You are an expert at summarizing long articles. Proceed to summarize the following text: 
tiny superconductors , accommodating only a few quantized magnetic flux lines , display a great variety of vortex structure different from the abrikozov vortex lattice in bulk superconductors @xcite . examples include vortex polygons and concentric vortex rings called `` vortex shells '' , which are stable configurations of the repulsive flux lines under the geometrical confinement via the influence of the screening current flowing along the sample edge . the issue of vortex matter confined into small superconductors has been studied theoretically and numerically for many years , focusing mainly on how the vortices are distributed in disk @xcite@xcite@xcite@xcite@xcite@xcite , square @xcite , and triangle shaped small superconductors @xcite . several studies reveal the field evolutions of the vortex structure in small superconductors , and argue which vortex configurations are energetically favorable and how the transition between different vortex states occurs . the obtained rule of shell filling and magic number configurations for consecutive new shells are relevant to phenomena observed in other systems , including the puzzling nucleation of vortices observed in rotating condensates of superfluid @xmath1he @xcite and cold dilute alkali - metal gases @xcite .    the experimental investigations for vortex states in small superconductors were initiated by hall magnetometer measurements @xcite@xcite , followed by a multiple - small tunneling junction measurement @xcite . kinks in the magnetization or jumps in the tunneling spectra mark transitions of vortex states in small superconducting dots . they evidence changes of vorticity @xmath2 ( the number of vortices ) , not actual distributions of vortices in the dots . the visualization of vortices in small superconductors was reported earlier in studies of the scanning superconducting quantum interference device ( squid ) microscopy @xcite@xcite and later in studies of the scanning hall probe microscopy @xcite@xcite , showing reasonable images of vortex configurations for @xmath2 up to 6 . more vortices have been imaged in recent bitter decoration studies as patterns of magnetic particles deposited over micrometer - sized dots of nb films @xcite@xcite and nb mesas@xcite . the experiments reveal the rule of the shell filling with magic numbers for @xmath2 up to 40 by combining vortex configurations visualized over many disks at different magnetic fields , although some features influenced by the bulk pinning and/or the roughness of the disks are involved . the scanning squid microscope technique , which we employed in this study , is an alternative and complement method to visualize vortices in the superconducting dots . by scanning a small pick - up loop over the dots , one can magnetically image the vortices without damaging the samples . this allows us to study systematically how the vortex configuration evolves with magnetic field in a specific sample . while the scanning squid microscope has attained the best magnetic sensitivity ( better than 5 @xmath3 ) , the spatial resolution is limited typically to a few micrometer due to the size of the pick - up loop ( @xmath4 10 @xmath5 m ) @xcite . this implies that the observable vortex density is low ( @xmath6 100 @xmath5 t ) and the sample becomes much larger than the loop size . the earlier studies were made on ybco @xcite and nb dots @xcite with @xmath4 50 @xmath5 m in size , which is decades larger than the penetration depth @xmath7 or the coherence length @xmath8 of the superconducting materials . in such large dots , the effect of the geometrical confinement is weak and ( sparse ) vortices form likely disordered configurations due to the dominant influence of the bulk pinning in the superconducting materials @xcite@xcite . if the influence of the bulk pinning is reduced by using weak pinning , superconducting materials , the situation can be different : the geometrical confinement dominates over the bulk pinning . moreover , when the dots become thin in thickness @xmath9 @xmath10 , the interaction between vortices becomes long ranged and it decays slowly with the vortex - vortex spacing @xmath11 as @xmath12 for @xmath13 @xcite , in contrast to the short ranged interaction characterized by the exponential decay @xmath14 for @xmath15 in the bulk superconductors . thus , even in the large dots the interplay between the vortex - vortex interaction and the geometrical confinement may lead to geometry - induced , symmetric configurations of the sparse vortices in the presence of the weak bulk pinning . here , we report on the direct observation