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superconducting mgb@xmath1 exhibits a number of rather peculiar properties , originating from the involvement of two sets of bands of different anisotropy and different coupling to the most relevant phonon mode @xcite . among them are pronounced deviations of the upper critical field , @xmath0 , from predictions of the widely used anisotropic ginzburg - landau theory ( aglt ) . apart from two - band superconductivity , mgb@xmath1 provides a link between low and high @xmath8 superconductors on a phenomenological level , particularly concerning vortex physics . in both high and low @xmath8 superconductors , for example , a phase transition of vortex matter out of a quasi - ordered `` bragg glass '' have been identified , with rather different positions in the @xmath5-@xmath6 plane . studying the intermediate mgb@xmath1 may help establishing a `` universal vortex matter phase diagram '' .    here , we present a torque magnetometry study of the anisotropic upper critical field , equilibrium magnetization , and the vortex matter phase diagram of single crystalline mgb@xmath1 @xcite . we will show direct evidence of a temperature dependence of the @xmath0 anisotropy , discuss strong indications of a difference between the anisotropies of the penetration depth and @xmath0 , and present the @xmath5-@xmath6 phase diagram for @xmath7 . single crystals were grown with a cubic anvil high pressure technique , described in this issue @xcite . three crystals were used in this study , labeled a , b , and c. sharp transitions to the superconducting state indicate a high quality of the crystals . an @xmath9 curve of crystal b with @xmath10 can be found in ref.@xcite . the torque @xmath11 , where @xmath12 is the magnetic moment of the sample , was recorded as a function of the angle @xmath13 between the applied field @xmath14 and the @xmath15axis of the crystal in various fixed fields @xcite . for measurements close to @xmath8 , in fields up to @xmath16 , a non - commercial magnetometer with very high sensitivity was used @xcite . for part of these measurements , a vortex - shaking process was employed to speed up the relaxation of the vortex lattice @xcite . crystal a was measured in this system . crystals b and c were measured in a wider range of temperatures down to @xmath17 , in a quantum design ppms with torque option and a maximum field of @xmath18 . for crystals b and c , @xmath19 measurements at fixed angles were performed in addition to @xmath20 measurements in fixed @xmath5 . early measurements on polycrystalline or thin film mgb@xmath1 samples with various methods and single crystals by electrical transport yielded values of the anisotropy parameter of the upper critical field @xmath21 in a wide range of values of @xmath22 @xcite . more recently , several papers reported a temperature dependence of the @xmath0 anisotropy , ranging between about @xmath23 at @xmath24 and @xmath25 close to @xmath8 @xcite . in this section , we present direct evidence of a temperature dependence of the @xmath0 anisotropy @xmath2 and discuss details of it s behaviour , comparing the torque data with numerical calculations @xcite . four angular torque dependences are shown in fig.[mgb2hc2raw ] . panels a ) and b ) correspond to measurements at @xmath26 . for fields nearly parallel to the @xmath27-axis , both curves are flat , apart from a small background visible in panel b ) . only when @xmath5 is nearly parallel to the @xmath28plane there is an appreciable torque signal . the curve can be interpreted in a straight - forward way : for @xmath5 parallel to the @xmath15axis the sample is in the normal state , while for @xmath5 parallel to the @xmath28plane it is in the superconducting state . the crossover angle @xmath29 between the normal and the superconducting state is the angle for which the fixed applied field is the upper critical field . from the existence of both superconducting and normal angular regions follows immediately that @xmath30 and @xmath31 . in panel c ) , on the other hand , the crystal is seen to be in the superconducting state for all values of the angle @xmath13 , and therefore @xmath32 . finally , the data in panel d ) show only a small background contribution @xmath33 form and angular regime of the deviation from a straight line are incompatible with a superconducting signal . therefore , the crystal is here in the normal state for any @xmath13 , and we have @xmath34 .    