PHYSICAL REVIEW B
VOLUME 57, NUMBER 1
1 JANUARY 1998-I
Vortex pinning by different angular distributions of columnar defects in Bi 2 Sr2 CaCu2 O8
S. Hébert, V. Hardy, G. Villard, M. Hervieu, Ch. Simon, and J. Provost
Laboratoire CRISMAT, UMR 6508 associé au CNRS, ISMRA et Université de Caen 6,
Boulevard du Maréchal Juin 14050 CAEN Cedex, France
~Received 12 June 1997!
Different configurations of columnar defects tilted with respect to c have been tailored in Bi 2 Sr2 CaCu2 O8
single crystals by controlling the orientation of a heavy-ion beam: One single direction of tracks yielding
parallel defects (n51), two coplanar directions of tracks (n52), and three directions of tracks symmetrically
distributed around the c axis (n53). The pinning efficiency has been systematically investigated by magnetic
measurements with the field applied along c. We found evidence for a low-T regime where the pinning
efficiency is nearly independent of n, and a high-T regime where it essentially increases with n. These features
are discussed in the frame of two different but not incompatible approaches: ~i! a static approach based on
accommodation effects, and ~ii! a dynamic approach based on elementary flux-creep processes. Finally, the
further comparison of these results with angular measurements performed for n51 strongly suggests that the
symmetry of the track distribution relative to the orientation of the applied field might play a major role in the
pinning efficiency of splayed configurations. @S0163-1829~98!01301-0#
I. INTRODUCTION
Columnar defects ~CD’s! can be introduced in high-T c
superconducting oxides by heavy-ion irradiation.1–4 These
defects were shown many times to yield spectacular improvements of vortex pinning, especially in single crystals
when the magnetic field H is applied along the c axis.5–9
While most of these studies dealt with CD’s parallel to c, it
was proposed by Hwa et al.10 and Le Doussal and Nelson11
that a dispersion in the orientation of the CD’s could further
increase the pinning efficiency. Beneficial effects are expected owing to the occurrence of an entangled vortex
ground state, and because of the related impediments to fluxcreep processes. Several studies of such splayed configurations were performed in Y-Ba-Cu-O.12–18 In all cases, a beneficial effect was observed for moderate angles of
disorientation ~around 10°) compared to the reference configuration with all the CD’s parallel to c. An optimal value
close to 5° was even reported.17
The compound Bi 2 Sr2 CaCu2 O8 ~Bi-2212! has received
less attention up to now. In tapes, high-energy protons were
used19 to create strongly splayed configurations of linelike
defects ~mean length '10 m m!, which yielded large enhancements of the pinning properties. More recently, similar
results were also reported in ~Bi,Pb!-2223/Ag tapes.20 Nevertheless, the angular distribution of these tracks bored by
fragments of Bi nuclei is ill-defined and, above all, a suitable
reference with tracks all parallel is missing. A more controlled situation can be reached by adjusting directly the orientation of a heavy-ion beam. By these means, Schuster and
co-workers14,15 studied crossed configurations consisting of
two families of CD’s inclined at 645° relative to c. On the
basis of magneto-optical measurements, they observed that
the critical current density for this splayed configuration is
around 50% larger than for the reference configuration
~tracks parallel to c with the same fluence! in the whole
range of temperature. It was concluded that flux motion in
Bi-2212 proceeds by depinning of single pancake vortices,
0163-1829/98/57~1!/649~7!/$15.00
57
which therefore requires nearly the same activation energy
for parallel and crossed defects.15
In a previous study involving various compounds,16 we
have investigated a splayed configuration with three directions inclined by u i from c and symmetrically distributed
around this axis. In Bi-2212, we observed for high-T and
low-H a substantial improvement for high-u i (575°) with
respect to the parallel configuration. With such high angles,
two important pinning parameters are changed when one
compares to the parallel configuration at the same ion fluence: ~i! the size of the column’s projection in the ab planes;
~ii! the planar density of the columns in the ab planes. Consequently, the occurrence of a true splayed effect as described in the theory10,11 cannot be safely distinguished from
possible effects just related to the track inclination. One way
to address this issue consists in studying the influence of the
number of track directions for a given u i . A first study by
magneto-optics has compared inclined parallel CD’s and
crossed CD’s in Bi-2212 for u i 545°.14
In the present work, three types of configurations have
been investigated corresponding to one, two, or three directions, which are labeled n51, n52, and n53, respectively.
