A comment on BCC crystalization in higher dimensions
aa r X i v : . [ h e p - t h ] J a n A comment on BCC crystalization in higherdimensions
S. Elitzur , B. Karni , E. Rabinovici Racah Institute of Physics, The Hebrew University,Jerusalem 91904, Israel
Abstract
The result that near the melting point three-dimensional crystalshave an octahedronic structure is generalized to higher flat non com-pact dimensions. [email protected] [email protected] [email protected] ontents In their paper ”Should All Crystals Be bcc? Landau Theory of Solidificationand Crystal Nucleation” [1] , S. Alexander and J. McTague review the Landautheory of solidification [2] which predicts that solids in three space dimensionsform a bcc(body centered cubic) structure and pointed out that indeed un-der certain circumstances many solids near their melting point exhibit thisbehavior. Landau’s theory of phase-transitions implies that an octahedronstructure in momentum space gives a global extremum to the free-energy,and thus is the preferred structure. This is true in three spatial dimensions.In two dimensions either triangular or honeycombed lattices are predicted toform depending on the sign of the appropriate order parameter. Solidificationoccurs when an inhomogeneous configuration becomes energetically favoredcompared to a homogeneous one. The condensing density waves allow vari-ous patterns of spontaneously broken translational and rotational invariance.For a review of somewhat more complex spontaneous breaking of space timesymmetries see i.e. [3].In string theory higher dimensional systems are natural, in fact ten dimen-sional supersymmetrical systems have a special role as being stable. Assumingthe microscopic theory allows the formation of solids in higher dimensions itis interesting to investigate if general structures can emerge generically. Intachyonic string backgrounds closed and open tachyon fields are also forcedto have a non-zero wave number, condensing this field leads to structures sim-ilar to that of solidification, however not necessarily in three dimensions [4].This motivates generalizing the study of Alexander and McTague to higherdimensions, which is the subject of this paper. Following their work we as-sume the structure of regular polytops and find that the result remains truein higher flat non compact dimension.In section 2 we review the work of S. Alexander and J. McTague and insection 3 we generalize it to more than three space dimensions.1
Review
In Landau’s theory of phase transition, one expands the free-energy,Φ = Φ + Φ + ... (1)Where Φ n is of order n in the density field ρ .The presence of a Φ term is essential for the solidification to occur. A Φ term provides stabilization.The second-order term, in momentum-space, is:Φ = Z d q d q A ( −→ q , −→ q ) δ ( −→ q + −→ q ) ρ −→ q ρ −→ q = Z d qA ( ~q ) ρ −→ q ρ − −→ q (2)With ρ ~q being the Fourier component of the density function, i.e. ρ ( ~r ) = π ) / R d qρ ~q e − i~q · ~r . The form of the function A ( ~q ) depends on themicroscopic theory. The delta-function ensures translational-invariance andthat A ( ~q ) is a function of a single ~q . Moreover isotropy implies that A ( ~q )depends actually only on the magnitude of ~q . Translational invariance willbe spontaneously broken for those class of functions A ( ~q ) for which the mag-nitude Q of the vector ~q does not vanish at the extremum. In such a groundstate the configuration ρ q gets support only on the momentum space sphereof this radius. The second order term is then proportional to R d Ω ~Q ρ ~Q ρ ~ − Q ,which equals to R d rρ ( r ), henceforth denoted by ρ .The magnitude of ρ isfixed by the fourth order term.In momentum-space the third-order term is:Φ = Z d q d q d q B ( −→ q , −→ q , −→ q , T ) δ ( −→ q + −→ q + −→ q ) ρ −→ q ρ −→ q ρ −→ q (3)Here T stand for all of the thermodynamical quantities on which B depends.Since the magnitude of the −→ q i ′ s is already fixed, one can re-write this termas: Φ = B Q ( T ) Z d Ω d Ω d Ω δ (cid:16) −→ Q + −→ Q + −→ Q (cid:17) ρ −→ Q ρ −→ Q ρ −→ Q (4)One sees that Φ gets a contribution only from configurations in which the −→ Q i ′ s form equilateral triangles. Thus rotational invariance is also sponta-neously broken. Since the position-space density ( ρ ( ~r )) is real, each configu-ration has to satisfy the equation ρ − −→ Q i = ρ −→ Q i † , which gives us ρ ( ~r ) = 1 √ n X i =1 (cid:18) ρ −→ Q i e i ~Q i ~r + ρ −−→− Q i e − i ~Q i ~r (cid:19) (5)2s possible density configurations. For such a configuration, with 2n valuesof q for which ρ q = 0, (if q is such a value, then necessarily -q is also sucha value), in order to maximize Φ , at a given ρ , all ρ −→ Q i ′ s should be of thesame magnitude. One obtains that n (cid:12)(cid:12)(cid:12)(cid:12) ρ −→ Q i (cid:12)(cid:12)(cid:12)(cid:12) = ρ , so the term ρ −→ Q ρ −→ Q ρ −→ Q isproportional to n − / . One can use this result to determine the configurationof the crystal. The momenta Q i for which ρ −→ Q i = 0 form some polyhedronin momentum-space. This polyhedron has exactly 2 n edges. The Φ termgets a contribution