Novel order parameter to describe the critical behavior of Ising spin glass models

高分子物理二、高聚物粘性流动有哪些特点？影响粘流温度T f的主要因素是什么？（8分）答：粘性流动的特点：1.高分子流动是通过链段的位移运动来完成的；2. 高分子流动不符合牛顿流体的流动规律；3. 高分子流动时伴有高弹形变。

影响T f的主要因素：1. 分子链越柔顺，粘流温度越低；而分子链越刚性，粘流温度越高。

2. 分子间作用力大，则粘流温度高。

3. 分子量愈大，愈不易进行位移运动，Tf越高。

4. 粘流温度与外力大小和外力作用的时间增大，Tf下降。

三、画出牛顿流体、切力变稀流体、切力变稠流体、宾汉流体的流动曲线，写出相应的流动方程。

（8分）答：牛顿流体η为常数切力变稀流体n < 1切力变稠流体n >1宾汉流体σy为屈服应力四、结晶聚合物为何会出现熔限？熔限与结晶形成温度的关系如何？答：1.结晶聚合物出现熔限，即熔融时出现的边熔融边升温的现象是由于结晶聚合物中含有完善程度不同的晶体之故。

聚合物的结晶过程中，随着温度降低，熔体粘度迅速增加，分子链的活动性减小，在砌入晶格时来不及作充分的位置调整，而使形成的晶体停留在不同的阶段上。

在熔融过程中，则比较不完善的晶体将在较低的温度下熔融，较完善的晶体需在较高的温度下才能熔融，从而在通常的升温速度下，呈现一个较宽的熔融温度范围。

2. 低温下结晶的聚合物其熔限范围较宽，在较高温度下结晶的聚合物熔限范围较窄。

五、测定聚合物分子量有哪些主要的方法？分别测定的是什么分子量？除了分子量外还能得到哪些物理量？聚合物分子量的大小对材料的加工性能和力学性能有何影响？（10分）答：端基分析法和渗透压测定的是数均分子量，光散射测定的是重均分子量，粘度法测定的是粘均分子量。

