Ginzburg-Landau理论
冶金热力学基础知识简介
第一章冶金热力学基础1.冶金反应的焓变和吉布斯自由能变计算2.化学反应等温方程式3.溶解组元的活度及活度系数4.有溶液参加反应化学反应等温方程式分析5.熔铁及其合金的结构6.铁液中组分活度的相互作用系数关系式7.铁液中元素的溶解及存在形式8.熔铁及其合金的物理性质绪论冶金过程，尤其是钢铁冶金过程是高温、多相、多组元的复杂物理化学反应体系，一般而言：温度：>1000℃，炼钢温度在1600℃，甚至1700℃；多相：包括气—液—固三相气相：大气、燃气、反应气体、金属及其化合物的蒸气；液相：金属液、渣液；固相：金属矿石、固体燃料、耐火材料；多组元：金属液、炉渣、燃料都不是纯物质，而是多组元物质。
冶金过程物理变化：熔化、溶解、吸附、脱气、分金属夹杂上浮、金属的凝固等；冶金过程化学反应：燃料燃烧反应、生成—离解反应、氧化—还原反应、脱硫反应、脱磷反应、脱氧反应、脱碳反应等。
对这样的复杂体系，冶金物理化学能做什么？运用物理化学基本原理及实验方法，冶金物理化学研究和分析冶金过程的基本规律，为探索高效、优质、绿色的冶金工艺过程提供理论依据。
冶金物理化学大致分为：冶金热力学——主要研究冶金过程（反应）进行的方向和限度，以及在复杂体系中实现意愿反应的热力学条件。
是以体系的状态（平衡态）为基础，以状态函数描述过程的可能性为基本分析方法，不涉及“时间”这个参数。
冶金动力学——主要研究冶金过程（反应）的机理和速率，以及确定过程的限制性环节和强化过程的措施。
工业过程是要在有限时间内完成反应产物的获得，光有“可能性”还不够，要有“实现性”，这就必然涉及过程（反应）的机理和速率。
冶金熔体——高温金属熔体和熔渣结构、性质及模型描述。
冶金电化学——高温电解反应、金属液熔渣多相反应的机理和描述。
应该说，正是冶金物理化学的发展，才使得冶金由“技艺”成为“工程”和含有“科学”分量。
相对而言，冶金热力学发展得较为成熟，但研究高温下多相复杂冶金反应很困难，许多热力学数据还不完整。
Critical Points of the Ginzburg--Landau Functional on Multiply-connected Domains
by the expression E (Zu; A) = 2 1 kru?iuAk2 +(r A?H0)2+ 2 (juj2 ?1)2 2 n0 1 Z (r A?H )2 ; (1–1) +2 0
0
We interpret u as a complex wave function, referred to as the order parameter, with juj2 representing the (probability) density of superconducting electrons (Cooper pairs), and A as the magnetic eld vector potential so that r A (third component) is the induced magnetic eld. The applied magnetic eld H0 is the magnetic eld that would be present in the absence of the superconductor, and a local minimum of E is obtained with u = 0 and r A = H0 everywhere in . The Ginzburg{Landau parameter is a temperature-independent nondimensional material number Rubinstein 1998]. A simpli ed version of 1{1 is treated in Bethuel et al. 1994]: 2 1Z E (u) = 2 kruk2 + 2 (juj2 ? 1)2 : (1–2) A mathematical study of minimizers of this functional has been carried out in Bethuel et al. 1994; Shafrir 1995; Comte and Mironescu 1995] as well as in papers to which these works refer. Much of that work seems to be directed toward developing intuition which might carry over to 1{1. In Neuberger and Renka 1997] we describe numerical experiments in which we located two families of critical points for 1{2 under boundary conditions of degree d, for d = 2; : : : ; 10. These were found in response to a question raised in Bethuel et al. 1994, problem 12, p. 139]. The critical points of E are solutions to the stationary Ginzburg{Landau equations | that is, the Euler equations associated with (1{1) | with the corresponding natural boundary conditions | a pair of coupled nonlinear PDE's with nonlinear boundary conditions Du et al. 1992; Neuberger and Renka 1999]. The equations can be uncoupled and the boundary conditions can be linearized, but the cost of the computational procedure is quite high Mu and Huang 1998]. A second approach to computing the critical points is to evolve the time-dependent Ginzburg{Landau equations until a steady state is
相场法数值 模拟
四、相场方程 (phase-field equations)
Ginzburg-Landau 方程 f 0 k (r , t ) F Lk k k Lk t k (r , t ) k
Cahn-Hilliard 方程
2、凝固-单相场变量
ϵ 2 * 2 F f 0 ( xB , , T ) (xB ) ( ) dr V 2 2
等温凝固，假设摩尔体积不变，即组成梯度项不考虑 ϵ则=0
均质自由能密度
f 0 ( xB , , T * ) f p ( xB , , T * ) g ( )
图1（a）性能不连续
（b）性能连续
N. Moelans, B. Blanpain , P. Wollants, "An introduction to phase-field modeling of microstructure evolution", CALPHAD -Computer Coupling of Phase Diagrams and Thermochemistry, 32, 268-294, 2008
相场量(phase-fields)
两相
多相
p相，相应的变量 k 在系统中任一点 r
1表示在固相中 0表示在液相中 0 1在固液界面
k 1
p
k
1，k 0, k
三、热力学势函数 (thermodynamic energy functional)
经典热力学
F ( xB ,k ) f 0 ( xB ,k ) 1 xB (r , t ) M M xB (r , t ) Vm t xB (r , t ) xB
超导现象的原因
超导现象的原因引言超导现象是指在超导材料中，在低温下电阻突然消失的现象。
自从发现超导现象以来，科学家们对其原因进行了深入研究。
本文将探讨超导现象的原因，并分析其产生的机制。
超导现象的定义超导现象是指某些物质在低温下，电阻突然变为零的现象。
这些物质被称为超导体，低温是指接近绝对零度的温度。
超导现象的发现在物理学领域起到了重要的推动作用，也在实际应用中有着广泛的应用。
