Yu V Mikhlin

A L Zhupievt

AN APPLICATION OF THE INCE ALGEBRAIZATION TO THE STABILITY OF NON-LINEAR NORMAL VIBRATION MODES

Keywords: normal vibration mode, stability, bifurcation, Ince algebraization

A normal vibration mode stability in conservative non-linear systems is investigated. The algebraization by !nee (transition from linear equations with periodic coefficients to equations with singular points) is used. The normal mode stability in homogeneous systems, whose potential is an even homogeneous function of the variables and systems close to the homogeneous one, is investigated. Eigenvalues and eigenfunctions are obtained. Conditions when a number of instability zones in a non-linear system parameters space are finite (finite zoning or finite-gap conditions) are also obtained.

INTRODUCTION

Normal (principal) vibration modes can be used to construct a general solution in the linear theory. In non-linear systems the "normal vibration modes" (or "normal vibrations") are defined as such free motions where all coordinates of the system vary equiperiodically, reaching the extreme equipotential surface simultaneously [START_REF] Moser | Lectures on Hamiltonian Systems[END_REF][START_REF] Rosenberg | Nonlinear vibrations of systems with many degrees of freedom[END_REF][START_REF] Manevich | The Method of Normal Oscillation for Essentially nonlinear Systems[END_REF][START_REF] Vakakis | Non-similar normal oscillations in a strongly non-linear discrete system[END_REF]. All position coordinates can be well defined from any one of them by X;=p;(xd (i=2,3, ... ) p;(x) being analytical functions.

Rosenberg [START_REF] Moser | Lectures on Hamiltonian Systems[END_REF][START_REF] Rosenberg | Nonlinear vibrations of systems with many degrees of freedom[END_REF] gets credit for being the first to introduce broad classes of essentially non-linear (not quasilinear) conservative systems allowing normal vibrations with rectilinear trajectories in a configurational space (similar normal modes). In his and other works, normal mode stability was computed by linearized, Mathieu-type analysis [START_REF] Moser | Lectures on Hamiltonian Systems[END_REF][START_REF] Rosenberg | Nonlinear vibrations of systems with many degrees of freedom[END_REF][START_REF] Rosenberg | On the geometristion of normal vibrations of nonlinear systems having many degree of freedom[END_REF][START_REF] Atkinson | The stability of the normal modes of nonlinear systems with polynomial restoring forces of high degree[END_REF][START_REF] Month | The stability of bifurcating periodic solutions in a two-degree-of-freedom nonlinear system[END_REF], but it cannot guarantee good precision.

The stability of the normal vibration modes is studied by deriving an approximation for the Poincare map by Month and Rand (via Birkhoff-Gustavson canonical transformation) [START_REF] Month | An application of the Poincare map to the stability of nonlinear normal modes[END_REF] and Pak [START_REF] Pac | On the stability behavior of bifurcated normal modes in coupled nonlinear systems[END_REF]. This method is presented as an alternative to the usuallinearized stability on Floquet-Liapunov theory. Some papers [START_REF] Pac | On the stability behavior of bifurcated normal modes in coupled nonlinear systems[END_REF][START_REF] Rand | The geometrical stability of nonlinear normal modes in two degree of freedom systems[END_REF] are based on Synge's stability in the kinematico-statical sense. Some examples were considered. In Pecelli and Thomas's works [START_REF] Pecelli | Normal modes, incoupling, and stability for a class of nonlinear oscillators[END_REF][START_REF] Pecelli | Stability of modes for a class of non-linear planar oscillators[END_REF] the Poincare map, Kolmogorov-Arnold-Moser theorems and Lame equation theory were used. The similar normal modes stability was investigated while the linearizing variational equations represent the Lame equation; criterions of a stability in the non-linear sense were obtained. The effect of a normal mode bifurcation is investigated in various papers, for example, [START_REF] Month | The stability of bifurcating periodic solutions in a two-degree-of-freedom nonlinear system[END_REF][START_REF] Rand | The geometrical stability of nonlinear normal modes in two degree of freedom systems[END_REF]. This effect and resonance neighborhood are investigated by numerical and analytical techniques in [START_REF] Vakakis | Normal modes and global dynamics of a two-degree-of-freedom non-linear system-!. Low energies; •-II. High energies[END_REF].

In this work, the algebraization by Ince [START_REF] Nee | Ordinary Differential Equations[END_REF] and transition from linear equations with periodic coefficients to equations with singular points (Section 2) will be used. The normal vibration modes stability of conservative homogeneous systems whose potential is an even homogeneous function of the variables and systems close to the homogeneous one will be investigated (Section 3). Eigenvalues and eigenfunctions will be obtained. Conditions when a number of instability zones in a system parameters space are finite (finite zoning or finite-gap conditions) will be also obtained (Section 4). Finally, in Section 5, the results relating to non-linear stability analysis [START_REF] Pecelli | Normal modes, incoupling, and stability for a class of nonlinear oscillators[END_REF] will be reduced.

TRANSITION FROM LINEAR EQUATIONS WITH PERIODIC COEFFICIENTS TO EQUATIONS WITH SINGULAR POINTS

Let us now describe what is called the algebraization by Ince, that is the transfer from equations with periodic coefficients to equations with singular points, following [START_REF] Nee | Ordinary Differential Equations[END_REF], together with the relevant change in the stability problem. There are many methods of algebraization but only one of them is shown in this item. In the analytical treatment of normal vibrations, the algebraization is performed by choosing a new independent variable associated with the undisturbed motion. The variable t may be replaced by x, a variable that defines motion along a rectilinear trajectory, or by the velocity x, or by the kinetic energy for this solution etc.

We consider now the Hill equation

y + (a 0 + 2a 1 cos2t + a 2 cos4t + ... )y = 0
Let us introduce a new independent variable z = cost. We obtain

d 2 y dy ( C1J ) (1-z 2 ) -2 -z -+ L b.z 2 " y = 0.
dz dz n=O This equation has regular singularities at z = ± 1 (the indices for singular points are 0 and 1/2), and an irregular singular point at infinity.

Two fundamental solutions may be separated for the vicinity of the singular point z = 1 [START_REF] Nee | Ordinary Differential Equations[END_REF]:

y 1 (1-z) = !1(1-z); y 2 (1-z) = J1-zfz(1-z),
where f 1 and f 2 are analytical functions that converge within a circle 11 -z I : : : : : ; 2.

Since the equation will not change if z is replaced by -z, the following solutions will be associated with the singular point z = -1:

Y1(1 +z)=fl(l +z); y 2 (1 +z)=Jl+zfz(1 +z), Here the analytical functions f 1 and f 2 converge within a circle 11 + z I :::::; 2.

Within a common region of the two circles of convergence IX, p, y, b are constant.

Y1(1-z) = 1XY1(1 + z) + PYz(1 + z) Yz(1 -z) = YY1 (1 + z) + brz(1 + z), Replacing z by -z, one obtains yd1 + z) = 1XY 1 (1-z) + py 2 (l -z) = (1X 2 + py)yi(1 + z) + P(IX + c5)y 2 (1 + z), rz(1 + z) = yyd1-z) + by 2 (l-z) =}'(IX+ b)y 1 (1 + z) + (py + b 2 )y 2 (1 + z) It follows that a 2 + fiy = fiy + b 2 = 1 and fi(IX + b) = y(a + b) = 0, with only two cases possible: (1) IX= b = ± 1, fi = y = 0; and (2) IX= -b, fir= 1 -IX 2 • Let us consider the case of IX = b = 1, fi = I' = 0. Here Y1(1-z) = Y1(1 + z), Yz(1-z) = Y2(1 + z).
The origin of coordinates, z = 0, is, however, a regular point of the equation, therefore no two independent even solutions can exist that are meaningful in the vicinity of the point z = 0. It follows that condition (1) must be rejected. The condition IX= c5 = -1, fi = y = 0 is not suitable, since it is associated with two distinct even solutions that are meaningful in the origin of coordinates. Hence, only condition (2) remains. We shall restrict ourselves to the cases which will bring us to the n-or 2n-periodic "boundary" solutions that separate the stability regions from the instability regions of the single equation with periodic coefficients [START_REF] Yakubovich | Linear Differential Equations with Periodic Coefficients, Vols. I and II[END_REF][START_REF] Magnus | Hill's Equation[END_REF].

(a) Let us assume that rJ. = -6 = ± 1, fJ = 0; y 1 (1-z) = ± ydl + z).

The solution is even at et. = 1 and odd at et. = -1; it is regular in any finite part of the plane. Since z = cost, at rJ. = 1 the solution may be presented as an even power expansion in cosines that are multiples of t, and et. = -1, in the form of an odd power expansion in cosines that are multiples of t.

