient $K$ at the fast
growth stage according to the method in VLT2001 . Note that the observation of
pressure oscillations evidences for the high magnitude of growth coefficient
$K$.

The crystal nucleation and its subsequent growth occur in the course of the
continuous flow of a liquid into the container. This is a consequence of the
decrease in external volume $V_{2}$ due to the compression of the bellows by
the engine. After nucleating the crystal and its growth, the direction of the
motor rotation is reversed, the volume of the external system increases, the
flow of helium comes from the container, and the crystal size reduces until it
completely melts.

The crystal volume is calculated from the pressure-time dependences $p_{1}(t)$
in the container. The crystals grow and melt under conserving the total mass
of helium in the container $M_{1}$ and the mass of gas $M_{2}$ in the external
pressure system. So, we have

$\displaystyle M_{1}+M_{2}=const,$ $\displaystyle
M_{1}=\rho^{\prime}V_{c}+\rho(p_{1})(V_{1}-V_{c}),$ $\displaystyle
M_{2}=p_{2}V_{2}(t)\frac{\mu_{4}}{R_{gas}T},$
$\displaystyle\dot{M_{1}}=-\dot{M_{2}}=\frac{p_{2}-p_{1}}{Z}.$ (1)

Index 1 refers to the parameters in the container and index 2 does to the
outer part. Here $p$ is the pressure, $V_{c}$ is the volume of the crystal,
$V_{1}$ is the internal volume of the container, $V_{2}$ is the internal
volume of the external pressure system at room temperature, $Z$ is the
impedance of the inlet capillary, $\rho$ and $\rho^{\prime}$ are the densities
of liquid and solid helium, respectively, $T$ is the gas temperature in outer
pressure system, $\mu_{4}$ = 4.0026 gm/mole and $R_{gas}$ = 8.314 J/mole*K.
The compressibility of helium gas at 25 bar differs from that of an ideal gas
by 1.1%. The third equation for the mass of gas $M_{2}$ is written with such
accuracy. In these experiments, the ratio of the volume of the crystal grown
at an initial supersaturation $\delta p_{0}\sim$20mbar to the internal volume
of the container is $V_{c}/V_{1}=\rho/\Delta\rho k_{l}\delta p_{0}\sim
10^{-3}\ll 1$. Taking this fact into account and transforming the equations,
we get

$\displaystyle\frac{dp_{1}}{dt}+\frac{\Delta\rho}{\rho}\frac{1}{k_{L}V_{1}}\frac{dV_{c}}{dt}=\frac{p_{2}-p_{1}}{\tau_{1}},$
$\displaystyle\frac{dp_{2}}{dt}=-\frac{p_{2}-p_{1}}{\tau_{2}}+\left(1+\frac{\tau_{1}}{\tau_{2}}\right)\dot{p}_{10},$
$\displaystyle\tau_{1}=Z\rho
V_{1}k_{L},\tau_{2}=Z\frac{\mu_{4}V_{2}}{R_{gas}T}$ (2)

where $k_{L}$ is the liquid helium compressibility and
$\Delta\rho=\rho^{\prime}-\rho$. The rate of increasing the fluid pressure in
the container $\dot{p_{1}}=$ const $=\dot{p}_{10}>0$ is determined by the rate
of changing the volume of the external system $\dot{V_{2}}<0$, remaining
constant in the course of the crystal nucleation and growth and reversing the
sign as the crystal starts melting. This gives us the initial conditions for
the pressures at the moment of crystal nucleation

$\displaystyle p_{1}(0)=\delta p_{0},p_{2}(0)=\delta
p_{0}+\tau_{1}\dot{p}_{10}.$ (3)

Integrating the second equation in Eq. (2) yields us the pressure variation
$p_{2}(t)$ in the external system

$p_{2}(t)=\frac{1}{\tau_{2}}\int\limits_{0}^{t}p_{1}(x)\exp\left(\frac{x-t}{\tau_{2}}\right)dx+\\\
+(\tau_{1}+\tau_{2})\left[1-\exp\left(\frac{-t}{\tau_{2}}\right)\right]\dot{p}_{10}+p_{2}(0)\exp\left(\frac{-t}{\tau_{2}}\right).$
(4)

Integrating the first equation (2), we obtain the temporal dependence of the
crystal volume

$\displaystyle
V_{c}(t)=\frac{\rho}{\Delta\rho}k_{L}V_{1}\left[p_{1}(0)-p_{1}(t)+\frac{1}{\tau_{1}}\int\limits_{0}^{t}(p_{2}-p_{1})dt^{\prime}\right].$
(5)

Note that formulas (4) and (5) are derived under the assumption that the
impedance $Z$ is constant during crystal growth. However, this assumption has
not been experimentally confirmed. For this reason, we do not calculate the
dependence of the crystal volume on time in this work.