of vortices in large thin disks of an amorphous ( @xmath16 ) moge film by the scanning squid microscopy . due to the weak bulk pinning of the amorphous film , we are able to observe ( quasi- ) symmetric configurations of polygons and concentric shells of vortices for @xmath2 up to 19 . the results illustrate not only how the vortex configuration evolves with the magnetic field , but also how the vortex polygons change in size and rotate with respect to the disk center . we present vortex images obtained in a disk with a pinning site and discuss how vortex configurations are altered by a pinned vortex . this paper is outlined as follows : after describing experimental details in sec . ii , we present vortex polygon states observed for small vorticities up to @xmath2= 5 in sec . iii . in sec . iv we present field evolutions of vortex shells observed for large vorticities @xmath17 6 . the influence of a pinning site on vortex configurations is discussed in sec . v. in sec . vi comparison with other experiment is made . our summary is given in sec . we used a commercial scanning squid microscope ( sqm-2000 , sii nanotechnology ) with a dc squid magnetometer made of nb / al - alo@xmath18/nb josephson junctions and an inductively coupled , pick - up loop of a nb film @xcite . the pick - up loop had 10 @xmath5 m in diameter with 2 @xmath5 m line width . the dc squid and the pick - up loop were integrated on a square si chip of 3 @xmath19 3 mm@xmath20 in size , which was mounted on a phosphor bronze cantilever . the sensor chip was tilted with respect to the sample stage by a shallow angle of @xmath4 10 degrees and a corner of the chip was softly in contact with the sample surface , in order to keep a short distance ( @xmath4 5 @xmath5 m at the minimum ) between the loop and the sample surface . this ensures safe operation of the scanning squid microscope with reasonably high spatial resolutions ( @xmath4 4 @xmath5 m ) . the whole assembly including the sensor chip and the sample stage was surround by a @xmath5-metal shield , resulting in a residual magnetic field ( ambient field ) of @xmath4 1 @xmath5 t . a coil wound in our sample stage allowed us to apply small magnetic field @xmath21 perpendicular to the sample . we scanned the sample stage in xy directions by dc stepper motors . in order to minimize the possible mechanical hysteresis effect , we took one scan direction from the left side to the right side of the image . the scanning speed was about 5 pixels / s and this resulted in an image of 64@xmath1964 pixels by typically 15 minutes . the temperature for the squid sensor was 2 - 3 k , while that for the sample stage was varied . we prepared weak pinning disks of an @xmath16-mo@xmath22ge@xmath23 ( @xmath24 78 @xmath25 ) thin film with a lithographic technique . after preparing a silicon substrate covered with a patterned - resist film , we sputtered the @xmath16-moge film with 0.22 @xmath5 m thickness on top . employing the lift - off technique , we patterned the film into arrays of disks with four different diameters @xmath26 of 20 @xmath5 m , 34 @xmath5 m , 56 @xmath5 m and 106 @xmath5 m . the spacing between the disks was chosen to be large enough ( @xmath27 200 @xmath5 m ) , so that the interdisk coupling can be excluded . in order to reduce possible damage during the scanning , all the disks were covered with 0.1 @xmath5 m thick sio@xmath28 film as a protective layer . the material parameters relevant to this study are as follows : the superconducting transition temperature @xmath29 is 6.0 k , the coherence length @xmath30 at @xmath31 is 4.5 nm , and the penetration depth @xmath32 at @xmath31 is 0.58 @xmath5 m ( @xmath33 ) @xcite . since the thickness is smaller than @xmath34 , the penetration depth is expanded to the effective one given by @xmath35 @xmath36 . for the present film @xmath37 3.1 @xmath5 m . normalizing the disk size by the effective penetration depth , we find @xmath38 6 - 35 @xmath39 , indicating large mesoscopic disks .    