(b) Let et. = -b = ± 1, 1 = 0, y 2 (1 -z) = ± y 2 (1 + z) at 11 ± z I < 2.
The solution is the product of y' 1 -z 2 by a regular function that changes its sign upon each revolution about the singular point z = 1 or z = -1. The regular function is even if 6 = 1 and odd if b = -1. Substituting z = cos t, one obtains an even power expansion in sines that are multiples of t(b = 1), and an odd power expansion in sines that are multiples of t(<> = -1).

Thus, for the Hill equation in algebraic form, "boundary" solutions of the Hill equation are obtained that separate the region in which all solutions are bounded from the one in which all solutions are unbounded.

Let us consider two examples which are special cases of the Hill equation.

Mathieu equation: ji + (6 + 2~: cos 2t)y = 0.

Substituting z = (sin t ) 2 , one obtains an algebraic form of

d 2 y dv , 4(1-z) dz 2 + 2(1 -2z) d~ + (6 + 2t:-4t:z)y = 0.
It has two regular singular points: z = 0 and z = 1 whose indices are 0 and 1/2, and an essential singular point z = oc. In view of the above, the solutions separating the stability region from the instability region should be sought in the following form: where each series converges in a domain containing both regular singular points. Using analytical or numerical methods, one can eventually obtain the Ince-Strett diagram. Here, a transfer to algebraic forms may be effected by a number of methods [START_REF] Bateman | Higher Transendental Functions[END_REF]. For example, substituting z = [sn(t, k)] 2 , one obtains the following equation with three finite regular singular points:

a (l)'7k k ~ ' k = 1, 3 ...
-+--+--+ -+

v=O d 2 y 1 (l 1 1 )dy hk~2 -n(n + 1)z dz 2 2 z z-l z-k 2 dz z(z-1)(z-k 2 )"
Algebraic forms of the Lame equation are used in a number of problems considered in this article. whose potential energy ll(x 1 , x 2 . . . . . x") is analytical.

Let us rotate the coordinate axes so that the axis x 1 is aligned with the rectilinear trajectory, with the remaining axes orthogonal to the trajectory. In the new coordinates, the trajectory is defined as xj = 0 (j = 2, 3, ... ,n), x 1 = x(t).

It was shown in [START_REF] Atkinson | The stability of the normal modes of nonlinear systems with polynomial restoring forces of high degree[END_REF] that upon such rotation of the axes in a system with two degrees of freedom the variational equations always "split". Denoting the variation along the rectilinear trajectory by u, and the variation in the orthogonal direction by v, we obtain:

. d u + -Ilx{x, O)u = 0, dx V+ Ilx,x,(X, O)v = 0.
Let us turn to a system with n degrees of freedom. Introducing a vector of variations U = (u 1 , u 2 , ... , u.f: we obtain a set of variational equations in the matrix form D + (UV)Vll = 0, or D + AU = 0.

(

) 2
Let us expand the matrix A is a series according to x:

where A; are symmetric matrices with constant coefficients. For a homogeneous system whose potential is a homogeneous function of variables to the power r + 1, equations ( 2) is "split" by the use of a non-degenerate transformation with constant coefficients. Indeed, the variational equations then have the form

D + A,U = 0,
and the symmetric matrix A, can be diagonalized usmg a non-degenerate orthogonal transformation [START_REF] Bellman | Introduction to Matrix Analysis[END_REF]. Let us separate out another case:

U + [A.

x" + Amxm]U = 0.

The two symmetric matrices A. and Am are always diagonalizable through one nondegenerate transformation [START_REF] Bellman | Introduction to Matrix Analysis[END_REF], provided one of them is positively defined; moreover, if the matrices are commuting, the transformation may be made orthogonal. In the literature [START_REF] Hsu | On a restricted class of coupled Hill's equations and some application[END_REF] one can find proof of the theorem on the diagonalization in the general system (2) under the conditions that all matrices A; commute between them and also have non-multiple proper values.

Let the set of variational equations decompose into n independent equations. One of these will govern the variations along the rectilinear trajectory, and the remaining equations will refer to the orthogonal directions. It is with the variations in the orthogonal directions that the orbital stability of normal vibrations is associated.

Let us introduce a new independent variable x. Every variational equation then will have the following form where the primes refer to the derivation with respect to x:

2Sy" + S'y' + G(x)y = 0, S =(a-x)(b-x)R. ( 3
)
Here R is an analytical function that has no real roots whose moduli are smaller than a or b.

As shown in the previous section, the problem of defining T-and 2T-periodic solutions of the variational equations (1: is the period of the coefficients in the equations) that separate the regions of stability and instability is reduced to solving Sturm-Liouville problems for functions that are regular at x = a and x = b or have a singularity of the form ~ or ~at these points.

If the potential energy IT is an even analytical function, the coefficients of the variational equations have a period which is twice as short as that of the normal vibrations. The kinetic energy is proportional to h -IT and is an even function of x too. However, S in ( 3) is proportional to the kinetic energy, therefore here b =a. Without loss of generality, one thus can select a scale such that S = (1x 2 )R. The problem of defining the boundaries of 4 the stability and instability regions by the first approximation is now decomposed into four Sturm~Liouville problems for even and odd functions that are regular at x = ± 1 or have a singularity of the form J1x 2 at these points.

Homogeneous systems

Let us consider variational equations for homogeneous systems with the potential energy IT= L (4)

:x1 + :t. 2 -+-•• . . . . . _ :1" = r + I :::t:, =0,1.2, ... .r + 1
Upon the diagonalization of the matrix A each of the equations can be written in the form

Here

v + qx'-1 v = 0 (q = const.) .x = -!1 2 x' , .X 2 = 2(h-Q 2 ~x;~} !1 2 = n(1, k 2 , . • . , k.).
Let us replace t by a new independent variable x. We obtain, instead of (5), or

t:".X 2 + V 1 .X + qXr-l V= 0 2r"(h -Q 2 x•+ 1 ) -v'!1 2 x' + qx'-1 v = 0 r + 1 Let us introduce another variable z = Q 2 x'+ 1 /(r + 1). We then have z(.1 -z)v_ + --- z v +AV = 0 [ r 3r + 1 J . <z r + 1 2(r + 1) z '
where . q

I. = 2(r + 1)!1 2 • (5) (6)
Relationship ( 6) is a hypergeometric equation with regular singular points z = 0 (the indices equal 0 and 1/(r + 1)) and z = 1 (the indices equal 0 and 1/2). The problem of stability and branching, as suggested above, is now much simpler. It essentially reduces to finding the values of the parameter i. such that, on having traversed a closed contour containing singular points 0 and 1, a solution is multiplied by + 1 or by -1 (Section 2). Such solutions are named degenerate solutions of the hypergeometric equation and are given in [START_REF] Bateman | Higher Transendental Functions[END_REF]. Their form is

v = z 11 •(1-z)i''q.(z),
where q.(z) is a polynome, Jll can take values of 0 and I /(r + I), J1 2 can take values of 0 and 1/2. Let us further denote the period of solution x(t) as T. By analogy with the Mathieu equation, let us denote the proper functions that are Gegenbauer polynomes as C 4 i(z) (the even T /2-periocic solution), C 41 + 1 (z) (the odd T -periodic solution), S 4 J+ 2 (z) (the even T -periodic solution), and S 4 J+ 3 (z) (the odd T /2-periodic solution). In the linear case, the trajectories of these solutions are the well-known Lissajou figures. We write the eigenvalues of these problems in the same order as above [START_REF] Zhupiev | Stability and branching of the normal modes of vibrations of the nonlinear systems[END_REF]:

i. 4 J =j[(2j + 1)(r + 1)-2], i, 4 J+ 1 = (2) + 1)[j(r + 1) + 1], A 4 J-2 = (2) + 1)[() + 1)(r + 1)-1],
). 4

J+ 3 = (j + 1) [(2) + 1)(r + 1) + 2] (j = 0, 1, 2, ... ).
Note that so simple a form for the spectrum of a stability problem is not very often obtainable for the non-linear case.

Homogeneous chain system

For the sake of definiteness, let us consider a homogeneous chain system with two degrees of freedom: [START_REF] Month | The stability of bifurcating periodic solutions in a two-degree-of-freedom nonlinear system[END_REF] An equation to determine the vibration forms is

f1 = ak' -xk (!X = a 22 f1 = a11 x = mz k = Xz) (1-kY(1+xk) a11' a1 1 ' m1' X1 (8)
An examination of this relationship shows [START_REF] Manevich | The Method of Normal Oscillation for Essentially nonlinear Systems[END_REF][START_REF] Zhupiev | Stability and branching of the normal modes of vibrations of the nonlinear systems[END_REF] that for positive values of the constraint parameter f1 there always exists one in phase vibration mode and one, two or three anti phase modes whose number depends on f1 and the homogeneity index r. For negative values of the constraint parameter f1 at any value of r there exists one anti phase and one or three inphase vibration modes.