A ratio of the time constants is independent of the impedance of the capillary
and determined by a ratio of the helium masses in the container and in its
outer part

$\displaystyle\frac{\tau_{1}}{\tau_{2}}=\frac{M_{1}}{M_{2}}k_{L}p_{phase}=0.069.$
(6)

Here $p_{phase}\cong 25$ atm is the phase equilibrium pressure. The time
constant $\tau_{1}\cong$ is 8.2 ms and $\tau_{2}\cong$ is 120ms. This is much
less as compared with the time constants in VLT2004  where these constants are
37s and 30s, respectively.

In the above equations the pressure $p_{1}$ is measured from the phase
equilibrium pressure for the flat interface. In the experiment the similar
situation is not realized due to the curvature of atomic rough regions of the
crystal surface and thus the pressure is always higher. Its smallest
distinction from the phase equilibrium pressure takes place at the moment of
starting the crystal melting. The crystal volume is maximal and the crystal
facets disappear as can be seen in Fig.2. The correction to the pressure can
approximately be estimated with calculating the curvature radius of the
crystal volume. In these experiments the correction to the pressure is about
0.05–0.1 mbar.

As is noted in the Introduction, the dependence of the crystal facet growth
rate on the supersaturation demonstrates a threshold different for the c\- and
a-facets. Even the equivalent crystal facets of a free-growing crystal have
the various growth rates. For these reasons, it is impossible to determine
such dependences for each crystal facet by measuring $p_{1}(t)$ alone. We will
use the kinetic growth coefficient $K$ averaged over the whole surface. The
dependence of surface growth rate $V_{surf}$ on the supersaturation, i.e.,
difference in the chemical potentials of the phases $\Delta\mu$, is
approximated by the linear function. The pressure supersaturation of the
crystal-liquid phase diagram in this region is determined by the pressure
difference in the container from the phase equilibrium pressure with the
correction for the curvature $R$ of the atomic rough surface segments (edges).

$V_{surf}=K\left(\frac{\Delta\rho}{\rho\rho^{\prime}}p_{1}-\frac{\alpha}{\rho^{\prime}R}\right).$
(7)

Here $\rho$ and $\rho^{\prime}$ are the densities of liquid and solid helium,
$\Delta\rho=\rho^{\prime}-\rho>0$, and $\alpha$ is the surface energy assumed
to be isotropic.

Under the constant flow of liquid into the container a drastic change in the
crystal facet kinetics is seen from the jump-like singularity of the
derivative in the $p_{1}(t)$ records. From the experimental data it follows
that the transition from the slow to the fast kinetics occurs faster than 40
$\mu$s UFN . The reverse transition, as it follows from the experimental
results presented below, takes place faster than the measurement step 80
$\mu$s. For this time, the rate of increasing the helium mass in the container
and the crystal surface area $S_{cryst}$ vary negligibly. The incoming mass
increases that of the crystal. We have

$\dot{M_{1}}\cong const\cong\Delta\rho
S_{cryst}V_{surf}\cong\frac{\Delta\rho^{2}}{\rho\rho^{\prime}}S_{cryst}Kp_{1}\sim
Kp_{1}.$ (8)

We get that in the stationary case, $p_{1}\sim 1/K$. After the fast-to slow
transition of kinetics, the supersaturation changes from small magnitude to
large one with the time constant $\tau_{1}$, see the first equation of system
(2).

The kink in the derivative from almost zero magnitude to the finite one means
that all the possibilities of rapid growth, i.e., with the c\- and a-facets,
are restricted. Note that, in order to stop the crystal growth, it is
sufficient to lose the mobility of three a-facets at angle $120^{\circ}$. If
the intermediate crystal facets continue to grow rapidly, their sizes will
reduce rapidly and the crystal shape will become closer to the triangular
prism. To summarize, the jump in the derivative of the $p_{1}(t)$ dependence
from the small to large magnitudes evidences unambiguously for blocking the
crystal growth with the c-facets and at least with three a-facets located at
angle $120^{\circ}$.

##  III Experimental results

###  III.1 Types of behavior $p_{1}(t)$ observed

In experiments on crystal growth in a metastable liquidUFN , two types of the
pressure curve behavior are observed after the crystal nucleation. In the both
cases the pressure drops to the phase equilibrium one but this occurs at very
different times. The crystals demonstrating the normal growth grow slowly and
for a long time. The facet kinetics remains unchanged in time and does not
differ from the kinetics of the crystal facets grown at low supersaturations.
The drastic drop in the pressure for the short time means the transition of
the crystal facets to the rapidly growing state like burst-like growth. The
start of the burst-like growth regime occurs with some delay with respect to
the moment of crystal nucleation. The higher the initial supersaturation
$\delta p_{0}=p_{1}(0)$ of the metastable liquid, the shorter the delay time.
Provided the facet kinetics is high, the excitation of the low-frequency
damped pressure oscillations is observed in the container. As the pressure
decreases to the phase equilibrium one for 20–100 ms, the facet kinetics
returns to the magnitudes typical for the normal slow growth of crystals.