in weak pinning materials , the distance @xmath40 between the pick - up loop and the sample surface is an important parameter to be controlled in order to obtain clear images of vortices . as demonstrated by plourde _ @xcite , the vortices may be dragged and/or shifted in positions by the pick - up loop during the scanning . this coupled motion of vortices can be reduced when @xmath40 is increased by lowering the sample stage from the contact point with the sensor chip . this procedure , however , results in a smeared image of vortices due to the spread of the magnetic flux above the sample surface @xcite . we could not resolve individual vortices when @xmath40 became more than the loop diameter ( @xmath41 @xmath5 m ) . hence , we set @xmath42 7 @xmath5 m as a typical distance . all the images presented in this study were taken in the field - cool measurement , where the magnetic field was applied in some temperatures ( @xmath410 k ) above @xmath29 , followed by cooling the samples in the magnetic field to temperatures ( @xmath4 3 - 5 k ) below @xmath29 where scanning squid measurements were done . this procedure results in the equilibrium distribution of vortices . because of the possible motion of vortices during the scanning , each image was taken in a single scan made on a fresh vortex state prepared by the field - cool procedure . no image taken by repeated scans is presented . first we focus on vortex configurations observed for small vorticities up to 5 . figs . 1(a)-1(f ) show selected vortex images observed in a 34 @xmath5 m disk . a color bar indicates the magnitude of the magnetic flux in the pick - up loop . these images display clearly individual vortices with reasonable spatial and magnetic resolutions @xcite . the imaged vortex size is larger than 2@xmath43 7 @xmath5 m ) due to defocusing effect originating from the large distance @xmath44 7 @xmath5 m ) between the loop and the sample surface . details of the field profile of vortices have been reported in refs . we find that the vortices form symmetric configurations with respect to the disk center . the vortex pattern evolves with field as follows ; after the meissner state where the magnetic field is expelled from the disk , a single vortex appears in the disk and it sits at the disk center ( fig . 1(a ) ) . then , two vortices are located symmetrically with respect to the disk center ( fig . 1(b ) ) , followed by the formation of a triangle ( fig . 1(c ) ) , a square ( fig . 1(d ) ) and a pentagon ( fig . 1(e ) ) of vortices in the disk . although some distortion is visible for the square , these symmetric configurations are nearly identical to those reported in the bitter decoration @xcite and the theoretical studies @xcite@xcite@xcite . an important remark for the observation is that some of the vortex patterns rotate with respect to the disk center . this feature is clearly visible in the images of the vortex pair shown in figs . 1 ( b ) and 1(f ) . in fig . 1(b ) , the vortex pair is aligned nearly with the horizontal direction of the image , while in fig . 1 ( f ) it rotates by @xmath4 45@xmath45 with respect to the disk center . other orientation of the pair is observed in other scan , indicating the rotational degree of freedom in the vortex configuration @xcite . similar rotation is observed for the triangle pattern , but not clearly for the square and the pentagon patterns . next we show how the number of vortices evolves with applied magnetic field . counting the number of vortices in the images taken at various magnetic fields , we plot the results in fig . 2(a ) . here , instead of the field strength @xmath21 , we use the magnetic flux @xmath46 in the disk normalized by the flux quantum @xmath47 . as observed , the vorticity @xmath2 shows a stair case behavior with field : after the meissner state ( @xmath48 0 ) , the single vortex state ( @xmath48 1 ) appears at @xmath49 2 , and it continues until @xmath49 4.5 , above which the vortex pair state ( @xmath48 2 ) appears . the vortex triangle ( @xmath48 3 ) , square ( @xmath48 4 ) and pentagon ( @xmath48 5 ) states appear at @xmath49 6.4 , 8.2 and 9.5 , respectively . we note that the stair case behavior is well below the condition of @xmath50 where the magnetic field penetrates fully the disk and this is represented by a solid line . the difference from the line becomes large on increasing field . at @xmath51 8 , for instance , @xmath52= 4 , which is half of @xmath53 . this indicates a large diamagnetic response of the disk and it implies the presence of the large screening current near the disk edge . the influence of the screening current upon vortex configurations is found by considering how the size of vortex polygons changes with magnetic field . as exemplified in the inset to fig . 