For the system in question the eigenvalue ;, of the hypergeometric equation [START_REF] Atkinson | The stability of the normal modes of nonlinear systems with polynomial restoring forces of high degree[END_REF] . r(1 + xk)(ak'-xk)

I.= . . x( 1 -k) (:1.k' + 1)
The plots of Yf = 4n-1 arctg({l• 2'/(r -1)) and ~ = arctg k shown in between the points whose numbers are from 4j + 1 to 4j + 2 and from 4j + 3 and 4j + 4 (j = 0, 1, 2, ... }. In the asymmetrical case(). i= 1} some points of branching turn into limit points.

Figure 2 shows plots of f1 versus arctg k for other values of the system parameters. Curve (e) corresponds to the limiting case r =

0, x = 1, a = 1; (f) r = 1/3, x = 1, a = 1; (g) 0 < r < 1, x > ct., x' > ct.; (h) 0 < r < 1, x < ct.,
x' < a. These plots give an idea of how great the number of vibration modes is at various r, a, x. Fig. 2. The dependence between the number of vibration modes and the parameters of the system [START_REF] Month | The stability of bifurcating periodic solutions in a two-degree-of-freedom nonlinear system[END_REF] correspond to r < I and various values of other system parameters.

In order to find those periodic solution branching points that do not affect the stability of normal vibrations, let us write a general solution of the variational equation ( 6):

V = C 1 U d z) + C 2 z 1 /(r + I) • U 5 ( z)
(in the vicinity of the singular point z = 0), For the eigenvalue ). = (r-1)/r such a branching periodic solution may be written in the explicit form:

v=D 1 u 2 (1-z)+D 2 j1-z-u 6 (1-z)

Systems approaching homogeneity

The results can be used for systems close to homogeneous ones. The periodic solutions separating the regions of stability and instability should be sought for the variational equations by perturbation theory. In the zeroth approximation these solutions are described by the eigenfunctions of a hypergeometric equation that correspond to the generating homogeneous system. From the conditions of the solutions periodicity, the stability shift points are determined in the further approximations.

In this section, however, only an example is given. Let us consider a symmetrical chain system whose potential contains x 1 and x 1 to the second and fourth power:

m.x1 + a 11 x 1 + a 13 xi + a 2 dx 1 -x 2 ) + a 23 (x 1 -x 2 ) 3 = 0, ( 9
)
The equation that governs the anti-phase vibration mode x 2 =x 1 and the variational equation that permits a judgement about the orbital stability of this vibration form are as follows:

x

+ x(y + 2px 2 ) = 0, v + v(o-+ f3x 2 ) = 0, all+ 2a 21 a 13 + 8a 23 all }'== ' p= ' er=-, m m m
or, with the primes denote the derivation with respect to u, [START_REF] Rand | The geometrical stability of nonlinear normal modes in two degree of freedom systems[END_REF] This is the Lame equation. Let us use the perturbation theory in order to find the first two points of stability shift (corresponding to the branching of normal vibrations and periodic solutions with the period of the generating solution but displaced in phase by half-period).

Let the system be quasilinear. We assume that p and f3 are small. Let us introduce a formal small parameter c; and multiply the non-linear terms by c;:

v"[y(1 -x 2 ) + c:p(1 -x 4 )] -v'(-;x + 2c:px 3 ) + v(o-+ c:f3x 2 ) = 0.
We use the following expansions:

t: = Vo + et:l + ... , •; = l'o + 8(! + ... ' a-= O"o + £0"1 + p=c;p 1 +e 2 p 2 + •••, f3=af3 1 +e 2 /3 2 +
In the zeroth approximation with respect to c;, one obtains

v~y 0 (1 -x 2 ) -Vo(oX + vao-o = 0.
In the first stability shift point, normal vibrations branch: v 0 = kx (here x = cos wt) and y 0 = a-0 . Therefore, in the zeroth approximation the linear system is uncoupled, that is a 21 = 0. We may readily check that it is at this point that the periodic solutions with the period of the generating solution which are displaced in phase by a half-period branch too: v 0 = kx 0 (here x 0 = sin wt, x 0 = j1x 2

). Consequently, in the zeroth approximation the equality l'o = a-0 holds for the second stability shift point as well.

Let us write the first approximation with respect to e:

v'{yo(1-x 2 ) -v'1}'oX + V10"o-vb(; 1 x + 2plx 3 ) + v 0 (o-1 + f31x 2 ) = 0 (for the first point), or l'o[v~(l-x 2 ) -v' 1 x + t: 1 ] = k[(( 1 -o-J)x + (2p 1 -f3dx 3 ].
The periodicity conditions are the conditions of orthogonality of the right-hand side with respect to the periodic solution of the homogeneous equation. Denoting a line integral along a closed contour (along the period) by f, one obtains for the first stability shift point

f[(y 1 -o-I)x 2 + (2p 1 -f3J)x 4 ]dt = 0 or f x 2 dx f x 4 dx (yl-o-J) . +(2pl-f3d J =0. ~ 1 -x 2 1 -x 2
After simple calculations one finds that Let us now consider a quasihomogeneous system. Assume that y and a are small. We introduce a formal small parameter c before each linear term:

v' [~>y(l -x 2 ) + p(l -x 4 )] -v' (~;yx + 2px 3 ) + v(ca + f1x 2 ) = 0.
We use the following expansions:

v=v 0 +cv 1 + •••, y=cy 1 + •••, a=ea 1 + P=Po+~>P1+ •••, P=Po+~>/11+
In the zeroth approximation with respect to c

v~po(l -x 4 ) -v 0 • 2px6 + vo • Px 2 = 0.
The branching of normal vibrations (the first stability shift point) v 0 = Cx provided that Po = 2p 0 . For the second point, that is the branching point of solutions v 0 = Cx the conditions will be Po = 6po.

In the first approximation with respect to c we write

v'{po(l-x 4 ) -v'1p 0 x 3 + v 1 {1x 2 -v~[p 1 (l-x 4 ) + yi(l-x 2 )] -v0(2plx 3 + }'1X) + v 0 (f31x 2 + a1) = 0.
From the periodicity conditions for the first and second points of stability shift we obtain, respectively:

f [x(2p1x 3 + Y1X)-x(/1 1 x 2 + at)x] dt = 0. for the second point, of stability shift.

FINITE ZONING CONDITIONS FOR A STABILITY PROBLEM BY THE FIRST APPROXIMATION

Finite number of orbital stability zones

Let us consider a situation when the number of orbital stability zones of normal vibration modes is finite (it is about a first approximation of stability and instability). Then all other instability zones are contracted to lines ("collapse"). This situation may be realized under certain conditions imposed upon the systems parameters. If these conditions are not satisfied, the number of instability zones will obviously be infinite (e.g. as on the well known Ince-Strutt diagram for the Mathieu equation). However, the importance of finite zoning conditions (or finite-gap conditions) depends on the fact that all the instability zones, except a finite number of them, are "narrow" if the values of the parameters are "close" to values which guarantee a realization of the finite zoning conditions when these zones are contracted to lines. It is interesting that sometimes the first parametric resonance zone may be "narrow" in this sense though it is usually considered as wide (e.g. for the Mathieu equation). Moreover, the finite zoning conditions are of interest for the problem of soli tons in non-linear wave systems.

So, let us consider a conservative system having two degrees of freedom, which may be described by the following equations

an 5.' ; + -~-= 0 (i = 1, 2) CX;
where n is the analytically positively determined potential energy.

(11)

One may assume the system to permit rectilinear normal vibration modes x 2 = kx1. Let us separate one of the modes and reduce it to x 2 = 0 by rotation of the coordinate axes. Then [START_REF] Pecelli | Stability of modes for a class of non-linear planar oscillators[END_REF] i=2 i=O i=3

where g;(x d are analytical functions. The condition of existence of the solution x 2 = 0 is obviously satisfied.