超导现象的起源超导现象起源于电子之间的相互作用。
当物质的温度降到一定程度时，电子之间的相互作用会导致电子的自发配对，形成所谓的“库珀对”。
这对电子可以在没有阻力的情况下通过超导体移动，从而导致电阻的消失。
超导现象的类型超导现象可以分为两种类型：Type I和Type II超导现象。
Type I超导现象Type I超导现象发生在临界温度以下的超导体中。
在这种情况下，超导态与正常态之间的相变是突然的，电阻值线性下降至零。
Type I超导材料的特点是磁场完全排斥，输运电流只能在外磁场的作用下才能够通过。
Type II超导现象Type II超导现象发生在临界温度以下的某些超导体中。
Type II超导体具有两个临界场：临界磁场Hc1和临界磁场Hc2。
当外磁场小于Hc1时，超导体处于完全超导态；当外磁场大于Hc2时，超导态会被磁场破坏，变为正常态。
而在Hc1和Hc2之间的外磁场范围内，超导体处于混合态，即部分区域处于超导态，部分区域处于正常态。
超导现象的机制超导现象的机制可以通过两个重要的理论来解释：BCS理论和Ginzburg-Landau理论。
BCS理论BCS理论是由约翰·巴丹和约翰·库珀等人于1957年提出的。
该理论基于量子力学和统计物理的原理，认为在超导体中，电子与声子之间的相互作用会导致电子自发配对，形成库珀对。
这些配对的电子不受散射的干扰，可以以零电阻的方式在超导体中移动。
BCS理论解释了Type I超导现象的发生，并成功预测了多种超导材料的临界温度。
吉布斯-亥姆霍兹公式
例4：试求标准压力下, 50 C 的过冷液体苯变为固 体苯的 S , 并判断此凝固过程是否可能发生.已知苯 的正常凝固点为 50 C, 在凝固点熔化热为 ：
f Hm 9940J mol1
液体苯和固体苯的定压摩尔热容分别为127和 123 J mol 1 。
解： -50C,苯（L） S ? -50C,苯（S）
)p
S (V )T
(
p T
)V
(
S p
)T
(V T
)p
利用该关系式可用实验可测偏微商来代替那些不易直接测 定的偏微商。
§2.9 △G的计算
一. 简单状态变化的定温过程的△G
因为dG SdT VdP
定温dT 0
所以dG VdP
2
G 1 VdP
对于理想气体 G 2 nRT dP nRT ln P2
melt
Tf
H
T Tb
3. 标准熵
4. 化学过程的熵变
在标准压力下，298.15 K时
Sm0
B
S
0 m
B
B
在标准压力下，反应温度T时
r Sm$ (T ) r Sm$ (298.15K)
T 298.15K
BCp,m (B)dT
B
T
§2.7 亥姆霍兹自由能与吉布斯自由能
S S1 S2 S3 35.62(J.K 1)
H2
H1
T2 T1
C
p
dT
268
9940 (123 127 )dT 278
9900 (J.mol 1)
Q H 9900 36.94(J.K 1) T T 268
材料成型基本原理课后答案解析
第一章习题1 . 液体与固体及气体比较各有哪些异同点？哪些现象说明金属的熔化并不是原子间结合力的全部破坏？答：（1）液体与固体及气体比较的异同点可用下表说明相同点不同点液体具有自由表面；可压缩性很低具有流动性，不能承受切应力；远程无序，近程有序固体不具有流动性，可承受切应力；远程有序液体完全占据容器空间并取得容器内腔形状；具有流动性远程无序，近程有序；有自由表面；可压缩性很低气体完全无序；无自由表面；具有很高的压缩性（2）金属的熔化不是并不是原子间结合力的全部破坏可从以下二个方面说明：①物质熔化时体积变化、熵变及焓变一般都不大。
金属熔化时典型的体积变化∆V m/V为3%~5%左右，表明液体的原子间距接近于固体，在熔点附近其系统混乱度只是稍大于固体而远小于气体的混乱度。
②金属熔化潜热∆H m约为气化潜热∆H b的1/15~1/30，表明熔化时其内部原子结合键只有部分被破坏。
由此可见，金属的熔化并不是原子间结合键的全部破坏，液体金属内原子的局域分布仍具有一定的规律性。
2 . 如何理解偶分布函数g(r) 的物理意义？液体的配位数N1、平均原子间距r1各表示什么？答：分布函数g(r) 的物理意义：距某一参考粒子r处找到另一个粒子的几率，换言之，表示离开参考原子(处于坐标原子r=0)距离为r的位置的数密度ρ(r)对于平均数密度ρo（=N/V）的相对偏差。
N1 表示参考原子周围最近邻(即第一壳层)原子数。
r1 表示参考原子与其周围第一配位层各原子的平均原子间距，也表示某液体的平均原子间距。
3．如何认识液态金属结构的“长程无序”和“近程有序”？试举几个实验例证说明液态金属或合金结构的近程有序（包括拓扑短程序和化学短程序）。
答：（1）长程无序是指液体的原子分布相对于周期有序的晶态固体是不规则的，液体结构宏观上不具备平移、对称性。
近程有序是指相对于完全无序的气体，液体中存在着许多不停“游荡”着的局域有序的原子集团（2）说明液态金属或合金结构的近程有序的实验例证①偶分布函数的特征对于气体，由于其粒子（分子或原子）的统计分布的均匀性，其偶分布函数g(r)在任何位置均相等，呈一条直线g(r)=1。
Generalized WDVV equations for B_r and C_r pure N=2 Super-Yang-Mills theory
a r X i v :h e p -t h /0102190v 1 27 F eb 2001Generalized WDVV equations for B r and C r pure N=2Super-Yang-Mills theoryL.K.Hoevenaars,R.MartiniAbstractA proof that the prepotential for pure N=2Super-Yang-Mills theory associated with Lie algebrasB r andC r satisfies the generalized WDVV (Witten-Dijkgraaf-Verlinde-Verlinde)system was given by Marshakov,Mironov and Morozov.Among other things,they use an associative algebra of holomorphic diffter Ito and Yang used a different approach to try to accomplish the same result,but they encountered objects of which it is unclear whether they form structure constants of an associative algebra.We show by explicit calculation that these objects are none other than the structure constants of the algebra of holomorphic differentials.1IntroductionIn 1994,Seiberg and Witten [1]solved the low energy behaviour of pure N=2Super-Yang-Mills theory by giving the solution of the prepotential F .The essential ingredients in their construction are a family of Riemann surfaces Σ,a meromorphic differential λSW on it and the definition of the prepotential in terms of period integrals of λSWa i =A iλSW ∂F∂a i ∂a j ∂a k .Moreover,it was shown that the full prepotential for simple Lie algebras of type A,B,C,D [8]andtype E [9]and F [10]satisfies this generalized WDVV system 1.The approach used by Ito and Yang in [9]differs from the other two,due to the type of associative algebra that is being used:they use the Landau-Ginzburg chiral ring while the others use an