Not sure if this is helpful Phil, but SCALEIT output includes various
definitions taken from the Willis and Prior book.
But then there is the problem of converting the amplitude B factors to
real space..
I attach my anisotropy notes..
It doesnt address the ? of sensible conventions!!
E
On 10/12/2011 02:55 PM, Phil Evans wrote:
I've been struggling a bit to understand the definition of B-factors,
particularly anisotropic Bs, and I think I've finally more-or-less got my head
around the various definitions of B, U, beta etc, but one thing puzzles me.
It seems to me that the natural measure of length in reciprocal space is d* =
1/d = 2 sin theta/lambda
but the "conventional" term for B-factor in the structure factor expression is
exp(-B s^2) where s = sin theta/lambda = d*/2 ie exp(-B (d*/2)^2)
Why not exp (-B' d*^2) which would seem more sensible? (B' = B/4) Why the
factor of 4?
Or should we just get used to U instead?
My guess is that it is a historical accident (or relic), ie that is the
definition because that's the way it is
Does anyone understand where this comes from?
Phil
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Scaleit:
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Anisotropic temperature factor (REFINE ANISOTROPIC) (default)
C * exp(-(h**2 B11 + k**2 B22 + l**2 B33 +
2hk B12 + 2hl B13 + 2kl B23))
T he anisotropic scale is applied to the derivative F as
(derivative scale)* exp( - (B11*h**2 + B22*k**2 + B33*l**2 + 2*(B12*h*k +
B13*h*l + B23*k*l) ) )
An equivalent form of the anisotropic temperature factor is where
beta11 = B11/(a*)**2 etc
exp(-0.25( h**2 * (a*)**2 * beta11 + k**2 * (b*)**2 * beta22 + l**2 * (c*)**2
* beta33
+ 2*k*l*(b*)*(c*)*beta23 + 2*l*h*(c*)*(a*) *beta31 + 2*h*k*(a*)*(b*)
*beta12))
(This means the Uij terms of an anisotropic temperature factor is equal to
betaij/(8*pi**2.)
For derivative : 1
beta matrix - array elements beta11 beta12 beta13
beta21 beta22 beta23,
beta31 beta32 beta33
-2.6094 0.0000 0.0000
0.0000 -2.6094 0.0000
0.0000 0.0000 -0.9378
+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Note: REFMAC5 outputs betaij/4
The isotropic equivalent is ::
exp(-B (sin**2(theta)/lamda**2) ) =
exp(-0.25( h**2 * (a*)**2 * B + k**2 * (b*)**2 * B + l**2 * (c*)**2
* B
+ 2*k*l*(b*)*(c*)*cosAS*B + 2*l*h*(c*)*(a*)*cosBS*B +
2*h*k*(a*)*(b*)*cosGS*B))
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REFMAC code:
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S1 = FLOAT(IHH(1))*RCELL(1)
S2 = FLOAT(IHH(2))*RCELL(2)
S3 = FLOAT(IHH(3))*RCELL(3)
S11 = S1*S1
S22 = S2*S2
S33 = S3*S3
S12 = 2.0*S1*S2
S13 = 2.0*S1*S3
S23 = 2.0*S2*S3
SBS = B_LS_ANISO_OVER(1)*S11 + B_LS_ANISO_OVER(2)*S22 +
& B_LS_ANISO_OVER(3)*S33 + B_LS_ANISO_OVER(4)*S12 +
& B_LS_ANISO_OVER(5)*S13 + B_LS_ANISO_OVER(6)*S23
EXPAN = EXP(-SBS)
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>From RWBROOK and SHELXL
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C
C PDB files contain anisotropic temperature factors as orthogonal Uo_ijs
multiplied by 10**4.
C The order is: Uo11 Uo22 Uo33 Uo12 Uo13 Uo23
C
C Shelx defines Ufn_ij to calculate temperature factor as:
C T(aniso_Ufn) = exp (-2PI**2 ( (h*ast)**2 Ufn_11 + (k*bst)**2 Ufn_22 +
(l*cst)**2 Ufn_33
C + 2hk*ast*bst*Ufn_12 + 2hl*ast*bst*Ufn_13+
2kl*bst*cst*Ufn_23)
C
C Note: Uo_ji == Uo_ij and Uf_ji == Uf_ij.
C
C 10**4*[Uo_ij] listed on ANISOU card satisfy the relationship:
C [Uo_ij] = [RFu]-1 [Ufn_ij] {[RFu]-1}T
C [Ufn_ij] = [RFu] [Uo_ij] {[RFu]}T
C where [Rfu] is the normalised [Rf] matrix read derived from the SCALEi
cards.
ie Rf11 Rf12 Rf13 = SCALE1(1) SCALE1(2) SCALE1(3)
Rf21 Rf22 Rf23 SCALE2(1) SCALE2(2) SCALE2(3)
Rf31 Rf32 Rf33 SCALE3(1) SCALE3(2) SCALE3(3)
and Rfu11 Rf12 Rfu13 = Rf11/FAC1 Rf12/FAC1 Rf13/FAC1
where FAC1 = SQRT(Rf11**2 +Rf12**2 +Rf13**2) etc.