分子量太低，材料的机械强度和韧性都很差，没有应用价值；分子量太高，熔体粘度增加，给加工成型造成困难。

七、解释下列现象（6分）：1. 尼龙6（PA6）室温下可溶于浓硫酸，而等规聚丙烯却要在130℃左右才能溶于十氢萘。

临界问题物理经典模型临界问题是物理学中的一个经典问题，指的是一个系统在某个特定的参数变化下由一个状态转变为另一个状态的临界点。

在物理学中，临界问题通常指的是相变问题，也就是当一个物质的温度、压力、密度等参数发生变化时，它会从一个相态转变到另一个相态的临界点问题。

临界问题在理论物理、统计物理等领域中有着广泛的应用和研究。

在物理学中，许多系统都具有临界性质，其中最典型的就是液体-气体相变和磁性相变。

这类系统的一般特点是在一个临界点附近，相变现象变得非常剧烈，因此需要有一种模型来描述相变过程。

在物理学中，一个经典的模型是Ising模型，它是用于描述磁性相变的一种模型，可以帮助我们更好地理解相变现象。

Ising模型最初由德国物理学家Ising于1925年提出，它是一种简化的模型，用于研究磁性材料中磁场引起的自旋取向改变现象。

该模型中的自旋只能取两个值：向上或向下，它们与周围自旋的相互作用会导致自旋的取向变化。

在Ising模型中，磁性相变通常发生在某个特定的临界温度下，当温度超过这个临界温度时，系统从有序到无序相变，从而导致磁性的消失。

除了Ising模型之外，还有其他一些经典模型用于描述相变现象，比如Potts模型、XY模型等。

这些模型都有着自己独特的数学形式和物理特性，可以帮助我们更好地理解相变过程。

在研究临界问题时，物理学家通常可以使用这些模型来解决问题，通过计算并预测相变的行为。

总的来说，临界问题是物理学中的一个重要领域，它涉及到相变、相态转变等多个方面。

通过使用适当的模型，我们可以更好地理解并预测这些现象，为相应的实际应用提供理论基础和指导。

a r X i v :c o n d -m a t /9503139v 1 27 M a r 1995Singularity of the density of states in the two-dimensional Hubbard model from finitesize scaling of Yang-Lee zerosE.Abraham 1,I.M.Barbour 2,P.H.Cullen 1,E.G.Klepfish 3,E.R.Pike 3and Sarben Sarkar 31Department of Physics,Heriot-Watt University,Edinburgh EH144AS,UK 2Department of Physics,University of Glasgow,Glasgow G128QQ,UK 3Department of Physics,King’s College London,London WC2R 2LS,UK(February 6,2008)A finite size scaling is applied to the Yang-Lee zeros of the grand canonical partition function for the 2-D Hubbard model in the complex chemical potential plane.The logarithmic scaling of the imaginary part of the zeros with the system size indicates a singular dependence of the carrier density on the chemical potential.Our analysis points to a second-order phase transition with critical exponent 12±1transition controlled by the chemical potential.As in order-disorder transitions,one would expect a symmetry breaking signalled by an order parameter.In this model,the particle-hole symmetry is broken by introducing an “external field”which causes the particle density to be-come non-zero.Furthermore,the possibility of the free energy having a singularity at some finite value of the chemical potential is not excluded:in fact it can be a transition indicated by a divergence of the correlation length.A singularity of the free energy at finite “exter-nal field”was found in finite-temperature lattice QCD by using theYang-Leeanalysisforthechiral phase tran-sition [14].A possible scenario for such a transition at finite chemical potential,is one in which the particle den-sity consists of two components derived from the regular and singular parts of the free energy.Since we are dealing with a grand canonical ensemble,the particle number can be calculated for a given chem-ical potential as opposed to constraining the chemical potential by a fixed particle number.Hence the chem-ical potential can be thought of as an external field for exploring the behaviour of the free energy.From the mi-croscopic point of view,the critical values of the chemical potential are associated with singularities of the density of states.Transitions related to the singularity of the density of states are known as Lifshitz transitions [15].In metals these transitions only take place at zero tem-perature,while at finite temperatures the singularities are rounded.However,for a small ratio of temperature to the deviation from the critical values of the chemical potential,the singularity can be traced even at finite tem-perature.Lifshitz transitions may result from topological changes of the Fermi surface,and may occur inside the Brillouin zone as well as on its boundaries [16].In the case of strongly correlated electron systems the shape of the Fermi surface is indeed affected,which in turn may lead to an extension of the Lifshitz-type singularities into the finite-temperature regime.In relating the macroscopic quantity of the carrier den-sity to the density of quasiparticle states,we assumed the validity of a single particle excitation picture.Whether strong correlations completely distort this description is beyond the scope of the current study.However,the iden-tification of the criticality using the Yang-Lee analysis,remains valid even if collective excitations prevail.The paper is organised as follows.In Section 2we out-line the essentials of the computational technique used to simulate the grand canonical partition function and present its expansion as a polynomial in the fugacity vari-able.In Section 3we present the Yang-Lee zeros of the partition function calculated on 62–102lattices and high-light their qualitative differences from the 42lattice.In Section 4we analyse the finite size scaling of the Yang-Lee zeros and compare it to the real-space renormaliza-tion group prediction for a second-order phase transition.Finally,in Section 5we present a summary of our resultsand an outlook for future work.II.SIMULATION ALGORITHM AND FUGACITY EXPANSION OF THE GRAND CANONICALPARTITION FUNCTIONThe model we are studying in this work is a two-dimensional single-band Hubbard HamiltonianˆH=−t <i,j>,σc †i,σc j,σ+U i n i +−12 −µi(n i ++n i −)(1)where the i,j denote the nearest neighbour spatial lat-tice sites,σis the spin degree of freedom and n iσis theelectron number operator c †iσc iσ.The constants t and U correspond to the hopping parameter and the on-site Coulomb repulsion respectively.The chemical potential µis introduced such that µ=0corresponds to half-filling,i.e.the actual chemical potential is shifted from µto µ−U412.(5)This transformation enables one to integrate out the fermionic degrees of freedom and the resulting partition function is written as an ensemble average of a product of two determinantsZ ={s i,l =±1}˜z = {s i,l =±1}det(M +)det(M −)(6)such thatM ±=I +P ± =I +n τ l =1B ±l(7)where the matrices B ±l are defined asB ±l =e −(±dtV )e −dtK e dtµ(8)with V ij =δij s i,l and K ij =1if i,j are nearestneigh-boursand Kij=0otherwise.The matrices in (7)and (8)are of size (n x n y )×(n x n y ),corresponding to the spatial size of the lattice.The expectation value of a physical observable at chemical potential µ,<O >µ,is given by<O >µ=O ˜z (µ){s i,l =±1}˜z (µ,{s i,l })(9)where the sum over the configurations of Ising fields isdenoted by an integral.Since ˜z (µ)is not positive definite for Re(µ)=0we weight the ensemble of configurations by the absolute value of ˜z (µ)at some µ=µ0.Thus<O >µ= O ˜z (µ)˜z (µ)|˜z (µ0)|µ0|˜z (µ0)|µ0(10)The partition function Z (µ)is given byZ (µ)∝˜z (µ)N c˜z (µ0)|˜z (µ0)|×e µβ+e −µβ−e µ0β−e −µ0βn (16)When the average sign is near unity,it is safe to as-sume that the lattice configurations reflect accurately thequantum degrees of freedom.Following Blankenbecler et al.