2(b ) , we draw a circle circumscribing the positions of vortices and estimate the diameter @xmath54 of the polygon ring . the results for this analysis made at different fields and different vorticities ( down to @xmath2= 2 ) are summarized in the main panel of fig . as observed , @xmath54 varies largely with @xmath2 . for example , @xmath54 at @xmath2= 2 is roughly 40 @xmath25 of the disk diameter , while that at @xmath2= 5 becomes expanded to roughly 65 @xmath25 of the disk diameter . focusing on the results at each vorticity , we find @xmath54 decreases with increasing field . the shrinkage of @xmath54 is observed to be 5 - 10@xmath25 of the disk diameter , depending on @xmath2 . the obtained results are in excellent agreement with the theoretical considerations for vortex polygons confined by the screening current near the disk edge @xcite@xcite . in the london limit , the diameter of a vortex polygon at a given magnetic field is written as @xmath55 where @xmath56 is the number of vortices on the polygon ring . the behaviors for @xmath56= 2 - 5 are represented by dotted curves in fig . our results for @xmath57 2 - 5 follow nicely those curves without any adjustable parameters . moreover , they are satisfactory for the condition that the polygon size should be below the maximum diameter above which the formation of the polygon ring is not stable . this is represented by a solid curve and it follows the condition @xcite given by @xmath58 thus , we are convinced that the vortex configurations observed in the 34 @xmath5 m disk are dominantly influenced by the geometrical confinement via the screening current near the disk edge . we note , to the best of our knowledge , that these quantitative agreements , together with the rotation of the vortex patterns in the same disk , are not reported in other experimental studies so far . for vorticities more than 6 , we observe the formation of concentric shell rings of vortices . since the vortices become dense and their magnetic images are overlapped each other , for clarity , we present here the results obtained in larger disks . 3(a)-3(c ) show vortex images for @xmath2= 6 - 8 observed in a disk with 56 @xmath5 m diameter . one can clearly see the shell formation of vortices . in the image of fig . 3(a ) , for example , a vortex sits at the center of the disk , while the others form a ring around . using the standard notation , this configuration is referred to as ( 1,5 ) , meaning that the inner shell has one vortex and the outer shell has five vortices . after the ( 1,5 ) configuration , we observe ( 1,6 ) and ( 1,7 ) configurations , which are shown in figs . 3(b ) and 3(c ) , respectively . namely , the increase of vorticity in the disk results in the growth of the outer shell ring . the growth of the inner shell occurs at even larger @xmath2 . 4 shows the field evolution of vorticity , together with vortex configuration , observed in the largest disk with 106 @xmath5 m diameter . the vortex shells with a single vortex at the center are observed for @xmath2= 6 , 7 , and 8 . this is consistent with the observation made on the 56 @xmath5 m disk discussed above . at @xmath2= 9 a vortex pair appears as the inner shell , while the others form the outer shell ring around , resulting in a ( 2,7 ) configuration . the pair configuration of the inner shell is also observed at @xmath2= 10 . at @xmath2= 11 three vortices form a triangle as the inner shell , resulting in a ( 3,8 ) configuration . the triangular inner shell is observed over some field range and persists up to @xmath2= 13 . in contrast , the appearance of 4 vortices as the inner shell occurs only at @xmath2= 14 . at @xmath2= 15 five vortices form a pentagon as the inner shell . the vortex shells with the pentagon inner ring appear in vorticities up to @xmath2= 17 above which three concentric shells are formed . these results follow mostly the rule of the shell filling obtained in theoretical and numerical studies @xcite@xcite . this includes the magic number configuration at @xmath2= 6 where the first formation of two concentric shells occurs , and also the formation of ( 2,7 ) at @xmath2= 9 where two configurations of ( 1,8 ) and ( 2,7 ) are energetically comparable but the latter one becomes the ground state for the large disk @xcite . however , some differences are found for large vorticities ( @xmath59 ) . the two shell configurations with 4 vortices in the inner ring are shown to appear for 13 @xmath60 15 , while we observe only the ( 4,10 ) configuration at @xmath61 . moreover , after the pentagon inner shell at @xmath48 16 the formation of three vortex shells is shown to occur at @xmath48 17 , which is not observed above . a physical origin for the differences would be found in actual vortex configurations observed in the largest disk . selected images are shown in figs . 3(d)-3(f ) . for clarity , the corresponding vortex configurations are sketched in figs . 3(g)-3(i ) . although some distortions are present , one can reasonably find quasi - symmetric shell formation of vortices even in the largest disk . this is exemplified by figs . 