from figure [ mgb2hc2raw ] we therefore have two limitations for the upper critical field anisotropy , hereafter called @xmath2 , without any detailed @xmath0 criterion , and without any model fits : @xmath35 these relations show that _ the upper critical field anisotropy @xmath2 of mgb@xmath1 can not be temperature independent_. as an immediate implication , the _ anisotropic ginzburg - landau theory _ ( aglt ) in it s standard form _ does not hold for mgb@xmath1_. the deviation is strong , within a change of temperature of about @xmath36 , @xmath2 changes , _ at least _ , by a fifth of it s value . although it is clear that aglt with it s effective mass anisotropy model can not describe the data measured at _ different _ temperatures consistently , the detail analysis of the @xmath13 dependence of @xmath0 we used is based on aglt . we will show that as long as we stay at a _ fixed _ temperature , aglt is able to describe @xmath37 remarkably well @xcite . although the location of @xmath29 , for example in fig . [ mgb2hc2raw]a ) , seems clear at first sight , this clarity disappears , when examining the transition region in a scale necessary for the precise determination of @xmath29 ( see fig .  1 in ref . @xcite ) . for an strict analysis , it is necessary to take into account that the transtion at @xmath0 is rounded off by fluctuations .    in sufficiently high fields , @xmath38 , the so - called `` lowest landau level '' ( lll ) approximation was used successfully to describe the effects of fluctuations around @xmath0 @xcite . in the case of the cuprates , the value of @xmath39 , and thus of the regime of applicability of the lll approximation , is a controversial issue ( see , e.g. , ref.@xcite for a theoretical discussion of the limits of the lll approximation ) . however , in the case of mgb@xmath1 , even using the theoretical criterion of ref . @xcite , which led to the high estimation @xmath40 in the case of yba@xmath1cu@xmath41o@xmath42 , we obtain an upper limit of @xmath43 . we therefore used a lll scaling analysis for the determination of @xmath44 or @xmath37 @xcite .    from the resulting @xmath37 curve , the anisotropy parameter @xmath2 is then extracted by an analysis with aglt , which predicts the angular dependence of the upper critical field to be @xcite @xmath45 we note that in the rescaling of the torque according to the lll fluctuations theory , the target parameter @xmath2 is used , which is obtained only later with eq . ( [ hc2_theta ] ) . therefore , scaling analysis and determination of @xmath2 with eq . ( [ hc2_theta ] ) had to be performed iteratively in order to self consistently find @xmath2 . however , the @xmath44 and @xmath37 points obtained with the scaling analysis depend not very strongly on the value of @xmath2 used in the scaling and the procedure converges rather fast . figure [ hc2crystc ] shows the angular dependence of @xmath0 of crystal b and c. the curves shown in the figure are fits of eq . ( [ hc2_theta ] ) to the data , showing that the angular dependence of @xmath0 is well described by aglt at both temperatures . on the other hand , the anisotropy parameter @xmath2 needed to describe the data with eq.([hc2_theta ] ) is temperature dependent , as is best seen in the inset . dependence of crystals b and c , at @xmath46 and @xmath47 . shown are results obtained both from @xmath20 measurements in fixed @xmath5 [ b at @xmath26 ( @xmath48 ) and at @xmath47 ( @xmath49 ) , c at @xmath26 ( @xmath50 ) and at @xmath47 ( @xmath51 ) ] and from @xmath19 measurements at fixed @xmath13 [ b at @xmath26 ( @xmath52 ) ] . inset : calculated curves from the main panel , scaled to the fitted value of @xmath53 . ]     vs.temperature @xmath6 . open symbols correspond to @xmath54 , full symbols to @xmath55 , from fits of eq . ( [ hc2_theta ] ) to the @xmath37 data . up triangles are from measurements on sample a ( with @xmath29 determined with a simple `` straight line crossing '' criterion ) and squares are from measurements on sample b , using a fluctuation analysis ( see text ) . full lines are theoretical calculated curves @xcite . b ) temperature dependence of the upper critical field anisotropy @xmath21 , determined from fits of eq . ( [ hc2_theta ] ) to @xmath37 . the full line is again from the theoretical calculation of ref . @xcite . ] the irreversible properties of the two crystals ( b and c ) are different in a pronounced way ( see secs . [ lockin ] and [ pesec ] ) , showing that they have a rather different defect structure . the good agreement both in value and angular dependence of @xmath0 of crystals b and c that is observable in fig . [ hc2crystc ] indicates that such differences in the defect structure do not influence the upper critical field much , at least in the region between @xmath46 and @xmath47 , and therefore can not influence our conclusion of a @xmath6 dependent @xmath0 anisotropy . small , but systematic , deviations from the angular dependence of @xmath0 according to eq . ( [ hc2_theta ] ) were observed only at temperatures close to @xmath8 . it may indicate that we are approaching @xmath39 in this region and the values of @xmath44 and @xmath37 obtained from the lll scaling analysis ( see above ) start to deviate from the mean field values . the good general approximation of @xmath37 by eq.