[1]the diagonal matrix elements of the equal-time Green’s operator G ±=(I +P ±)−1accurately describe the fermion density on a given configuration.In this regime the adiabatic approximation,which is the basis of the finite-temperature algorithm,is valid.The situa-tion differs strongly when the average sign becomes small.We are in this case sampling positive and negative ˜z (µ0)configurations with almost equal probability since the ac-ceptance criterion depends only on the absolute value of ˜z (µ0).In the simulations of the HSfields the situation is dif-ferent from the case of fermions interacting with dynam-ical bosonfields presented in Ref.[1].The auxilary HS fields do not have a kinetic energy term in the bosonic action which would suppress their rapidfluctuations and hence recover the adiabaticity.From the previous sim-ulations on a42lattice[3]we know that avoiding the sign problem,by updating at half-filling,results in high uncontrolledfluctuations of the expansion coefficients for the statistical weight,thus severely limiting the range of validity of the expansion.It is therefore important to obtain the partition function for the widest range ofµ0 and observe the persistence of the hierarchy of the ex-pansion coefficients of Z.An error analysis is required to establish the Gaussian distribution of the simulated observables.We present in the following section results of the bootstrap analysis[17]performed on our data for several values ofµ0.III.TEMPERATURE AND LATTICE-SIZEDEPENDENCE OF THE YANG-LEE ZEROS The simulations were performed in the intermediate on-site repulsion regime U=4t forβ=5,6,7.5on lat-tices42,62,82and forβ=5,6on a102lattice.The ex-pansion coefficients given by eqn.(14)are obtained with relatively small errors and exhibit clear Gaussian distri-bution over the ensemble.This behaviour was recorded for a wide range ofµ0which makes our simulations reli-able in spite of the sign problem.In Fig.1(a-c)we present typical distributions of thefirst coefficients correspond-ing to n=1−7in eqn.(14)(normalized with respect to the zeroth power coefficient)forβ=5−7.5for differ-entµ0.The coefficients are obtained using the bootstrap method on over10000configurations forβ=5increasing to over30000forβ=7.5.In spite of different values of the average sign in these simulations,the coefficients of the expansion(16)indicate good correspondence between coefficients obtained with different values of the update chemical potentialµ0:the normalized coefficients taken from differentµ0values and equal power of the expansion variable correspond within the statistical error estimated using the bootstrap analysis.(To compare these coeffi-cients we had to shift the expansion by2coshµ0β.)We also performed a bootstrap analysis of the zeros in theµplane which shows clear Gaussian distribution of their real and imaginary parts(see Fig.2).In addition, we observe overlapping results(i.e.same zeros)obtained with different values ofµ0.The distribution of Yang-Lee zeros in the complexµ-plane is presented in Fig.3(a-c)for the zeros nearest to the real axis.We observe a gradual decrease of the imaginary part as the lattice size increases.The quantitative analysis of this behaviour is discussed in the next section.The critical domain can be identified by the behaviour of the density of Yang-Lee zeros’in the positive half-plane of the fugacity.We expect tofind that this density is tem-perature and volume dependent as the system approaches the phase transition.If the temperature is much higher than the critical temperature,the zeros stay far from the positive real axis as it happens in the high-temperature limit of the one-dimensional Ising model(T c=0)in which,forβ=0,the points of singularity of the free energy lie at fugacity value−1.As the temperature de-creases we expect the zeros to migrate to the positive half-plane with their density,in this region,increasing with the system’s volume.Figures4(a-c)show the number N(θ)of zeros in the sector(0,θ)as a function of the angleθ.The zeros shown in thesefigures are those presented in Fig.3(a-c)in the chemical potential plane with other zeros lying further from the positive real half-axis added in.We included only the zeros having absolute value less than one which we are able to do because if y i is a zero in the fugacity plane,so is1/y i.The errors are shown where they were estimated using the bootstrap analysis(see Fig.2).Forβ=5,even for the largest simulated lattice102, all the zeros are in the negative half-plane.We notice a gradual movement of the pattern of the zeros towards the smallerθvalues with an increasing density of the zeros nearθ=πIV.FINITE SIZE SCALING AND THESINGULARITY OF THE DENSITY OF STATESAs a starting point for thefinite size analysis of theYang-Lee singularities we recall the scaling hypothesis forthe partition function singularities in the critical domain[11].Following this hypothesis,for a change of scale ofthe linear dimension LLL→−1),˜µ=(1−µT cδ(23)Following the real-space renormalization group treatmentof Ref.[11]and assuming that the change of scaleλisa continuous parameter,the exponentαθis related tothe critical exponentνof the correlation length asαθ=1ξ(θλ)=ξ(θ)αθwe obtain ξ∼|θ|−1|θ|ναµ)(26)where θλhas been scaled to ±1and ˜µλexpressed in terms of ˜µand θ.Differentiating this equation with respect to ˜µyields:<n >sing =(−θ)ν(d −αµ)∂F sing (X,Y )ν(d −αµ)singinto the ar-gument Y =˜µαµ(28)which defines the critical exponent 1αµin terms of the scaling exponent αµof the Yang-Lee zeros.Fig.5presents the scaling of the imaginary part of the µzeros for different values of the temperature.The linear regression slope of the logarithm of the imaginary part of the zeros plotted against the logarithm of the inverse lin-ear dimension of the simulation volume,increases when the temperature decreases from β=5to β=6.The re-sults of β=7.5correspond to αµ=1.3within the errors of the zeros as the simulation volume increases from 62to 82.As it is seen from Fig.3,we can trace zeros with similar real part (Re (µ1)≈0.7which is also consistentwith the critical value of the chemical potential given in Ref.