Muon Anomalous Magnetic Moment in the Supersymmetric Models with and Without Right
Drell-Hearn-Gerasimov Sum Rule for the Nucleon in the Large-N_c Limit
a rXiv:h ep-ph/995334v113May1999Drell-Hearn-Gerasimov Sum Rule for the Nucleon in the Large-N c Limit Thomas D.Cohen and Xiangdong Ji Department of Physics University of Maryland College Park,Maryland 20742(UMD PP#99-108DOE/ER/40762-181May 1999)Abstract We show that the Drell-Hearn-Gerasimov sum rule for the nucleon is entirely saturated by the ∆resonance in the limit of a large number of colors,N c →∞.Corrections are at relative order 1/N 2c .Typeset using REVT E XThe Drell-Hearn-Gerasimov(DHG)sum rule for a spin-1/2fermion relates its anomolous magnetic moment to an integral over the spin-dependent photo-production cross section[1]. Define the total photo-production cross sectionsσP(ν)andσA(ν),where P and A label the spins of the photon being parallel and antiparallel,respectively,to the spin of the fermion andνis the photon energy.The DHG sum rule says,∞νin dνσP(ν)−σA(ν)M2.(1) whereκand M are the anomalous magnetic moment(dimensionless)and mass of the fermion,respectively.For the case of the nucleon it is useful to break the sum rule up into isoscalar and isovector components.It has been known empirically for a long time that the isoscalar part of the DHG sum rule for the nucleon is almost entirely dominated by the∆-resonance contribution.The theoretical expectation for the sum(i.e.the right-hand side(RHS)of eq.(1))is219µb, whereas the∆-resonance contribution to the left-hand side(LHS)integral is240µb[2].To the best of our knowledge,there hasn’t been a solid theoretical explanation for this from fundamental principles of quantum chromodynamics(QCD).In this short note,we show that such phenomenon is expected in QCD in the limit of a large number of colors(N c→∞) [3,4].During the past several years,there has been significant progress in understanding the baryon properties in the large-N c rge-N c consistency of the theory has been ex-ploited to show that it has a contracted SU(2n f)spin-flavor symmetry,where n f is the number of quarkflavors[5,6].For example,the symmetry is required for cancellations among Feynman diagrams with different intermediate states in a hadronic description of scattering in order to preserve the unitarity bound.The spin-flavor symmetry implies that there is a tower of states with I=J which are degenerate in the large-N c limit;the lowest two states are identified with the nucleon and the∆-resonance.It has also been shown the splitting between the lowest states in this tower(eg.the nucleon and∆)scales as1/N c [7].The symmetry also has the feature that certain hadronic matrix elements are related to each other in the large-N c limit.For example,the ratio of the axial-current matrix element in the nucleon to the transition matrix element between the nucleon and the∆isfixed in the large-N c limit.Moreover,there are cases,such as the axial matrix elements,where the leading correction to this ratio occurs at relative order1/N2c[6].As pointed out by Broniowski[8],the same underlying symmetry principle is responsible for understanding large-N c consistency of low-energy sum rules such as those of Addler-Weisberger[9]and Cabibbo-Radicati[10].In this paper,we follow Broniowski’s observation and point out that the∆-dominance in the isoscalar part of the DHG sum rule is required by a similar large-N c consistency and reflects the above-mentioned spin-flavor symmetry. Moreover,we will show that the∆dominance is valid up to relative order1/N2c.Let us do some large-N c counting for the electromagnetic couplings in order to understand the large-N c consistency of the DHG sum rule.Consider two quarkflavors:up and down. The proton(neutron)is then made of(N c±1)/2up and(N c∓1)/2down quarks.We adopt a scheme of assigning electric charges to the quarks such that the proton and neutron have the usual charges one and zero.Thus,e u(d)=±1/2+1/2N c.Other charge-assignment schemes are possible,such as keeping the quark chargesfixed at2/3and−1/3.We will notexplore them here because they do not affect the main conclusion of the paper.In the large N c-limit,the quark matrix elements in the nucleon have the following simple N c-counting rules[6]:i 1 ∼N c,i τi ∼1,i σi ∼1,iτi σi ∼N c.(2)where σi and τi are the spin and isospin matrices of the i-th quark.According to the above, the N c-counting for the electromagnetic couplings of the nucleon goes asisovector charge coupling e V∼1,isoscalar charge coupling e S∼1,isovector magnetic couplingµV∼N c,isoscalar magnetic couplingµS∼1/N c.(3) The preceding counting rules appear to be results of the simple quark model.In fact,they are general results of the group theory—the quark model language is serving as a simple device to do the group theory conveniently.The isovector magnetic coupling scales as N c.This leads to a rather perverse formal large-N c limit.As N c→∞,the electromagentic energy of the nucleon goes asαem N2c. Keepingαem at order1,we see that the electromagnetic energy will eventually become nonperturbative and overwhelm the strong interaction contribution which is order N c.In practice we wish to study a regime close to the real world in which electromagnetism is, indeed,perturbative.A simple way to do this is to takeαem N c to be independent of N c and <<1,which is well satisfied in the real world.Now we are ready to examine the large-N c consistency of the DHG sum rule.First, consider the RHS of Eq.(1)in the large-N c limit.According to the previous discussion,its leading dependence in our expansion isαem N2c becauseκV/M∼µV∼N c.Second,it is easy to see that the integrand of the spin-dependent cross section in the LHS of the DHG sum rule(Eq.