algebra of holomorphic differentials.For the A,D,E cases this difference in approach is negligible since the two different types of algebras are isomorphic.For the Lie algebras of B,C type this is not the case and this leads to some problems.The present article deals with these problems and shows that the proper algebra to use is the onesuggested in[8].A survey of these matters,as well as the results of the present paper can be found in the internal publication[11].This paper is outlined as follows:in thefirst section we will review Ito and Yang’s method for the A,D,E Lie algebras.In the second section their approach to B,C Lie algebras is discussed. Finally in section three we show that Ito and Yang’s construction naturally leads to the algebra of holomorphic differentials used in[8].2A review of the simply laced caseIn this section,we will describe the proof in[9]that the prepotential of4-dimensional pure N=2 SYM theory with Lie algebra of simply laced(ADE)type satisfies the generalized WDVV system. The Seiberg-Witten data[1],[12],[13]consists of:•a family of Riemann surfacesΣof genus g given byz+µz(2.2)and has the property that∂λSW∂a i is symmetric.This implies that F j can be thought of as agradient,which leads to the followingDefinition1The prepotential is a function F(a1,...,a r)such thatF j=∂FDefinition2Let f:C r→C,then the generalized WDVV system[4],[5]for f isf i K−1f j=f j K−1f i∀i,j∈{1,...,r}(2.5) where the f i are matrices with entries∂3f(a1,...,a r)(f i)jk=The rest of the proof deals with a discussion of the conditions1-3.It is well-known[14]that the right hand side of(2.1)equals the Landau-Ginzburg superpotential associated with the cor-∂W responding Lie ing this connection,we can define the primaryfieldsφi(u):=−∂x (2.10)Instead of using the u i as coordinates on the part of the moduli space we’re interested in,we want to use the a i .For the chiral ring this implies that in the new coordinates(−∂W∂a j)=∂u x∂a jC z xy (u )∂a k∂a k )mod(∂W∂x)(2.11)which again is an associative algebra,but with different structure constants C k ij (a )=C k ij(u ).This is the algebra we will use in the rest of the proof.For the relation(2.7)weturn to another aspect of Landau-Ginzburg theory:the Picard-Fuchs equations (see e.g [15]and references therein).These form a coupled set of first order partial differential equations which express how the integrals of holomorphic differentials over homology cycles of a Riemann surface in a family depend on the moduli.Definition 6Flat coordinates of the Landau-Ginzburg theory are a set of coordinates {t i }on mod-uli space such that∂2W∂x(2.12)where Q ij is given byφi (t )φj (t )=C kij (t )φk (t )+Q ij∂W∂t iΓ∂λsw∂t kΓ∂λsw∂a iΓ∂λsw∂a lΓ∂λsw∂t r(2.15)Taking Γ=B k we getF ijk =C lij (a )K kl(2.16)which is the intended relation (2.7).The only thing that is left to do,is to prove that K kl =∂a mIn conclusion,the most important ingredients in the proof are the chiral ring and the Picard-Fuchs equations.In the following sections we will show that in the case of B r ,C r Lie algebras,the Picard-Fuchs equations can still play an important role,but the chiral ring should be replaced by the algebra of holomorphic differentials considered by the authors of [8].These algebras are isomorphic to the chiral rings in the ADE cases,but not for Lie algebras B r ,C r .3Ito&Yang’s approach to B r and C rIn this section,we discuss the attempt made in[9]to generalizethe contentsof the previoussection to the Lie algebras B r,C r.We will discuss only B r since the situation for C r is completely analogous.The Riemann surfaces are given byz+µx(3.1)where W BC is the Landau-Ginzburg superpotential associated with the theory of type BC.From the superpotential we again construct the chiral ring inflat coordinates whereφi(t):=−∂W BC∂x (3.2)However,the fact that the right-hand side of(3.1)does not equal the superpotential is reflected by the Picard-Fuchs equations,which no longer relate the third order derivatives of F with the structure constants C k ij(a).Instead,they readF ijk=˜C l ij(a)K kl(3.3) where K kl=∂a m2r−1˜C knl(t).