(a) 50 40 30 20 10 0 Magnetic field ( T) (d) (b) 50 40 30 20 10 0 (e) (c) 100 50 0 (f) (g) (h) (i) (b) 50 40 30 20 10 0 (e) 30 20 10 0 Distance ( m) (g) 10 0 0 Ratio (%) (f) 10 20 (d) (a) 50 40 30 20 10 0 Magnetic field ( T) (c)$^{5}$ 10 15 20 25 Fit Data 30 20 10 0 0 Magnetic field ( T) Position ( m) 2 FIG. 1: (color online) Scanning SQUID microscope images of vortices at (a) 1 µ T and (b) 10 µ T for a MgB 2 single crystal, the dimensions of which are approximately 80 × 60 µ m $^{2}$. Dotted lines show the edges of the sample. A scale bar corresponds to 10 µ m. The integral of the magnetic field within a solid line in (a) gives the flux ∼ φ 0 . (c) The cross section of the vortex image along a broken line in (a) (open circles) and the model fit (solid line)(see text). (d) Scanning SQUID microscope image of vortices at 10 µ T in a Nb film with a thickness of 200 nm ( T$_{c}$ =9.1 K). (e) The 2D multi- fits of the SQUID image in (b) to numerical calculations by a London model (see text). (f) and (g) The distribution of the intervortex distance derived from (e) and (d), respectively. A solid line in (g) represents a fit to the Gaussian function. An arrow indicates the intervortex distance calculated from ( φ 0 /B ) 1 / $^{2}$. FIG. 2: (color online) Scanning SQUID microscope images of vortices at (a) 10 µ T, (b) 20 µ T, and (c) 50 µ T for a MgB 2 single crystal. A scale bar corresponds to 10 µ m. (d)-(f) The 2D multi-fits of the SQUID images in (a)-(c) by a London model (see text). (g)-(i) The locations of vortices in (d)-(f) obtained by the 2D multi-fits of the images. Broken lines show the edges of the sample. Figure 1(c) shows a fit of the cross section of a vortex image in Fig. 1(a) by the numerical simulations. We obtain λ =0.01-0.13 µ m ( z =1.7 ± 0.1 µ m, where z is the distance between the sample surface and the pickup loop) from the fit. The z value was determined by the fit of a vortex image in a Nb film (known parameter λ =50 nm) mounted on a sample holder with a MgB$_{2}$ crystal. The range of λ is a residual based on the error bar in the z value which was assigned by doubling of the χ value in the fit in a Nb film (a doubling of the χ value only gives statistical errors). Due to the error it is difficult to specify how close our λ value is to the band structure calculations $^{6}$. However, first-principles calculations 6,22 of the penetration depth in the clean limit fall on our λ value within the error bar. derived by numerically minimizing the two-gap GL free energy of two vortices with a variational procedure $^{18}$. The minimization gives the interaction consisting of short-range repulsion and weak long-range attraction, which is similar to that reported in Ref. 18. An equation of motion for a vortex i is F i = F vv i + F T i = η v $_{i}$, where F vv i is the vortex-vortex interaction term we calculate above, F T i the thermal stochastic force which satisfies 〈 F T i ( t ) 〉 =0 and 〈 F T i ( t ) F T j ( t $^{′}$) 〉 = 2 ηδ$_{ij}$ δ ( t − t $^{′}$) k$_{B}$T , and η the viscosity, where we take a value esti- mated in Ref. 5. The systems with no pinning cen- ters were initially prepared in a high temperature molten state and then annealed with two million temperature steps. 19 We made the system stable during 2000 time steps in each step of temperature. In type-2 superconductors with the weak pinning, the Abrikosov lattice is often seen in field cooling experi- ments even if a vortex density is quite low: 15 the vortex lattice made of vortices strongly interacting with each other at temperatures very close to T$_{c}$ is frozen with de- creasing temperature. Vortex patterns at very low vortex densities in a MgB$_{2}$ single crystal, which has the weak pinning, 17 seem to be different from that in the conven- tional weak pinning system. At B =1 µ T vortices are located far away from each other.[Fig. 1(a)] However, with increasing applied field vortices come closer to each other to form compact groups as shown in Figs. 1(b) and 1(e), which is never seen in superconductors with the weak pinning, such as MoGe films 15 and NbSe$_{2}$ sin- gle crystals $^{5}$. Figures 1(a) and 1(b) show a scanning SQUID micro- scope images of vortices at very low vortex densities for a MgB$_{2}$ single crystal. The size of a vortex in the image does not correspond straightforwardly to the penetration depth λ but rather to a combination of λ and the extent of a stray field emanating from a vortex at the sample surface. In order to estimate λ it is necessary to compare the cross section of the vortex image with appropriate numerical calculations where the extent of the stray field is taken into account. 15 We calculate numerically the spa- tial distribution of a magnetic field from a vortex above the sample surface, using a London model $^{15,20,21}$.