(1))generically asαem N0c.Indeed,in the quark representation,σP−σA∼αem i σi ,(4)which according to Eq.(2),scales likeαem N0c.So,how does the DHG sum rule work if the two sides have different large-N c behavior? The answer lies in the∆-resonance contribution.While the generic contribution to the LHS is O(αem N0c),the∆contribution,as a member of the same tower of states as the nucleon, is special.In the large-N c limit,the∆-resonance is the only state excitable with a photonof energy of order1/N c.Moreover,the∆-nucleon transition magnetic moment,like the nucleon’s isovector magnetic moment is order N c.Overall,the∆contribution to the spin-dependent cross section turns out to be quite special:it is of orderαem N c.Therefore,the ∆-resonance contribution to the DHG sum rule can be of orderαem N2c.Indeed,a straightforward calculation yields the following∆-photoproduction cross sec-tion,2π2αemµ2γ∆Nσ∆(ν)=δ(ν−∆)∆.(6)M2In the large N c limitµγN∆=κV,and so the above contribution matches exactly the leading N c contribution from the RHS of Eq.(1).Since the isoscalar anomalous magnetic moment κS/M is of order1/N c,the relationµγ∆N=κV is in fact correct up to O(N c)(κV is O(N2c)).The large-N c consistency of the isovector component of the DHG sum rule is simple:The RHS of the sum rule scales likeαemκVκS/M2∼αem.The isovector spin-dependent cross section goes likeαem i σi τi /N c∼αem;the N c counting is completely consistent.The Drell-Hearn-Gerasimov sum rule is derived under the condition that N c=∞.How-ever,if one starts with QCD with N c=∞,then the low-energy theorem for the Compton scattering amplitude changes since the∆is now a massless excitation.The Compton am-plitude,S1(ν,Q2),is defined according toTµν=i d4ξe iξ·q P S|T Jµ(ξ)Jν(0)|P S=−iǫµναβqαSβS1(ν,Q2)+...,(7) where Sαis the polarization vector of the nucleon.Including the∆-resonance contribution, wefindS1(0,0)=−(2κSκV+κ2S)/M2.(8) The leading-order nucleon-pole contribution is exactly cancelled by the∆-resonance pole. This cancellation reflects the contracted SU(4)symmetry.On the LHS of the DHG,the∆-resonance contribution is absent because it is forbidden by energy-momentum conservation.It is significant to notice that the∆dominates the DHG sum rule up to corrections of orderαem1/N2c.The easiest way to see this is to compare the two sides of the sum rule. The right-hand side scales asαem N2c while the generic contributions to the left-hand side are orderαem N0c.One concludes that the∆contribution must therefore account for both theαem N2c andαem N c contribution on the left-hand side.In fact,the lack of a correction at relative order1/N c is easily understood in another way.As noted above,there are cases, such as the axial-vector current,where the ratios of nucleon matrix elements and nucleon-∆transition matrix elements arefixed with corrections at relative order1/N2c[6].The isovector magentic coupling of a photon has the same structure(in terms of the couplingsto quarks)as the axial current.Thus the ratio of the nucleon isovector magnetic moment and the nucleon-∆magnetic transition moment is alsofixed by the group structure up to relative corrections of order1/N2c.To summarize,we argued that the DHG sum rule is saturated by the∆resonance in the large-N c limit up to correction terms of relative order1/N2c.The result reflects the contracted spin-flavor SU(4)symmetry in the large-N c QCD with twoflavors.Turning the argument around,the contracted SU(4)symmetry indicates that the spin-dependent photonucleon cross section must be of orderαem N0c.We thank C.-W.Kao and J.Osborne for discussions about the subject.This work is supported by funds provided by the U.S.Department of Energy(D.O.E.)under cooperative agreement DOE-FG02-93ER-40762.REFERENCES[1]S.D.Drell and A.C.Hearn,Phys.Rev.Lett.16(1966)908;S.B.Gerasimov,Sov.J.Nucl.Phys.2(1966)430.[2]I.Karliner,Phys.Rev.D7(1973)2717;R.L.Workman and R.A.Arndt,Phys.Rev.D45(1992)1789.[3]G.t’Hooft,Nucl.Phys.B72(1974)461.[4]E.Witten,Nucl.Phys.B160(1979)57;B156(1979)269.[5]J.Gervais and B.Sakita,Phys.Rev.Lett.52(1984)87.[6]R.Dashen,E.Jenkins,and A.Manohar,Phys.Rev.D49(1994)4713;D51(1995)3697.[7]E.Jenkins,Phys.Lett.B315(1993)441;Phys.Rev.D54(1996)4515;E.Jenkins and R.F.Lebed,Phys.Rev.D52(1995)282.[8]W.Broniowski,Nucl.Phys.A580(1994)429.[9]A.Adler,Phys.Rev.140(1965)736;W.I.Weisberger,Phys.143(1966)1302.[10]N.Cabibbo and L.A.Radicati,Phys.Lett.19(1966)697.[11]X.Ji and J.Osborne,to be published.。
GRE阅读材料练习：鸟类拥有磁性感知能力新
GRE阅读材料练习：鸟类拥有磁性感知能力为了关心大家坚持学习，高效备考，我整理供应了“GRE阅读材料练习：鸟类拥有磁性感知力量”，欢迎阅读参考！更多相关讯息请关注我！Birds can navigate by the Earth's magnetic field.How they do it is still a mystery.Where would people be without magnetic compasses?The short answer is: lost.By giving human beings a sixth sense—an ability to detect the hitherto invisible magneticfield of the Earth—the compass proved one of the most important inventions ever.鸟类能够利用地球磁场导航。
机理尚不明确人类没有指南针会怎样?很简洁：迷失方向。
指南针给了人类第6感，使人能辨别地球无形的磁场，成为最重要的创造之一。
It let sailors navigate without sight of the night sky.And that led to the voyages of discovery, trade and conquest which created the politicalgeography of the modern world.Imagine, then, what animals which had their own, built-in compasses could achieve.They might spend their summers doingthe English Season in Glyndebourne or Henley, and then overwinter in the warmth of Mombasa. They might strike out, like intrepid pioneers, from Angola to Anchorage.They might even, if truly gripped by wanderlust and a hatred of the darkness, live in near-perpetual daylight by migrating from Pole to Pole.And that is just what some birds do.海员不用观看夜空便可以辨识方向。