Figure 1: The crystals selected by the growth type dependent on
supersaturation $\delta p_{0}$ at which the crystal nucleates. The solid curve
separates the regions of normal and abnormal growth described earlier in
review UFN . Experimental results are shown by symbols. The grey squares refer
to the normal growth crystals, see Section 3.1. The grey circles are the
crystals grown normally with the pressure splashes in the growth process. The
triangles are the crystals for which the pressure $p_{1}$ jumps are observed
after the normal growth stage. The empty circles indicate the crystals showing
the rapid growth and oscillation after drastic pressure drop. The diamonds
show the result of the articleRF-1 .

Figure 1 shows the boundary between the crystals that have the features in the
$p_{1}(t)$ record characteristic for the burst-like growth regime and the
crystals that have no such features in a set of previous experimental data UFN
. The same figure shows the symbols corresponding to the temperature and
supersaturation $\delta p_{0}$ at the moment of crystal nucleation according
to the results of this work. It can be seen that the separation of the
$T-\delta p_{0}$ diagram into the normal and abnormal growth regions is
conserved. This Section shows the measurement results for the crystals grown
on a tip and having no contact with the container wall. For this reason, we
restrict ourselves to recording $p_{1}(t)$ for 0.3–0.5 s after the crystal
nucleation.

Note that our classification of crystal growth process by time into two
stages, as is discussed in Sec. 3.1, is most suitable for the burst-like
growth mode. After the crystal facets acquire the fast growth kinetics, the
crystal growth occurs at almost constant helium mass in the container. The
further crystal growth and the supersaturation magnitude are now governed by
the crystal facet kinetics and the liquid inflow from the external system
under increasing the helium mass in the container.

###  III.2 Growth of faceted crystals that do not exhibit the burst-like
growth mode. T=0.83 K

An example of the pressure change in the container after crystal nucleation is
shown in Fig. 2. Several characteristic segments are visible on the record.
After nucleating the crystal, the pressure drops rapidly over $\sim$3 ms,
section A-B. The crystal growth occurs at almost constant mass of helium in
the container. This simplifies the system of equations (2). The average
kinetic growth coefficient is ${K\approx 2.3\cdot 10^{-3}}$ s/cm. This value
is in agreement with the data obtained previously for the free-growing
crystals, see Fig. 4 UFN .

It is seen that the pressure does not drop to the phase equilibrium pressure.
The reason lies in its small magnitude. It can be seen from relation (7) that
the crystal growth continues as long as the pressure in the liquid exceeds the
contribution from the curvature of the edges. In the previous experiments in
the containers with internal volume 10–30 times larger this effect is
negligible. The growth of a spherical crystal at various initial
supersaturation taking into account the curvature of the surface is considered
in Appendix I. Figure 10 shows the experimental data on the pressure
$p_{stop}$, at which the growth of the crystal stops at point B. The
experimental points lie above the model calculation. This means that the shape
of the crystals differs from the spherical one.

The crystal growth regime changes at the B-C-C’ site. The quasi-stationary
growth continues due to the liquid flow into the container. The
supersaturation is determined by the balance between the fluid flow and the
crystallization rate. As the crystal grows, the surface area of the crystal
increases, the pressure in the external system decreases with the time
constant $\tau_{2}$, and, as a result, the supersaturation in the container
decreases.

Figure 2: Recording of pressure variations during the crystal growth at 0.83
K. The images, obtained in an optical container, illustrate the change in the
crystal shape from growth to melting. The right-hand graph in the inset shows
the pressure oscillations that occur after the complete melting of the
crystal.

In the interval C’-D, the motor rotation is reversed (Fig.2, middle insert).
The outer volume increases, the helium flow is directed out from the
container, and the crystal melts. This inset shows the change in the crystal
shape from growth to melting. The illustration is obtained in our previous
experiments in the optical container, see UFN . During melting, the crystal
shape is determined by the most mobile atomically-rough segments of the
surface. The edges are significantly reduced and the crystal takes a drop-like
shape. Since the curvature of the crystal surface is now less than the
curvature of the rounded edges, the pressure drops by the amount corresponding
to their difference.

During the melting process, the supersaturation in the container is almost
constant, section D-E. Near complete melting, the pressure starts to rise.
This is due to increasing the curvature of the crystal surface with decreasing
its size, as follows from Eq. (7). After disappearing the crystal, low-
frequency pressure oscillations at frequency18.9 Hz and decrement 0.66 are
observed in the container. The solutions of system (2) after crystal melting
have a relaxation character. Consequently, in order to obtain the pressure
oscillations, the inertial factor must be introduced into the equations. The
natural oscillation frequencies of the liquid in the container and the gas in
the external system have frequencies much higher than the frequency observed.
The only factor unaccounted for in Eqs. (2) is the helium mass in the filling
capillary connecting the container and the external pressure system. The
container-capillary-external volume system is similar to the Helmholtz
resonator. The difference is that when the helium flows in the capillary, the
restoring force arises from the both sides of the capillary. This model is
discussed in Appendix II. The estimate is in the satisfactory agreement with
the experimental parameters.