([hc2_theta ] ) is in agreement with recent calculations @xcite . however , the calculations predict small deviations at low temperatures @xcite , rather than close to @xmath8 . our experimental limitation of fields up to @xmath56 may prevent the observation of deviations from eq . ( [ hc2_theta ] ) at low temperatures .    the upper critical fields parallel and perpendicular to the layers obtained with the scaling analysis and eq .  ( [ hc2_theta ] ) are shown in fig .  [ hc2mgb2sumup]a ) . results obtained for two crystals measured in two magnetometers are depicted as different symbols . the @xmath57dependence of @xmath53 is in agreement with ( isotropic ) calculations by helfand _ @xcite , with @xmath58 . on the other hand , @xmath59 exhibits a slight positive curvature near @xmath8 . these features are common to highly anisotropic ( layered ) superconductors . although mgb@xmath1 as a whole is rather isotropic , superconductivity is dominant on the quasi-2d bands , which may well account for the different @xmath6 dependence of @xmath53 and @xmath60 . this may also be the origin of the positive curvature of @xmath0 observed in other measurements of bulk , thin film and single crystal mgb@xmath1 @xcite . due to the lack of low @xmath6 data and the @xmath61 dependence , only an estimation @xmath62 can be given . the anisotropy data [ fig . [ hc2mgb2sumup]b ) ] show that @xmath2 _ systematically _ _ decreases _ with increasing temperature , from @xmath63 at @xmath64 to @xmath65 at @xmath66 . from the experimental data shown in fig . [ hc2mgb2sumup ] we estimate @xmath67 , while at zero temperature , @xmath2 may become as large as @xmath68 . comparing our data with the data reported by other authors @xcite , we note that electrical transport measurements @xcite yield too high values of @xmath53 @xcite . all bulk measurements ( torque @xcite , magnetization @xcite , thermal conductivity @xcite , and specific heat @xcite ) agree well on the @xmath69 dependence and value . concerning @xmath59 , and consequently @xmath61 , however , reported values differ from each other . exchanging the samples between different groups could help clarifying , whether the discrepancies of @xmath59 values are mainly due to sample differences or due to differences in the experimental methods employed . very recently , @xmath59 , @xmath69 , and @xmath61 , have been calculated for mgb@xmath1 @xcite . the fermi surface was modeled as consisting of two separate sheets , approximated as simple spheroids , but with average characteristics taken from first principles calculations . the result of these calculations are compared with our experimental data in fig.[hc2mgb2sumup ] . very good agreement is seen for the upper critical field perpendicular to the layers ( @xmath70 ) . qualitatively , calculations and experiment also agree well for the upper critical field parallel to the layers ( @xmath71 ) . this shows that the essential source of the deviations of the upper critical field from aglt predictions is captured with a simple effective two band model , while further details of the fermi surface and superconducting gap are negligible . in fig . [ hc2mgb2sumup ] , we see good quantitative agreement between experimental data and the theoretical curve between @xmath72 and @xmath73 .    the deviations at lower @xmath6 may , on the one hand , be due to a decreased the accuracy of our analysis because the field limitation of @xmath56 restricts the angular range where @xmath0 data could be obtained . this can lead to deviations larger than the estimated error bars , especially since the theoretical calculations indicate deviations of the @xmath74 dependence from the prediction of eq.