[22])as the lattice size increases,which allows us to examine only the scaling of the imaginary part.Table 1presents the values of αµand 1αµδ0.5±0.0560.5±0.21.3±0.3∂µ,as a function ofthe chemical potential on an 82lattice.The location of the peaks of the susceptibility,rounded by the finite size effects,is in good agreement with the distribution of the real part of the Yang-Lee zeros in the complex µ-plane (see Fig.3)which is particularly evident in the β=7.5simulations (Fig.4(c)).The contribution of each zero to the susceptibility can be singled out by expressing the free energy as:F =2n x n yi =1(y −y i )(29)where y is the fugacity variable and y i is the correspond-ing zero of the partition function.The dotted lines on these plots correspond to the contribution of the nearby zeros while the full polynomial contribution is given by the solid lines.We see that the developing singularities are indeed governed by the zeros closest to the real axis.The sharpening of the singularity as the temperature de-creases is also in accordance with the dependence of the distribution of the zeros on the temperature.The singularities of the free energy and its derivative with respect to the chemical potential,can be related to the quasiparticle density of states.To do this we assume that single particle excitations accurately represent the spectrum of the system.The relationship between the average particle density and the density of states ρ(ω)is given by<n >=∞dω1dµ=ρsing (µ)∝1δ−1(32)and hence the rate of divergence of the density of states.As in the case of Lifshitz transitions the singularity of the particle number is rounded at finite temperature.However,for sufficiently low temperatures,the singular-ity of the density of states remains manifest in the free energy,the average particle density,and particle suscep-tibility [15].The regular part of the density of states does not contribute to the criticality,so we can concentrate on the singular part only.Consider a behaviour of the typedensity of states diverging as the−1ρsing(ω)∝(ω−µc)1δ.(33)with the valueδfor the particle number governed by thedivergence of the density of states(at low temperatures)in spite of thefinite-temperature rounding of the singu-larity itself.This rounding of the singularity is indeedreflected in the difference between the values ofαµatβ=5andβ=6.V.DISCUSSION AND OUTLOOKWe note that in ourfinite size scaling analysis we donot include logarithmic corrections.In particular,thesecorrections may prove significant when taking into ac-count the fact that we are dealing with a two-dimensionalsystem in which the pattern of the phase transition islikely to be of Kosterlitz-Thouless type[23].The loga-rithmic corrections to the scaling laws have been provenessential in a recent work of Kenna and Irving[24].In-clusion of these corrections would allow us to obtain thecritical exponents with higher accuracy.However,suchanalysis would require simulations on even larger lattices.The linearfits for the logarithmic scaling and the criti-cal exponents obtained,are to be viewed as approximatevalues reflecting the general behaviour of the Yang-Leezeros as the temperature and lattice size are varied.Al-though the bootstrap analysis provided us with accurateestimates of the statistical error on the values of the ex-pansion coefficients and the Yang-Lee zeros,the smallnumber of zeros obtained with sufficient accuracy doesnot allow us to claim higher precision for the critical ex-ponents on the basis of more elaboratefittings of the scal-ing behaviour.Thefinite-size effects may still be signifi-cant,especially as the simulation temperature decreases,thus affecting the scaling of the Yang-Lee zeros with thesystem rger lattice simulations will therefore berequired for an accurate evaluation of the critical expo-nent for the particle density and the density of states.Nevertheless,the onset of a singularity atfinite temper-ature,and its persistence as the lattice size increases,areevident.The estimate of the critical exponent for the diver-gence rate of the density of states of the quasiparticleexcitation spectrum is particularly relevant to the highT c superconductivity scenario based on the van Hove sin-gularities[25],[26],[27].It is emphasized in Ref.[25]thatthe logarithmic singularity of a two-dimensional electrongas can,due to electronic correlations,turn into a power-law divergence resulting in an extended saddle point atthe lattice momenta(π,0)and(0,π).In the case of the14.I.M.Barbour,A.J.Bell and E.G.Klepfish,Nucl.Phys.B389,285(1993).15.I.M.Lifshitz,JETP38,1569(1960).16.A.A.Abrikosov,Fundamentals of the Theory ofMetals North-Holland(1988).17.P.Hall,The Bootstrap and Edgeworth expansion,Springer(1992).18.S.R.White et al.,Phys.Rev.B40,506(1989).19.J.E.Hirsch,Phys.Rev.B28,4059(1983).20.M.Suzuki,Prog.Theor.Phys.56,1454(1976).21.A.Moreo, D.Scalapino and E.Dagotto,Phys.Rev.B43,11442(1991).22.N.Furukawa and M.Imada,J.Phys.Soc.Japan61,3331(1992).23.J.Kosterlitz and D.Thouless,J.Phys.C6,1181(1973);J.Kosterlitz,J.Phys.C7,1046(1974).24.R.Kenna and A.C.Irving,unpublished.25.K.Gofron et al.,Phys.Rev.Lett.73,3302(1994).26.D.M.Newns,P.C.Pattnaik and C.C.Tsuei,Phys.Rev.B43,3075(1991);D.M.Newns et al.,Phys.Rev.Lett.24,1264(1992);D.M.Newns et al.,Phys.Rev.Lett.73,1264(1994).27.E.Dagotto,A.Nazarenko and A.Moreo,Phys.Rev.Lett.74,310(1995).28.A.A.Abrikosov,J.C.Campuzano and K.Gofron,Physica(Amsterdam)214C,73(1993).29.D.S.Dessau et al.,Phys.Rev.Lett.71,2781(1993);D.M.King et al.,Phys.Rev.Lett.73,3298(1994);P.Aebi et al.,Phys.Rev.Lett.72,2757(1994).30.E.Dagotto, A.Nazarenko and M.Boninsegni,Phys.Rev.Lett.73,728(1994).31.N.Bulut,D.J.Scalapino and S.R.White,Phys.Rev.Lett.73,748(1994).32.S.R.White,Phys.Rev.B44,4670(1991);M.Veki´c and S.R.White,Phys.Rev.B47,1160 (1993).33.C.E.Creffield,E.G.Klepfish,E.R.Pike and SarbenSarkar,unpublished.Figure CaptionsFigure1Bootstrap distribution of normalized coefficients for ex-pansion(14)at different update chemical potentialµ0for an82lattice.The corresponding power of expansion is indicated in the topfigure.(a)β=5,(b)β=6,(c)β=7.5.Figure2Bootstrap distributions for the Yang-Lee zeros in the complexµplane closest to the real axis.(a)102lat-tice atβ=5,(b)102lattice atβ=6,(c)82lattice at β=7.5.Figure3Yang-Lee zeros in the complexµplane closest to the real axis.(a)β=5,(b)β=6,(c)β=7.5.The correspond-ing lattice size is shown in the top right-hand corner. Figure4Angular distribution of the Yang-Lee zeros in the com-plex fugacity plane Error bars are drawn where esti-mated.(a)β=5,(b)β=6,(c)β=7.5.Figure5Scaling of the imaginary part ofµ1(Re(µ1)≈=0.7)as a function of lattice size.αm u indicates the thefit of the logarithmic scaling.Figure6Electronic susceptibility as a function of chemical poten-tial for an82lattice.The solid line represents the con-tribution of all the2n x n y zeros and the dotted line the contribution of the six zeros nearest to the real-µaxis.(a)β=5,(b)β=6,(c)β=7.5.。

