We have studied the order–disorder transition in high quality MgB2 single crystals, using a torque magnetometry combined with a ‘vortex shaking’ technique. In the wide range of temperature T, field H and the H direction, we succeed in obtaining reversible magnetization curves Mrev(T, H) by shaking the pinned vortices. Especially at low temperatures below 25K and high fields, where the irreversible magnetization curve exhibits the peak effect due to the order–disorder transition, it is found that the peak is transformed into the clear step in Mrev(H). Similar step-like behavior is also observed in the temperature dependence of magnetization Mrev(T). These results give direct evidence that the order–disorder transition, which is hidden by the large hysteresis of magnetization, has the nature of first-order transition.

Measurements of reversible magnetization have been performed on high quality MgB2 single crystals using torque magnetometry combined with the 'vortex shaking' technique. At high temperatures above 26 K, a step like decrease in the magnetization curve is observed at H-m with and without shaking. This anomaly can be ascribed to the first order vortex lattice melting transition. At low temperature below 25 K, where the irreversible magnetization curve reveals the peak effect, we succeed in suppressing the hysteresis completely by shaking and observing the clear step in the reversible magnetization curve at the field H* for the peak effect. The entropy change, estimated from the step of the reversible magnetization at H-m is 2-3 times larger than that at H*.

We have performed the detailed measurements of temperature dependence of magnetization M(T) on MgB2 single crystals in a wide range of field H using torque magnetometry. In a field region, where the peak effect in M(H) is observed, we find that M(T) of the warming branch increases monotonously with T and exhibits a small step-like increase at T1∗, which is related to the order–disorder transition, while M(T) of the cooling branch, showing the larger values than that in warming branch below T1∗, drops suddenly toward the warming branch at T2∗. We ascribe this anomalous behavior of M(T) of the cooling branch to a supercooling phenomenon of disordered phase, which transforms to the ordered state at T2∗.

Anisotropic behavior of pinning properties in MgB2 and Mg0.88Al0.12B2 single crystals has been studied using torque τ (=M x H) magnetometry as a function of magnetic field H and angle θ between the H direction and the c axis. When the direction of H is apart from the ab plane, we do not observe remarkable pinning effect. On the other hand, as θ approaches 90°, the hysteresis in τ(H) and τ(θ) curve suddenly increases and shows the sharp maximum at θ = 90°, which remind us of the so-called intrinsic pinning. However, it is found that this pinning effect is irrelevant to the change of anisotropy γ with temperature in MgB2, and is strongly enhanced in Mg0.88Al0.12B2 in spite of the decrease of γ with Al substitution. These results may indicate another plane-like pinning mechanism, which is extrinsic and become effective by Al substitution.