###  III.3 Growth of crystals demonstrating the burst-like growth mode

####  III.3.1 T=0.73K

Figure 3 shows the pressure variation in the container after the crystal
nucleation. In the lower record (a) when the initial supersaturation is
approximately a half of the boundary magnitude $\delta p_{c}$, the $p_{1}(t)$
dependence is similar to the pressure drop during the normal crystal growth,
see Fig.2. The kinetic growth rate, estimated from the initial pressure drop,
is $K=2.5\cdot 10^{-3}$ s/cm. No specific features are observed.

Figure 3: The pressure records for the crystal growth at $T=0.73$ K. The
symbol N and the arrows indicate the stage of normal crystal growth.

Plot (b) shows the crystal growth near the boundary separating the normal and
fast growth regions. At first, the crystal grows at the kinetic growth rate
$K=1.8\cdot 10^{-3}$ s/cm typical for the normal crystal growth at this
temperature. This region is highlighted by the arrows and denoted with symbol
N. Later, the pressure starts decreasing, as in the previous plot, and over
time $\sim$50 ms we observe a drastic pressure drop, revival to the pressure
before its jump, and the drop once more after which the crystal continues
growing without any features like in the previous plot. The drastic pressure
drop indicates a rapid acceleration in the crystal surface growth. Then, there
is a return to the slow growth kinetics and, after a while, another drastic
pressure drop. Eventually, the transition from the normal to burst-like growth
is observed twice in the growing crystal at supersaturation $\sim$2 mbar being
a half of the boundary supersaturation $\delta p_{c}$ at this temperature.

We give above the records of the pressure for the crystals nucleated in the
anomalous region. After nucleation the crystal grows at the normal rate
accompanied by the gradual pressure drop. In the inserts these regions are
indicated by the arrows. Then the crystal goes over to the burst-like growth
state and grows quicker than for 0.6 ms with the pressure drop to the phase
equilibrium pressure. The inserts show the rapidly decaying pressure
oscillations evidencing for the high magnitude of kinetic growth coefficient.

The plot (c) shows that the pressure increases for the time interval $\sim$35
ms. Later, we have a pressure drop to the phase equilibrium pressure,
succeeded by the next increase in the pressure to about 80% of initial
supersaturation $\delta p_{0}$. Then, there is a monotonous pressure drop with
the concurrent jumps of characteristic shape. The drastic drop in pressure is
followed by its rise. When the supersaturation drops to $\sim$2 mbar, the
concurrent pressure jumps disappear.

Plot (d) shows the pressure variation during the growth of a crystal nucleated
at the starting pressure $\delta p_{0}\sim 12$ mbar. This is similar in kind
to the previous plot. Note that the pressure jump appears at $\sim$ 60ms and
similar to that accompanying the crystal nucleation and exciting the pressure
oscillations as well. After this, for $\sim$12 ms, the pressure
supersaturation is small, i.e., the kinetic growth coefficient is large and,
correspondingly, the crystal is in an anomalous state during this time
interval.

####  III.3.2 T=0.66K

After the crystal nucleation below the critical supersaturation curve ${\delta
p_{0}<\delta p_{c}(T)}$, a slow growth stage is observed with kinetic
coefficient ${K=6.9\cdot 10^{-4}}$ s/cm. This is followed by a monotonous
pressure drop, in which we can see one pressure jump due to the transition of
the facets to an anomalous rapid growth state. The facet kinetics then returns
to the normal slow kinetics. Plot (b) shows the crystal growth near the
$\delta p_{c}(T)$ curve, which proceeds with kinetic growth rate $K=8.6\cdot
10^{-4}$ s/cm typical for normal crystal growth at this temperature. The
further decrease in pressure occurs with several jumps, indicating the
transition of the normal growth regime to the burst-like growth regime and the
subsequent restoration of slow kinetics. In the region of anomalous growth
(c-d), after the initial jump from the value $\delta p_{0}$ to the phase
equilibrium pressure, numerous pressure jumps are visible. These jumps appear
at the supersaturations below $\sim$3 mbar and stop when the pressure drops
below $\sim$1mbar.

Figure 4: The pressure records during the crystal growth at T = 0.66 K.

####  III.3.3 T=0.57 K

Plot (a) shows the pressure variation accompanying the growth of a crystal
nucleated near the boundary between the normal and anomalous regions. The
crystal growth occurs with kinetic coefficient $K=2.4\cdot 10^{-3}$ s/cm. In
$\sim$280 ms after the crystal nucleation at supersaturation of the order of
1mbar the pressure jumps are observed continuing for $\sim$200 ms. For the
crystals nucleated in a metastable liquid within the range $\delta
p_{0}=1.7-2.4$ mbar, this is a typical picture. The difference is that the
pressure jumps begin to appear on the $p_{1}(t)$ record with an arbitrary
delay relative to the moment of crystal nucleation.

Figure 5: The pressure records during the crystal growth at T = 0.57K.