a r X i v :c o n d -m a t /0003311v 1 17 M a r 2000Coulomb gap in a model with finite charge transfer energy.S.A.Basylko 1,P.J.Kundrotas 2,3,V.A.Onischouk 1,2,E.E.Tornau 2,4and A.Rosengren 21Joint Institute of Chemical Physics of Russian Academy of Sciences,117977Kosygin Str.4,Moscow,Russia 2Department of Physics/Theoretical physics,Royal Institute of Technology,SE–10044Stockholm,Sweden3Faculty of Physics,Vilnius University,Sauletekio al.9,LT–2040,Vilnius,Lithuania4Semiconductor Physics Institute,Goˇs tauto 11,LT–2600Vilnius,Lithuania(Received )The Coulomb gap in a donor-acceptor model with finite charge transfer energy ∆describing theelectronic system on the dielectric side of the metal-insulator transition is investigated by means of computer simulations on two-and three-dimensional finite samples with a random distribution of equal amounts of donor and acceptor sites.Rigorous relations reflecting the symmetry of the model presented with respect to the exchange of donors and acceptors are derived.In the immediate neighborhood of the Fermi energy µthe the density of one-electron excitations g (ε)is determined solely by finite size effects and g (ε)further away from µis described by an asymmetric power law with a non-universal exponent,depending on the parameter ∆.PACS numbers:71.23.-k,71.30.+h,71.45.GmI.INTRODUCTIONDoping of solids might lead to drastic qualitative changes in their properties.The metal-insulator tran-sition (MIT)is a spectacular manifestation of this.The understanding of the driving forces of the MIT is a long-standing problem.In the early seventies,the prediction 1was made that on the dielectric side of the MIT the long-range Coulomb interactions deplete the density of one-electron excitations (DOE)g (ε)near the Fermi energy µ.Further,analytical calculations of g (ε)with Coulomb correlation taken into consideration have been performed on the metallic side of the MIT.Altshuler and Aronov 2showed that for the metallic case g (ε)in three dimensions has a cusp-like dependence g (ε)∼|ε−µ|1/2near µ.This was later confirmed in electron tunneling experiments for amorphous alloys 3and granular metals 4.On the insulating side of the MIT charge transport occurs via inelastic electron tunneling hopping between states localized on the impurity sites with one-electron energies close to µ.Mott 5demonstrated that at low temperatures electrons seek accessible energy states by hopping distances beyond the localization length,lead-ing to a hopping conductivity σ(T )∼exp(−T 0/T )νwith T 0being a characteristic temperature depending on lo-calization length and with the hopping exponent ν=1/4for the non-interacting case in three dimensions.Efros and Shklovskii 6(ES)argued that the ground state of a system with long-range Coulomb interactions is stable with respect to one-particle excitations only if g (ε)in the vicinity of µhas the symmetric shapeg (ε)∼|ε−µ|D −1(1)with the universal exponent D −1depending only on the dimensionality D of the system.In particular,ES pre-dicted that in D =3g (ε)=3e 2 3(ε−µ)2,where χis the dielectric constant and e is the electron charge.Because g (ε)vanishes only at ε=µ,this is called a “soft”Coulomb correlation gap with a width ∆ε∼e 3(N 0/χ3)1/2,where N 0is the DOE far away from µ.The power law (1)gives 7a hopping exponent ν=D/(D +3)at low temperatures,so for three-dimensional system with long-range Coulomb interactions ν=1/2.The intriguing hypothesis about universality of (1)has stimulated further theoretical research,both analytical 8and numerical 9–13.To establish the hypothesis (1)Efros 14used the ground-state stability conditions for lo-calized electrons (LES)with respect to charge transferεj −εi −e 2localized on all the sites of a D-dimensional lattice andthe negative charge from k×N acceptors is uniformlysmeared over the lattice sites so that each site i has acharge e(n i−k),where n i=1if a donor on the site i isionized and n i=0if a donor is neutral.Disorder in thismodel is ensured by introducing randomly distributedone-site potentials.Monte Carlo simulations12on verylarge specimens of the lattice d-a model,however,havegiven rise to doubts about the universality of the g(ε)behavior.Another hint about possible non-universal behavior ofg(ε)has come from the intriguing and still not com-pletely unfolded problem whether the so called spin-glassphase does exist in the classical d-a model(see,e.g.Ref.18–20).Grannan and Yu18studied the classical three-dimensional d-a model with k=0.5but with the totalacceptor charge uniformly distributed over donor sitesas in the lattice d-a model.In this case,the classical d-amodel is equivalent to a model of Ising spins,localized onrandomly distributed sites,with pairwise Coulomb inter-actions,a model in which a transition into the spin-glassstate was found18to occur at non-zero temperature.Itwas then concluded that such a transition should existin all d-a models(with and without smearing of negativecharge,defined on a lattice or on a continuous sample)aswell because of the Efros universality hypothesis.Voitaand Schreber20,however,have shown that the spin glasstransition does not exist in the lattice d−a model14.Besides,in recent work by one of us19it was unequivo-cally demonstrated that the ground state of the classicald-a model and that of the model studied in Ref.18arequalitatively different.An analysis of histograms H[Qαβ]of the so called overlaps Qαβ=12 i=j n a(i)n a(j)2 k=l(1−n d(k))(1−n d(l))− i,k(1−n d(k))n a(i)r a−d ik +1r a−aij++1r d−dkl,(4)whereεa(i)is the one-electron excitation(OEE)energy for the acceptorsεa(i)≡δE(n a,n d)r a−aij− k1−n d(k)r a−aij≥0,(8)whereε1(0)a(i)denotesεa(i)if n(i)=1(0).The stabilityconditions with respect to the other three manners of thecharge transfer are obtainable in the similar manner.The relation(8)implies thatεa’s for the neutral ac-ceptors are,in general,larger thanεa’s for the chargedacceptors.Furthermore,the pair of neutral and chargedacceptors might be located on any distance and thereforein the thermodynamic limit the chemical potential for theacceptors(i.e.an energy level which separates the ener-gies of the neutral and charged acceptors)is determinedasµa=min{ε0a(i)}=max{ε1a(i)}.(9)Alike,there exist the chemical potentialµd for the donorsas well.Moreover,the stability relations with respect tothe ionization and recombination lead toµa=µd=µ.(10)Despite thefinite size of samples we investigated,the re-lation(10)with the chemical potentials calculated from(9)is valid within the limits of accuracy of our calcula-tions(see Sect.III).A macroscopic state of the sample R is characterizedby degree of acceptor ionizationC a(R)=1N iδ(ε−εa(i))(12)and by the corresponding DOE g d(εd,R)for the donors.Note,that for thefinite samples(especially for the rel-ative small systems we were able to investigate)C a(R),g a(εa,R)and g d(εd,R)depend essentially on the partic-ular implementation R of the spatial distributions of thedonor and acceptor sites(if a sample would be big enoughall quantities would be self-averaging).Therefore,inorder to obtain reliable results,one has to work withthe quantities C a≡ C a(R) ,g a(ε)≡ g a(εa,R) andg d(ε)≡ g d(εd,R) ,where ... denotes the average overa number of R’s.Note,that the values g a(d)(εa(d),R)dεobtained for independent R’s are scattered according tothe Gaussian distribution with the mean g a(d)(ε)dεandthe standard deviationB.Acceptor-donor symmetryLet us rewrite the energy(3)in terms of the OEE energies(5)E(n a,n d)=12 kεd(k)(1−n d(k))−∆2.