3(d ) and 3(g ) , where three vortices form a triangle in the inner shell surrounded by 8 vortices as the outer shell ring around . focusing on vortex configurations in the inner shell , we find the formation of a pair , the triangle and a pentagon , of which patterns are similar to those observed for small vorticities ( @xmath62 5 ) given in fig . however , this is not the case for a square . as shown in figs . 3(e ) and 3(h ) , the vortices in the inner shell form a strongly deformed square like a rhombus . as discussed below , this originates from the effect of the bulk pinning quenched on the @xmath16-moge film . we often observe a vortex ( or vortices ) close to the rim of the disk for large vorticities , which could be attributed to possible roughness of the disk boundary . this is visible in the configurations of ( 4,10 ) ( figs . 3(e ) and 3(h ) ) and ( 1,5,13 ) ( figs . 3(f ) and 3(i ) ) . as denoted by an open circle(s ) in the corresponding sketch , a vortex ( vortices ) appears close to the lower - right rim of the disk . more rim vortices are occasionally observed when the field - cool procedure is repeated in the same magnetic field , indicating extra vortices trapped in the rim during the field - cool process . if we neglect the rim vortex ( vortices ) , the differences between the experimental observation and the theoretical prediction in vortex configurations found for @xmath63 would be partly explained . in fig . 3(h ) , the rim vortex becomes a part of the outer shell ring . if we neglect the vortex , the configuration becomes ( 4,9 ) . this is consistent with the first configuration for 4 vortices in the inner shell given theoretically @xcite@xcite . similarity is applicable to the ( 1,5,13 ) configuration shown in fig . 3(i ) , where two rim vortices are recognized . thus , the configuration could be reduced to ( 1,5,11 ) , which is consistent with the magic number configuration for three vortex shells @xcite@xcite . further studies would be needed to clarify the origin and the role of the rim vortices on the vortex configurations in details . repeating field - cool measurements over some field range , we observe configurations with different vorticities at the same magnetic fields @xcite . an example obtained on the 34 @xmath5 m disk is given to the inset to fig . although we take the same procedure of the field cooling , both the @xmath2= 5 polygon and @xmath2= ( 1,5 ) shell are observed at the fields of @xmath64 10.4 and 10.9 , resulting in some field crossover in the transition between the @xmath2= 5 and @xmath2= ( 1,5 ) states . similar crossover may be present in every step in the stair - case behavior of @xmath2 given to fig . 2(a ) and fig . out of over 20 disks measured in this study , we found that some disks have a relatively strong pinning site(s ) @xcite , although no defect is visible under an optical microscope . the pinning center(s ) is probably introduced during the sample preparation and it originates from the slight compositional inhomogeneity or subtle roughness in the disk . the influence of the pinning site(s ) on vortex configurations depends highly on where it is present in the disk . as an example , we present the results ( for @xmath2 up to 13 ) obtained on a 56 @xmath5 m disk with a pinning site near the disk center . 5(a)-5(i ) show selected vortex images observed in the disk . we find the pinning site from the image at @xmath2= 1 shown in fig . 5(a ) , where a vortex is unnaturally off - centered in the disk @xcite . this is against the geometrical confinement and it should be attributed to the influence of the pinning center . thus , we identify the vortex position as the pinning site and mark a red dot on the image . no more pinning center is found in the disk . the rest of the images indicate clearly how other vortices are distributed in the presence of the vortex trapped at the pinning site as marked by the red dot . 5(b ) shows the image of two vortices in the disk : one of them sits at the pinning site , while the other appears in the opposite side with respect to the disk center . we do not observe any orientational changes in the pair even when the magnetic field is varied . similarity is applicable to the image of a triangle configuration given to fig . 5 ( c ) . thus , the influence of the pinning site fixes the vortex patterns and it breaks the rotational degree of freedom in the vortex configurations .    the deformation of vortex polygons occurs for vorticities more than 4 . fig . 5(d ) shows the image of 4 vortices in the disk . the vortices form a rhombus - shaped configuration , where the diagonal distance between the pinned vortex and the diagonal vortex is shorter than that for the other two vortices . this is similar in shape to the disordered square in the inner shell observed in the largest disk ( see figs . 