Graphs (b-d) show the pressure behavior in the container after nucleation and
growth of crystals in the burst-like growth mode. The fast growth after the
crystal nucleation results in the damped pressure oscillations. After decay of
the oscillations for 5–20 ms, the pressure is close to the phase equilibrium
one. Then the supersaturation increases. This leads to numerous pressure
surges, reflecting abrupt transitions of the crystal facet kinetics from the
burst-like growth regime to the normal slow one. Note that during this process
the supersaturation is small as compared with the initial supersaturation
$\delta p_{0}$ and does not exceed $\sim$2 mbar.

####  III.3.4 T=0.50K

The records in Fig.6 clearly show the different modes for the transition of
the crystal growth kinetics from normal to anomalous and vice versa. The fast
crystal growth after nucleation leads to a sharp drop in pressure. The
observed pressure oscillations indicate a high magnitude of kinetic growth
coefficient $K$. Taking into account the remarks of Sec. 2, we can determine
only the lower boundary for the coefficient according to the methodology of
paper VLT2001 , $K>0.1$ s/cm.

Figure 6: The pressure records during the crystal growth at $T=0.50K$. The
dashed line in graph (c) represents the calculated pressure change $p_{2}(t)$
in the external system.

At the end of the oscillations, the pressure increases. In plot (a), during
the growth of a crystal nucleated at supersaturation 2.6mbar, quasiperiodic
pressure jumps with amplitudes $\sim$2 mbar are observed. They decrease in
magnitude over time until they stop at $\sim$170 ms after the crystal
nucleation. The crystal growth pressure record, shown in graph (b),
demonstrates the similar pressure behavior. The jumps with amplitude of
$1.13\pm 0.44$ mbar and the interval between them $2.35\pm 0.56$ ms are
observed for $\sim$80 ms. The supersaturation after the sharp pressure drop is
close to zero, $p_{min}=0.25\pm 0.23$ mbar. Then the nature of the process
changes. Instead of frequent and numerous jumps with the amplitude not
exceeding $\sim$1.5 mbar, 5 jumps are observed with amplitude up to 3.5 mbar.
The same picture is observed in the pressure record during the crystal growth
nucleated at $\delta p_{0}=8.2$ mbar in the graph (c). The frequent jumps of
small amplitude stop after 80 ms after the crystal nucleation. Then ten jumps
of large amplitude are visible. During the crystal growth the pressure in the
external system drops in spite of the continuing decrease in its volume by the
compression of the bellows. The pressure change $p_{2}(t)$, calculated
according to equations (2-5), is shown by the dashed curve.

Record scan (c) is shown in Fig. 7. The time counting on all graphs starts
from the moment of crystal nucleation. The bottom graph shows the record of
pressure oscillations at the end of the burst-like growth. Let us estimate the
averaged kinetic growth coefficient $K$ by the maximum oscillation amplitude.
From the calculation of work VLT2001  it follows that this amplitude changes
noticeably from the maximum at $K>1$ s/cm to approximately zero at $K\sim
0.03$ s/cm. At values of $K<0.03$ s/cm, oscillations are not excited and the
pressure drops monotonically. The initial oscillation amplitude is $\sim$3
mbar, which, according to the method VLT2001 , gives an estimate as $K\approx
0.12$ s/cm. This value is consistent with the measurements on for the large
crystals, see review UFN  Fig.4.

Figure 7: The scan recording Fig. 6c from bottom to top: pressure oscillations
after crystal nucleation in a metastable liquid and abnormal growth;
quasiperiodic pressure surges; chaotic jumps; quasi-periodic jumps of large
amplitude.

The next plot shows the quasi-periodic pressure jumps in the range 10-30 ms.
The average interval between them is $1.08\pm 0.24$ ms and the amplitude of
the jumps is $0.67\pm 0.16$ mbar. In this mode the overpressure after the jump
exceeds noticeably the phase equilibrium pressure $p_{1min}=0.66\pm 0.15$
mbar. The following plot shows that the nature of the pressure jumps changes
later. Within 30-80 ms the jumps occur irregularly, amplitude of several ones
reaches 1.5 mbar, and the pressure after the jumps is close to the phase
equilibrium pressure.

The next mode is observed within interval 80–170 ms. The crystal is in the
state with the normal growth kinetics for a long time. During this time the
pressure in the container increases to magnitudes 2–4 mbar. Then, the jump in
the crystal facet growth kinetics from normal to anomalous is observed and
accompanied by the rapid crystal growth and exciting the pressure oscillations
in the container. The plot shows that the pressure oscillations take place
near the phase equilibrium pressure $p_{phase}$. This means that for this
time, the kinetic growth coefficient of the crystal surface remains large and
typical for the burst-like growth mode. The average magnitudes of each
oscillation period differ from $p_{phase}$ within $0.04\pm 0.06$ mbar. This
means that according to relation (8), the kinetic growth coefficient of the
crystal surface remains large and indicative in favor of the burst-like growth
mode.

The burst-like growth mode continues for a few milliseconds. Then, at the end
of the oscillations, the pressure in the container starts to rise rapidly. As
is noted in Sec. 2, a kink in the time derivative $p_{1}(t)$ indicates that
the c facets and at least three a facets at angle $120^{\circ}$ become
inactive. From the width of the derivative jump it is possible to estimate the
minimum transition time from the anomalous to normal kinetics for these facets
as about 100$\mu$s.