([hc2_theta ] ) at low @xmath6 @xcite . on the other hand , @xmath0 at low temperatures depends on the shape of the fermi surface in rather subtle manner , and the model fermi surface used for the calculations @xcite may be too simple for a quantitatively correct description at low @xmath6 . the deviations at higher @xmath6 may be due to the limitations of the lll scaling approach in low fields , and or due to the influence of disorder , which is not accounted for in the calculations . close to @xmath8 , non - locality is not important , and consequently , aglt is expected to hold even if this is not the case at lower @xmath6 . despite of this , fig . [ hc2mgb2sumup ] clearly indicates that the variation of @xmath2 with temperature is the strongest close to @xmath8 . therefore , in mgb@xmath1 , aglt seems to have a very limited range of applicability indeed . an alternative method to obtain the anisotropy parameter @xmath75 of a superconductor , used often and with success in the case of cuprates @xcite , consists of measuring the torque , as a function of angle , well below @xmath0 , and analyzing the data with a formula developed by kogan @xcite : @xmath76 where @xmath77 , @xmath78 is the effective mass anisotropy , @xmath79 is the in - plane penetration depth , @xmath80 is the volume of the crystal , @xmath81 is the flux quantum , and @xmath82 is a constant of the order of unity depending on the vortex lattice structure . equation ( [ tau_rev ] ) is valid in the limits of fields @xmath83 and not too close to @xmath8 . a further restriction is that eq . ( [ tau_rev ] ) describes the reversible torque only . to obtain the true reversible torque , we employed a vortex - shaking process @xcite . in the investigated field and temperature region , the shaked torque was found to be well reversible .     in @xmath84 [ a ) ] and @xmath85 [ b ) ] . shown together with the measured values ( open circles ) are best fits of eq . ( [ tau_rev ] ) ( full curves ) . ]    in fig . [ mgb2taurev ] , normalized torque @xmath86 vs angle @xmath13 curves , measured in different fields at @xmath47 , are compared . increasing @xmath5 from @xmath87 [ panel a ) ] to @xmath85 [ panel b ) ] leads to an unexpectedly large shift of the maximum torque towards @xmath88 , which may indicate an increase of the anisotropy @xmath3 with increasing @xmath5 . this is confirmed by the analysis of the data with eq . ( [ tau_rev ] ) . the best agreements of the equation with the data are obtained for @xmath89 in @xmath84 and @xmath90 in @xmath85 ( full curves in fig . [ mgb2taurev ] ) . although descriptions with @xmath90 in @xmath84 or @xmath89 in @xmath85 are also possible without obvious discrepancies to the data , the corresponding qualities of the fit as expressed by the parameter @xmath91 are worse by more than an order of magnitude in both cases .    from the analysis of reversible torque data for crystal a measured in the range of fields and temperatures from @xmath92 to @xmath93 and from @xmath94 to @xmath95 @xcite , a few points are worth to be emphasized : 1 . ) @xmath3 is field dependent , increasing nearly linearly from @xmath87 in zero field to @xmath96 in @xmath93 . no clear @xmath6 dependence is visible between @xmath94 and @xmath95 . 3 . ) the effective anisotropy @xmath3 , as obtained from the analysis with eq.([tau_rev ] ) is different from the @xmath0 anisotropy @xmath2 .    especially concerning point 3 . ) , it is important to recognize that , also theoretically , the anisotropy @xmath3 is not necessarily the same as the @xmath0 anisotropy @xmath2 . when aglt is not applicable , the anisotropies of the penetration depth , @xmath97 , and of the upper critical field , @xmath2 , differ in general . calculations of @xmath97 of mgb@xmath1 @xcite indeed found values much lower than the upper critical field anisotropy values . there is also experimental support for a low @xmath97 @xcite .        in eq . ( [ tau_rev ] ) , @xmath3 appears twice , and in a first approximation @xcite , the appearance outside of the logarithm can be thought of as due to the @xmath98 anisotropy , while the appearance in the logarithm is linked to the @xmath0 anisotropy . a corresponding calculation with different ( fixed ) @xmath97 and @xmath2 yields @xcite a field dependent ( common ) effective anisotropy @xmath3 similar to the experimental observations . field dependent point - contact spectroscopy @xcite and specific heat @xcite measurements indicate that the small gap disappears in fields of the order of @xmath99 , i.e. , superconductivity in the ( 3d ) @xmath100 fermi sheets is rapidly suppressed by even low fields , whereas it persists in the ( 2d ) @xmath101 sheets up to much higher fields . this should result in an increase of the effective ( bulk ) anisotropy with increasing @xmath5 . further studies are needed for a complete understanding of the detailed interplay of the effects described above .    in the `` unshaked '' torque data of crystals a and b , a pronounced peak in the irreversible torque for field alignments close to @xmath102 , was observed ( see , e.g. , upper inset of fig .  5 of ref.