Pinning properties versus temperature and magnetic field
were studied by superconducting quantum interference device ~SQUID! magnetometry and for different u i ’s up to 75°.
For n51, one deals with a set of inclined parallel CD’s so
there are no defect crossings. For n52 and n53, crossed
CD’s are installed in geometries corresponding to 1 and 3
irradiation planes, respectively.
The plan of the paper is as follows: In Sec. II some experimental details are given about the irradiation and measurement procedures. The main experimental results about
the influence of n are presented in Sec. III and discussed in
Sec. IV. It will be shown that the basic data can be accounted
for by dynamical arguments as described in the splayed
model, but can also be interpreted within the frame of a
purely static approach based on accommodation effects.
Some experimental points involving angular measurements
649
© 1998 The American Physical Society
650
S. HÉBERT et al.
57
FIG. 1. Schematic representations of the investigated configurations, corresponding to 1, 2 or 3 directions of irradiation. All the
tracks are inclined by u i from the c axis. In all cases, the total
fluence is the same and the measurements were performed with H
applied along c.
are then presented in order to compare the reliability of each
interpretation. A conclusion is drawn in Sec. V.
II. EXPERIMENTAL DETAILS
The Bi 2 Sr2 CaCu2 O8 ~Bi-2212! single crystals were
grown by a self-flux method.21 The T c values of the samples
used in this study were close to 83 K and decreased by 2 K
after irradiation. Their thickness is around 20 m m.
The single crystals were irradiated at room temperature by
6-GeV Pb ions at the national heavy-ion accelerator GANIL
~Caen, France!. These ions lead to the creation of continuous
amorphous tracks in the whole thickness of such thin crystals. The mean amorphous diameter of these columnar defects is around 90 Å. In the present experiment, the samples
were mounted on inclined platforms in such a way that their
c axis was tilted by u i from the ion-beam direction. For n52
and n53, the crystals were rotated once or twice, respectively, in order to yield the track configurations sketched in
Fig. 1.
Such sets of track distributions were achieved for three
values of u i : 45°, 60°, and 75° ~note that samples with n52
are missing for u i 575°). In all cases, the total fluence ~measured perpendicular to the ion beam! was 531010 ions cm22.
It means that ~i! the total volume of CD’s, V CD, is the same
whatever n and u i ; ~ii! the planar density of CD’s, n CD , is
the same whatever n but depends on u i : n CD5 f t * cos u i .
The experimental uncertainty on the determination of the
fluence is about 15%. For each u i , the samples corresponding to the different n values were irradiated together on the
same main sample holder which ensures that the fluence was
exactly the same for these samples.
The pinning properties of each sample were investigated
before and after irradiation by SQUID magnetometry with
the magnetic field H always applied along the c axis. Hysteresis loops were recorded with a waiting time of 1 min
after each field change. The field and temperature dependences of the persistent current J p for this characteristic time
of 1 min were derived from the Bean model. The procedures
used to extract the low-field part of J p (H) and to ensure a
reliable comparison between all the samples are detailed in
Ref. 16.
III. RESULTS
Let us first consider the case u i 575°. Figure 2 shows the
J p (H) curves at 1 min for the samples (n51; u i 575°) and
FIG. 2. Field dependence of the persistent current J p at 1 min
for u i 575° at 40 K ~a! and 60 K ~b!. Two samples have been
studied for n51 ~filled and empty circles! as well as for n53
~filled and empty diamonds!.