Anomalous Hall Effect and Magnetic Monopoles in Momentum-Space
a r X i v :c o n d -m a t /0310232v 1 [c o n d -m a t .s t r -e l ] 10 O c t 2003Anomalous Hall Effect and Magnetic Monopoles inMomentum-SpaceZhong Fang 1,2∗,Naoto Nagaosa 1,3,4,Kei S.Takahashi 5,Atsushi Asamitsu 1,6,Roland Mathieu 1,Takeshi Ogasawara 3,Hiroyuki Yamada 3,Masashi Kawasaki 3,7,Yoshinori Tokura 1,3,4,Kiyoyuki Terakura 81Spin Superstructure Project (SSS),ERATO,Japan Science and Technology Corporation (JST),AIST Tsukuba Central 4,Tsukuba 305-8562,Japan;2Institute of Physics,Chinese Academy of Science,Beijing 100080,China;3Correlated Electron Research Center (CERC),AIST Tsukuba Central 4,Tsukuba 305-8562,Japan;4Department of Applied Physics,University of Tokyo,7-3-1,Hongo,Bunkyo-ku,Tokyo 113-8656,Japan;5DPMC,University of Geneva,24,quai Ernest-Ansermet,1211Geneva 4,Switzerland;6Cryogenic Center,University of Tokyo,2-11-16Bunkyo-ku,Tokyo 113-0032,Japan;7Institute for Materials Research,Tohoku University,Sendai 980-8577,Japan 8Research Institute for Computational Sciences (RICS),AIST Tsukuba Central 2,Tsukuba 305-8568,JapanEfforts tofind the magnetic monopole in real space have been made in cosmic rays and in accelerators,but up to now there is nofirm evidence for its existence due to the very heavy mass∼1016GeV.However,we show that the magnetic monopole can appear in the crystal-momentum space of solids in the accessible low energy region(∼0.1−1eV)in the context of the anomalous Hall effect. We report experimental results together withfirst-principles calculations on the ferromagnetic crystal SrRuO3that provide evidence for the magnetic monopole in the crystal-momentum space.In1931,Dirac[1]postulated the existence of magnetic monopole(MM)searching for the symmetry between the electric and magneticfield in the laws of electromagnetism.The singularity of the vector potential is needed for this Dirac magnetic monopole.Theoretically, the MM was found[2,3]as the soliton solution to the equation of the non-Abelian gauge theory for the grand unification.However its energy is estimated to be extremely large ∼1016GeV,which makes its experimental observation difficult.In contrast to this MM in real space,one can consider its dual space,namely the crystal momentum(k-)space of solids,and the Berry phase connection[4]of Bloch wavefunctions.This magnetic monopole in momentum space is closely related to the physical phenomenon,i.e.,the anomalous Hall effect(AHE)in ferromagnetic metals.The AHE is a phenomenon where the transverse resistivityρxy in ferromagnets contains the contribution due to the magnetization M in addition to the usual Hall effect.The conventional expression forρxy isρxy=R0B+4πR s M(1)where B is the magneticfield,R0is the usual Hall coefficient,and R s is the anomalous Hall coefficient.This expression implicitly assumes that the additional contribution is pro-portional to the magnetization,and is used as an experimental tool to measure the mag-netization as a function of temperature.Especially it is extensively used in the studies of ferromagnetic semiconductors with dilute magnetic impurities,which are the most promis-ing materials for the spintronics[5].However,the mechanism of AHE has been controversial for long[6,7,8,9,10].The key issue is whether the effect is intrinsic or extrinsic and how to treat the impurity and/or phonon scatterings.Karplus and Luttinger[6]was thefirst to propose the intrinsic mechanism of AHE,where the matrix elements of the current operatorsare essential.Other theories[7,10]attribute the AHE to the impurity scattering modified by the spin-orbit interaction,namely the skew scattering[7]and/or the side-jump mecha-nism[10].These extrinsic mechanisms are rather complicated and depends on the details of the impurities as well as the band structure.Nevertheless,all those conventional theories [6,7,8,9,10]for AHE derives Eq.1,as they are based on the perturbative expansion in the spin-orbit coupling(SOC)λand the magnetization M,i.e.,R s∝λ.Recently the geometrical meaning of the AHE of the intrinsic origin[6]has been recog-nized by several authors[11,12,13,14].The transverse conductivityσxy can be written as the integral of the Berry phase curvature(gaugefield)over the occupied electronic states in crystal momentum space(Eq.5below).Magnetic monopole corresponds to the source or sink of the gaugefield/curvature defined by this Berry phase connection.Therefore the AHE can be the directfingerprint of the MM in crystal momentum space.Note here that the presence of the time-reversal symmetry results inσxy=ρxy=0in the d.c.limit from the very generic argument,and the group theoretical condition for the nonzeroσxy is equivalent to that offinite ferromagnetic moment(see Supporting Online Material(SOM)).Therefore we need the ferromagnets to studyσxy even though the Berry phase connection is more universal and exists even in nonmagnetic materials.In this paper,we show by detailedfirst principles band calculation combined with the transport,optical,and magnetic measurements,that the observed unconventional behavior of the anomalous Hall effect and Kerr rotation in metallic ferromagnet SrRuO3is of intrinsic origin and is determined by the MM in k-space.The conventional expression Eq.1is invali-dated by the experimental data presented below,showing the non-monotonous temperature dependence including even a sign change.SrRuO3with the perovskite structure is an itinerant(metallic)ferromagnet.There are 4t2g electrons with the low spin configurations.The4d orbitals of SrRuO3are relatively extended and the