@xcite ) . it is tempting to ascribe this feature , also observed by other authors @xcite , to `` intrinsic pinning '' , in analogy to observations on strongly anisotropic cuprate superconductors . however , the observation of such `` intrinsic pinning '' in mgb@xmath1 is rather counter - intuitive , since the `` intrinsic pinning '' is mostly determined by the ratio of the @xmath27-axis coherence length to the separation of the superconducting layers , which is much larger in mgb@xmath1 than in the case of cuprates in the region where `` intrinsic pinning '' is commonly observed by torque magnetometry ( see , e.g. , @xcite ) . the apparent paradox is resolved by further measurements : torque measurements on crystal c with the same conditions show no sign of `` intrinsic '' pinning for @xmath102 @xcite , indicating the _ extrinsic _ origin of the feature . the most likely cause of the peak in the irreversible torque for @xmath102 is a small amount of stacking faults . it may indicate the presence of some stacking faults in crystals a and b , while they would seem to be absent in crystal c @xcite . in low @xmath8 superconductors , such as nbse@xmath1 , the order - disorder transition is signified experimentally by a peak effect ( pe ) in the critical current density @xcite . we observed such a pe by torque measurements on mgb@xmath1 single crystals b and c , both in @xmath20 and @xmath19 measurements , as can be seen in fig.[peakmgb2 ] . in sec . [ lockin ] , we have noticed that crystals b and c behave quite differently for @xmath102 , which may be due to the presence of a small number of stacking faults in the former one . the presence of the pe in two crystals with such pronounced differences strongly indicates that the pe , or rather it s underlying mechanism , is an intrinsic feature of mgb@xmath1 . a study with a `` minor hysteresis loop '' technique on crystal b @xcite revealed a history dependent critical current density in the pe region , compatible with and expected for the behaviour at the order - disorder transition .    , vs field , at @xmath103 . shown are representative curves , measured at angles ( from left to right ) @xmath104 , @xmath105 , @xmath106 , @xmath107 , @xmath108 , @xmath109 , @xmath110 , and @xmath111 . horizontal lines indicate zero and the criterion of @xmath112 chosen for the determination of the irreversibility line . for the curve measured at @xmath113 , peak maximum and onset are indicated ( see text ) . ] figure [ perawtheta ] shows the irreversible part @xmath114 of the torque , scaled by @xmath115 , vs field , at @xmath103 for various angles . the scaling was chosen to minimize the angle and field dependence intrinsic to the torque . since the peak is not visible at all temperatures and angles as well as in fig.[peakmgb2 ] , onsets and maxima were determined from irreversible torque curves as those shown in fig.[perawtheta ] . @xmath116 was defined as the field , where the irreversible torque starts to deviate from a straight line behaviour , as indicated in the figure for the curve measured at @xmath117 . @xmath116 , defined in this way , is close to @xmath118 as indicated in fig.[peakmgb2]b ) . however , we note that with the determination of onsets and maxima from the irreversible torque , the fine details of the differences in the field increasing and decreasing branch of the hysteresis loops are lost .    . a ) torque @xmath119 vs @xmath5 , measured at an angle of @xmath120 between the @xmath27-axis of the sample and the applied field . due to the anisotropy , the location in fields of the pe is much higher than for @xmath7 , but the angular scaling is straight forward @xcite . b ) magnetization @xmath121 vs @xmath5 , for @xmath7 , determined by squid magnetometry . c ) absolute magnetic susceptibility @xmath122 vs @xmath123 , measured with an ac amplitude @xmath124 and a frequency of @xmath125 . the pe visible in panels a ) and c ) is indicated by arrows . ] it can be seen in fig . [ perawtheta ] that the height of the peaks varies in a pronounced way with the angle @xmath13 . one possible explanation for this behaviour is an interaction of the peak effect with stacking faults @xcite . although the presence and the location of the peak effect are not affected by stacking faults , the extent of hysteresis may be . the difference of how pronounced the peaks of crystals b and c are [ see fig . [ peakmgb2]b ) ] supports such a scenario . the location in higher fields of the peak effect in crystal c indicates that there is less point - like disorder present in this crystal than in crystal b. however , the smaller ratio @xmath126 in crystal c , compared to crystal b , is difficult to explain with only one sort of disorder . individual strong pinning , e.g. , by sparse stacking faults , should be much more efficient in the disordered phase than in the bragg glass with it s nearly perfect ordered lattice @xcite . if the peak height observed in crystal b is affected by stacking faults , a more pronounced pe close to @xmath102 is natural , since the pinning efficiency of stacking faults , similar to twin boundary pinning , is strongly direction - dependent @xcite .    