(n53; u i 575°) at 40 and 60 K. Two samples have been
investigated in both configurations. At low temperatures up
to about 50 K, the J p (H) curves for n51 and n53 are
almost superimposed, whereas a clear shift appears at higher
temperatures. In this high-T range illustrated by Fig. 2~b!,
the shapes of the J p (H) curves are very different for n51
and n53. The curves merge at very low fields but the falloff
in J p as H increases is much steeper for n51. For n53, the
precipitous fall of J p as H is increasing appears to bump
against a plateaulike feature which leads to a clear upward
shift of the irreversibility field H irr . Note that this shape for
n53 is similar to what is observed for classical configurations of CD’s parallel to c.
The low-field range where the curves n51 and n53 are
close vanishes rapidly as T increases. Actually, the crossover
between the above-mentioned low-T and high-T regimes
corresponds to a rapid shrinking of this field range: The
curves for n51 and n53 are still superimposed up to H irr
for T550 K, whereas they merge only at H50 for T570 K.
A low-T regime where the J p (H) curves do not depend
on n was also observed for u i 545° and u i 560°. Figure 3
illustrates for both angles the high-T regime with J p (H) at
60 K for n51, 2, and 3. As in Fig. 2~b! one observes a clear
shift between n51 and n53. The variation of shape in
J p (H) is even very similar to what is observed for u i 575°.
Nevertheless one can note that a narrow plateaulike feature is
visible for n51 in Figs. 3~a! and 3~b!. The curves for n52
roughly lie in an intermediate position between n51 and
n53. However one can note that n52 is closer to n53 than
to n51 if one considers the H irr values.
To investigate the influence of relaxation on the J p (H)
curves, hysteresis loops were registered with a waiting time
t w of 150 min at each measuring point. The J p (H) at 60 K
for (n51; u i 575°) and (n53; u i 575°) are displayed in
57
VORTEX PINNING BY DIFFERENT ANGULAR . . .
651
fects in the values of the persistent current at this temperature. Accordingly, the shift of J p (H) between n51 and
n53 can essentially be ascribed to a difference in the dominating activation energies. By contrast, in the low-T regime
where J p (n51)'J p (n53), the S 1 min(1000 G! values were
found to be very close in both configurations ~the difference
is less than 10% at 50 K, for instance!.
IV. DISCUSSION
FIG. 3. Field dependence of the persistent current J p at 1 min
and T560 K for u i 560° ~a! and u i 545° ~b!. In both cases, the
three configurations displayed in Fig. 1 were studied: n51
~circles!, n52 ~squares!, and n53 ~diamonds!.
Fig. 4 for t w 51 min and t w 5150 min. First, one observes
that the shape of the curves does not vary with t w . At
t w 5150 min, there is still a plateaulike feature for n53 and
an exponential-like decrease of J p (H) for n51. However it
must be noted that the J p (H) curves for n51 and n53
remain separated down to lower fields for t w 5150 min. Second, the normalized relaxation rates are found to be lower for
n53 than for n51. This is displayed in the inset of Fig. 4
considering the relaxation parameter S5 @ 2d lnJ p /d lnt # . A
closer inspection of these curves shows that the shape of
J p (H) essentially reflects the field dependence of @ 1/S # 1 min ,
which demonstrates the strong implication of dynamical ef-
The main outlines of the present study can be summarized
as follows:
~i! existence of a low-T regime where the J p (H) curves
do not depend on n.
~ii! existence of a high-T regime where the pinning efficiency roughly increases with n, especially if one considers
the irreversibility fields.
Some previous studies with inclined CD’s have shown
that accommodation effects can play a major role when the
field is not applied along the tracks.22 Accordingly, the results of the present study will be analyzed in the frame of
two approaches:
~i! Dynamic approach.
The involved dynamic effects are those predicted in the
model of splay.10,11 In the presence of CD’s, the starting
point of the flux motion is the generation of kinks, i.e., segments of a vortex line lying in the ab planes and connecting
two neighboring CD’s. The subsequent stage, which costs
nearly no additional energy in the case of parallel CD’s, is
the sliding of these kinks along the defects. The presence of
crossed CD’s will oppose this easy spreading of the kinks.