The change in the time duration $t_{state}$ of the normal and abnormal state
is based on plots 6b, 6c and 7 and is shown in Fig. 8.

Figure 8: The duration $t_{state}$ for the normal and abnormal state of the
crystal facets during their growth at T = 0.50 K. The left plot shows the data
from the record in Fig. 6b for the crystal nucleated at supersaturation 4.5
mbar, Fig. 6b. The right-hand plot shows the results for the crystal nucleated
at the initial supersaturation 8.2 mbar, Figs. 6c and 7. The filled symbols
represent the duration for the normal kinetics of the crystal facets. The
empty symbols represent the lifetime of the abnormally fast kinetics of the
crystal facets.

It can be seen that the duration of the normal state kinetics changes
negligibly in time and lies within the range 2–4 ms. The lifetime of the
abnormal state is longer in average as compared with that of the normal state
and increases in time.

###  III.4 Anomalies in the kinetics of normal crystal facets.

The results of subsections 3.3.1–3.3.4 are obtained during a single growth of
crystals nucleated in a metastable liquid. To answer the question whether the
formation of an anomalous phase in normal crystals is possible, the following
experiments are performed. The crystals are nucleated on a tip at the voltage
providing the crystal nucleation without noticeable potential barrier. Then
the crystal growth stops. After that the crystal melts to the small size. The
measurement starts after this procedure. The helium flow into the container is
the same as in the previous experiments.

Figure 9: The pressure recording during the normal crystal growth at T = 0.66
K. The motor that produces the pressure in the external system starts at the
moment $t=0$. The downward arrow in the lower plot labels the time of the
motor reversal and the beginning of the crystal melting. The horizontal arrows
denote the time intervals expanded above on the scheme. The upward arrows show
the pressure jumps due to the transition of crystal facet growth to the burst-
like mode.

An example of the $p_{1}(t)$ dependence is shown in Fig.9. It can be seen that
no singularities are observed before the pressure supersaturations $\sim 0.2$
mbar. In the range 0.2–0.6 mbar the pressure jumps appear randomly over the
record. This shows that when the necessary supersaturation is produced, the
facets of normally grown crystal go over to the burst-like growth state RF-1 ;
RF-2 .

##  IV Discussion

###  IV.1 The boundary between the regions of anomalous and normal crystal
growth. The temperature effect.

In Fig. 1 we show the boundary separating the regions of normal and abnormal
growth in the $\delta p_{0}-T$ coordinates. This boundary is conditional. The
point is that the drastic pressure drop, meaning the transition of the crystal
facets to the burst-like growth, is observed some time after the crystal
nucleation. Prior to this transition, the crystal grows in the normal manner
with a gradual pressure drop in the container. This fact is well described in
the review UFN , Fig.7–10. Also, it can be seen from the pressure records in
this series of experiments (see Fig. 3 and the insets in the c and d plots).
Because of this time delay, the magnitude of the pressure jump is smaller than
the supersaturation at which the crystal nucleates. If we mark exactly the
jump amplitudes in the $\delta p_{0}(T)$ plot, the boundary moves in the
downward direction.

As it follows from the pressure records performed in the range 0.5–0.73 K, the
pressure jumps, indicating the onset of the burst-like growth regime, have the
different amplitudes varying from a few fractions of millibar to the
magnitudes close to the initial pressure supersaturation. Nevertheless, there
is a general tendency for the temperature effect on the formation of abnormal
growth. Such transitions at 0.83 K are not observed either at the growth stage
after nucleation in a metastable liquid, or during the subsequent growth of
the crystal for longer than 1 s (Fig. 2). As it concerns temperature 0.73 K,
the crystal growth occurs free from abnormal state of the crystal facets for
0.4 s if the supersaturation of crystal nucleation does not exceed $\sim 2$
mbar (Fig. 3a). Increasing the initial supersaturation to $\sim 4$ mbar
results in observing two pressure jumps at the crystal growth stage. The
pressure jump at 0.66 K is observed as well as on the pressure records during
the growth of the crystal nucleated at supersaturation $\sim 2$ mbar (Fig.
4a). For the lower temperatures, we detect the numerous pressure jumps on the
$p_{1}(t)$ records of crystal growth with the same initial supersaturation,
see Figs. 5a and 6a.

To summarize, the higher temperatures prevent from the formation of an
anomalous state at the helium crystal facets. The boundary which approximately
divides the $\delta p_{0}-T$ plane into the normal and abnormal growth regions
reflects this fact in kind. The question how sharply the temperature
suppresses the formation of the burst-like growth regime in the range 0.73 -
0.83 K remains open.

###  IV.2 The abrupt nature of the transition from the fast kinetics of facet
growth to the normal one.

Section 2 presents a criterion for the moment of the fast-to-slow transition
in the crystal facet kinetics. In the lack of visual control of the crystal
shape, one can only assert that the fast crystal growth is blocked by the c
facets and at least by three a facets at angle 120∘ at the moment in the
derivative $dp_{1}/dt$ discontinuity.