on the other hand , the peak height can be influenced by the natural angle dependence of the torque , despite the scaling made . this is because the @xmath127 is only an approximation , which is not appropriate for all angles @xmath13 , in a superconductor with pronounced anisotropy ( see also ref . @xcite ) . the angular dependence of the _ onsets and maxima _ of the pe tracks the one of @xmath0 , i.e. , it follows eq.([hc2_theta ] ) @xcite . this indicates that the pe ( or rather it s underlying mechanism ) is a feature for all directions of the applied field , and not just of the angular region where it is readily discernible . to directly check the situation for @xmath7 and @xmath102 , where torque measurements are not possible , squid and ac susceptibility @xcite measurements were performed . ( crystal b ) for @xmath7 . data points shown are from the projection of the torque vs field data presented in ref . @xcite and the projection of additional @xmath20 measurements in fixed @xmath5 . the dashed line is a calculation @xcite of the @xmath128 dependence ( cf . [ hc2mgb2sumup ] ) , the dotted lines are guides for the eye . the different phases of vortex matter are labeled ( see text ) . ]    in fig .  [ pedifftech ] , we compare measurement curves obtained on crystal b at @xmath129 , using different experimental techniques . torque measurements performed at an angle of @xmath120 show [ fig . [ pedifftech]a ) ] a clearly discernible pe located in a field of about @xmath130 . scaled with eq . ( [ hc2_theta ] ) to @xmath7 , this corresponds to @xmath131 to @xmath132 . as can be seen in fig.[pedifftech]b ) , there is no sign of a pe observable in squid data in this field region . generally , no sign of a peak effect was observed by squid magnetometry at any temperature , for both field directions . this is likely due to insufficient sensitivity of the squid . in ac susceptibility data [ fig . [ pedifftech]c ) ] , on the other hand , a pe _ is _ visible for @xmath7 in the appropriate field region . a report of the ac susceptibility results will be published elsewhere @xcite . a pe in mgb@xmath1 was also reported recently by other authors , in the case of @xmath7 from transport data @xcite and ac susceptibility with a local hall probe @xcite , in the case of @xmath102 from transport data @xcite . the phase diagram for @xmath7 obtained from torque magnetometry , based on both @xmath20 and @xmath19 measurements and the angular scaling of eq . ( [ hc2_theta ] ) is presented in fig.[mgb2phdhc ] . the magnitude of the peaks is reduced quickly by increasing the temperature , and above @xmath133 , the pe is no longer discernible in the torque data . this is due to the decreased sensitivity of the torque magnetometer in lower fields and due to thermal smearing of the effective pinning potential . in a recent report of low frequency ac susceptibility measurements @xcite , the peak effect was observed for @xmath7 at temperatures up to about @xmath73 , and interpreted in terms of the order - disorder transition as well . in ref.@xcite , the transformation of the pe into a `` step - like '' ac susceptibility is reported for the temperature interval between @xmath134 and @xmath135 , and interpreted as a signature of thermal melting . in our case , no step - like feature in the reversible torque was observed in the continuation of the pe . it should be emphasized , that thermal melting so far below @xmath0 would be at odds @xcite with theoretical expectations @xcite . the equilibrium order - disorder transition , which corresponds to @xmath136 @xcite , is located in fields of about @xmath4 in crystal b and in about @xmath137 in crystal c. the peak effect observed in other crystals by transport was reported to be located even closer to @xmath0 @xcite . these differences are natural for a disorder - induced phase transition in crystals with varying degrees of disorder . form and location of the pe observed in mgb@xmath1 resembles results obtained on nbse@xmath1 single crystals with varying degrees of disorder @xcite , but are rather different from the order - disorder transition in cuprate superconductors @xcite . in summary , studying the anisotropic superconducting state properties of mgb@xmath1 revealed a strong temperature dependence of the upper critical