The two most relevant phenomena should be ~1! individual
kink pinning at the shortest distance between pairs of crossed
CD’s @Fig. 5~a!#, and ~2! entanglement effects preventing the
kinks between parallel CD’s from moving easily @Fig. 5~b!#.
~ii! Static approach.
Here one considers the vortex ground state and more particularly the occurrence of accommodated arrangements of
vortices on the tracks, allowing for a minimization of the
Gibbs energy. Such accommodation effects imply the creation of kinks in the low-temperature phase. Above a crossover temperature, the vortex state changes into another configuration where the vortices just intersect locally the CD’s.22
This crossover for n51 is sketched in Fig. 6. The pinning
efficiency in these different vortex states is supposed to increase with the relative length of vortex line strongly pinned
in CD’s.
A. Dynamic interpretation
FIG. 4. Field dependence of the persistent current J p at T560 K
and u i 575° for n51 ~circles! and n53 ~diamonds!, at two characteristic times: t w 51 min ~empty symbols! and t w 5150 min
~filled symbols!. The inset shows the field dependence of @ 1/S # 1 min
in the same conditions for n51 ~circles! and n53 ~diamonds!.
In such an interpretation, one can state that the effect of n
is clearly visible only above a crossover temperature because
high temperatures enhance the influence of flux creep on the
persistent current J p . In other respects, it can be noted that
the major effect which occurs between n51 and n.1 corresponds precisely to the introduction of defect crossings
supposed to notably impede the relaxation. In this interpretation, it must be specified that the relevant crossings are
regions where the close paths of two differently angled
tracks stimulate the development of a kink, rather than true
defect-defect intersections. Krusin-Elbaum et al.17 even
652
S. HÉBERT et al.
57
increases with n ~note that pinned vortex kinks are all parallel for n52 contrary to n53). These phenomena of angular
selection and vortex entanglement combine to slow down
more efficiently the kink dynamics for n53 than for n52.
A weakness of this dynamical approach concerns the description of the low-T regime. In their study, Schuster et al.14
did not report any influence of temperature on the comparison of activation energies between n51 and n52. In our
case, the relaxation measurements performed for n51 and
n53 at u i 575° have yielded similar relaxation rates in the
low-T regime. It is only at high temperatures that
S(n51).S(n53) leading to J p (n51),J p (n53). The
problem is that different temperature dependences of the activation energies for single or double kinks are not theoretically expected.
B. Static interpretation
FIG. 5. Sketch of vortex arrangements on crossed tracks for
n52. Tracks are drawn as cylinders and vortices as bold lines. In
~a! both kinks are pinned at the smallest interdistance between the
involved crossed defects. Note that these pinned kinks are all parallel for n52. In other respects, the motion of kinks connecting
parallel defects can be impeded by differently angled tracks ~b!.
The configuration ~b! could stem from ~a! by simple flux cuttingflux reconnecting processes.
pointed out that such intersections can rather promote creep
by easing the nucleation of kinks.
In a previous comparative study of configurations n51
and n52 ( u i 545°) by magneto-optics, Schuster et al.14 reported a much higher pinning in the latter case. This was
ascribed to the fact that creep for n52 must proceed via
kink-pair nucleation in the bulk, while single kinks requiring
a smaller activation energy are effective in depinning for
n51. For crossed defects, several kinds of vortex excitations
can be considered, as it is detailed in Ref. 14. Among them,
the development of a kink pair between parallel defects
should be one of the most efficient for flux creep.
In our experiments, the observed difference between n52
and n53 could be related to such basic mechanisms. For
instance, in each ab plane, the probability for two adjacent
CD’s to belong to the same family is (1/n). Consequently,
the probability for a double-kink excitation to yield an effective vortex transport between parallel CD’s decreases as n
increases. Moreover, the probability for a kink sliding between parallel CD’s to cross a track or another vortex kink
FIG. 6. Schematic representations of the totally adjusted state
~a! and the ‘‘straight’’ state ~b! in the case n51.