bandwidth is large compared with the Coulombic interaction.The rela-tivistic SOC is also large in4d electrons because of the heavy mass(of the order of0.3eV for Ru atom).High-quality single crystal is available with the residual resistivity of the order of10µΩcm.These aspects make this system an ideal candidate to observe the AHE due to the k-space gaugefield.Stoichiometric SrRuO3(bulk and thinfilm)and Ca-doped (Sr0.8Ca0.2RuO3thinfilm)single crystals were prepared(see SOM for the details of sample preparation).The magnetization curve(Fig.1A)of SrRuO3film is quite similar to that ofbulk single crystal,except the Curie Temperature T c(∼150K)is slightly lower than that of bulk(∼160K),due to the strain effects.On the other hand,the isovalent Ca-doping sup-presses the T c and magnetization M dramatically.All of the samples were used for the trans-port measurement in the d.c.limit.As seen from Fig.1C,theρxy changes non-monotonously with temperature including even a sign change.Such behavior is far beyond the expectation based on the conventional expression Eq.1.In addition to the transport measurement,the frequency(ω)dependent conductivities(Fig.2)were measured for SrRuO3film by optical method(see SOM).Except the strong structures around3.0eV,which are mostly due to the charge transfer from O-2p to Ru-4d,sharp structures are also observed for both the real and the imaginary part ofσxy(ω)below0.5eV.Those low energy sharp structures cannot befitted by extended Drude analysis.The lower the energy is,the stronger the deviations fromfitting are.We will show,by combination with thefirst-principles calculations,that those unconventional behavior actually originate from the singular behavior of MM in the momentum space.Now we turn to some details of the theoretical analysis.Berry phase is the quantal phase acquired by the wavefunction associated with the adiabatic change of the Hamiltonian [4,15].Let|n(α)>be the n-th eigenstate of the Hamiltonian H(α)withα=(α1,....,αm) being the set of parameters,the Berry’s connection is the overlap of the two wavefucntions infinitesimally separated inα-space,i.e.,<n(α)|n(α+∆α)>=1+∆α·<n(α)|∇α|n(α)>=exp −i∆α·a n(α) .(2) where the vector potential a n(α)is defined by a n(α)=i<n(α)|∇α|n(α)>.Although the concept of Berry phase has broad applications in physics[15],its relevance to the band structure in solids has been recognized only recently and in limited situations such as the quantum Hall effect under strong magneticfield[16]and the calculation of electronic polar-ization in ferroelectrics[17,18].In this case the parameterαis the crystal momentum k.For the Bloch wavefunctionψn k ( r)=e i k· r un k( r)with n denoting the band index and un kbeing the periodic part,the vector potential for the Berry phase a nµ( k)isa nµ( k)=i<un k |∂of xµ’s includes the gaugefield Fµν=∂kµa nν−∂kνa nµas[xµ,xν]=−iFµν.(4)which leads to the additional(anomalous)velocity−i[xµ,V(x)]=−Fµν∂V(x)/∂xνbeing transverse to the electricfield Eν=−∂V(x)/∂xν.Therefore the transverse conductivityσxy is given by sum of this anomalous velocity over the occupied states as[16]σxy= n,k n F(εn(k))b z(k).(5) where b z(k)=F xy(k)and n F(ε)=1/(eβ(ε−µ)+1)(β:inverse temperature,µ:chemical potential)is the Fermi distribution function.Hence the behavior of gaugefield b z(k)in k-space[19]determines that ofσxy.One might imagine that it is a slowly varying function of k,but it is not the case.Fig.3B is the calculated result for b z(k)in the real system SrRuO3.It has a very sharp peak nearΓ-point and also ridges along the diagonals.The origin for this sharp structure is the(near)degeneracy and/or the band crossing,which act as MM.Consider the general case where two band Hamiltonian matrix H(k)at k can be written as H(k)= µ=0,1,2,3fµ(k)σµwhereσ1,2,3are the Pauli matrices andσ0is the unit matrix.We can consider the mapping from k to the vector f(k)=(f1(k),f2(k),f3(k)),andthe contribution toσxy from the neighborhood of this degeneracy region is given by the solid angle dΩf for the infinitesimal dk x dk y integrated over k.Therefore the gaugeflux near the MM,namely the degeneracy point f=0is singularly enhanced(as shown in Fig.3).(See SOM for more details).We studied the behavior ofσxy by thefirst-principles calculations(see SOM for meth-ods).The calculated density of states(DOS)is not so different between the cases with and without SOC(Fig.4A),while theσxy should be very sensitive to the Bloch wavefunctions and depends on the Fermi level position and the spin-splitting(magnetization)significantly, as predicted by above discussions.The behavior ofσxy as a function of the Fermi-level position was obtained by using a small broadening parameter for the lifetimeδ(=70meV) (Fig.4B).By shifting the Fermi level,not only the absolute value but also the sign ofσxy is found to change.The sharp and spiky structures are just the natural results of singular behavior of MM(Fig.3).For the case without any shift of Fermi level,we obtain the value ofσxy=−60Ω−1cm−1,which has the same sign as and is comparable with the experimental value(∼−100Ω−1cm−1).Such a spiky behavior should be also reflected in theωdependentσxy,especially for the low energy range with longer lifetime,while be suppressed at higher activation energy with shorter lifetime.As shown in Fig.2,for theωdependence of optical conductivity,the high energy part(>0.5eV),which is dominated by the p−d charge transi-tion peak,is usual,and can be well reproduced by our calculations.While the observed peak structure ofσxy(ω)below0.5eV is the clear demonstration of the predicted spiky behavior. The spectra below0.2eV is not measured due to the