(20) Thus,the Fermi energy of our model system in the ther-modynamic limit is a fundamental quantity depending only on the energy of charge transfer from an acceptor to a donor.III.METHODA.Algorithm of energy minimizationWe start from a random allocation of N donor andN acceptor sites in the continuous D-dimensional sys-tem(generate a sample R)with the density n=1,sothat the system has a linear size L=N1/D and then charge randomly chosen C a×N both donors and accep-tors(usually we take C a=0.7),i.e.generate an initial microscopic state(IMS)(n a,n d)of the sample R.Fur-ther,we search for such microscopic state(n0a,n0d)which obeys the stability conditions(8)with respect to the four mechanisms of the charge transfer allowed in our model. We used an algorithm which is an extension of the al-gorithm proposed in Ref.9to the case∆=∞.The algorithm consists of the three main steps.In order to save computer time,first,we look for pairs a0−a−(d0−d+)for which the“crude”stability relation ∆ε≡ε0a(d)−ε1a(d)>0is violated.Then,the energy of the system is decreased by transferring an electron be-tween such pair of sites for which∆εhas its minimal non-positive value.This process is repeated until a state is reached,in which∆ε>0for all possible a0−a−and d0−d+pairs(step I).In the similar manner,we further minimize the energy of the system with respect to the “true”stability relations(8)for the charge transfer be-tween the a0−a−and d0−d+pairs(step II).And,finally, in the step III we diminish the energy of the system with respect to the stability relations for ionization and recom-bination processes.Since ionization and recombination processes change the degree C a of the system ionization, each time after one of these processes takes place dur-ing calculations,we go back to the step II.Repeating the steps II and III,wefinally arrive at a microscopic state(n0a,n0d)for which all four stability conditions are fulfilled.We name the procedure(n a,n d)→(n0a,n0d)via above steps I,II and III as“a single descent”.It should be noted,however,that the state(n0a,n0d)isnot necessarily the ground state of the sample R since for the ground state,in general,not only the simplest re-lations(8)with only pairs of sites included,but the more complicated relations involving quadruplets,sextets,etc. of sites have to be fulfilled.Therefore,the state(n0a,n0d) (after Ref.9)hereafter will be referred to as the pseudo-ground state(PGS)of the sample R.Then,two questions naturally arise:How close the PGS and the ground state of the given sample are and how this may influence the output of our calculations?In order to answer thefirst question,we calculate and analyze the histograms H for the so-called overlapsQαβ=1with different IMS(n a,n d).If two PGS’s are identical then Qαβ=1.We calculated for the D=2systemwith N=500at∆=0the mean Q(R)= Qαβ αβfor the sequence of100PGS’s generated by single descentsfrom the different IMS of the same sample R.We further acquire Q(R)for100different samples and obtain that the mean¯Q≡ Q(R) R=0.96.It means that in PGS generated by the single descent only20acceptors out of 500are,in average,in the“wrong”states compared to those in the true ground state of the sample.In order to evaluate how the“erroneousness”of PGS influences the outcome of our calculations we perform an analysis of ground states obtained by means of the so called multirank descents.Descent of rank m comprises of a consequence of the single descents on the same sam-ple with different IMS when calculations are stopped af-ter the lowest observed PGS energy repeats m times.We calculate¯Q(all other parameters were the same as de-scribed in the previous paragraph,where actually the case m=0was explored)for descents with different ranks m=5,10,15and found that,for instance,for m=15(which implies drastic increase in the compu-tation time)¯Q=0.990.g a(ε)and g d(ε)obtained from the PGS’s generated by means of the single descents and by means of descents with m=10,say,do not differ within the limits of statistical errors.So,we conclude, that reliable results can be obtained by means of single descents already,thereby saving a lot of computer time and resources.B.Finite-size effectsDue to constraints in computer resources,the largest samples,we were able to deal with,comprise up to N= 2000donor and N=2000acceptor sites(L∼45for D=2and L∼12for D=3).Such relative small sizes of the samples investigated might influence the outcome of calculations.Detailed analysis offinite size effects on the results obtained will be presented in Section IV and here we want to make two remarks about inherentfinite size effects in the model system considered.First,as follows from(8),the energiesε0a for the neutral acceptors andε1a for the charged ones infinite samples at T=0cannot be further away than(L×√D)−1(22) Of course,the same holds for donors as well.The rela-tion(22)gives the estimation how close toµdata on the energy spectrum are,in principle,obtainable from the calculations onfinite samples.Secondly,as follows from(5)the energiesεa andεd for thefinite samples are sensitive to the location of the donor and acceptor sites.Therefore,the Fermi energyµforfinite samples does differ,in general,from sample to0.00020.0004ε−µg(ε−µ)FIG.1.Density of one electron excitations g a(ε−µ) in the vicinity of the Fermi energyµobtained for the two-dimensional model(3)with N=1500at∆=0(cir-cles),2(squares),4(diamonds)and10(triangles).Data points presented in thefigure are calculated as the average over10.000(∆=0),5.100(∆=2),3.700(∆=4)and2.200 (∆=10)different samples.Insert shows double logarithmic plot of g a(ε−µ)forε>µin the regionε−µ 0.05.sample.A straightforward averaging of g(ε)over differ-ent samples might thus lead to a distortion of the g(ε) shape especially in the region where the Coulomb gap is observed.In order to avoid this undesired effect,we used a trickfirst proposed in Ref.9.During accumulation of the results for g(ε)we added together g(ε)for the same values ofε−µ(R)rather than for the same values ofε. Hereµ(R)denotes the Fermi energy for afinite sample R calculated asµ(R)=10.010.030.10.310−710−610−510−410−710−610−510−40.010.030.10.3(a)(c)(b)(d)∆ = 0∆ = 0∆ = 4∆ = 4123123123123|ε − µ|g(ε−µ)FIG.2.Density of one electron excitations g a (ε−µ)for ε>µ(a,b)and ε<µ(c,d)obtained for the two-dimensional model (3)at ∆=0(a,c)and 4(b,d),with N =500(curves numbered 1),1000(2)and 1500(3).The dashed lines are least-squares power-law fits g a (ε−µ)∼|ε−µ|γwith γ=0.9(a),0.55(b),0.98(c)and 0.78.Data presented in the figure are calculated as the average over 10.000different samples (except the case N =1500and ∆=4with the average over 3700different samples).width of the Coulomb gap ∆εand the energy scale in our model E 0=e 2n 1/D /χare of the same order of mag-nitude.Fig.1shows g a (ε−µ)in the vicinity of the Fermi energy µobtained for the two-dimensional samples with N =1000and various values of ∆.As it is seen,g a (ε−µ)depends considerably on ∆except for a narrow window |ε−µ| 0.05,where all data merge into some “uni-versal”curve symmetric with respect to µ,the curve which can be anticipated to obey the Efros universal-ity hypothesis (1).However,a double-logarithmic plot of the “universal”g a (ε−µ)(insert in the Fig.1),reveals that the behavior of g a (ε−µ)in the “universality”region is not even a power law.The width of this “universality”region is comparable to the width of the region where g a (ε−µ)=0due to the finite size effects (for the data presented in Fig.1relation (22)gives |ε−µ|<0.011),so it is plausible to suggest that the “universal”behavior of g a (ε−µ)is governed by the finite-size effects.This is clearly demonstrated in Fig.2where g a (ε−µ)are shown for several sizes of the samples investigated.The εwindow where finite size effects are severe,shrinks considerably with increasing N for all values of ∆we investigated.For instance,g a (ε−µ)for N =500and N =1000at ∆=0(see Fig.2a,c)merge when |ε−µ| 0.2while corresponding curves for N =1000and N =1500are indistinguishable already at |ε−µ| 0.1.The statistical noise observed for the curves in Fig.2is quite small even close to µand hence,the influence of insufficient large statistics on the results obtained is ex-0246810∆0.20.40.60.81γFIG.3.The exponent γof the power law g a (ε−µ)∼|ε−µ|γas a function of the charge-transfer energy ∆.The data are obtained from least-squares fits of g a (ε−µ)for the two-dimensional model (3)with N =1500within the region 0.2 |ε−µ| 0.7.Circles represent the positive values of ε−µwhile diamonds stand for the negative values of ε−µ.Lines are guides to the eye.cluded.Note,that the “universal”behavior of g (ε)in the vicinity of µobtained for the classical d −a model (see Fig.3in Ref.11)is most likely due to the finite size effects as well.In the region |ε−µ| 0.2,where the curves for all N collapse into a single curve (and where we believe the thermodynamic limit is reached),the behavior of g a (ε−µ)is described by a power law g a (ε−µ)∼|ε−µ|γ.The deviation from the power-law observed far away from µ(|ε−µ| 0.7)is due to the boundaries of the Coulomb gap which,as was mentioned above,are ∼1in units of E 0.One can see from a comparison of the data shown in Fig.2for different ∆,that the exponent γdepends considerably on ∆.Furthermore,values of γin the region ε−µ>0and those in the region ε−µ<0differ as well with this difference increasing with increasing ∆.The data for γobtained for the two-dimensional MCDAM are summarized in Fig.3where a significant deviation of γfrom the value D −1predicted by the hypothesis (1)is observed at all values of ∆investigated except for the case ∆=0when γ≈1within the limits of statistical accuracy.Note,that the deviation of γfrom its predicted value grows monotonically with increasing ∆.At ∆=10where the features of the MCDAM are expected to be nearly the same as those of the classical d −a model with all the acceptors being ionized (indeed,the degree of the acceptor ionization C a ∼0.9for the two-dimensional MCDAM at ∆=10,see Fig.6below)the deviation from the Efros exponent is very large.The main results for g a (ε−µ)obtained for the three-ε−µ00.00040.0008g(ε−µ)FIG.4.Density of one electron excitations g a (ε−µ)in the vicinity of the Fermi energy µobtained for the three-dimensional model (3)with N =1000at ∆=0(cir-cles),2(squares),4(diamonds)and 10(triangles).Data points presented in the figure are calculated as the av-erage over 10.000different samples.Inserts show dou-ble-logarithmic plots of g a (ε−µ)at ∆=2,for N =500(curves numbered 1),1000(2)and 2000(3),in the regions ε>µ(a)and for ε<µ(b).The dashed lines in the in-serts are least-squares power-law fits g a (ε−µ)∼|ε−µ|γwith γ=1.16(a),1.29(b),0246810∆0.511.5γFIG. 5.The exponent γof the power lawg a (ε−µ)∼|ε−µ|γas a function of the charge-transfer energy ∆.The data are obtained from least-squares fits of g a (ε−µ)for the three-dimensional model (3)with N =1000within the region 0.4 |ε−µ| 0.8.Circles represent the positive values of ε−µwhile diamonds stand for the negative values of ε−µ.Lines are guides to the eye.0246810∆0.70.80.91C aFIG.6.The degree of acceptor ionization C a as a function of the charge-transfer energy ∆.The data are obtained for the model (3)in two (circles)and three (diamonds)dimensions with N =500as an average over 1000different samples.The solid lines are third-degree polynomial fits.dimensional MCDAM are summarized in Figs.4and 5.It is seen,that the behavior of g a (ε−µ)in three di-mensions does not differ qualitatively from the behavior of g a (ε−µ)in two dimensions.Some quantitative dif-ferences observed arise from the fact that at given N (the parameter which determines the amount of com-puter memory needed for the calculations)the linear size of a two-dimensional sample with a given density of sites is larger than that of a three-dimensional sample with the same density of sites and thereby,the finite size effects for three-dimensional samples with given N are more pro-nounced compared to those for the two-dimensional sam-ples with the same N .For example,the lower boundary of the region where g a (ε−µ)can be described by the power law |ε−µ|γshifts towards larger |ε−µ| 0.4values (see inserts in Fig.4).Remarkably,the exponent γdoes not reach the value D −1predicted by the uni-versality hypothesis (1)even at ∆=0(Fig.5).Unlike g a (ε−µ)in the vicinity of the Coulomb gap,the density of ionized acceptors C a (11)describes the state of the entire sample and therefore reaches the thermody-namic limit much faster than g a (ε−µ).This allows us to obtain quite accurate results for C a from data on a relatively small amount of samples with N =500only.Fig.6shows the variations of C a with ∆both for two and three dimensions.In three dimensions almost all accep-tors become ionized (C a ∼1)rather soon while for two dimensions even for the largest ∆investigated around 10%of the acceptors remain neutral.So,one can say,that the three-dimensional MCDAM at ∆ 7reduces already to the classical d −a model.It is known that the classicalTABLE I.The means ¯µand standard deviations ∆µoftheFermienergy calculatedforthethree-dimensionalmodel(3)withN =1000and various ∆.∆¯µ∆µ2ε.I.e.,sites with energies ε1i ∈[−ε,0]cannot be inside a D-dimensional sphere of radius R sp =12εDεg (ε′)dε′(24)where S (D )is the volume of a D-dimensional sphere with the radius equal to unity.Since V sp cannot exceed the total volume V of a sample (V =N at n =1)we arrive at the inequalityεg (ε′)dε′≤(2ε)DS (D )|ε|D −1(26)The universality hypothesis (1)then is a limit case of (26).The density of sites with energies ε1i ∈[−ε,0]indeed decreases when ε→0,so the assumption (24)for the spheres with finite radii seems to be plausible.However,simultaneously R sp →∞and consequently the plausibility of the assumption (24)and thereby of the hypothesis (1)becomes questionable.And finally,the universality hypothesis (1)can be also obtained as the asymptotic behavior of a non-linear in-tegral equation for g (ε)as ε→0,the equation which,in turn,is heuristically obtained from the stability condition (2).The derivation of this integral equation (given,for example,in Ref.17)is based on the implicit assump-tion that the sites with charged donors are randomly distributed in space according to the Poisson statistics.However,it was unequivocally demonstrated in computer studies of the Coulomb gap 11that charged donor sites with energies close to µtend to form clusters (Ref.11,Fig.6).We conclude that g a (ε−µ)in the region of the Coulomb gap in model (3)has a power law behavior for all energies down to µand that the universality hypothesis of Efros (1)is questionable.Note,that our results are in contra-diction not only to the universality hypothesis (1),but to the inequality (26)as well.Up to now,all exponents found are in good agreement with this inequality.E.g.in Ref.12specimens of 40000and 125000sites for two-and three-dimensional samples were investigated in the Efros’lattice model 14and the power law g a (ε−µ)∼|ε−µ|γwas found with γ=1.2±0.1and γ=2.6±0.2for two and three dimensions,respectively.The main conclusionTABLE II.Some donor–acceptor pairs for which the difference between the donor and acceptor energy levels does not exceed 10meV.E g,E v and E c are,respectively,the energy gap,the top of the valence band and the bottom of the conductivity band. If the solubilities of both donor and acceptor are known,the parameter E0is calculated using the data for the less soluble of the pair.Donor Acceptor Solubility,cm−3E0,meV E j,meV∆,meVmin max min maxGe(E g=740meV,χ=15.9)S no reliable data E c−2964 Ni4.8×10158×1015 1.5 1.8E c−300ACKNOWLEDGEMENTSThis research was supported by The Swedish Natural Science Council and by The Swedish Royal Academy of Sciences.。