(3.4)The D l ij are defined byQ ij=xD l ijφl(3.5)and we switched from˜C k ij(a)to˜C k ij(t)in order to compare these with the structure constants C k ij(t). At this point,it is unknown2whether the˜C k ij(t)(and therefore the˜C k ij(a))are structure constants of an associative algebra.This issue will be resolved in the next section.4The identification of the structure constantsThe method of proof that is being used in[8]for the B r,C r case also involves an associative algebra. However,theirs is an algebra of holomorphic differentials which is isomorphic toφi(t)φj(t)=γk ij(t)φk(t)mod(x∂W BC2Except for rank3and4,for which explicit calculations of˜C kij(t)were made in[9]we will rewrite it in such a way that it becomes of the formφi(t)φj(t)=rk=1 C k ij(t)φk(t)+P ij[x∂x W BC−W BC](4.3)As afirst step,we use(3.4):φiφj= Ci·−→φ+D i·−→φx∂x W BC j= C i−D i·r n=12nt n2r−1 C n·−→φ+D i·−→φx∂x W BCj(4.4)The notation −→φstands for the vector with componentsφk and we used a matrix notation for thestructure constants.The proof becomes somewhat technical,so let usfirst give a general outline of it.The strategy will be to get rid of the second term of(4.4)by cancelling it with part of the third term,since we want an algebra in which thefirst term gives the structure constants.For this cancelling we’ll use equation(3.4)in combination with the following relation which expresses the fact that W BC is a graded functionx ∂W BC∂t n=2rW BC(4.5)Cancelling is possible at the expense of introducing yet another term which then has to be canceled etcetera.This recursive process does come to an end however,and by performing it we automatically calculate modulo x∂x W BC−W BC instead of x∂x W BC.We rewrite(4.4)by splitting up the third term and rewriting one part of it using(4.5):D i·−→φx∂x W BC j= −12r−1 D i·−→φx∂x W BC j= −D i2r−1·−→φx∂x W BC j(4.6) Now we use(4.2)to work out the productφkφn and the result is:φiφj= C i·−→φ−D i2r−1·r n=12nt n D n·−→φx∂x W BC j +2rD i2r−1·rn=12nt n −D n·r m=12mt m2r−1[x∂x W BC−W BC]j(4.8)Note that by cancelling the one term,we automatically calculate modulo x∂x W BC −W BC .The expression between brackets in the first line seems to spoil our achievement but it doesn’t:until now we rewrote−D i ·r n =12nt n 2r −1C m ·−→φ+D n ·−→φx∂x W BCj(4.10)This is a recursive process.If it stops at some point,then we get a multiplication structureφi φj =r k =1C k ij φk +P ij (x∂x W BC −W BC )(4.11)for some polynomial P ij and the theorem is proven.To see that the process indeed stops,we referto the lemma below.xby φk ,we have shown that D i is nilpotent sinceit is strictly upper triangular.Sincedeg (φk )=2r −2k(4.13)we find that indeed for j ≥k the degree of φk is bigger than the degree ofQ ij5Conclusions and outlookIn this letter we have shown that the unknown quantities ˜C k ijof[9]are none other than the structure constants of the algebra of holomorphic differentials introduced in [8].Therefore this is the algebra that should be used,and not the Landau-Ginzburg chiral ring.However,the connection with Landau-Ginzburg can still be very useful since the Picard-Fuchs equations may serve as an alternative to the residue formulas considered in [8].References[1]N.Seiberg and E.Witten,Nucl.Phys.B426,19(1994),hep-th/9407087.[2]E.Witten,Two-dimensional gravity and intersection theory on moduli space,in Surveysin differential geometry(Cambridge,MA,1990),pp.243–310,Lehigh Univ.,Bethlehem,PA, 1991.