In the normal vibration mode the system behaves like a system with one degree of freedom .x + Dx,(x, 0) = 0 (x = xd [START_REF] Vakakis | Normal modes and global dynamics of a two-degree-of-freedom non-linear system-!. Low energies; •-II. High energies[END_REF] with the energy integral x l + D(x, 0) =h. [START_REF] Nee | Ordinary Differential Equations[END_REF] The orbital stability of normal mode x 2 = 0 is related to y variations orthogonal to the trajectory. The variational equation is:

Y + Yllx,x,(X, 0) = 0. ( 15
)
The function nx,x,(x(t), 0) is polynomial with respect to X and periodical with respect tot. The finite zoning criterion for equations as [START_REF] Yakubovich | Linear Differential Equations with Periodic Coefficients, Vols. I and II[END_REF] was given in [START_REF]Solitons[END_REF][START_REF] Zakharov | Solitons Theory. Inrerse Scattering Method[END_REF] by Novikov. Namely, periodic potentials u(t) of the Schri:idinger equation tj; + [eu(t)] tj; = 0 may have just n finite restricted zones (instability zones) if the equation

.dx 2 zdt-x =u-e admits of a solution D-(1/2)2 X = ---- z (16)
where z is a polynomial in e of degree n with variable factors (z = I,;=o &k(t)ek), D being a constant. Then the equation for the z polynomial may be written down as:

z -4(u -e)i -2uz = 0. ( 17
)
Comparing [START_REF] Yakubovich | Linear Differential Equations with Periodic Coefficients, Vols. I and II[END_REF] and ( 16) one may conclude that u-e =nx,x,(x, 0). Taking into consideration the initial expression [START_REF] Pecelli | Stability of modes for a class of non-linear planar oscillators[END_REF] for potential energy D(x 1 , x 2 ), we have e = 2e 0 .

Finite zoning conditions

Let us consider an algebraic form of the equation ( 17) (algebraization in the Ince sense [START_REF] Nee | Ordinary Differential Equations[END_REF]) to obtain finite zoning conditions as some conditions imposed on factors of potential energy ll(x 1 , x 2 ) expansion. Namely, let us introduce a new independent variable x instead of t. Taking the following equalities into account

d d d 2 d 2 •2 d .. d 3 d 3 •3 3 d 2 • .• d ... dt = dx X, dt2 = dx 2 X + dx X, dt 3 = dx 3 X + dx 2 XX + dx X' x 2 = 2(h -ll(x, 0)), x = -llx(x, 0), x: = -llx(x, O)x
[the last three equalities were obtained from ( 13) and ( 14)]. One may have an equation with regular singularities instead of equation ( 17) with periodic coefficients:

2z'"(h-ll(x, 0)) + 3z"(-llx(x, 0)) + z'[-llxx(x, 0) + 4llx,x,(x, 0)] a + 2z -;-[llx x (x, 0)] = 0 uX 2 2
where a prime symbol denotes differentiation with respect to x.

(

) 18
Then then-zoning condition means the solution of equation ( 18) to be a polynomial in e, the coefficients being functions of x:

• z = L 1Xk(x)ek. k=O (19)
Substituting ( 19) into [START_REF] Bellman | Introduction to Matrix Analysis[END_REF] and grouping the terms with equal powers of e, one may have a problem in the eigenvalues a;, e;.

Characteristic classes of analytical potential

This problem is absolutely boundless in general, so let us describe results for three characteristic classes of analytical potentials. Moreover, we should confine ourselves to the values n = 0, 1, 2. 4.3.1. a 2 -# 0, a 4 -# 0, e 0 -# 0, e 2 -# 0, the rest of the factors in [START_REF] Pecelli | Stability of modes for a class of non-linear planar oscillators[END_REF] being zero. Note, that equation [START_REF] Pecelli | Normal modes, incoupling, and stability for a class of nonlinear oscillators[END_REF] in this case has linear and cubic terms with respect to x 1 ,x 2 .

Grouping the terms with identical powers of e, then grouping the terms with identical powers of x, we may obtain the finite zoning condition: [START_REF] Bateman | Higher Transendental Functions[END_REF] As shown in this section, the equation in variations is reduced to the Lame equation in this case, and the finite zoning conditions are known [START_REF] Pecelli | Normal modes, incoupling, and stability for a class of nonlinear oscillators[END_REF][START_REF] Bateman | Higher Transendental Functions[END_REF]. Hence, let us discuss other potentials for which finite zoning conditions were determined for the first time in [START_REF] Zhupiev | Conditions for finiteness of the number of instability zones in the problem of normal vibrations of non-linear systems[END_REF]. 4.3.2. a 2 -# 0, a 4 -# 0, a 6 -# 0, e 0 -# 0, e 2 -# 0, e 4 -# 0, the rest of the factors in [START_REF] Pecelli | Stability of modes for a class of non-linear planar oscillators[END_REF] being zero. Equation [START_REF] Pecelli | Normal modes, incoupling, and stability for a class of nonlinear oscillators[END_REF] in this case contains the terms of the first, third and fifth powers of

x 1 , x 2 •
Grouping the terms with identical powers of e, then grouping the terms with identical powers of x, we may obtain the finite zoning conditions: e 4 = 4(n + 1)na 6 ; e 2 = 2(n + 1)na 4 ; 4a 2 a 4 a 6 -a~ + 8ha~ = 0 (21) 4.3.3. a 2 -# 0, a 3 -# 0, a 4 -# 0, e 0 -# 0, e 1 -# 0, e 2 -# 0, the rest of the factors in [START_REF] Pecelli | Stability of modes for a class of non-linear planar oscillators[END_REF] being zero. Equation [START_REF] Pecelli | Normal modes, incoupling, and stability for a class of nonlinear oscillators[END_REF] in this case contains the terms of the first, second and third powers of xl> x 2 • Grouping the terms with identical powers of e, then grouping the terms with identical powers of x, we may obtain the finite zoning conditions: [START_REF]Solitons[END_REF] The finite zoning conditions (20) for case 4.3.1 is valid for any positive integer n, but conditions (21) for the case 4.3.2 and ( 22) for the case 4.3.3 were obtained only for n = 0, 1, 2.

One may repeat all the calculations for n > 2 selecting the terms with senior powers of x to realize that the first two equations in ( 21) and ( 22) are valid again.

Eigenfunctions and eigenvalues

Now let us discuss the calculation of eigenfunctions and eigenvalues for a finite zoning case. At the same time conditions ( 20)-( 22) will be obtained in another way for any n.

Let us transform the variational equation ( 15) to an algebraic form introducing a new independent variable x instead oft. Then equation [START_REF] Yakubovich | Linear Differential Equations with Periodic Coefficients, Vols. I and II[END_REF] will be 2y"(hn(x, 0)) + y'(-Ox(x, 0)) + yllx,x,(x, 0) = 0. This is an equation of Fuchs class for the polynomial potentials given above, whose exponents of the finite singularities equal 0 or 1/2 (the singularities here are zeros of the expression h -O(x, 0) and x = oc ). Hence, one may search all the eigenfunctions as (the number of the eigenfunctions are 2n + 1 for the case of n-zoning):

y=p(x) or y=Jx-~R(x)
where p(x) and R(x) are analytical functions and x = ~ is a singular point of the equation.

Indeed, that way one may find all the eigenfunctions and eigenvalues for the potentials I, II, III when n = 0, 1, 2.

Let us consider another approach. First, let us write down the variational equations in the algebraic form for the three types of potentials:

I: 2y"(h-a 2 x 2 -a 4 x 4 ) + y'( -2a 2 x-4a 4 x 3 ) + y(e + 2e 2 x 2 ) = 0 (23)
II: 2y"(h-a 2 x 2 -a 4 x 4 -a 6 x 6 ) + y'( -2a 2 x-4a 4 x 3 -6a 6 x 5 ) + y(e + 2e 2 x 2 + 2e 4 ."1A) = 0

Ill: 2y"(h-a 2 x 2 -a 3 x 3 -a 4 x 4 ) + y'(-2a 2 x-3a 3 x 2 -4a 4 x 3 ) + y(e + 2e 1 x + 2e 2 x 2 ) = 0
Then let us write the known Lame equation in an algebraic form

y"(z 2 -a 2 )(z 2

-b 2 ) + y'z(2z 2 -a 2 -b 2 ) -y[n(n + l)z 2 -1-] = 0 (26)
(finite zoning for n = 0, 1, 2, 3, ... ).

Equation (26) coincides with (23) if

(the last equality is the finite zoning condition for n = 0, 1, 2, ... ). Connecting (26) and ( 23), one may note that the coefficient of y" in (26) has the meaning of the kinetic energy (h -n (x, 0)) for periodic motion x(t). Kinetic energy vanishes twice per period at the amplitude values. Without loss of generality one may assume x = ± a. Then one may assume a > 0, I 2 I ::::; a and either b 2 < 0 or b 2 > a 2

.

All the eigenfunctions and eigenvalues of the Lame equation for n = 0, 1, 2 may be easily found by substitution into (26) of polynomials or polynomials multiplied by J a 2 Note, that the finite zoning conditions [START_REF]Solitons[END_REF] for any positive integer n are the consequences of these expressions. This shift makes it possible to find all eigenfunctions and eigenvalues for the equation ( 25) from the functions obtained for the Lame equation. Equation ( 24) may be reduced to the Lame equation by a quadratic law transformation z = x 2 a if (then the number of singularities being changed) h/a 6 = 2a(a 2 -b 2 ), a 2 ja 6 = 5a 2 -b 2 , a 4 /a6 = -4a, e6ja 6 = 4[a 2 n(n + 1)-le], e 2 ja 6 = -8an(n + 1), e 4 ja 6 = 4n(n + 1).