The angular distribution of CD’s may have a strong influence on the accommodation ability of the vortices to these
defects. As a basic assumption of this approach, it is supposed that the mean direction of each vortex line must be
parallel to the direction of the applied magnetic field, except
at very low fields when the alignment term in the Gibbs free
energy becomes much smaller than the free energy itself.22
In Bi-2212 with (n51; u i 545°) we have previously22
compared J p (H,T) curves in aligned (H i CD ) and
misaligned orientations ~including H i c). In this comparison,
one considered curves versus H i , the component of the field
along c. In accordance with other authors,23 we found evidence for a low-T regime where the J p (H i ) curves are superimposed whatever the orientation, and a high-T regime
where J p for (H i CD) exceeds that for (H i c), except close to
H50. We interpreted this crossover in temperature by a
change of the vortex state in the misaligned orientation: As T
increases, the decrease of the pinning energy may yield a
transition from a totally adjusted state towards a ‘‘straight’’
state as shown in Fig. 6. On the other hand, a linear pinning
holds up to higher temperatures in the aligned orientation.
According to this model, a rearrangement of the vortices
leading to a less pinned state is expected for n51 around
T550 K. For n>2, this ‘‘disaccommodation’’ can be delayed due to the more symmetric distribution of the CD’s
around the direction of H. In terms of symmetry elements,
the track distribution has a twofold axis for n52 and a threefold axis for n53, both parallel to H. Consequently, for
n.1, the vortex lines can zigzag essentially between tracks
of different orientations to maintain a mean orientation along
c whereas, for n51, they have to make long kinks in the ab
planes ~see Fig. 7!. With crossed defects, a vortex runs along
one track until it meets a close differently angled track which
reorientates it towards c. Note that the vortex line can take
advantage of the random track distribution to choose small
kinks. Moreover, it must be underlined that true defectdefect intersections are not rare events ~see the Appendix!. In
addition to a reduction of the total length of kinks for n.1,
the elastic energy related to the angle between successive
vortex segments is also reduced. Indeed, this characteristic
angle equal to ( p /21 u i ) for n51 decreases to (2 * u i ) for
n.1.
57
VORTEX PINNING BY DIFFERENT ANGULAR . . .
FIG. 7. Schematic representation of the accommodated state for
n52. A vortex line is shown as a succession of segments ~bold
lines! located in tracks and connected by small kinks ~filled circles!
perpendicular to the irradiation plane. The parameter u 0 is the characteristic distortion of the vortex around its mean direction ~dashed
line! which remains parallel to H.
653
FIG. 8. Curves of persistent current versus the field component
parallel to c at T550 K, u i 545°, and Ft51011 cm 22 in three
cases: n51 with H i c ~circles!, n53 with H i c ~diamonds!, and
n51 with the field applied along the tracks ~triangles!.
V. CONCLUSION
The difference between n52 and n53 could also be related to the energetic term coming from the deformation of
the vortex lines. The characteristic amplitude u 0 of the vortex distortion is related to the ‘‘mean free path’’ of a line
along one track, i.e., the mean distance d L between two defect crossings. For instance, if one considers only true defectdefect intersections, it is shown in the Appendix that d L is
shorter for n53 than for n52 at large u i (>45°), leading to
a smaller energy cost for n53 than for n52. Since this
energetic contribution works against accommodation, it
could explain that J p falls down at lower fields for n52 than
for n53.
Another experimental feature consistent with accommodation effects is the merging of the J p (H) curves at H50
whatever n. Indeed, as H decreases towards zero, the mean
vortex direction can progressively deviate from the orientation of H towards that of the CD’s, allowing for a strong
linear pinning on the tracks in all cases, even for n51. Such
a ‘‘flux-flop’’ phenomenon was previously invoked in angular studies with n51 to interpret the isotropy of pinning
observed near H50.24,25 Another result of angular measurements for n51 strongly supports the relevancy of accommodation effects when studying splayed configurations. Hysteresis loops were recorded for H i c and (H i CD ) in a sample
irradiated at u i 545° along one direction (n51) with
f t51011 ions cm 22 . Curves of persistent current in the
high-T regime are shown in Fig. 8 as a function of H i , the
field component along c. Also reported is the J p (H) curve
for a splayed configuration (n53; u i 545°; f t51011 cm 22 )
measured with H i c. It appears that the difference between
the aligned and misaligned orientations for n51 is exactly of
the same nature that the difference between n51 and n53
for H i c. It must be underlined that very close J p (H i ) curves
are obtained for (n53; H i c) and (n51; H i CD), though
there are no defect crossings in the latter case. Accordingly,
the orientation of the applied field versus the symmetry elements of the track distribution appears to be a more determinant parameter than the existence or not of crossed tracks, at
least in this case.