from each face of that polyhedron which is an equilateraltriangle. Φ is then proportional to (cid:16) N e (cid:17) − / · N t , where N e is the number ofedges of the polyhedron, and N t is the number of faces that are equilateraltriangles. The regular, convex, three-dimensional polyhedron that maximizesthe Φ term, is the octahedron [1]. Although two tetrahedra give the sameresult, this is essentially the same configuration, i.e. the momentum vectorsthat generate one generate the other. In higher dimensions, the same ideas apply. One considers the regular, convexpolytopes in higher dimensions. Postponing for a moment the discussion ofthe case of a four-dimensional space, which is a bit more complicated, wediscuss five or more dimensions. It is known [5] that in five dimensions ormore there are only three regular polytopes, all convex: The n-simplex, then-hypercube, and its dual the n-cross polytope, which is the n-dimensionalanalog of the octahedron. The free energy corresponding to each polytopeconfiguration depends on the number of elements in its one-skeleton (edges),and on the number of elements of its two-skeleton (two-dimesional faces) thatare equilateral triangles. The hypercube does not contribute in this case, sinceits two-skeleton does not include any triangles, equilateral or otherwise, sothe value of Φ for it is zero. For the n-simplex the number of elementsin its one-skeleton is given by (cid:16) n +12 (cid:17) [5] and the number of elements in itstwo-skeleton is given by (cid:16) n +13 (cid:17) [5]. All the elements of the two-skeleton areequilateral triangles, so we should consider all of them in our calculation.One should notice, though, that a single n-simplex does not contain anypair of opposite edges, so two n-simplexes have to be considered, which gives N e = 2 · (cid:16) n +12 (cid:17) elements in the one-skeleton and N t = 2 · (cid:16) n +13 (cid:17) elements in3he two-skeleton. The Φ term is then proportional to (cid:18) N e (cid:19) − / · N t = n + 12 ! − / · · n + 13 ! = 2 / n − n ( n + 1)) / (6)Turning our attention now to the cross-polytope, we have 2 · (cid:16) n (cid:17) elementsin the one-skeleton [5], and 2 · (cid:16) n (cid:17) elements in the two-skeleton [5], all ofwhich are equilateral triangles. This will give a value of (cid:18) N e (cid:19) − / · N t = · n !! − / · · n ! = 4( n − q n ( n −
1) (7)Dividing these two results, we get that the ratio between the value of Φ forthe cross-polytope and the value of Φ for the simplex is √ · n − n − · (cid:18) n + 1 n − (cid:19) / (8)which is greater than one for n greater than three. So we see that for fivedimensions, or more, the cross-polytope, which is the n-dimensional analogueof the octahedron, is the preferred momentum-space configuration in thesedimensions as well. In the special case of four dimensions one has six convexregular polytopes [5]. A direct check shows that here also the cross-polytope,called a 16-cell in four dimensions, is preferred. Thus the result holds for anydimension greater than two. Since (7) is a rising function of n for n >
2, ahigher dimensional polytope will always be preferable to a lower dimensionalone, ruling out liquid-crystals as a possible structure for a high-dimensionalcrystal, for the case studied here of a scalar order parameters. Tensorial orderparameters allow more structure already at n = 3. To determine the structureof the resulting lattice in position-space, notice that the vertices of the n crosspolytope are of the form e i , 1 ≤ i ≤ n , with ± ± i = j , and 0 in all others, i.e. a vector of theform , , ..., ± |{z} i’th place , , , ..., ± |{z} j’th place , , ..., (9)Thus, the lattice in momentum-space is generated by n linearly-independentsuch vectors. The reciprocal lattice, in position-space, is the set of vectorswhere scalar products with the momentum lattice vectors are integers. Itis generated by vectors with ± in all their entries. This is exactly a b.c.c.lattice. 4 Conclusions
We have shown that the result that near melting a bcc lattice is preferred, istrue in any number of flat non compact dimensions.Several issues require further study. One is how to generalize this analysisto compact dimensions of various topologies. The global structure of the man-ifold will further constrain the allowed configurations requiring in some casesto deal with commensurability aspects. Another is the case of configurationsthat form polytopes that are not convex or not regular, as may occur on thesphere. It is possible that such a configuration, which would be preferable tothe cross-polytope, exists. Such a configuration will not form a crystal, butit is possible, if it has sufficient symmetry, that it will form a quasi-crystal.
Acknowledgments
We thank David Mukamel for many discussions.The work of E. Rabinovici is partially supported by the Humbodlt foundationand the American-Israeli Bi-National Science Foundation.The work of S. Elitzur and E. Rabinovici is partially supported by a DIPgrant H, 52, the Einstein Center at the Hebrew University, and the IsraelScience Foundation Center of Excellence.