The fastest changes in the derivative $dp_{1}/dt$ are observed in the quasi
periodic and stochastic sequences of pressure jumps with the small amplitude
and duration, see plots 4–7. The negative value of derivative $dp_{1}/dt$ in
the burst-like growth state is replaced with the positive one. The typical
duration of such changes is one or two measurement points, that is, no more
than 80-160 $\mu$s.

The $p_{1}(t)$ dependence in a sequence of long alternating anomalous and
normal states of the crystal facets differs from those considered above (Fig.
7, upper plot). The transition of the crystal facets to the burst-like growth
mode results in the rapid growth of the crystal. This, in turn, induces the
pressure oscillations inside the container. This problem is discussed in
Ref.VLT2001  in detail. According to the result of calculations, the
occurrence of oscillations is determined by the value $K$, which should exceed
the magnitude 2-5 s/m. The surface growth rate is also determined by
supersaturation. As follows from the measurementsUFN , this velocity is much
less than the speed of sound in a liquid and does not exceed 5m/s. The average
pressure magnitudes during this time are close to the phase equilibrium
pressure. The end of oscillations is accompanied by a drastic increase in
pressure, indicating the transition of the crystal facets to the normal state
with the slow kinetics. The estimate of the transition time interval yields
the magnitude shorter than 200 $\mu$s.

Note that this magnitude is close to the time of the normal-to-anomalous
growth transition studied earlier. The video shows that the transition time to
the fast growth state for all crystal facets does not exceed 40 $\mu$s, Fig.12
UFN . The change in the facet kinetics from anomalous to normal and vice versa
occurs for the time no longer than tens of microsecond. This is significantly
smaller as compared with the time of changing the external factors such as
temperature and fluid flow into the container. From this point of view the
phase transition in the crystal facet kinetics has a discontinuous character.

###  IV.3 Alternative explanation for the ultraslow growth of the defect-free
c facet.

In experiments RF-1 ; RF-2  it is found with the aid of optical method that
before the start of the burst-like growth mode, the defect-free c-facet grows
at a very low rate $(1-5)\cdot 10^{-8}$ cm/s. Taking the data scatter into
account, we see that the mobility of the crystal facet is constant in the
range 2–20mK and then starts to decrease as the temperature increases. The
theoretical explanation of this effect is proposed in paper AM  where the
vacancy mechanism of crystal growth is considered.

The discovery of the crystal growth regime accompanied by the pressure jumps
of small amplitude and duration makes it possible to propose an alternative
explanation for the ultraslow growth of the defect-free c facet. The short-
term transition of the crystal c facet to the burst-like growth state
corresponding to such facet growth rate produces a small pressure jump. The
pressure meter with a long time constant averages the pressure jumps out. This
is not visible on the $p(t)$ record. The optical technique based on digitizing
the CCD camera image is inertial as well and does not register any single tiny
shift of the crystal surface. Then, the ultraslow c facet growth observed
results from averaging the sequence of small facet growths during its short-
term transition to the burst-like growth state.

Within the framework of this model the ultraslow growth is determined by the
same physical mechanism of unknown nature which the burst-like growth mode
demonstrates at high pressure supersaturations. From the plots of the
cumulative pressure distribution of burst-like growth instability RF-2
(Fig.12) and their approximation, it follows that the probability of small
amplitude pressure jumps is exponentially small. The transition of the crystal
facet to abnormal state is rare. The growth in such single transition is
small.

The model is qualitatively confirmed by comparing the temperature variation
for the cumulative distribution of the pressure burst-like growth instability
RF-2  (Fig. 13) with the temperature behavior of the ultraslow growth rate
(Fig. 11). As the temperature increases, the median of the distribution shifts
to higher supersaturations. Then, according to the logic of the model
proposed, the probability reduces for appearing the short-term low-amplitude
transitions at this distribution tail. Hence, it follows that the ultraslow
growth rate should decrease as the temperature grows. This conclusion is
consistent with the experimental data demonstrating the reduction in the
ultraslow growth rate as the temperature increases.

##  V Conclusion.

The development of the experimental technique has significantly extended the
prospects for studying the mysterious phenomenon, namely, drastic change in
the crystal facet growth rate by several orders of the magnitude. All the
previous studies are carried out for a short time from the moment of crystal
nucleation in the metastable liquid to the completion of crystal growth, that
is, for $\sim$ 0.5 –1 ms. At present, in one experiment the numerous
transitions of the crystal facet growth kinetics from normal to anomalous and
vice versa are observed, providing us an opportunity to extend the research
methods. This result is partially facilitated by the fact that the mobility of
crystal facets in the normal state has a minimum in the range 0.5 – 0.8 K.
Therefore, the change in the crystal facet kinetics becomes readily
distinguishable.

