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*To*: ccp4bb@dl.ac.uk*Subject*: [ccp4bb]: Summary: anisotropic ellipsoids*From*: Norbert Straeter <strater@chemie.fu-berlin.de>*Date*: Fri, 9 Mar 2001 11:31:32 +0100 (CET)*In-Reply-To*: <200103090920.JAA32818@dlpx1.dl.ac.uk>*Sender*: owner-ccp4bb@dl.ac.uk

*** For details on how to be removed from this list visit the *** *** CCP4 home page http://www.dl.ac.uk/CCP/CCP4/main.html *** This is a summary to the following question I posted yesterday: >According to many textbooks the first three of the thermal parameters >U11 U22 U33 U12 U13 and U23 describe the displacements along the >perpendicular principal axis of the ellipsoid and the latter three give >the orientation of the principal axes with respect to the unit cell axes. >However, I can't find anywhere how U12 U13 and U23 (apparently as >direction cosines) exactly describe the orientation of the ellipsoid, say >in a cartesian system. First of all, my question was based on the false assumption that U11, U22 and U33 are the components along the principal axes of the ellipsoid. The text on page 533 of Glusker et al. "Crystal structure analysis for chemists and biologists" led me to that conclusion, although the example on page 536 indicates that things are not as simple as that. U11, U22 and U33 are the <u2> values along the reciprocal cell axes a*, b* and c*, respectively (e.g. Drenth, page 94). The principal axes of the thermal ellipsoid can be obtained from the U values via a principal axes transformation. This is described e.g. in Giacovazzo et al., p. 75 ff. and 148 (don't rely on the index), in the ORTEP manual, and in some of the hints cited below. Thanks to all who replied: From: Manfred Buehner <buehner@biozentrum.uni-wuerzburg.de> From: Eleanor J. Dodson <ccp4@ysbl.york.ac.uk> From: Nicholas M Glykos <glykos@crystal2.imbb.forth.gr> From: Witek Kwiatkowski <witek@octopus.salk.edu> From: "Papiz, MZ (Miroslav)" <M.Z.Papiz@dl.ac.uk> From: Phraenquex VD <loretta@scripps.edu> From: M.D.Winn <M.D.Winn@dl.ac.uk> Following are parts of the replies (in arbitrary order): --------------------------------------------------------------------- Not really. U11 U22 U33 U12 U13 and U23 are the tensor elements in the chosen coordinate system (usually orthogonal). Only if the coordinate system is rotated to the principal axes system are U11 U22 U33 the displacements along the principal axes, in which case U12 U13 and U23 are zero. U12 U13 and U23 only define the orientation of the principal axes in the very loose sense that their non-zero-ness indicates the deviation between the orthogonal and principal axes systems. Hmmm, quick flip through ortep manual, if that's what you're using. There seem to be some definitions in terms of orientations (which I've never used) but not couched in terms of U elements .... ------------------------------------------------------------------------- Would the equation on page 327 of Vol.II of the international tables (for the anisotropic thermal parameters) be of help to you ? ps In case you can't find this volume, the equation is : fr = exp{-(b11*h^2 + b12*hk + b13*hl + b22*k^2 + b23*kl + b33*l^2)} * fr0 where fr0 is the scattering factor for atom r at rest, fr is the scattering factor of the anisotropically-treated atom, and b11,b12,... are the atomic anisotropic thermal parameters of the atom. ------------------------------------------------------------------- Apologies for repeating the obvious, but I thought that what you was looking for was a qualitative answer to your question. If you are up to calculating things, page 76 of "Fundamentals of Crystallography" by Giacovazzo et al., has a worked (numerical) example for calculating the principal axes of an atomic thermal ellipsoid starting from (b11,b12,...). -------------------------------------------------------------------------- Is this any help? Taken from $CLIBS/rwbrook.f C PDB files contain cartesian anisotropic temperature factors as orthogonal Us. C Stored as U11 U22 U33 U12 U13 U23 C The anisotropic temperature factors can be input/output to this routine C as orthogonal or as crystallographic Us. C C Shelx defines Uf to calculate temperature factor as: C T(aniso_Uf) = exp (-2PI**2 ( (h*ast)**2 Uf_11 + (k*bst)**2 Uf_22 + ... C + 2hk*ast*bst*Uf_12 +..) C C Note: Uo_ji == Uo_ij and Uf_ji == Uf_ij. C C [Uo_ij] listed on ANISOU card satisfy the relationship: C [Uo_ij] = [RFu]-1 [Uf_ij] {[RFu]-1}T C C where [Rfu] is the normalised [Rf] matrix read from the SCALEi cards. C see code. [ROu] == [RFu]-1 C C T(aniso_Uo) = U(11)*H**2 + U(22)*K**2 + 2*U(12)*H*K + ... C where H,K,L are orthogonal reciprocal lattice indecies. ( EJD: I think????) C C Biso = 8*PI**2 (Uo_11 + Uo_22 + Uo_33) / 3.0 C C [Uf(symm_j)] = [Symm_j] [Uf] [Symm_j]T C ------------------------------------------------------------------ i think to change into an orthonormal system you use U = [A .b. AT] where B is the usual anisotropic Dsip. tensor A is the cholensky factor and AT is its transpose A has the property G = (AT.A) where G is the real space metric tensor g = a**2, a*b*cos(g), a*c*cos(b) a*b*cos(G), b**2, B*C*cos(a) a*c*cos(b), b*c*cos(a), c**2 the eignvalues and eigenvectors oF u are the principal axes (PA) of the elipsoides. I guess the angles they make with a, b, and c are the cosine of the vector. so with zero off diagonal terms the PA's are along the cell axes (in an orthoganal cell) and with non zero values this tells you how to rotate these Pa's from the cell axes. -------------------------------------------------------------------- exp(X^T U X) (where X is a vector from atom's equilibrium position X^T is a transpose of this vector and U is an Uij matrix or anisotropic displacement factor tensor if you prefer) is a probability function that describes the relative probability of finding nucleus displaced by the vector X from equilibrium. and equation X^T U X = C describes an ellipsoidal surface on which such probability is constant. What you are asking is finding principal axis of this quadratic form - can be solved via the Lagrange multipliers. --------------------------------------------------------------------- -Norbert

**References**:**Re: [ccp4bb]: anisotropic ellipsoids***From:*"M.D.Winn" <M.D.Winn@dl.ac.uk>

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