[3]R.Dijkgraaf,H.Verlinde,and E.Verlinde,Nucl.Phys.B352,59(1991).[4]G.Bonelli and M.Matone,Phys.Rev.Lett.77,4712(1996),hep-th/9605090.[5]A.Marshakov,A.Mironov,and A.Morozov,Phys.Lett.B389,43(1996),hep-th/9607109.[6]R.Martini and P.K.H.Gragert,J.Nonlinear Math.Phys.6,1(1999).[7]A.P.Veselov,Phys.Lett.A261,297(1999),hep-th/9902142.[8]A.Marshakov,A.Mironov,and A.Morozov,Int.J.Mod.Phys.A15,1157(2000),hep-th/9701123.[9]K.Ito and S.-K.Yang,Phys.Lett.B433,56(1998),hep-th/9803126.[10]L.K.Hoevenaars,P.H.M.Kersten,and R.Martini,(2000),hep-th/0012133.[11]L.K.Hoevenaars and R.Martini,(2000),int.publ.1529,www.math.utwente.nl/publications.[12]A.Gorsky,I.Krichever,A.Marshakov,A.Mironov,and A.Morozov,Phys.Lett.B355,466(1995),hep-th/9505035.[13]E.Martinec and N.Warner,Nucl.Phys.B459,97(1996),hep-th/9509161.[14]A.Klemm,W.Lerche,S.Yankielowicz,and S.Theisen,Phys.Lett.B344,169(1995),hep-th/9411048.[15]W.Lerche,D.J.Smit,and N.P.Warner,Nucl.Phys.B372,87(1992),hep-th/9108013.[16]K.Ito and S.-K.Yang,Phys.Lett.B415,45(1997),hep-th/9708017.。
二元合金非等温凝固枝晶生长的相场法模拟
－６９８·ＳｅＤ２００３ＦＯＵＮＤＲＹ的界面区域，溶质不易向液相中扩散，因此整个区域的溶质浓度比较高。
圈３￡＝０．１６Ⅲ酬铷非等沮氍矗时的蕾虚分布田Ｆｉｇ３ＴｈｅｔＢ咖茸ｒａ抽地ｆ酶岫ｓｆｏｒｎｏｆ一∞啦卿ｍ－ｌ∞ｌｉｄｉｆｉ龉ｔｉｏｌｌａｔ睁：ｏ．１６璐二元合金非等温凝固枝晶生长的相场法模拟作者：龙文元， 蔡启舟， 陈立亮， 魏伯康作者单位：龙文元(华中科技大学模具国家重点实验室,湖北,武汉,430074;南昌航空工业学院材料科学与工程系,江西,南昌,330034)， 蔡启舟,陈立亮,魏伯康(华中科技大学模具国家重点实验室,湖北,武汉,430074)刊名：铸造英文刊名：FOUNDRY年，卷(期)：2003,52(9)被引用次数：21次1.Wheeler A A;Ahmad N A;Boettinger W J查看详情 1995(07)2.Wheeler A A;Boettinger W J;McFadden G B Phase-field model for isothermal phase transitions in binary alloys[外文期刊] 19923.Beckermann C;LI Q;Tong X Microstructure evolution of equiaxed dendritic growth 2001(02)4.Tong X;Beckermann C;Karma A Velocity and shape selection of dendritic crystals in a forced flow[外文期刊] 2000(01)5.Charach C h;Fife P C Phase-field models of solidification in binary alloys:capillarity and solute trapping effects 19996.Murray B T;Wheeler A A;Glicksman M E Simulations of experi mentally observed dendritic growth behavior using a phase-field model[外文期刊] 1995(3/4)7.Wang S L;Sekerka R F;Wheeler A A Thermodynamically consistent phase-field models forsolidification 19938.Wheeler A A;Murrary B T;Schaefer R J Computation of dendrites using a phase field model 19939.Seol D J;Oh K H;Cho J W Phase-field modeling of the thermo-meehanical properties of carbon steels [外文期刊] 2002(9)10.Ode M;Kim S G;Kim W T Numerical predicttion of the secondary dendrite arm spacing using a phase-field model[外文期刊] 2001(04)11.Suzuki T;Ode M;Kim S G Phase-field model of dendritic growth[外文期刊] 2002(Part 1)12.Ode M;Suzuki T;Kim S G;Kim W T Phase-field model for solidification of Fe-C alloys 2000(01)13.Machiko ODE;Suzuki T Numerical simulation of initial microstructure evolution of Fe-C alloys using a phase-field model[外文期刊] 2002(04)14.Kim S G;Kim W T Phase-field modeling of rapid solidification 200115.Kim S G;Kim W T;Suzuki T Phase-field model for binary alloys[外文期刊] 199916.Boettinger W J;Warren J A The Phase-field method:simulation of alloy dendritic solidification during recalescence 1996(03)17.Warren J A;Boettinger W J Prediction of dendritic growth and microsegregation patterns in abinary alloy using the phase-field method[外文期刊] 1995(02)18.Kobayashi R Modeling and numerical simulations of dendritic crystal growth 19931.袁训锋.丁雨田.YUAN Xunfeng.DING Yutian强制对流作用下多晶粒生长的相场模拟[期刊论文]-材料导报2011,25(4)2.段珍珍.孙大千.朱松.邱小明.DUAN Zhen-zhen.SUN Da-qian.ZHU Song.QIU Xiao-ming Ti/ZrN/瓷界面微观结构和力学性能[期刊论文]-材料工程2008(9)3.刘晶峰.赵紫玉.蹇崇军.江开勇.LIU Jing-feng.ZHAO Zi-yu.JIAN Chong-jun.JIANG Kai-yong枝晶生长的相场法数值模拟[期刊论文]-华侨大学学报（自然科学版）2010,31(1)4.严卫东.刘汉武.杨爱民.熊玉华.刘林Al-Cu合金等轴枝晶组织形成的模拟及计算机可视化[期刊论文]-铸造技术2001(6)5.孙伟.周彩滨.艾红军开窗减压术后修复效果观察[期刊论文]-中国医科大学学报2008,37(2)6.宫文彪.孙大千.孙喜兵.刘威.GONG Wen-biao.SUN Da-qian.SUN Xi-bing.LIU Wei等离子喷涂纳米团聚体粉末的熔化特性研究[期刊论文]-材料热处理学报2007,28(4)7.