Inclined CD’s tilted by an angle u i from the c axis have
been introduced in Bi-2212 single crystals with three different configurations: Only one direction of tracks yielding parallel defects (n51), two coplanar directions of tracks
(n52), three directions of tracks symmetrically distributed
around the c axis (n53). The pinning efficiency in these
three cases has been systematically investigated as a function
of temperature and magnetic field. The persistent current J p
was derived from magnetic measurements, with the field always applied along c.
The central result of this study is the existence of two
regimes of temperature: At low temperatures, the J p (H)
curves are nearly superimposed whatever n, whereas they
clearly shift at high temperatures. In this high-T regime, the
pinning efficiency essentially increases with n. These features were systematically observed for the three studied
angles, namely u i 545°, 60°, and 75°, with a crossover temperature around 50–60 K in all cases.
Two kinds of explanations are proposed to interpret the
role of n: ~i! a static approach based on accommodation effects and ~ii! a dynamic approach based on elementary fluxcreep processes. In the static approach, the increased pinning
efficiency for n.1 is basically related to the symmetry of
the track distribution relative to the direction of the applied
field. In the dynamic approach, this increased pinning efficiency arises from the existence of the crossings themselves
which impede the motion of the vortex kinks.
Actually, these two interpretations cannot be clearly decorrelated. In particular, the effects of kink nucleation and kink
pinning, which are involved in the dynamical interpretation,
should directly affect the persistent current in an accommodated ground state.17 Nevertheless, some experimental facts
suggest that static accommodation effects must play a major
role. First, the existence of a crossover temperature above
which the influence of n appears can be accounted for more
easily in the frame of accommodation effects. Indeed, since
the pinning energy decreases as T increases, one expects that
the configuration n51 disaccommodates above a characteristic temperature. Subsequently, this change of ground state
leads to an easier vortex creep. In our study as a function of
654
57
S. HÉBERT et al.
T and H, we indeed observed that the averaged activation
energies measured by relaxation measurements are very
close for n51 and n53 in the low-T regime, while they
diverge clearly in the high-T regime. Secondly, the striking
similarity between the J p (H i ) curves for (n53; H i c) and
(n51; H i CD) strongly suggests that the modifications of
the dynamics related to the defect crossings are not predominant in this case.
Finally, this study shows that the pinning enhancement
previously observed16 for (n53; u i 575°) compared to a
reference with all the tracks parallel to c ~same damaged
volume! takes place in a regime where J p (n53)'J p (n51).
This suggests that the inclination of CD’s in itself improves
a pinning parameter. The increase of the effective track size
in the ab planes is one possibility which deserves to be studied in more detail.
ACKNOWLEDGMENTS
FIG. 9. Mean distance between defect-defect intersections as a
function of u i for Ft5531010 cm 22 and R550 Å: simulations for
n52 ~squares!, simulations for n53 ~diamonds!, calculation for
n52 ~full line!, and calculation for n53 ~dashed line!.
We thank C. Poirier for making the sample holders used
in the irradiation experiments.
According to the symmetry of the track distribution displayed in Fig. 1,
D u ~ u i ;n52 ! 52 u i ,
APPENDIX: DENSITY OF DEFECT-DEFECT
INTERSECTIONS
The density of defect-defect intersections has been estimated by analytical approximations and computer simulations as a function of the different irradiation parameters.