technical difficulty,but it is clear that even sharper structure should be there because the d.c.limit Re(σxy)≈−100Ω−1cm−1has the opposite sign(the Im(σxy)at d.c.limit should go back to zero).Such a low energy behavior is well represented by our calculations,providing further evidence for the existence of magnetic monopoles.Now it is straightforward to understand the results of transport measurement forσxy. Here we attribute the temperature(T)dependence ofσxy to that of the magnetization M(T).As the results of k-space integration over occupied states,the calculatedσxy is non-monotonous as a function of magnetization(Fig.1D).With the reduction of spin-splitting, the calculatedσxy,after the initial increase,decreases sharply,then increases and changes sign becoming positive,andfinally goes down again,capturing the basic features of the ex-perimental results.Even more surprisingly,by converting the measuredρxy versus T curves shown in Fig.1C into theσxy versus M curves shown in Fig.1D,now they all follow the same trend,and match with our calculations as long as the experimental data is available. Those curves are measured for different samples(with different saturation moments),they all follow the same rule qualitatively,and could be simply explained by the reduction of mag-netization[21](see Fig.1A).Note,however,that the comparison between the experiments and calculations should be semi-quantitative,because the results are sensitive to the lattice structures.The calculatedσxy for thefictitious Cubic structure shows a strong deviation from that obtained for orthorhombic structure,and it changes the sign to be positive at low temperature(large M).Therefore more accurate information on the structure is needed to obtain the quantitative result.However,such a sensitivity does not affect our main results, i.e,the non-monotonous behavior ofσxy.Even the calculations for Cubic structure show such a behavior,and may be used as a guide of possible deviation.The results and analysis presented here should stimulate and urge the reconsideration of the electronic states in magnetic materials from the very fundamental viewpoint.For example,the magnetic monopole is accompanied by the singularity of the vector potential,i.e.,Dirac string[1].As shown by Wu-Yang[22]this means that more than two overlapping regions have to be introduced,in each of which the gauge of the wavefunction is defined smoothly.This means that one cannot define the phase of the Bloch wavefunctions in a single gauge choice when the magnetic monopole is present in the crystal momentum space. This leads to some nontrivial consequences such as the vortex in the superconducting order parameter as a function of k[23],and many others are left for future studies.[1]P.A.M.Dirac,Proc.Roy.Soc.London133,60(1931).[2]G.t’Hooft,Nucl.Phys.B79,276(1974)[3] A.M.Polyakov,JETP Lett.20,194(1974).[4]It has been recognized in the original paper by M.V.Berry(Proc.R.Lond.A392,45(1984))that the degeneracy point in the parameter space acts as a magnetic monopole where the gaugefield is enhanced.[5]H.Ohno,SCIENCE,281,951(1998).[6]R.Karplus,J.M.Luttinger,Phys.Rev.95,1154(1954).[7]J.Smit,Physica21,877(1955);Physica24,39(1958).[8]W.Kohn,J.M.Luttigner,Phys.Rev.108,590(1957).[9]J.M.Luttigner,Phys.Rev.112,739(1958).[10]L.Berger,Phys.Rev.B2,4559(1970).[11]M.Onoda,N.Nagaosa,Phys.Rev.Lett.90206601(2003).[12]T.Jungwirth,Q.Niu,A.H.MacDonald,Phys.Rev.Lett.88,207208(2002).[13]Y.Taguchi,Y.Ohara,H.Yoshizawa,N.Nagaosa,Y.Tokura,Science291,2573(2001).[14]J.Sinova,T.Jungwirth,J.Kucera,A.H.MacDocnald,Phys.Rev.B67,235203(2003)[15] A.Shapere,F.Wilczek,Geometric Phases in Physics,(World Scientific,1989).[16] D.J.Thouless,M.Kohmoto,M.P.Nightingale,M.den Nijs,Phys.Rev.Lett.49,405(1982).[17]R.M.Martin,Phys.Rev.B5,1607(1972).[18]R.D.King-Smith and D.Vanderbilt Phys.Rev.B.47,1651(1993).[19]Note that this gaugefield is distinct from that of the magneticfield B( r)in real space,althoughthey are analogous to each other.In the presence of B( r),the covariant momentum operator πµis given byπµ=−i∂x+eAµ( r)where B=∇× A.This leads to the commutation relationµ[πx,πy]=−ie(∂x A y−∂y A x)=−ieB z etc.,and to the Lorenz force due to the magneticfieldk)and B( r),are dual to each other,and theB.Therefore these two gaugefields,i.e.,bµ(presence of the one does not necessarily mean that of the other.[20]M.Shikano,T.K.Huang,Y.Inaguma,M.Itoh,T.Nakamura,Solid State Comm.90,115(1994).[21]More Ca-doped samples with different concentrations have been measured.They all follow thesame trend,and are not shown here due to the space.[22]T.T.Wu,C.N.Yang,Phys.Rev.D12,3845(1975).[23]S.Murakami,N.Nagaosa,Phys.Rev.Lett.90,057002(2003).Supporting Online MaterialSupporting Online TextReferences0.511.5Magnetization M (µB /Ru)−200−150−100−5050σx y (Ω−1c m −1)0.01.02.0M (µB /R u )Single Crystal Film Ca−doped film100200300ρx x (µΩc m )Single Crystal Film Ca−doped film 050100150200T (K)−0.8−0.400.40.8ρx y (µΩc m )Single Crystal Film Ca−doped film AB CDFIG.1:Measured temperature dependence of the (A)magnetization M ,(B)longitudinal resistivity ρxx ,and (C)transverse resistivity ρxy for the single crystal and thin film of SrRuO 3,as well as for the Ca doped Sr 0.8Ca 0.2RuO 3thin film.The corresponding transverse conductivity σxy is shown in(D)as a function of the magnetization,together with the results of first principle calculations for cubic and orthorhombic structures [20].In our calculations,the change of magnetization is taken into account by the rigid splitting of up and down spin bands.As the transverse conductivity should vanish with M at high temperatures,the calculated σxy is