field anisotropy @xmath2 , and indicated a difference of the anisotropies of the penetration depth and the upper critical field . these findings , which imply a breakdown of the standard form of the widely used anisotropic ginzburg - landau theory in mgb@xmath1 , can be explained by superconductivity in this compound involving two band systems of different dimensionality , in accordance with microscopic studies . a pronounced peak effect in the magnetic hysteresis is a signature of an `` order - disorder '' transition of vortex matter , similar to transitions in both high @xmath8 cuprate and low @xmath8 superconductors . despite the intermediate importance of thermal fluctuations in mgb@xmath1 , the phase diagram resembles quite closely the one of the low @xmath8 superconductor nbse@xmath1 . on the other hand , chances of a proper identification of mainly thermally induced melting at higher temperatures are better in mgb@xmath1 , due to the increased thermal fluctuations . we thank v.  g. kogan and b.  batlogg for enlightening and stimulating discussions . this work was supported by the swiss national science foundation , by the european community ( contract ica1-ct-2000 - 70018 ) , by the polish state committee for scientific research ( 5 p03b 12421 ) , and by the swiss federal office bbw ( 02.0362 ) . note that there are two torque conventions used in the literature , leading to opposite signs of the torque shown , but to the same physical consequences . the convention used here ( and in refs.@xcite ) is that the diamagnetism is expressed by a negative value of the magnetic moment . the other convention , used in refs.@xcite , is that the diamagnetism is expressed solely by the direction of the ( positive ) magnetic moment vector . l. lyard , p. samuely , p. szabo , c. marcenat , t. klein , k.  h.  p. kim , c.  u. jung , h .- s . lee , b. kang , s. choi , s .- lee , l. paulius , j. marcus , s. blanchard , a.  g.  m. jansen , u. welp , and w.  k. kwok , cond - mat/0206231 . this justifies our aglt based analysis _ a posteriori_. an only small deviation of the @xmath138dependence of @xmath139 from aglt predictions ( with a @xmath6 dependent anisotropy ) was also found in numerical calculations @xcite .                            in a very recent study , kogan [ v. g. kogan , cond - mat/0207688 ] performed detailed calculations of the torque density from the free energy of the london model with different anisotropies @xmath2 and @xmath97 . the obtained result is more complicated than the first approximation used in ref . @xcite . x ray investigations of crystals b and c are in progress . stacking faults have been observed in other samples of mgb@xmath1 , see , e.g. , y.  zhu , l.  wu , v.  volkov , q.  li , g.  gu , a.  r. moodenbaugh , m.  malac , m.  suenaga , and j.  tranquada , physica c * 356 * , 239 ( 2001 ) . see , e.g. , s.  s. banerjee , n.  g. patil , s. ramakrishnan , a.  k. grover , s. bhattacharya , p.  k. mishra , g. ravikumar , t.  v.  c. rao , v.  c. sahni , m.  j. higgins , c.  v. tomy , g. balakrishnan , and d.  m. paul , phys . b * 59 * , 6043 ( 1999 ) .    in the bragg glass , the strong elastic forces disfavour non - collective pinning . similar ideas were discussed in the context of cuprate superconductors with twin boundaries . see a.  i. larkin , m.  c. marchetti , and v.  m. vinokur , phys . . lett . * 75 * , 2992 ( 1995 ) . similar observations were made on a yba@xmath1cu@xmath41o@xmath42 crystal containing two twin boundaries . see w.  k. kwok , j.  a. fendrich , c.  j.  v. der beek , and g.  w. crabtree , phys . lett . * 73 * , 2614 ( 1994 ) .
the angular and temperature dependence of the upper critical field @xmath0 in mgb@xmath1 was determined from torque magnetometry measurements on single crystals . 
 the @xmath0 anisotropy @xmath2 was found to decrease with increasing temperature , in disagreement with the anisotropic ginzburg - landau theory , which predicts that the @xmath2 is temperature independent . 
 this behaviour can be explained by the two band nature of superconductivity in mgb@xmath1 . 
 an analysis of measurements of the reversible torque in the mixed state yields a field dependent effective anisotropy @xmath3 , which can be at least partially explained by different anisotropies of the penetration depth and the upper critical field . 
 it is shown that a peak effect in fields of about @xmath4 is a manifestation of an _ order - disorder _ phase transition of vortex matter . 
 the @xmath5-@xmath6 phase diagram of mgb@xmath1 for @xmath7 correlates with the intermediate strength of thermal fluctuations in mgb@xmath1 , as compared to those in high and low @xmath8 superconductors .