Hence the finite zoning conditions [START_REF] Zhupiev | Stability and branching of the normal modes of vibrations of the nonlinear systems[END_REF] for any positive integer n may be obtained. The eigenfunctions and eigenvalues are transformed accordingly.

It is remarkable that the first zone of parametric resonance is being contracted to a line in the case 11 when the finite zoning conditions are valid. Indeed, the eigenfunctions of equation [START_REF] Zhupiev | Conditions for finiteness of the number of instability zones in the problem of normal vibrations of non-linear systems[END_REF] corresponding to the boundaries of the finite zone of instability, e.g. for n = 1 (b 2 < 0), are (Fig. 5)

Yt = C(x 2 -a), y 2 = C "/a 2 -(x 2 -a) 2 .
For a periodic mode: -2a:::; x:::; 2a. But the functions y 1 (x) and y 2 (x) turn to zero within the interval of [ -2a, 2a], i.e. their period with respect tot is T /2 (T being the period of x(t), x being changed within these limit (on a half of the period). Functions corresponding to the first zone of parametric resonance should have the period ofT).

If conditions (20)- [START_REF]Solitons[END_REF] are not strictly met, the number of instability zones will be infinite. However all of them are "narrow" in the sense mentioned except the selected n zones, since the zones are contracted to lines when the finite zoning conditions are valid. This is also true for the first zone of parametric resonance which is usually assumed to be "wide".

Note once again that we will obtain the finite zoning conditions (20)-( 22) for an arbitrary positive integer n transforming the Lame equation into variational equations as mentioned above (parameters being changed continuously).

Pendulum systems

Let us consider another class of non-linear systems-pendulum systems with the potential

V= -a cosx 1 + x~(b + d cosx) + x~g(x 1 , x 2 ).
Linearization around the normal mode of vibration x 2 = 0, x 1 = x(t) (the equation of motion

x + a sin x = 0) gives the variational equation y + 2(b + d cos x) y = 0. In this case a solution x(t) may be either periodic or aperiodic (oscillations and rotation of the pendulum). Introducing a new independent variable z = :X 2 /2, one may transform the variational equation to:

2b -2b--(z-h) " + ! ' [! + 1 + 1 J + y a = 0 (27) y 2y z z-(h +a) z-(h-a) 4z[z-(h +a)] [z-(h-a)]
being the Lame equation [START_REF] Bateman | Higher Transendental Functions[END_REF]. Here the finite zoning conditions are:

2d -= n(n + 1) (n = 0, 1, 2, ... ) (28) a
(n being a number of finite zones of instability within the space of system parameters).

General procedure of transformation

Let us describe the general procedure of transformation from the finite zoning Lame equation (26) to the variational equation ( 15) in an algebraic form. One should multiply (26) by an arbitrary function q(z) and perform the transformation z = f(x). Comparing the equation obtained with equation ( 15), one should have the following conditions:

2(h-TI(x, 0))(!') 2 = (F -a 2 )(F -b 2 )q(f), 2(h-TI(x, O))f" -Tix(x, O)f' = f(2jl -a 2 -b 2 )q(f), Dx,x,(X, 0) = -[n(n + l)F-A.]q(f) (n = 0, 1, 2 ... ). . f . f TIX X (X, 0) d h Excludmg the unction q( ) = 2 _ n(~ + 1 )f 2 , one woul ave 2(h-TI(x, 0)) [2-n(n + l)jl] (f') 2 = Dx,x,(x, 0)(! 2 -a 2 )(jl-b 2 ), [2(h-TI(x, O))f" -Dx(x, O)f'] [A.-n(n + 1)FJ = f(2F -a 2 -b 2 ) Dx,x,(X, 0).
These equations may be used to solve the following problem. Let us assume the transformation z = f(x). Then one should select the value of potential energy TI(x 1 , x 2 ) to satisfy the equalities. The conditions imposed upon the coefficients of potential energy are the finite zoning conditions for a positive integer n.

Note that stability of some modes of oscillations has been investigated in [START_REF] Pecelli | Stability of modes for a class of non-linear planar oscillators[END_REF] for a particle on a plane. In many cases the stability problem may be reduced to analysis of the finite zoning Lame equation.

The results presented in this paragraph are not obvious. Suffice it to mention the possibility of the .first zone of parametric resonance being contracted to a line.

NON-LINEAR STABILITY OF NORMAL VIBRATIONS

A problem of non-linear stability of normal vibrations in conservative systems may be solved only after the corresponding linear problem has been solved. The non-linear analysis is applicable to a system class, such that the variational equations are reduced to hypergeometric equations or Lame equations, etc. The problem of non-linear stability of normal vibrations was also discussed by Pecelli and Thomas [START_REF] Pecelli | Normal modes, incoupling, and stability for a class of nonlinear oscillators[END_REF][START_REF] Pecelli | Stability of modes for a class of non-linear planar oscillators[END_REF] (using an approach first published by Arnol'd and Mozer) for a system with two degrees of freedom and a Hamiltonian

i,j=O i,j= 0 may be constructed. On substituting these series into (30•), one is able to find the coefficients aij(T), bii(T). In this case, a The series (31) define the solution P(T) of (30). What is known as Poincare transformation is P(2n). Thus comparison is made between the values of x(O), y(O) and x(2n), y(2n). Stability is ensured when the linearized Poincare transformation P 1 (2n) defined by the coefficients a 10 , a 01 , h 10 , b 01 has eigenvalues /.T with a modulus equal to unity and when the condition If we now consider normal vibrations that are stable in the first approximation (which implies that £\ 2 < 4, for in this case the eigenvalues of the linearized transformation have moduli equal to unity) and assume A -:p 0, -1 and y =1= 0, it follows from the Arnold-Mozer-Ruisman theorem [26,[START_REF] Zigel | Lectures on Celestial Mechanics[END_REF] that the normal form of vibrations possesses orbital stability in a 2n-dimensional phase space. It is evident that in the system parameter space, the stability regions A = 0, A = -1 and"/ = 0 have a smaller dimension than the instability region A 2 ;?: 4.

Once again it is highlighted that it is only the knowledge of the solution of the linearized variational equation that is necessary to calculate the values of A and y that define the instability regions obtained when taking into account non-linear terms.

This section deals with isoenergetic stability (for fixed values of the energy h) of normal vibrations; however, it was shown [26] that complete stability takes place when R = <l>(x, y) is continuously dependent on the energy h.

More detailed analysis to a specific Hamiltonian of the type is given in [START_REF] Pecelli | Normal modes, incoupling, and stability for a class of nonlinear oscillators[END_REF].

The problem of non-linear stability of normal vibrations was also discussed by Month and Rand [START_REF] Month | An application of the Poincare map to the stability of nonlinear normal modes[END_REF].

3 .

 3 THE SIMILAR NORMAL MODES STABILITY BY THE FIRST APPROXIMATION OF HOMOGENEOUS SYSTEMS AND SYSTEMS CLOSE TO THE HOMOGENEOUS ONE 3.1. Splitting of variational equations Let us consider variational equations for rectilinear vibration modes x; = k;x 1 (i = 2, 3, ... , n) in a conservative system of the form .. (. dx; an . ) m;Xi + llx = 0 X i = -, llz = -;-, I = 1,2, ... ,n ,

Fig. 1

 1 allow one to find the relation between the number of vibration modes and the parameters of the chain system. Curve (a) corresponds to the case of r = 3, a = 1; (b) r = 7, ct. = 1; (c) r = 3, ct. = 0; (d) r = 3, a = 1, 2 (the value of x = 1). Each first point of stability shift for r = 3 is marked with a circle o. and for r = 7, with a cross x . A figure shown near a point corresponds to the number of the eigenvalue ;•i• The ranges in which vibration modes are unstable lie

Fig

  Fig. I. The dependence between the number of vibration modes and the parameters of the system (7). Curves correspond to the cases r = 3. 7; j = 0, 1, 2. Points of stability shift are marked by circles and crosses. Numbers of the eigenvalue ;,, are shown near the points.

  1t

(

  in the vicinity of the singular point z = 1), C 1 ,C 2 ,D 1 ,D 2 being arbitrary constants, u 1 (z), u 2 (1-z), u 5 (z), u 6 (1-z) known hypergeometric functions [20]. Let us denote the matrix that relates C, C 2 to D 1 , D 2 as N: One can formulate denumerably many boundary problems ( JN -1 JN )P = I ( p = 1, 2, . . . ), I = 11 ~ _ ~ 11 in order to define the branching periodic solutions traversing the singular points z = 0 and z = 1 p times.