multiplied by the additional M/M 0(M 0=1.5µB )factor,which does not affect its behavior except at the very vicinity of Tc.012345Frequency ω (eV)0100020003000σx x (Ω−1c m −1)Calc.OrthorhombicCubic 012345Frequency ω (eV)Exp. (10K)−200204060R e (σx y ) (Ω−1c m −1)0.00.5 1.0−200204060I m (σx y ) (Ω−1c m −1)0.00.51.02.0 3.04.0Frequency ω(eV) 2.0 3.0 4.0Frequency ω (eV)× 4× 4FIG.2:Calculated (left panels)and measured (right panels)longitudinal (σxx )and transverse (σxy )optical conductivity of SrRuO 3film.The measurements were performed at low temperature (10K).The calculations were done for both the orthorhombic single crystal structure and the hypothetic cubic structure by keeping the average Ru-O bond length.The experimental σxy are shown by multiplying a factor of 4.The quantitative comparison between the experiments and calculations about the absolute value of σxy would require more accurate structure information (see text part).Nevertheless,the clear peak structures for the low energy σxy is the demonstration of monopolesFIG.3:de-generate.H ( k )= µ=0,1,2,3µµ1,2,30from k to the vector f ( k )=(f 1( k ),f 2( k ),f 3( k ))=f ( k )(cos ϕ f sin θ f,sin ϕ f sin θ f ,cos θ f ),the contribu-tion to σxy from the neighborhood of degeneracy region can be given by the f-space solid angle d Ω f =[∂(θ f,ϕ f )/∂(k x ,k y )]sin θ f dk x dk y =dϕ f sin θ f dθ f for the infinitesimal dk x dk y integrated over k .The solid angle corresponds to the flux from the monopole at f = 0.(See SOM).(B)Calculated flux distribution in k -space for t 2g bands as a function of (k x ,k y )with k z being fixed at 0for SrRuO 3with Cubic structure.The sharp peak around k x =k y =0and the ridges along k x =±k y are due to the near degeneracy of d yz and d zx bands because of the symmetry reasons as explained in the online supporting text.−1−0.500.51Fermi Level Position (eV)−400−2000200400σx y ( Ω−1c m −1)−10−505Energy (eV)−4−224D O S (S t a t e s /e V f .u .)without SOC with SOC (a)(b)FIG.4:The calculated (A)density of states (DOS)and (B)σxy as functions of Fermi level position for the orthorhombic structure of single crystal SrRuO 3.The Fermi level is shifted rigidly relative to the converged solution,which is specified as zero point here.The sharp and spiky structure of σxy is the demonstration of the singular behavior of magnetic monopoles.SUPPORTING ONLINE MATERIALMaterials and Experimental MethodsBoth stoichiometric SrRuO3and Ca-doped(Sr0.8Ca0.2RuO3)thin(500˚A)single crystal epitaxialfilms were grown on the(001)surface of SrTiO3(STO)single crystal substrate by pulsed laser deposition(PLD)employing KrF laser pulses(100mJ)focused on the polycrys-talline target,while SrRuO3bulk single crystal were prepared using aflux method(SrCl2flux).The quality of those samples were confirmed by the X-ray diffraction.The bulk sin-gle crystal is orthorhombic,with a=3.911˚A,b=3.936˚A,c=3.922˚A.Thefilms are coherently strained by the SrTiO3substrate,yielding a tetragonal distortion in the[001]direction.As the results,the out-of-plane lattice constants offilms are elongated(c=3.950˚A for SrRuO3film and c=3.931˚A for Ca-dopedfilm),and the magnetic easy axis is perpendicular to the film plane.For the transport measurements offilms,they were patterned in Hall bar geometry us-ing conventional photo-lithography and Ar ion dry-etching.The Hall resistivityρH was measured together with the longitudinal resistivityρxx as a function of temperature under applied magneticfield.The anomalous resistivityρxy was determined after subtraction of the ordinary Hall contribution from the measuredρH,and the transverse conductivityσxy was estimated as−ρxy/ρ2xx.In addition to the transport measurement,the frequency(ω)de-pendent conductivities were measured for SrRuO3film by optical method.The longitudinal part of the optical conductivityσxx(ω)was obtained from Kramers-Kronig transformation of normal reflectivity from0.08eV to40eV,while the transverse partσxy(ω)was deduced from magneto-optical Kerr spectra.The spectra were measured by a polarization modu-lation method.For the energy region of0.7–4eV,ordinary measurement system with grating monochromator and photo-elastic modulator(CaF2window,modulation frequency of56kHz)was used for the measurement.For the energy region below0.8eV,Fourier transform infrared(FT-IR)spectrometer with rapid scan mode(scan speed0.16cm/s)was utilized for the measurement.The light from the FT-IR interferometer was,atfirst,polar-ized by wire-grid polarizer(BaF2substrate).Then the polarization was modulated by ZnSe photo-elastic modulator(modulation frequency of50kHz)and focused by BaF2lens(focal length of150mm).The light reflected from the sample was detected by photovoltaic typeHgCdTe detector.By this system we can measure magneto-optical spectra down to0.2eV with accuracy higher than0.01degree.Some Details of First-Principles CalculationsFirst-principles calculations ofσxy are quite challenging,requiring the combination of sev-eral modern calculation techniques and powerful supercomputer sources.The plane-wave pseudopotential calculations[1]were performed based on the local spin density approxima-tion(LSDA),that give good descriptions for the electronic and magnetic properties of this compound[2,3].The spin-orbital coupling(SOC)is treated self-consistently by using the relativistic fully separable pseudopotentials[4]in the framework of non-collinear magnetism formalism.The inter-band optical transitions are calculated from the converged Kohn-Sham eigen states by using the Kubo formula[5,6].The analytical tetrahedron method[7]has been used for the accurate k-space integration,and the convergence has been checked care-fully.Thefinite life-time broadeningδhas been estimated from the experimental residual resistivity and extended Drude analysis ofσxx.The validity of our calculation techniques has been well demonstrated in the explanation of anisotropic optical data in Ca2RuO4[8].