李自军.魏伯康非热处理灰铸铁活塞环的铸造技术[期刊论文]-中国铸造装备与技术2007(1)8.蔡启舟.魏伯康.田中雄一等温淬火温度对奥贝球铁水脆化行为的影响[期刊论文]-材料热处理学报2004,25(6)9.朱兆军.米国发.尹冬松.王宏伟.曾松岩.ZHU Zhao-jun.MI Guo-fa.YIN Dong-song.WANG Hong-wei.ZENG Song-yan氮对Ti-6Al合金的铸态组织与性能的影响[期刊论文]-哈尔滨工业大学学报2006,38(5)10.苏润刚.艾红军.战德松义齿磁性固位体中两种不锈钢的细胞毒性研究[期刊论文]-口腔医学2003,23(5)1.许林.郭洪民基于宏微观耦合三维模型的铝合金凝固模拟[期刊论文]-特种铸造及有色合金 2011(4)2.杨恩娜.张雅静.赵文娟.丁桦金属凝固微观组织的相场模拟研究进展[期刊论文]-铸造技术 2010(11)3.余海洪.骆珊.龙文元.王东成对流作用下二元合金枝晶生长的相场法模拟[期刊论文]-热加工工艺 2010(18)4.吕冬兰.龙文元.夏春.潘美满.万红强迫对流影响合金凝固过程枝晶生长的数值模拟[期刊论文]-特种铸造及有色合金 2009(11)5.冯力.王智平.路阳.朱昌盛.肖荣振多元合金多晶粒的枝晶生长非等温相场模拟[期刊论文]-铸造 2009(5)6.王承志.张玉妥.曹秀丽相场法模拟Al-Cu合金连续冷却枝晶生长[期刊论文]-沈阳理工大学学报 2009(3)7.吕冬兰.龙文元.夏春.万红.潘美满相场法模拟对流作用下铝合金的枝晶生长[期刊论文]-铸造技术 2009(2)8.于志生.刘平.龙永强基于Ginzburg-Landau理论的相场法研究进展[期刊论文]-热加工工艺 2008(16)9.何可龙.龙文元各向异性对过冷熔体中枝晶生长影响的相场法模拟[期刊论文]-铸造工程 2007(2)10.白锐.周志敏.宋协青.杜娜液相线铸造法半固态金属组织特征的相场模拟研究[期刊论文]-铸造 2006(5)11.龙文元.蔡启舟.魏伯康.陈立亮相场法模拟多元合金过冷熔体中的枝晶生长[期刊论文]-物理学报 2006(3)12.路阳.王帆.朱昌盛.王智平等温凝固多晶粒生长相场法模拟[期刊论文]-物理学报 2006(2)13.路阳.王帆.朱昌盛.王智平.贾伟建基于KKS模型二元合金等温凝固过程的相场法模拟[期刊论文]-兰州理工大学学报 2006(1)14.单洪彬金属和金属玻璃基复合材料的凝固和热应力计算模拟[学位论文]硕士 200615.邱万里.蔡启舟.龙文元.魏伯康用相场法模拟Fe-0.5mol%C合金枝晶生长的相关参数优化[期刊论文]-现代铸铁2005(4)16.龙文元.蔡启舟.魏伯康.陈立亮二元合金非等温凝固过程的相场法模拟[期刊论文]-特种铸造及有色合金2005(2)17.龙文元.蔡启舟.魏伯康.陈立亮合金凝固过程再辉现象的数值模拟[期刊论文]-铸造 2005(11)18.庄建平.龙文元.方立高.张丽攀过冷熔体定向凝固过程枝晶生长的相场法模拟[期刊论文]-热加工工艺2005(12)19.邱万里.蔡启舟.龙文元.魏伯康相场法模拟Al-4.5%Cu枝晶生长参数的优化[期刊论文]-铸造设备研究 2005(1)20.庄建平.龙文元.方立高.张丽攀铝合金定向凝固过程枝晶生长的相场法模拟[期刊论文]-南昌航空工业学院学报（自然科学版） 2005(3)21.龙文元.方立高.张丽攀.陈乐平相场法模拟铝合金凝固过程的枝晶演变[期刊论文]-国外金属加工 2005(2)22.白锐半固态合金液相线铸造组织相场模拟研究[学位论文]硕士 2005本文链接：/Periodical_zz200309013.aspx。
Magnon Bose condensation in symmetry breaking magnetic field
a r X i v :c o n d -m a t /0607711v 2 [c o n d -m a t .s t r -e l ] 1 S e p 2006Magnon Bose condensation in symmetry breaking magnetic field S.V.Maleyev,V.P.Plakhty,S.V.Grigoriev,A.I.Okorokov and A.V.Syromyatnikov Petersburg Nuclear Physics Institute,Gatchina,Leningrad District 188300,Russia E-mail:maleyev@.spb Abstract.Magnon Bose condensation (BC)in the symmetry breaking magnetic field is a result of unusual form of the Zeeman energy that has terms linear in the spin-wave operators and terms mixing excitations which momenta differ in the wave-vector of the magnetic structure.The following examples are considered:simple easy-plane tetragonal antiferromagnets (AFs),frustrated AF family R 2CuO 4,where R=Pr,Nd etc.,and cubic magnets with the Dzyaloshinskii-Moriya interaction (MnSi etc.).In all cases the BC is important when the magnetic field is comparable with the spin-wave gap.The theory is illustrated by existing experimental results.1.Introduction Magnon Bose condensation (BC)in magnetic field was intensively studied in spin singlet materials (see for example [1]and references therein).In this case magnons condens in the field just above the triplet gap.In this paper we consider magnon BC that appears in the symmetry breaking magnetic field.The theoretical discussion is illustrated by experimental observation of this BC in frustrated antiferromagnet (AF)Pr 2CuO 4and cubic helimagnets MnSi and FeGe.To clarify our idea we begin with consideration of conventional AFs.In textbooks two limiting cases are considered.First,the magnetic field is directed along the sublattices.In this case the system remains stable up to the critical field H C =∆,where ∆is the spin-wave gap.Then the first order transition occurs to the state in which the field is perpendicular to sublattices (spin-floptransition).Second,the field is perpendicular to initial staggered magnetization.The system remains stable but the spins are canted toward the field by the angle determined by sin ϑ=−H/(2SJ 0),where J 0=Jz ;J and z are the exchange interaction and the number of nearest neighbors,respectively.At H +2SJ 0the spin-flip transition