3(e ) and 3(h ) ) . a unique configuration of vortices is observed in the image at @xmath2= 6 . as shown in fig . 5(f ) , one can see two rows of three vortices like a dice pattern of six with the pinned vortex in the middle of the upper row . this pattern is stable against some field change and no shell pattern is observed at @xmath2= 6 . a precursor of the formation of the rows can be seen in the image at @xmath2= 5 ( fig . 5(e ) ) , where the vortices form a pentagon with deformation inward by the pinned vortex ( the top of the pentagon ) and higher right three vortices try to be aligned . the formation of the ( upper ) row remains in the image at @xmath2= 7 ( fig . thus , the pinning site near the disk center triggers off the formation of the vortex row and stabilizes the dice like vortex pattern at @xmath2= 6 .    because of the dice pattern of vortices at @xmath2= 6 , the first formation of two concentric shells occurs at @xmath2= 7 . this is larger than @xmath2= 6 where the magic number configuration for two concentric shells is observed in the `` pin - free '' disks . thus , the shell filling rule is altered by the pinning site near the disk center .    for @xmath65 7 , the vortex shells are observed with the pinned vortex being a part of the inner shell ring . as in the image of fig . 5(h ) , the pinned vortex is involved in the formation of a pair as the inner shell , while the other vortices form the outer vortex ring around . the corresponding sketch given in fig . 5(k ) allows us to identify the configuration as ( 2,6 ) . to our knowledge , this configuration has not been reported in other studies so far . for larger vorticities , we observe two shell configurations of ( 2,7 ) , ( 2,8 ) and ( 3,8 ) which are similar to those observed in the largest disk . it is interesting to show the image at @xmath2= 13 given in fig . 5(i ) , where 4 vortices form the inner shell surrounded by 9 vortices as the outer shell . as sketched in fig . 5(l ) , the vortices in the inner shell form a rhombus - shaped , deformed square , while the outer shell vortices form a ring around . again , focusing on the inner shell , we find that the deformation from the regular square is weaker than that observed at @xmath2= 4 ( fig . 5(d ) ) and the effect of the pinning site is opposite ; namely , the pinned vortex and the diagonal vortex gives the longer diagonal of the rhombus . thus , the vortex square is stretched by the influence of the pinning site . we note that this effect depends highly on where the pinning site is present and how large the square polygon is . the regular square may appear if the pinning site is located on the circumference of the ring defined by the regular square vortices . all the vortex configurations observed in the disk can be characterized by a line symmetry , defined by the pinned vortex and the disk center * o * , as sketched in fig . drawing the line in the sketch of fig . 5(l ) , for instance , one can find that the vortex configuration is divided into halves that are nearly the mirror configurations of each other . similarity is applicable to the other configurations including unique ones at @xmath2= 6 and 8 , although slight deviation is present at @xmath2= 5 and 7 . note that the odd ( even ) number of vortices lies on the line when @xmath2 is odd ( even ) . in the `` pin free '' disks no unique line symmetry characterizing all the vortex patterns with different vorticities appears . thus , we are convinced that the line symmetry in the vortex configurations is induced by the off - centered pinning site in the disk . a physical reason for the symmetry is not simple since the vortex patterns are determined by subtle balance of many competing interactions , including the interaction with the bulk pinning , the repulsive interaction between vortices , and the interaction with the disk boundary . to clarify this issue , a numerical simulation study would be needed . let us estimate roughly the pinning force @xmath66 for the vortex at the pinning site . we find that at 4.5 @xmath5 t , just below the step field between @xmath48 1 and @xmath48 2 states , the disk has only a single vortex , but it appears at the disk center . this is distinct from the observation made at lower fields of 3.5 @xmath5 t and 4.0 @xmath5 t , where the vortex appears at the off - centered pinning site ( fig . these facts indicate the important influence of the force on the vortex due to the geometrical confinement , and suggest that this force should be comparable to the pinning force in between 4.0 @xmath5 t and 4.5 @xmath5 t. a model described in refs . @xcite , @xcite and @xcite shows that the force @xmath67 of vortex interactions with the shielding current