σxy and Magnetic Monopole in Momentum SpaceHere we present more details on the contribution to the transverse conductivityσxy from the k region where two bands are nearly degenerate.Enhancement of the gaugefield b z( k)occurs when more than two bands are close energetically,which corresponds to the magnetic monopole and shows itsfingerprint in anomalous Hall effect(AHE)and Kerr rotation as described below[9].Consider the general case where two band Hamiltonian matrix H( k)at k can be written as H( k)= µ=0,1,2,3fµ( k)σµwhereσ1,2,3are the Pauli matrices andσ0is the unit matrix.Then we can consider the mapping from k to the vectorf( k)=(f1( k),f2( k),f3( k))=f( k)(cosϕfsinθf,sinϕfsinθf,cosθf)as shown in Fig.3A ofthe main text.Then H( k)can be easily diagonalized to obtain the two eigenvaluesε±( k)= f0( k)±f( k).Calculating Eq.5of the main text in this case,we obtain the contribution to σxy from these two bands asσ2−bands xy =e2× ∂ϕ f∂k y−∂ϕ f∂k x sinθ fe2=(S2| f|3due to the monopole at f= 0over the infinitesimal surface in f-space corresponding toagain hasthe small square dk x dk y in k-space(Fig.3A of the main text).Thereforeσ2−bandsxythe geometrical meaning in f space.This gaugefield strongly depends on k in the(near)degeneracy case,i.e.,when f( k)is near the monopole.There are two cases for the(near)degenerate bands.One is the accidental degeneracy[10,11]where the three equations f1( k)=f2( k)=f3( k)=0are satisfied at k= k0.Nearthis accidental band crossing,one can expand as f a( k)= bαab(k b−k0b)where a,b=1,2,3.In this case the f-space can be identified with the k-space,and the gaugefield distributionin k-space around k= k0is similar to Eq.S1above replacing f by k− k0.Although thereoccurs no singularity inσxy atµ=ε±( k0)due to the cancellation between the positive andnegative k z−k z0,the nonlinear dependence of f a( k)gives rise to the strongµ-dependenceofσxy slightly away from this energy.The other class of(near)degeneracy is due to the symmetry,where the k-group has the irreducible representation with the dimensions morethan2.As an example,one can consider the simplest tight binding model of t2g orbitalsd yz and d zx on the cubic perovskite structure,which is relevant to the SrRuO3discussedin the main text.In this case the H( k)for these two bands with up-spin is given byf0( k)=−2t1cos k z−t1(cos k x+cos k y),f1( k)=2t2sin k x sin k y,f2( k)=−λM,f3( k)=−t1(cos k x−cos k y),where t1,t2are the effective intra-and inter-orbital transfer integrals respectively,λis the spin-orbit coupling(SOC)constant,and M is the magnetization.WhenλM=0,there occurs the degeneracy along the line k=(0,0,k z).Furthermorewhen t2=0,there occurs the degeneracy along the plane k x=±k y.Considering the caset1>>t2>>λM,which is relevant to SrRuO3,the gaugefield b z( k)with k z=0has alargest peak at k x=k y=0and is enhanced along the lines k x=±k y,which is actuallyseen in the realistic calculation for SrRuO3shown in Fig.3B of the main text.In this casethere occurs no cancellation of b nz( k)from the integral over k z.Although the discussion above is applicable to any(near)degeneracy,the(singular)gauge field from different band crossings in f-space cancel out in the presence of the time-reversal symmetry and/or the absence of the SOC.The former prohibits thefiniteσxy.In the absence of SOC,up-spin and down-spin bands are decoupled,and each of them can be represented by the Hamiltonian matrix without the time-reversal symmetry breaking.In ferromagnets with the SOC,the singular behavior ofσxy by changing some parameters such as the chemical potential and the magnetization is thefingerprint of the monopoles in k-and/or f-space.[1]Z.Fang,K.Terakura,J.Phys.:Condens.Matt.14,3001(2002).[2]I.I.Mazin,D.J.Singh,Phys.Rev.B56,2556(1997).[3] D.J.Singh,J.Appl.Phys.79,4818(1996).[4]G.Theurich,N.A.Hill,Phys.Rev.B64,73106(2001).[5] C.S.Wang,J.Callaway,Phys.Rev.B9,4897(1974).[6]Z.Fang,N.Nagaosa,K.Terakura,Phys.Rev.B67,35101(2003).[7]mbin,J.P.Vigneron,Phys.Rev.B29,3430(1984).[8]J.H.Jung,Z.Fang,J.P.He,Y.Kaneko,Y.Okimoto,Y.Tokura,Phys.Rev.Lett.91,056403(2003).[9]M.Onoda,N.Nagaosa,J.Phys.Soc.Jpn.71(1)19(2002).In this paper,the magneticmonopole in momentum space has been discussed in relation to AHE for a simple two-dimensional model.[10]V.Neumann,E.Wigner,Physik.Zeits,30,467(1929).[11] C.Herring,Phys.Rev.52,365(1937).。
关于磁化方法的物理题
关于磁化方法的物理题英文回答：Magnetic Materials and Magnetization Methods.Magnetic materials are substances that can be magnetized when exposed to a magnetic field. Magnetization is the process of aligning the magnetic moments of the atoms or molecules in a material in a certain direction. There are different methods for magnetizing materials, each with its own advantages and disadvantages.Types of Magnetization Methods:Saturation Magnetization: This method involves exposing a material to the maximum possible magnetic field strength to align all of its magnetic moments.Remanent Magnetization: After a material has been saturated, it will retain some of its magnetization evenwhen the magnetic field is removed. This residual magnetization is called remanence.Hysteresis: The relationship between the applied magnetic field strength and the magnetization of a material is often not linear. This nonlinearity is called hysteresis, and it can affect the magnetic properties of the material.Demagnetization: This