occurs to the ferromagnetic state.To the best of our knowledge the first consideration of the symmetry breaking field was performed theoretically in [2]in connection with experimental study of the magnetic structure of the frustrated AF R 2CuO 4,where R=Pr,Nd,Sm and Eu [3,4].In these papers the non-collinear structure was observed using the neutron scattering in the field directed at angle of δ=450to the sublattices.It was found in[2]that in inclinedfield the Zeeman energy has unusual form with terms which are linear in the spin-wave operators and term mixing magnons which momenta differ in the AF vector k0.As a result the BC arises of the spin-waves with momenta equal to zero and±k0.Similar situation exists in cubic helimagnets MnSi etc.[5].If thefield is directed along the helix wave-vector k the plain helix transforms into conical structure and then the ferromagnetic spin state occurs at criticalfield H C.But if H⊥k the magnon condense with momenta zero,±k,±2k etc.This leads to the following observable phenomena:i)a transition to the state with k directed along thefield at H⊥∼H C1=∆√S/2(a l−a+l),l=1,2 and a l(a+l)are Bose operators.As a result the Zeeman energy has unusual formH Z=H a+q+k0a q+iϑN,i.e. these operators has to be considered as classical variables as in the Bogoliubov theory of the BC in dilute Bose gas.Due to thefirst term in(2)we must consider the operatorsa±k0and a+∓kas classical variables too.Minimizing the full Hamiltonian with respectto these variables we obtainE=(∆2sin22ϕ)/(16J0)−S2J0ϑ2−(H H⊥)2/[4J0(∆2(ϕ,H)],(3)where thefirst term is the energy of the square anisotropy.In cuprates with S=1/2ithas quantum origin and arises due to pseudodipolar in-plane interaction[9].The secondterm is the energy of the spin canting in perpendicularfield.The last term is the BCenergy and∆2(ϕ,H)=∆2cos4ϕ+H2⊥−H2 is the spin-wave gap in thefield[2].This contribution becomes important at H∼∆.The spin configuration is determined byd E/dϕ=0and equilibrium condition d2E/dϕ2≥0.This theory was verified by neutrons scattering[10,11].In diagonalfield H (1,1,0)the spin configuration in frustrated Pr2CuO4is governed by Eq.(3)and the intensityof the(1/2,1/2,−1)is given by I∼1+sin2ϕ[2].Neglecting the BC term we getsin2ϕ=−(H/H C)2,where H C=∆.As a result at H→H C we obtain I∼H C−H. But very close to H C the BC term becomes important and we have a crossover to I∼(H C−H)1/2.It is clearly seen infigure2.This crossover was observed in[10,11].3.Frustrated AFsIn frustrated R2CuO4AFs there are two copper spins in unit cell belonging to different CuO2planes(see inset infigure1).From symmetry considerations these spins do not interact in the exchange approximation.The orthogonal spin structure is a result of the interplane pseudodipolar interaction(PDI)[2,3]and the ground state energy is given by∆2E=on T we obtain H C≃7.8T andγC≃5.30that is in stronger disagreement wit the experiment.The experimentally obtained anglesαandγat T=18K andδ=9.50are shown infigure4[12].The transition to the collinear state withα∼−450andγC∼200 was observed.Again the non-BC theory can not explain the experimental data.For example it givesγC≃2.50.Explanation of all these experimental data using the BC theory will be given elsewhere.4.BC in helimagnetsIn helimagnets MnSi etc.Dzyaloshinskii-Moriya interaction(DMI)stabilizes the helical structure and the helix wave-vector has the form k=SD[ˆa׈b]/A,where D is the strength of the DMI,A is the spin-wave stiffness at momenta q≫k,ˆa andˆb are unit orthogonal vectors in the plane of the spin rotation.The classical energy depends on thefield component H along the vector k and the cone angle of the spin rotation is given by sinα=−H/H C,where H C=Ak2is the criticalfield of the transition into ferromagnetic