4 For

 4 the second stability shift point, the periodicity condition is f[v0(ylx + 2plx 3 ) -vo(al + P1x 2 )]xdt = 0 Here v 0 = Cx = C sin wt, x =cos wt. After simple rearrangements one obtains 1 -4

f [x(2plx 3 +

 3 Y1X)-x(/11x 2 + at)x] dt = 0.

2 -b 2 orFig. 4 .Fig. 5 .

 2245 Fig. 4. Stability chart of the Lame equation if the parameter values close to values which guarantee a realization of the finite zoning conditions.

H = 1 (==

 1 Yi + y~) + TI(xb Xz), x 1 , x 2 being positional variables, y 1 , y 2 the corresponding impulses. Assume that the system allows the normal form of vibrations x 2 = y 2 = 0. What we have in view is the isoenergetic stability of the vibration form. Introducing, following[START_REF] Pecelli | Normal modes, incoupling, and stability for a class of nonlinear oscillators[END_REF], the variables R, r, x, y by a substitution Y1 = J2R sin r, x 1 = J2R cosr, x 2 = x, y 2 = y, one obtains R(x, y) from the equation H =h. (29) 408 Yu. V. Mikhlin and A. L. ZhupievIf T is a new independent variable, one obtains[START_REF] Whittaker | Analytical Dynamics of Particles and Rigid Bodies[END_REF] Setting x(O) = >< 1 , y(O) = x 2 , expansions

2 - 4 / 2 ,

 242 ).k -:p 1 holds for k = 1, 2,3, 4 [26, 27]. The eigenvalue problem l a 1o(2n)-A a0 d2n) I= 0 b 10 (2n) b 01 (2n)-A implies that;,= A± v' A where A= (a 10 (2n) + b 01 (2n)). It is assumed here that the solutions of the homogeneous equations (32) are so normalized that a 01 (2n)b 10 (2n) = 1. The above limitations on the eigenvalues are satisfied with A 2 < 4, A ':1 0, -1. Under these conditions, a change of variable (x, y) = S(~, '1) is possible that reduces the Poincare transformation to what is known as the Birkhoff normal form [26, 27], N = S 1 PS. By introducing the complex variable ~ = ~ + il], one obtains, instead of P(2n), the transformation ~' = A.(ei' 1 ' + 0 4 where 0 4 refers to the terms of the fourth-order with respect to (and S' .

  10 , a 01 , b 10 , h 01 are solutions of the linear variational equation

	Z=AZ,	(32)
	wherein
	Z= 11:11•

Ao =a,

The following intervals of the A. axis correspond to instability zones: (-oo, A. 0 ], [..1. 1 , . . 1. 2 ], [..1. 3 , . . 1. 4 ]. Figures 3 and4 present a stability chart of the Lame equation for n = 3 and n # 3 but "close" to 3.

Other variational equations may be also be reduced to the Lame equation (26). Equation ( 25) may be reduced to the Lame equation by shifting along the real axis (the number of singularities being constant and their symmetry about the origin of coordinates being restored)

a 3 /a 4 = 4J1, eo/a 4 = -A.+ J1 2 n(n + 1), eifa 4 = 2Jln(n + 1), e 2 /a 4 = n(n + 1).


Stéphane Grenier

V Kiryukhin

S.-W Cheong

B G Kim

J P Hill

K J Thomas

Jean-Marc M Tonnerre

Yves Joly

U Staub

V Scagnoli

Observation of orbital ordering and Jahn-Teller distortions

I. INTRODUCTION

Crystallization of charge carriers in doped transitionmetal oxides may exhibit charge and orbital orderings. As the insulating phase competes with metallic ferromagnetism or superconductivity, it is essential to determine precisely which electronic orderings take place to help identify the correlations that induce the phase transitions and stabilize each state. However, the characterization of an electronic crystal remains an experimental challenge. Conventional crystallography is difficult as charge and orbital orderings correspond to a small spatial variation of the electronic density with a weak scattering amplitude for x-rays or electrons. For example, orbital ordering of the 3d 4 configuration in x hole-doped manganites concerns 1 -x electron per chemical unit. In fact, recent neutron, x-ray and electron crystallographic studies have shown that it is also a strenuous task to determine the space group and the amplitudes of the atomic displacements related to the electronic phase transitions. 1,2 Experimental efforts aimed at directly detecting periodic electronic structures relevant to transport and magnetism are undertaken utilizing complementary techniques. In the past few years, Resonant X-ray Diffraction (RXD) has been invaluable in the characterization of charge and orbital ordering. This technique combines crystallographic and spectroscopic informations, exploiting its sensitivity to crystallographic sites, chemical el-ements and spectral density of states, as illustrated in the following. In this paper we address the experimental characterization of the periodic electronic configuration on Mn atoms in manganites within complex (large) unit cells, at high doping, thereby distinguishing between two models recently proposed. In previous studies, it was shown that at half-doping a Wigner-crystal (WC) model describes the electronic organization on the Mn atoms. 3 At higher doping the WC model was also proposed along another model, so-called "bi-stripe" model. 4 Here, the use of RXD provides direct evidence of orbital ordering on the 3d -like band in Bi 1-x Ca x MnO 3 (x=0.69) with a pattern of electronically equivalent Mn atoms corresponding to a WC model. We show that an alternative model based on a bi-stripe organization of equivalent Mn atoms can be ruled out. The WC model is also strongly supported at higher doping (x = 4 5 ). We show that several RXD spectra are satisfyingly reproduced with a crystallographic model based on the displacement patterns observed in a smaller WC unit cell (x = 2

3 ).

II. METHODS

Resonant X-ray Diffraction was performed at the L IIIand L II -edges and at the K-edge of the Mn atoms. These edges correspond to the momentary (Γ ∼ 10 -15 s) transition of a 2p electron to an unoccupied 3d orbital, and of a 1s electron to an unoccupied 4p orbital, respectively. The transition is induced by the absorption followed by the re-emission of a photon with the energy that separates the excited states from the ground state. By sweeping the incident photon energy through the edges, the spectrum of the density of unoccupied states is probed, the spectrum being weighted by the crystallographic structure factor. So, these energy-scans modulate the amplitude of the scattering factor of the resonant atoms, and therefore the intensity of the diffraction peaks.

With the diffraction vector chosen to correspond to the propagation vector of some periodic electronic features, these resonances provide two sensitivities relevant to the present study. The first technique is sensitive to the spectrum of the ordered states that are active in the physical properties of the system. Specifically, it allows the observation of the orbital ordering in manganite. It is achieved at the Mn L II/III -edges as it directly probes the first unoccupied states that derive from the atomic Mn 3d, 5-10 these states being hybridized Mn 3d and O 2p states. For the present study, L-edge data were collected at the SIM beamline of the Swiss Light Source using the RESOXS ultrahigh vacuum chamber. 10,11 The energy resolution was about 300 meV. The σ or π incident polarizations (perpendicular to and within the scattering plane, respectively) were chosen thanks to two Apple undulators.

The second technique utilizes the Mn K-edge and is sensitive to the 4p and higher p states, about 10 eV above the Fermi level. While these states are neither active in determining the physical properties, nor reflecting significantly the active states, they do depend strongly on the surrounding atomic configuration due to band structure effects. 5,[12][13][14][15][16] One can therefore take advantage of the difference in the spectrum of p states on Mn atoms that sit in different surrounding structures. At the 1s → 4p resonance, this difference can result in a scattering contrast equivalent to ∼ 5 electrons for Mn atoms sitting in different crystallographic sites 16 , be the difference in the electronic density as small as one-tenth of an electron. The maximum contrast is realized when inequivalent atoms scatter with opposite phase, if their distance is at least 1/4 of the wavelength λ. In principle the sensitivity to the different crystallographic sites can be exploited at either the L-edges or the K-edge. However, λ is about 19 Å at the L-edges; the available reciprocal space is reduced to typically one reflection only. The short wavelength at the K-edge (λ ∼ 2 Å) allows several Bragg reflections and the maximum contrast between inequivalent sites can be realized. As shown below, the K-edge spectra measured on superstructure reflections that are weak off-resonance are extremely sensitive to the subtle displacements of all the atoms of the unit cell as a result of the interference of the resonant contrast (strong on superstructure peak) with Thomson scattering (very weak but information rich). K-edge data were taken at X22C beamline of the National Synchrotron Light Source (Brookhaven National Laboratory). A Ge(111) monochromator was used providing an energy resolution of 3 eV. The incident polar-ization was σ.

In order to help interpret the K-edge resonant spectra, numerical simulations of the data were performed with the FDMNES code. 17 This program calculates the density of states in a mono-electronic approximation within the full multiple scattering formalism and calculates the matrix elements between the core electronic state 1s 2 4p 0 and the excited states 1s 1 4p 1 . Because the main signal measured at the K-edge corresponds to rather delocalized states, 10 eV above the Fermi level, the calculations are performed without consideration of any particular 3d distribution. 12 In fact, we adopt a mono-electronic formalism where correlations are treated by a Hedin-Lundquist potential. No calculations of the L-edges spectra were attempted. The calculation of both the ground state and the excited states involving the strongly correlated 3d band is beyond the scope of the present paper; dedicated studies have been presented recently. 6,9,18 In this paper, qualitative, yet definitive, conclusions have been drawn on the pattern of the orbital ordering, but not at the quantitative level for the electronic structure itself.