and edge for a given magnetic field is written as @xmath68\cdot\frac{r}{d_d/2}\ ] ] where @xmath11 is the distance of the vortex position from the disk center and @xmath69 n / m ) is the unit of the force for a vortex per unit length . using the distance ( @xmath70 12 @xmath5 m ) for the off - centered vortex , we find the pinning force as @xmath71 1.8 @xmath72 . this result , together with our observation where the pinning site accommodates only one vortex , are consistent with the numerical simulation made at the weak pinning force of @xmath73 2 @xcite . further measurements are made on other disks with a pinning site near the disk edge . it turns out that the pinned vortex near the disk edge gives only some distortion on the vortex ring(s ) . no unique configuration is observed except for a highly off - centered vortex configuration at @xmath2= 1 . these facts indicate that the pinning site near the disk center provides more strong influence upon the vortex configurations than that near the disk edge . comparing the results in the present experiment to those found in the bitter decoration experiment @xcite , we find that both experiments share the general features of vortex configurations including the shell filling rule and the magic number , although magnetic fields and disk sizes are different in decades . let us shortly discuss the experimental conditions made on these studies . in the decoration experiment the symmetric vortex shells were observed in smaller disks with 1 - 5 @xmath5 m diameter at higher magnetic fields ( @xmath4 mt ) . due to the small disk size , the geometrical confinement is strong and it dominates over the influence of the strong bulk pinning in the nb film . meanwhile , in the present experiment , the disk size is @xmath4 10 times larger and the field range is two decades smaller . the corresponding average vortex spacing is @xmath4 10 @xmath5 m , which is larger than the effective penetration depth . thus , the interaction between vortices as well as the influence of the edge current are weaker , but they remain effective in the vortex configurations because the bulk pinning in the amorphous film is weak . this situation is validated in the observation of a reasonably ordered configuration of vortices in @xmath16-moge thin films at a low field of 10 @xmath5t@xcite . moreover , normalizing the disk size by the effective penetration depth , we find that our disk size ( 6 - 35 @xmath74 ) is not much different from that of 9 - 46 @xmath74 for the decoration experiment . these facts not only link two experiments made in different sizes and different field ranges , but also confirm that our observation is made in the large _ mesoscopic _ regime . we presented the scanning squid microscope images of vortices observed in the weak pinning , @xmath16-moge thin disks with different diameters . the observed images illustrate clearly geometry - induced , ( quasi-)symmetric configurations of vortices for @xmath2 up to 19 .    for small vorticities up to 5 , the vortices form the symmetric polygons with respect to the disk center . the polygon size grows with @xmath2 , while it shrinks by the enhancement of the screening current . some of the polygons rotate clearly with respect to the disk center , indicating the rotational degree of freedom in the vortex configurations . the influence of the geometrical confinement is evident in the quantitative agreements between the field evolution of the polygon size and the theoretical considerations in the london limit .    for large vorticities more than 6 , the vortices form the concentric shell rings . the field evolutions of the inner and outer shell rings follow the rule of the shell filling obtained in the theoretical and numerical studies . some differences are found in vortex configurations for large vorticities ( @xmath75 13 ) observed in the largest disk , and they could be explained partly by the influence of the weak bulk pinning and/or the edge roughness in the disk . we also presented the vortex images obtained in the disk with the off - centered pinning site . the results reveal the pinning induced , novel configurations of vortices : in addition to the deformed polygons , we observed unique vortex configurations of the dice pattern at @xmath2= 6 and ( 2,6 ) at @xmath2= 8 . these lead to the pinning induced changes in the shell filling rule and the magic number . all the configurations observed in the disk are nearly symmetric with respect to the line defined by the pinned vortex and the disk center . n. k. thanks o. narikiyo , h. kawai , k. kadowaki , b. j. baelus , and v. r. misko for useful discussion . this work was supported partly by the grant in aid for scientific research from mext ( the ministry of education , culture , sports , science and technology ) , japan .    