process involves reducing the magnetization of a material by exposing it to a magnetic field in the opposite direction of its original magnetization.Factors Affecting Magnetization:The magnetization of a material depends on several factors, including:Material Properties: The composition, microstructure, and magnetic susceptibility of the material influence its ability to be magnetized.Magnetic Field Strength: The strength of the applied magnetic field determines the degree of magnetization.Temperature: High temperatures can reduce the magnetization of a material.Mechanical Stress: Applying mechanical stress to a material can affect its magnetic properties.Applications of Magnetization:Magnetization has various applications in different fields, such as:Data Storage: Magnetic recording is used in hard disk drives and magnetic tapes to store data.Magnetic Resonance Imaging (MRI): MRI scanners use powerful magnets to create images of the inside of the human body.Electric Motors and Generators: Magnets are used inelectric motors to convert electrical energy into mechanical energy and in generators to convert mechanical energy into electrical energy.Magnetic Separation: Magnets can be used to separate magnetic materials from non-magnetic materials.中文回答：磁性材料和磁化方法。
微纳米流动和核磁共振技术
微纳米流动和核磁共振技术英文回答：Microfluidics and nuclear magnetic resonance (NMR) are two important technologies that have revolutionized various fields of science and engineering.Microfluidics refers to the study and manipulation of fluids at the microscale level, typically in channels or chambers with dimensions ranging from micrometers to millimeters. It allows precise control and manipulation of small volumes of fluids, enabling a wide range of applications such as chemical analysis, drug delivery systems, and lab-on-a-chip devices. Microfluidic devices are often fabricated using techniques such as soft lithography, which involve the use of elastomeric materials to create microchannels and chambers.NMR, on the other hand, is a powerful analytical technique that utilizes the magnetic properties of atomicnuclei to study the structure and dynamics of molecules. It is based on the principle of nuclear spin, which is the intrinsic angular momentum possessed by atomic nuclei. By subjecting a sample to a strong magnetic field and applying radiofrequency pulses, NMR can provide information about the chemical composition, molecular structure, and molecular interactions of the sample. NMR has diverse applications in fields such as chemistry, biochemistry, medicine, and materials science.Microfluidics and NMR can be combined to create powerful analytical tools for studying various biological and chemical systems. For example, microfluidic devices can be used to precisely control the flow of samples and reagents, while NMR can provide detailed information about the composition and structure of the samples. This combination has been used in the development ofmicrofluidic NMR systems, which allow rapid and sensitive analysis of small sample volumes. These systems have been applied in areas such as metabolomics, drug discovery, and environmental monitoring.中文回答：微纳米流体力学和核磁共振技术是两种重要的技术，已经在科学和工程的各个领域引起了革命性的变化。
Unit 2 Language and Mind Part B高二英语(上外版2020选择性必修二)
Reading B words and expressions
Alzheimer's disease
New expressions
n. 阿尔茨海默病， 老年期痴呆最常见的一种类型
New expressions
magnetic adj. 有磁性的,有吸引力的
drive someone to distraction 使某人极为苦恼
A very clingy child can drive a parent to distraction. 依赖性太强的孩子会逼得父母心烦意乱。
versus
New expressions
prep. (表示两队或双方对阵)对; (比较两种不同想法、选择等)与…相对; 对抗
magnetic field
磁场
magnetic separation 磁性分离
magnetic resonance imaging 磁共振成象
The secondary current sets up a magnetic field inside the tube. 二次电流在管子内部会产生磁场。
consider
New expressions
v. 考虑; 把(某人,某事)看作…, 认为(某人,某事)如何; 细想
all things considered 综合各方面来看
After a year we will sympathetically consider an application for reinstatement. 一年后我们会对复职申请予以考虑和批准。
New expressions consideration n. 考虑,考察; 照顾,关心
New Value of the α3 Electron Anomalous Magnetic Moment
a( (Fig. 1; 1990)
= 0.905 l(86) .
(14)
a6C
~M6C
+ L2~M4
2L2~M4b
+ klsm4 (M2 [I] —M2 ) + ( 584, + I4»— 2I4, — + 2L282)M2, + 2L2
At
= At(2) (n/vr) + At(4) (a/n)
+
A,
(6)
(n/~)
3
+
A,
(8)
(n/~)
+
representing 1, 7, 72, and 891 Feynman diagrams, respecPreviously reported values tively, have been evaluated. of these coefficients are [3]
= AM6g —282AM4b
+ (2I4, +
B2
(L2
—2L2B2)M2,
(5)
a6~
= AM6s +
A~,
= 4.46(20)
X 10
(2)
Thus far the first four coefficients in the perturbation expansion of the mass-independent A~ term
a6g
.
where m„m~, and m are the masses of the electron, muon, and tau, respectively. A2 and A3 as well as the contributions of the hadronic and weak interactions are very small and known with sufficient accuracy. They add up to [3]
Fe-based amorphous,soft magnetic composites
（3）Those who filled electron shell of the total magnetic moment are zero. Electronic magnetic moment on only unfilled electron shells have total magnetic moment of the unpaired atoms contribute;
（4）All the magnetic moments of the ferromagnetic material mainly by the contribution of electron spin rather than electron orbital motion of contribution.
A soft magnetic material mainly those magnetized easily repeated, and the external magnetic field is removed, a material easy demagnetization. 2. Soft magnetic materials features:
(1)Composition of matter of elementary particles (electrons, protons, neutrons, etc.) have an intrinsic magnetic moments (spin magnetic moment) （2） Micro-current magnetic moment (orbital moment) due to the motion of electrons generated atoms, as well as protons and neutrons in the nucleus of the produced.