1. Analytical approximations
The starting point of this analysis stems from the Poissonian nature of the track distribution: For a given fluence Ft,
the sum of the overlapped regions yield the same volumic
fraction whatever the track distribution, in particular whatever u i and n.
For u i 50 ~all the tracks parallel!, this volumic fraction
can be exactly calculated.26 The fraction of amorphized area
perpendicular to the ion beam is
~ S a /S ! 512exp~ 2 P ! ,
V over /V5N over3 v over .
~A3!
N over( u i ;n) can thus be derived from Eqs. ~A2! and ~A3!
by estimating v over( u i ; n).
For crossed defects, the averaged value of this elementary
volume of intersection, denoted v cross , is directly related to
the relative angle D u between the two crossing tracks:
v cross'4R 3 /sin~ D u ! .
~A4!
G
113 cos~ 2 u i !
.
4
V over /V5 ~ V over /V ! i 1 ~ V cross /V ! .
~A5b!
~A6!
According to Eq. ~A2!,
~ V over /V ! i 5n3
~ P/n ! 2
.
2
~A7!
From Eqs. ~A2!, ~A6!, and ~A7!,
~ V cross /V ! 5 ~ P 2 /2 ! 3 ~ 121/n ! .
~A8!
Considering (V cross /V)5N cross3 v cross , one derives
N cross~ u i ;n ! '2RFt 2 ~ 121/n ! sinD u ~ u i ;n ! .
~A2!
This volumic fraction can also be written by combining the
density of overlapping events N over with the mean characteristic volume of overlap v over :
F
It must be taken into account that a part of the total overlapped volume occurs within each family of parallel tracks.
Accordingly, the right-hand side term of Eq. ~A3! should be
divided in two contributions: a first one related to the overlaps between parallel tracks, denoted (V over /V) i , and a second one involving crossed tracks, denoted (V cross /V). Rewriting Eq. ~A3!,
~A1!
where P is the nominal porosity: P5S 1 Ft, S 1 is the individual track area: S 1 '4R 2 .
For the present experimental conditions (Ft'531010
cm22; R'50 Å!, it turns out that P!1 and almost all the
overlaps involve just two tracks. Consequently, the fraction
of overlapped volume can be approximated by
V over /V' P211exp~ 2 P ! ' P 2 /2.
D u ~ u i ;n53 ! 5arccos
~A5a!
~A9!
An important parameter directly related to N cross is the
mean distance d L between crossings. Assuming that for
n53 triple crossings are rare compared to double ones, the
same relation holds for both values of n:
d L ~ u i ;n ! 5Ft/ @ 23N cross~ u i ;n !# .
~A10!
Finally, the influence of n can be summarized as follows:
3
4
d L ~ u i ;3 ! 5 d L ~ u i ;2 ! 3 3 f ~ u i ! ,
with
~A11!
VORTEX PINNING BY DIFFERENT ANGULAR . . .
57
f ~ ui!5
4
A3
3
A
sin2 ~ 2 u i !
23 cos2 ~ 2 u i ! 22 cos~ 2 u i ! 15
.
f ( u i ) is lower than 1 for u i .50° and rapidly decreases as
u i increases towards 90°. For instance f ( u i 575°)'0.55.
2. Computer simulations
Independently of the previous statistical estimates, Monte
Carlo calculations were performed. One crossing is counted
as soon as two tracks touch. Nevertheless a crossing involving overlapped parallel tracks is just counted once.
For n53, a distinction is made between triple and double
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655
crossings. It was found that the triple crossings represent
about 2–3 % of the total density of crossings. This validates
the approximation made in the first part.
In each case, many track configurations have been considered. Each of these simulations involved at least 1000 tracks.
Very good agreement with Eq. ~A9! were found about the
influence of R and Ft. The results about the angular dependence of d L in the conditions of our experiments
(Ft'531010 cm 22 ; R'50 Å! are displayed in Fig. 9 for
n52 and n53. The theoretical estimates of Eq. ~A10! are
also directly reported in this figure without any adjustable
parameters. One can note a very good agreement between the
simulations and the calculations for n52 as well as for
n53.
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