corresponding author : n. kokubo , center for research and advancement in higher education , kyushu university , 4 - 2 - 1 , moto - oka , nishi - ku , fukuoka , fukuoka 819 - 0395 , japan ; e - mail : kokubo(at)rche.kyushu - u.ac.jp . a. a. abrikosov , sov . jetp * 5 * , 1174 ( 1957 ) . a.i . buzdin and j.p . brison , phys . a * 196 * , 267 ( 1994 ) . p. a. venegas and e. sardella , phys . b * 58 * , 5789 ( 1998 ) . j. j. palacios , phys . lett . * 84 * , 1796 ( 2000 ) . b. j. baelus , l. r. e. cabral , f. m. peeters , phys . rev . b * 69 * , 064506 ( 2004 ) . l. r. e. cabral , b. j. baelus , and f. m. peeters , phys . b * 70 * , 144523 ( 2004 ) . v. r. misko , b. xu , and f. m. peeters , phys . b * 76 * , 024516 ( 2007 ) . l. f. chibotaru , a. ceulemans , v. bruyndoncx , v. v. moshchalkov , nature * 408 * , 833 ( 2000 ) . l. f. chibotaru , a. ceulemans , v. bruyndoncx , and v. v. moshchalkov , phys . lett . * 86 * , 1323 ( 2001 ) . a. k. geim , i. v. grigorieva , s. v. dubonos , j. g. s. lok , j. c. maan , a. e. filippo , f. m. peeters , nature * 390 * , 259 ( 1997 ) . a. k. geim , s. v. dubonos , j. j. palacios , i. v. grigorieva , m. henini , j. j. schermer , phys . lett . * 85 * , 1528 ( 2000 ) . a. kanda , b. j. baelus , f. m. peeters , k. kadowaki , y. ootuka , phys . * 93 * , 257002 ( 2004 ) . y. hata , j. suzuki , i. kakeya , k. kadowaki , a. odawara , a. nagata , s. nakayama and k. chinone , physica c * 388 - 389 * , 719 ( 2003 ) . t. nishio , s. okayasu , j. suzuki , k. kadowaki , physica c * 412 - 414 * , 379 ( 2004 ) . t. nishio , q. chen , w. gillijns , k. de keyser , k. vervaeke , and v. v. moshchalkov , phys . b * 77 * , 012502 ( 2008 ) . m. r. connolly , m. v. milosevic , s. j. bending , j. r. clem and t. tamegai , euro phys . lett . * 85 * , 17008 ( 2009 ) .      h. j. zhao , v. r. misko , f. m. peeters , v. oboznov , s. v. dubonos and i.v . grigorieva , phys . b * 78 * , 104517 ( 2008 ) . grigorieva , w. escoffier , v. r. misko , b. j. baelus , f. m. peeters , l. y. vinnikov , and s. v. dubonos , phys . rev . lett . * 99 * , 147003 ( 2007 ) . j. r. kirtley , m. b. ketchen , k. g. stawiasz , j. z. sun , w. j. gallagher , s. h. blanton , and s. j. wind appl . * 66 * , 1138 ( 1995 ) . we estimate @xmath7 with the dirty - limit expression described in p. h. kes and c. c. tsuei , phys . b * 28 * , 5126 ( 1983 ) . the coherence length @xmath76 is estimated by @xmath77 with the second critical field @xmath78 at @xmath79 0 . b. l. t. plourde , d. j. van harlingen , n. saha , r. besseling , m. b. s. hesselberth , and p. h. kes , phys . b * 66 * , 054529 ( 2002 ) . v. g. kogan , v. v. dobrovitski , j. r. clem , y. mawatari and r. g. mints , phys . rev . b * 63 * , 144501 ( 2001 ) . the rms flux noise for our system referred to the pick - up loop is @xmath4 5 @xmath80 per pixel in the images . f. tafuri , j. r. kirtley , p.g . medaglia , p. orgiani , and g. balestrino , phys . lett . * 92 * , 157006 ( 2004 ) . t. nishio , s. okayasu , j. i. suzuki , n. kokubo , and k. kadowaki . b * 77 * , 052503 ( 2008 ) . we note that the observed orientations are some of the metastable states stabilized by subtle uneven boundary confinement due to the edge roughness and/or possible weak bulk pinning in the disk . otherwise , during the scanning , the orientation can be aligned along the scan direction of the pick - up loop due to the weak coupling between the loop and the vortices . we emphasize that the pinning site accommodates only one vortex and others are distributed around in the disk . this is distinct from phenomena observed in the micrometer - sized nb mesas , where multi - fluxons are observed near the strong pinning site @xcite .
direct observation of vortices by the scanning squid microscopy was made on large mesoscopic disks of an amorphous moge thin film . 
 owing to the weak pinning nature of the amorphous film , vortices are able to form geometry induced , ( quasi-)symmetric configurations of polygons and concentric shells in the large disks . 
 systematic measurements made on selected disks allow us to trace not only how the vortex pattern evolves with magnetic field , but also how the vortex polygons change in size and rotate with respect to the disk center . 
 the results are in good agreement with theoretical considerations for mesoscopic disks with sufficiently large diameter . 
 a series of vortex images obtained in a disk with a pinning site reveals a unique line symmetry in vortex configurations , resulting in modifications of the shell filling rule and the magic number .    # 
 1@xmath0#1    2