晶体结构题一（2004年全国高中学生化学竞赛决赛6分）氢是重要而洁净的能源。

要利用氢气作能源，必须解决好安全有效地储存氢气问题。

化学家研究出利用合金储存氢气，LaNi5是一种储氢材料。

LaNi5的晶体结构已经测定，属六方晶系，晶胞参数a=511 pm,c＝397 pm,晶体结构如图2所示。

⒈从LaNi5晶体结构图中勾画出一个LaNi5晶胞。

⒉每个晶胞中含有多少个La原子和Ni原子？⒊LaNi5晶胞中含有3个八面体空隙和6个四面体空隙，若每个空隙填人1个H原子，计算该储氢材料吸氢后氢的密度，该密度是标准状态下氢气密度(8.987×10-5 g·m-3)的多少倍？（氢的相对原子质量为1.008；光速c为2．998×108 m·s-1；忽略吸氢前后晶胞的体积变化）。

解：⒈晶胞结构见图4。

⒉晶胞中含有1个La原子和5个Ni原子⒊计算过程：六方晶胞体积：V=a2csin120°=(5.11×10-8)2×3.97×10-8×31/2/2=89.7×10-24cm3氢气密度是氢气密度的1.87×103倍。

二. (2006年全国高中学生化学竞赛决赛理论试题1)在酸化钨酸盐的过程中，钨酸根WO42-可能在不同程度上缩合形成多钨酸根。

多钨酸根的组成常因溶液的酸度不同而不同，它们的结构都由含一个中心W原子和六个配位O原子的钨氧八面体WO6通过共顶或共边的方式形成。

在为数众多的多钨酸根中，性质和结构了解得比较清楚的是仲钨酸根[H2W12O42]10-和偏钨酸根[H2W12O40]6-。

在下面三张结构图中，哪一张是仲钨酸根的结构？简述判断理由。

(a) (b) (c)解：提示：考察八面体的投影图，可以得到更清楚地认识。

三．(2006年全国高中学生化学竞赛决赛理论试题4)轻质碳酸镁是广泛应用于橡胶、塑料、食品和医药工业的化工产品，它的生产以白云石（主要成分是碳酸镁钙）为原料。

固体物理习题及解答⼀、填空题1. 晶格常数为a 的⽴⽅晶系 (hkl)晶⾯族的晶⾯间距为a该(hkl)晶⾯族的倒格⼦⽮量hkl G 为 k al j a k i a h πππ222++ 。

2. 晶体结构可看成是将基元按相同的⽅式放置在具有三维平移周期性的晶格的每个格点构成。

3. 晶体结构按晶胞形状对称性可划分为 7 ⼤晶系，考虑平移对称性晶体结构可划分为 14 种布拉维晶格。

4. 体⼼⽴⽅（bcc ）晶格的结构因⼦为 []{})(ex p 1l k h i f S hkl ++-+=π，其衍射消光条件是奇数=++l k h 。

5. 与正格⼦晶列[hkl]垂直的倒格⼦晶⾯的晶⾯指数为 (hkl) ，与正格⼦晶⾯（hkl ）垂直的倒格⼦晶列的晶列指数为 [hkl] 。

6. 由N 个晶胞常数为a 的晶胞所构成的⼀维晶格，其第⼀布⾥渊区边界宽度为a /2π，电⼦波⽮的允许值为 Na /2π的整数倍。

7. 对于体积为V,并具有N 个电⼦的⾦属, 其波⽮空间中每⼀个波⽮所占的体积为 ()V /23π，费⽶波⽮为 3/123?=V N k F π。

8. 按经典统计理论，N 个⾃由电⼦系统的⽐热应为 B Nk 23，⽽根据量⼦统计得到的⾦属三维电⼦⽓的⽐热为 F B T T Nk /22，⽐经典值⼩了约两个数量级。

9．在晶体的周期性势场中，电⼦能带在布⾥渊区边界将出现带隙，这是因为电⼦⾏波在该处受到布拉格反射变成驻波⽽导致的结果。

10. 对晶格常数为a 的简单⽴⽅晶体,与正格⽮R =a i +2a j +2a k 正交的倒格⼦晶⾯族的⾯指数为 (122) , 其⾯间距为 .11. 铁磁相变属于典型的⼆级相变，在居⾥温度附近，⾃由能连续变化，但其⼀阶导数（⽐热）不连续。

13．等径圆球的最密堆积⽅式有六⽅密堆（hcp ）和⾯⼼⽴⽅密堆（fcc ）两种⽅式，两者的空间占据率皆为74%。

14. ⾯⼼⽴⽅（fcc ）晶格的倒格⼦为体⼼⽴⽅（bcc ）晶格；⾯⼼⽴⽅（fcc ）晶格的第⼀布⾥渊区为截⾓⼋⾯体。

金兹堡朗道方程
金兹堡-朗方程是一种非线性偏微分方程，是一类描述超导现象的非线性抛物型方程组.它是由Ginzburg和Landau在Landau二级相变理论的基础上,综合了超导体的电动力学、量子力学和热力学性质,提出的一个描述超导的唯象模型1937年，俄罗斯的列夫•达维多维奇•朗道(LevDavidovich Landau)提出了外磁场下超导中间态的结构模型，用超导体正常态层与超导态层交替共存的分层结构揭示超导现象的本质。

1950年，俄罗斯的维塔利•金兹堡(Vitaly Ginzburg)与朗道合作，在朗道提出的二级相变理论基础上提出了一个从宏观角度描述超导现象的数学模型(金兹堡-朗道方程，也称G-L方程)：超导电子并非单独存在，电子之间可能有关联的最长距离称为它们的相干长度(受到外界扰动后超导恢复所需的空间尺度)。

由G-L方程可推导出超导体在外界磁场强度接近超导体的临界磁场强度(能破坏超导态的磁场强度)时的临界行为：外界磁场并非完全不能进入超导体，而是穿透进入超导体表面，足够强的外磁场则可进入超导体内破坏超导态而恢复成正常态。

超导性的唯象理论，是结合了超导体的电动力学、量子力学和热力学特性，为超导相变给予热力学解释而提出的。

此理论可以由Bcs理论得到，它能相当好地描述第Ⅱ类超导体的磁学性能。