In principle RXD is also sensitive to charge ordering, that is to differing net electronic densities on different Mn sites. This sensitivity comes from the shift of the core level energies due to different screening by the electronic densities, and from different densities of the unoccupied states. In mixed-valent oxides, the measured difference in density is much smaller than the formal valence. Nazarenko et al. 19 and Joly et al. 20 have recently evaluated the charge density segregation in Fe 3 O 4 and NaV 2 O 5 . They found 0.12 and 0.04 electrons, respectively, from a first-principles fit to the experimental data (instead of one electron). Herrero-Martin et al. 21 explored an empirical method and claimed a segregation of ±0.08 e -in charge-ordered manganites. In fact, most of the K-edge signal is due to modulations of the 4p states whereas at the L-edge the charge ordering wave vector is not accessible for the systems studied. Therefore in the present study we don't consider further the charge ordering.

Here we report results in the Bi 1-x Ca x MnO 3 system with two high dopings, x = 0.69 and x = 0.8. These systems have an electronic phase transition at T = 220 K and at T = 165 K, respectively. The transitions have been interpreted as a charge and orbital ordering on different types of Mn atoms, with significant difference in the orbital occupancies, following a pattern in the unit cell that has yet to be characterized. The same phase transitions have been studied mostly in the La 1-x Ca x MnO 3 system. However, to the best of our knowledge, no single crystal has been grown for x ≥ 0.5 preventing thourough RXD studies. In contrast, it is possible to grow highly doped (Bi, Ca) single crystal, allowing us to perform RXD measurements on the weak superstructure reflections with a good counting statistics. Single crystals were grown by the flux technique and characterized by hard x-ray diffraction. When refering to the reflections, we use the Pbnm orthorhombic notations for the unit cell above the electronic phase transition.

III. RESULTS

Figure 1 shows the spectra taken at the L III and L II edges of the sample Bi 0.31 Ca 0.69 MnO 3 for the reflection (0δ0), where δ = 0.31. This Q is expected to correspond to the orbital ordering reflection; it is the only superstructure reflection observable at these edges. RXD spectra for two incident polarizations are shown. Because there was no polarization analysis on the scattered photons, the two measurements correspond to σ → σ + π and π → σ + π channels, the σ polarization is directed along the a axis, as shown in Figure 2. Significantly, the two channels have not only the same energy lineshape, but also the same intensity all over the L III -edge. The energy lineshape is the same also at the L II -edge but we attribute the difference in intensity to a slight misaligment of the optical path along the beamline for the two incident polarizations. In the following, we assume that the two channels give equal scattering. This result leads to the conclusion that the scattering channel is mainly rotated, corresponding to σ → π and π → σ, respectively. This is because the π → π channel is weighted by the polarization factor P compared to the σ → σ channel, P = cos 2 2θ B . With θ B = 32.13 deg. at 650 eV, P ∼ 19%. Moreover, different matrix elements are probed for the σ → σ and π → π channels which should give rise to different energy lineshapes in the manganite systems. 16 Instead, the same matrix elements are probed for the two rotated channels. 16 The absence of variation in the lineshape and even in the intensity suggests that the two unrotated channels are negligible, and that the scattering is purely rotated, that is σ → π and π → σ respectively. These results are consistent with all the resonant measurements performed on the reflections with wave vectors corresponding to the propagation vectors of the orbital ordering. 10,16,22,23 In the discussion, we will conclude that one of the proposed models (the bi-stripe) is ruled out, and that instead the Wigner-Crystal type structure is strongly supported.

Figure 3 shows the K-edge resonant spectra of the (0 2 + δ 0) for three manganites at various dopings, including the same Bi 0.31 Ca 0.69 MnO 3 sample used for the L-edge experiment. This reflection is the first superstructure reflection from the main Bragg reflection (0 2 0) (it is the fifth order of the reflection measured at the Ledge). For the three systems, the energy lineshapes are similar and the scattering off-resonance, if any, is at least two orders of magnitude smaller than on resonance. For the two other systems, it was found that this reflection corresponds to the propagation vector of oxygen octahedra having the same distortion (Jahn-Teller distortion), and that the scattering was purely rotated (σ → π). 16 Here, the incident polarization was σ and no polarization analyser was used. Similar energy lineshapes may be an indication that similar matrix elements of the scattering tensor are probed, and that the scattering channel may also be purely rotated for the Bi 0.31 Ca 0.69 MnO 3 system, as deduced from the L-edge data.

Figure 4 shows K-edge resonant spectra of main and superstructure reflections for Bi 0.2 Ca 0.8 MnO 3 in the OO phase. The spectra for (1 1 + 2δ 6) and (2 2 + 2δ 6) are strikingly reminiscent of all the published spectra interpreted as Jahn-Teller distorted and undistorted octahedra diffracting with opposite crystallographic phase, in half-doped manganites. 14,16 As in those studies, the main scattering channel is σ → σ as deduced from the numerical simulations discussed below. The strong variation in the lineshape of the superstructure reflections is a clear indication of the contrast -presumably maximum -realized between crystallographically different Mn sites.

IV. DISCUSSION

The present study is aimed at determining the pattern of the orbital ordering in highly doped manganites and distinguishing between different electronic patterns proposed recently. The analysis relies on the direct probe of the electronic structure of a subset of the Mn atoms, and on the sensitivity to distortions of octahedra associated with the orbital ordering of the (formal) 3d 4 configuration.

The crystallographic structure of LaMnO 3 shows Jahn-Teller distortions of the O octahedra consisting of long and short Mn-O bonds. 24 This peculiar structure is associated with orbital ordering of the 3d 4 configuration. This ordering has not yet been directly probed experimentally but it is obtained by first principle calculations 25,26 . As the system is doped, various crystallographic and magnetic structures appear. At halfdoping, say for La 0.5 Ca 0.5 MnO 3 , the unit cell is doubled with a checkerboard organization of distorted and undistorted octahedra 2,3,16 . This structure also reflects a recently observed orbital ordering [7][8][9][10] . At higher doping, two electronic models were proposed: the "Wigner crystal" and the "bi-stripe" models. The WC model consists of 3d 4 atoms placed as far apart as possible in the (ab) plane, whereas the bi-stripe model places equivalent Mn closer to each other and forms stripes perpendicular to the orbital ordering direction. 4 These two different patterns of orbital ordering lead to different considerations over the importance of electron-lattice coupling and Coulomb repulsion in defining the physics of the system.

Electron diffraction and high-resolution imaging were first interpreted according to the bi-stripe model, 27 but image simulations showed that the WC model was more likely 28 . The crystallographic structure of La 0.33 Ca 0.66 MnO 3 was obtained by Radaelli et al. using neutron scattering. 4 The refinements reached the lowest minimum with the WC structure "within the limits of the sets of constraints imposed". If the ligand-metal displacements associated with the orbital ordering are clearly detected by crystallography, the standard method only resolves the supercell with great difficulty. Even for the relatively simpler half-doped compound, various crystallographic structures, corresponding to various orbital order patterns, were proposed. 2 The L-edge RXD spectra presented in Figure 1 can differentiate the two proposed models. At the diffraction vector (0 δ 0), with δ = b/Λ and Λ the orbital order supercell size along b, the atoms separated by Λ/2 scatter in opposite phase, that is, with the maximum contrast. For the bi-stripe model these atoms are electronically and crystallographically different. This should allow all the scattering channels to occur at the resonance, in particular the σ → σ and π → π scatterings, like for the socalled charge-order reflections observed at half doping. 16 In fact, their electronic structures being different, these Mn atoms can be considered as different x-ray scatterers at the resonance, like two different elements. Instead, our measurements show that the scattering in the σ → σ and π → π channels is forbidden, that the scattering is rotated, of the type σ → π. Therefore, we find that the atoms are equivalent, electronically and crystallographically. The scattering is only due to the same electronic anisotropy on each equivalent site, exactly as observed and explained in previous orbital ordering studies. 10,16,29 This deduction applies to every couple of Mn atoms of the unit cell separated by Λ/2: this pattern of equivalent Mn corresponds to the WC model.

There is, however, a limitation in the determination of the whole orbital ordering pattern. Here, the direct observation of orbital ordering occurs only from sites that are Jahn-Teller distorted; it is the anisotropy of the electronic structure of a subset of the Mn atoms that allows the scattering, whereas isotropic electronic structures are "transparent". Therefore, the set of data is not complete in order to determine the pattern of the orbital ordering in the whole unit cell, where supposedly undistorted octahedra separate the distorted octahedra. One would need to realize an opposite phase shift between distorted and undistorted octahedra, which is prevented in many manganite systems by the too large wavelength at the L-edge.