An important qualitative result is setting the fact that the reverse
transition from the fast to slow facet kinetics occurs simultaneously or with
an insignificant delay. So far, it has been known that the kinetics becomes
normal in $\sim$0.1 s after the growth of the crystal from the metastable
liquid in the burst-like growth mode. Currently, the conditions imposed on the
physical mechanism for the transition in the growth kinetics are determined.
One should look for a phenomenon that can induce the fast facet growth
kinetics simultaneously or with a slight delay and stop it as well.

Our measurements demonstrate a variety of crystal growth regimes with
switching the kinetics. The quasi periodic pressure jumps are observed in the
kinetics without the drops to the phase equilibrium pressure, stochastically
occurring transitions, formation of a sequence of the long time periods of the
normal growth states resulting in the long-term states with the high growth
kinetics. So far, it has not been possible unambiguously to link the onset of
certain regimes with the magnitude of supersaturation, previous history of
crystal growth, growth rate, or crystal size.

The discovery of new quality, namely, drastic return of the crystal facet
growth kinetics to the normal one, opens new field of research in the
mysterious burst-like crystal growth phenomenon.

###### Acknowledgements.

The author is grateful to V. V. Dmitriev for the possibility of performing
these experiments at Kapitza Institute for Physical Problems RAS. The author
is also grateful to V. V. Zavyalov for supporting this work, S.N.Burmistrov
for helpful comments and V. S. Kruglov for interest to the work.

Appendix I

Let us consider the spherical crystal growth in the closed volume at pressure
supersaturation $\delta p_{0}$. In this case the rate $\dot{R}=V_{surf}$ from
equation (7) and the parameter $\alpha$ should be replaced by $2\alpha$. The
first equation of system (2) and equation (7) describe the crystal growth
taking the curvature of crystal surface into account. The growth stops when an
additional pressure $p_{stop}$ due to the surface curvature equals the liquid
overpressure. This moment is shown by the arrow B on the experimental record
in Fig. 2. In Fig. 10 the dependences are shown for a ratio $p_{stop}/p_{0}$
as a function of the initial overpressure obtained from the calculation and
experimental data.

Figure 10: The ratio of the pressure of crystal growth deceleration to the
initial supersaturation at 0.83 K.

All experimental points lie above the curve calculated. This means that the
edge curvature is larger than that obtained with calculating for the spherical
crystal. Using pressure $p_{stop}$, we calculate the curvature of the crystal
edges and reconstruct the crystal shape. It is assumed that the crystal shape
is correct and the ratio of the growth rates of the basic and lateral facets
equals 1:2.

Appendix II

Let us consider container with a liquid and connected to the external system
filled with gas at room temperature. We have also the capillary of constant
cross-section and the constant mass of helium in it. When mass $\Delta m$
flows through the capillary into the container, the pressure in the container
increases by $\delta p_{1}$ while in the external system it decreases by
$\delta p_{2}$. The ratio of the restoring forces in the given geometry is
$\frac{\delta p_{1}}{\delta p_{2}}\cong-16$. Thus, the main restoring force is
generated by compressing the liquid in the container. Neglecting the response
from the external system, we obtain the equation for velocity $v$ of
oscillations in the capillary

$\rho SL_{eff}\frac{d^{2}v}{dt^{2}}+\rho
S^{2}Z\frac{dv}{dt}+\frac{S^{2}}{k_{L}V_{1}}v=0.$ (9)

Here $S$ is the cross-sectional area of the capillary and $L_{eff}$ is its
effective length. The parameters $Z$ and $L_{eff}$ are determined by the
frequency and attenuation of oscillations. In the containers with the large
internal volume, as can be seen from the equation, the role of energy
dissipation enhances and the solutions become of relaxation type. For this
reason, such oscillations are not observed in the experiments with the optical
containers of volume 4 - 10 cm3.

## References

  * (1) A. F. Andreev, A. Ya. Parshin, Sov. Phys. JETP 48, 763 (1978).
  * (2) H. Alles, S. Balibar and A.Ya. Parshin, Rev. Mod. Phys. 77, 317 (2005).
  * (3) V. L. Tsymbalenko, Ukr. Low Temp. Phys. 21, 120 (1995)
  * (4) J. P. Ruutu, P. J. Hakonen, A. V. Babkin, A. Ya. Parshin, J. S. Penttila, J. P. Saramaki and G. Tvalashvili, Phys. Rev. Lett. 76, 4187 (1996).
  * (5) V. L. Tsymbalenko, Phys. Lett. A 211, 177 (1996).
  * (6) V. L. Tsymbalenko, Physics-Uspekhi 58, 1059 (2015).
  * (7) V. L. Tsymbalenko, JETP 92, 1214 (2004).
  * (8) V. L. Tsymbalenko, JETP Letters 113, 30 (2021).
  * (9) V. L. Tsymbalenko, Phys. Lett. A 274, 223 (2000)
  * (10) V. L. Tsymbalenko, JETP 92, 1024 (2001)
  * (11) A. V. Babkin, P. J. Hakonen, A. Ya. Parshin, J. P. Ruutu, G. Tvalashvili, JLTP 112, 117 (1998)
  * (12) A. F. Andreev, L. A. Melnikovsky, JETP 93, 1261 (2001).