state[5].However at H⊥≪H C rotation of the helix axis toward thefield direction and the second harmonic2k of the spin rotation were observed[6-8].Both phenomena are related to the magnon BC in perpendicularfield[5].The linear and mixing terms appear in the Zeeman energy in much the same way as it was discussed above:H Z=(H a−i H b)2the real form of the BC energy is not so simple.It is determined by nonlinear interactions but consideration of this problem is out of the scope of this paper.As a result the perpendicular susceptibility is proportional to1/(∆2−H2⊥/2)and 2k harmonic appears.The last was observed by neutron scattering[6-8].Intensities of corresponding Bragg satellites have the formI±∼[∆2/(∆2−H2⊥/2)]2[1∓(kP)]δ(q∓2k),(7) where P is the neutron polarization.If H⊥→∆√5.ConclusionsWe discuss a few examples of the magnon BC in symmetry breaking magneticfield.BC appears due to unusual terms in Zeeman energy.Obviously this phenomenon is very general and can be observed in other ordered magnetic systems.Effects related to the BC has to be more pronounced in thefield of order of the sin-wave gap.6.AcknowledgmentsThis work is partly supported by RFBR(Grants03-02-17340,06-02-16702and00-15-96814),Russian state programs”Quantum Macrophysics”,”Strongly Correlated electrons in Semiconductors,Metals,Superconductors and Magnetic Materials””Neutron Research of Solids”,Japan-Russian collaboration05-02-19889-JpPhysics-RFBR and Russian Science Support Foundation(A.V.S.).References1Crisan M,Tifrea I,Bodea D and Grosu I2005Phys.Rev.B721644142Petitgrand D,Maleyev S,Bourges P and Ivaniv A1999Phys.Rev.B5910793D.Petitgrand,Moudden A,Galez P and Boutrouille P1990J.Less-Common Metals164-165768 4ChattpodhayaT,Lynn J,Rosov N,Grigereit T,Barilo S and Zigunov D1994Phys.Rev.B49 99445Maleyev S2006Phys.Rev.B731744026Lebech B,Bernard J and Feltfoft1989J.Phys.:Condens.Matter161057Okorokov A,Grigoriev S,Chetverikov Yu,Maleyev S,Georgii R,B¨o ni P,Lamago D,EckerslebeH and Pranzas P2005Physica B3562598Grigoriev S,Maleyev S,Chetverikov Yu,Georgii R,B¨o ni P,Lamago D,Eckerlebe H and Pranzas P2005Phys.Rev B721344029Yildirim T,Harris A,Aharony A and Entin-Wohlman O1995Phys.Rev.B521023910Plakhty V,Maleyev S,Burlet P,Gavrilov S and Smirnov O1998Phys.Lett.A25020111Ivanov A and Petitgrand D2004J.Mag.Mag Materials272-27622012Plakhty V,Maleyev S,Gavrilov S,Bourdarot F,Pouget S and Barilo S2003Europhys.Lett.61 53413Ivanov A,Bourges P and Petitgrand D1999Physica B259-261879FiguresaFigure 1.Spin configuration in the field.Full and dashed arrows correspond to zero and nonzero field,respectively.Addition spin canting in H ⊥is shown by broken arrows.Inset:spin configuration in neighboring planes of frustrated AF.Figure 2.Log-Log plot of the (1/2,1/2,−1)Bragg intensity in diagonal field,h ∼(H C −H ).Figure3.Thefirst order transition in thefield directed along b axis.Calculated intensities for the spinflop configurations when spins are perpendicular to thefield (white arrows).Figure4.Field dependence of anglesαandγatδ=9.50.)10a H mT =)50b H mT =)150c H mT=Figure 5.Bragg reflections in the field along (1,1,0).a)Four strong spots corresponds to ±(1,1,1)and ±(1,1,−1)reflections.Weak spots are the double Bragg scattering.b)The 2k satellites appear.c)The helix vector is directed along the field.。
landau 方程
landau 方程
Landau方程是指Landau-Lifshitz方程，是描述物质中磁矩热力学行为的方程。
这个方程是由列昂尼德·朗道和Evgeny
Lifshitz在20世纪中期提出的。
Landau-Lifshitz方程描述了磁矩在外磁场和温度梯度下的行为，并在固态物理学和磁性材料研究中得到了广泛的应用。
Landau方程可以用来描述铁磁性材料中磁矩的行为。
在外磁场作用下，磁矩会发生定向，而在温度梯度下，磁矩也会受到影响。
Landau-Lifshitz方程可以描述这些影响，并且可以用来研究磁性材料在不同条件下的行为。
除了描述磁性材料的行为外，Landau方程还可以用来研究相变现象和相界的性质。
相变是物质在外界条件改变时从一种状态到另一种状态的转变，比如固液相变、液气相变等。
Landau方程可以描述相变的一些性质，比如临界温度、临界压力等。
总的来说，Landau方程是一个重要的物理方程，它在研究磁性材料、相变现象等方面有着广泛的应用，对于理解物质的性质和行为具有重要意义。