To conclude on this analysis of the L-edge data, we can affirm that the orbital ordering was directly evidenced, that all the Mn atoms separated by Λ/2 are electronically equivalent, that it contradicts the bi-stripe model and supports the WC model.

The (0 2+δ 0) RXD spectra presented in Figure 3 were taken at the K-edge. It corresponds to the same propagation vector as the orbital ordering, like before, but with sensitivity to the distortions in the surrounding crystallographic structure of the Mn atoms. The similar energy lineshape for the three spectra of the same (0 2 + δ 0) reflection calls for the same conclusion drawn in previous studies for undoped and half-doped manganites: 14,16 similar to the 3d-sensitive L-edge data, the scattering is here due to the anisotropic p electronic states on Mn atoms that are electronically and crystallographically equivalent, specifically the Mn atoms sitting in the Jahn-Teller distorted octahedra. These data confirm the presence of distortions of the O octahedra accompanying the orbital ordering, preventing from differentiating a purely electronic from a purely structural origin of the phase transition.

The K-edge data presented in Figure 4 also strongly support the WC model. A structure based on the WC model was built in order to demonstrate that it can reproduce the scattering of the six K-edge RXD spectra. The model has the same symmetries and follows the same displacement directions (not the same displacement amplitudes) occuring in the La 0.33 Ca 0.66 MnO 3 structure, in the Pbnm space group. 4 In this x = 0.66 system, the WC model was found, and a trend of displacements can be identified when compared to displacements refined at x = 0.5. 3 We used the same trend for the directions of the atomic displacements and the Pbnm space group to generate a WC model within a larger unit cell, at x = 0.8. Table I gives the atomic positions used; the scheme in Fig. 5 gives the directions of the displacements in the plane of the orbital ordering. The main ingredients are the tilt of the octahedra (displacements of the apical oxygens), the JT distortions (displacements of the planar oxygens) and the cation and Mn displacements tranverse to the orbital ordering propagation vector, in opposite directions for the two halves of the unit cell (see Fig. 5). The resonant scattering factors were calculated as described above, in the σ → σ channel, using the dipolar approximation (i.e. quadrupolar s → d transitions were ignored), with this structure having the lattice parameters we found by x-ray diffraction: a=5.386, b=26.720 and c=7.449 Å. Notably, the calculations of the 2δ superstructure reflections reproduce the energy lineshapes (figure 6). These results are a strong support for the structure because these reflections are extremely sensitive to the pattern of the JT-octahedra. 16 Moreover, the relative intensities of the reflections, and the ratio between onresonance and off-resonance intensities are comparable over six orders of magnitude. In Figure 7 the calculations are instead performed with less distorted JT octahedra in which the planar oxygens atoms undergo the same displacements than those of the undistorted octahedra (the antidistortive order almost vanishes). Comparing with Figure 6, the main and δ superstructure reflections undergo almost no change. The sensitivity to the displacements is much more pronounced on the 2δ reflectionsat the Mn K-edge resonance -in intensity and in energy lineshapes. The 2δ reflections evidence the changes in the band structure due to the atomic displacements of the JT octahedra. Programs are currently developed in order to use such RXD spectra to constrain the conventional crystallographic methods, in broad analogy with the Multiwavelength Anomalous Method used in biological crystallography. Such code being in the developpment stage 19 , we used a "test and try" method, manually fitting the few crystallographic parameters, only guided by the previous work by Radaelli et al. We suggest that the resulting structure shall be used as input to a standard refinement to neutron or x-ray diffraction data in order to retrieve a more precise structure.

FIG. 7: Illustration of the sensitivity of the resonant spectra to displacements of oxygen atoms. The calculations are performed for the same positions than in table I, except for the oxygens Op1 and Op2 which undergo the same displacements than Op3, Op4 and Op5. The positions are (0.765, 0.05, 0.035) for 0p1, and (0.265, 0.05, -0.035) for Op2, with the effect of reducing the distortion of the Jahn-Teller octahedra. Remarkably, only on-resonance scattering of 2δ superstructure reflections is affected, in intensity and in energy lineshape.

V. CONCLUSION

The L-edge and K-edge experiments combined are the strong evidence for the Wigner-crystal model in the Bi x Ca 1-x MnO 3 system at high doping. Orbital ordering corresponding to a Wigner-crystal model was directly observed by Mn L-edge diffraction in the Bi 0.31 Ca 0.69 MnO 3 system with equivalent Mn atoms separated by a distance of Λ/2 along the orbital ordering direction, ruling out any bi-stripe models. The orbital ordering is accompanied by distortions of the O octahedra. A crystallographic structure with the same propagation vector for equivalent Mn octahedra was obtained after manual fit to the K-edge resonant x-ray diffraction spectra of Bi 0.2 Ca 0.8 MnO 3 . The structure follows the same displacement pattern as the La 0.33 Ca 0.66 MnO 3 system. We conclude that the Wigner-crystal pattern of electronic order is robust with high hole doping in the Bi x Ca 1-x MnO 3 system.

VI. AKNOWLEDGMENTS

This work was in part performed at the Swiss Light Source of the Paul Sherrer Institute, where we benefitted from the excellent support of the beamline staff on X11MA-SLS. Work performed at BNL was supported by US Dept of Energy, Division of Materials Science, under contract No. DE-AC02-98CH10886. We thank A. Borissov from Rutgers University for his help during the experiment at the NSLS. This work was also supported by the NSF grant DMR-0093143.

FIG. 1 :FIG. 2 :

 12 FIG.1: L II/III -edge Resonant X-ray Diffraction spectra of the (0 δ 0) reflection (δ = 0.31) for Bi0.31Ca0.69MnO3, at T = 200K taken with two polarizations of the incident photon, π and σ (without absorption correction). Inset, longitudinal Q-scan.

FIG. 3 :

 3 FIG.3: K-edge Resonant X-ray Diffraction spectra of the (0 2 + δ 0) reflection for various doped manganites in their orbital ordered phase (without self-absorption correction). Similarities in the spectra indicate the presence and the same specific arrangement of Jahn-Teller distorted octahedra. The slight shifts (about 2 eV) in the resonance is due to various interatomic distances, and differing energy calibrations on the beamline.

FIG. 6 :

 6 FIG.6: Calculation of the resonant spectra presented in Fig.4using the structure for Bi0.2Ca0.8MnO3 presented in table I and the FDMNES code17 . The intensities are normalized to the maximum intensity calculated for the (116) reflection. Straight lines are the intensities with self-absorption correction 30 , to be compared with Fig 4; dash lines are the normalized intensities without the absorption correction.

TABLE I :

 I Atomic positions for a constructed Wignercrystal model for Bi0.2Ca0.8MnO3, in the Pbnm space group, according to displacement directions obtained for La0.33Ca0.66MnO3.4

		atoms	x	y	z
		Mn1	0.018 0.1	0
		Mn2	0.018 0.3	0
		Mn3	0	0.5	0
		Bi/Ca1 -0.005 0.005 0.25
		Bi/Ca2 0.0075 0.205 0.25
		Bi/Ca3 0.0075 0.405 0.25
		Bi/Ca4 -0.0175 0.605 0.25
		Bi/Ca5 -0.0175 0.805 0.25
		Oa1 -0.065 0.1 0.25
		Oa2 -0.065 0.5 0.25
		Oa3 -0.065 0.9 0.25
		Oa4 -0.065 0.3 0.25
		Oa5 -0.065 0.7 0.25
		Op1	0.745 0.054 0.035
		Op2	0.24 0.046 -0.035
		Op3	0.265 0.15 -0.035
		Op4	0.265 0.25 -0.035
		Op5	0.265 0.35 -0.035
	p1
	a
	p2
	p3	p4	p5
	1	2	3
			5b
	FIG. 5: (Color online) Model of a crystallographic structure
	for the orbital order phase at x = 0.8 doping. The arrows indi-
	cate the displacements from the undistorted structure, those
	have the same directions in the crystallographic structure pro-
	posed for La0.33Ca0.66MnO3. The small circles represent pla-
	nar oxygen atoms. Larger circles are Mn atoms; shaded circles
	are the Jahn-Teller Mn atoms; the labels corresponds to the
	atoms listed in table I. Squares represent the cations. The
	arrows indicate the directions (not the amplitude) of the dis-
	placements.

µ * where µ * = µMn+0.2µBi+0.8µCa+3µO 0.2µBi+0.8µCa+3µO the absorption coefficients are calculated using the FDMNES code for the Mn atoms, and the Cromer-Liberman tables for the other atoms.