For conventional SCALEi FAC1 = a*, etc but I am not sure if it is always
true..
If it is and you convert
[Uf_ij] = [RF] [Uo_ij] {[RF]}T where [Rf] is the SCALEi matrix without
normalisation
then
T(aniso_Uf) = exp (-2PI**2 ( (h)**2 Uf_11 + (k)**2 Uf_22 +(l)**2 Uf_33 +
2hk*Uf_12 +..)
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
Do you want structure factor equations in reciprocl space with anisotropic B
(or U values) values? Then no questions marks there.
For scaling You are right. You just calculate structure factors as you would
do normally with formfactors also of course and then apply
contribution of anistropic scale factor. I.e.
F_scaled = k exp(-SBS) F_unscaled
F_uscale = sum_overatoms f_iatom exp(-2pi i (h xf + k yf + l zf)
Garib
========================================================================
Structure factor for atoms with anisotropic U values
F(hkl) = sum_overatom f_atom(hkl) T(U_atom,hkl) exp(-2pi i (h xf + k yf + z yf)
T(U_atom,hkl) = exp(-(U11 h2 + U22 k2 + U33 l2 + 2 U12 hk + 2 U13 hl + 2 U23
kl)) = exp(-(hkl) U hkl^T)
hkl is a row vector. If you write it as a column vector then transpose should
be on the right side.
There might be constant or so somewhere there.
PDB definition of ANISOU record
ANISOU
Overview
The ANISOU records present the anisotropic temperature factors.
Record Format
COLUMNS DATA TYPE FIELD DEFINITION
----------------------------------------------------------------------
1 - 6 Record name "ANISOU"
7 - 11 Integer serial Atom serial number.
13 - 16 Atom name Atom name.
17 Character altLoc Alternate location
18 - 20 Residue name resName Residue name.
22 Character chainID Chain identifier.
23 - 26 Integer resSeq Residue sequence number.
27 AChar iCode Insertion code.
29 - 35 Integer u[0][0] U(1,1)
36 - 42 Integer u[1][1] U(2,2)
43 - 49 Integer u[2][2] U(3,3)
50 - 56 Integer u[0][1] U(1,2)
57 - 63 Integer u[0][2] U(1,3)
64 - 70 Integer u[1][2] U(2,3)
73 - 76 LString(4) segID Segment identifier, left-justified.
77 - 78 LString(2) element Element symbol, right-justified.
79 - 80 LString(2) charge Charge on the atom.
Details
* Columns 7 - 27 are identical to the corresponding ATOM/HETATM record.
* The anisotropic temperature factors (columns 29 - 70) are scaled by a factor
of 10**4 (Angstroms**2) and are presented as integers.
* The anisotropic temperature factors are stored in the same coordinate frame
as the atomic coordinate records.
* ANISOU values are listed only if they have been provided by the depositor.
Examples :
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1) P212121
34.44 48.95 72.62 90.00 90.00 90.00
Refmac:
Overall : scale = 0.276, B = -34.671
Partial structure 1: scale = 0.928, B = 25.925
Overall anisotropic scale factors
B11 = 0.95 B22 = -0.40 B33 = -0.55 B12 = 0.00 B13 = 0.00 B23 = 0.00
-----------------------------------------------------------------------------
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
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SFCHECK
Anisotropic distribution of Structure Factors:
Ratio of Eigen values : 1.0000 0.7735 0.7615
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Phaser:
Principal components of anisotropic part of B affecting observed amplitudes:
eigenB (A^2) direction cosines (orthogonal coordinates)
6.392 1.0000 0.0000 0.0000
-3.043 -0.0000 1.0000 -0.0000
-3.349 -0.0000 0.0000 1.0000
Anisotropic deltaB (i.e. range of principal components): 9.741
--------------
ABSOLUTE SCALE
--------------
Scale factor to put input Fs on absolute scale
Wilson Scale: 2.16908
Wilson B-factor: 26.499
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Ctruncate Anisotropic scaling:
Anisotropic scaling (orthogonal coords):
| -0.1543 0.0000 0.0000 |
| 0.0000 0.0718 0.0000 |
| 0.0000 0.0000 0.0824 |
Anisotropic U scaling (fractional coords):
| -1.300e-04 8.387e-22 5.753e-21 |
| 8.387e-22 2.998e-05 -1.601e-21 |
| 5.753e-21 -1.601e-21 1.563e-05 |
Anisotropic B scaling (fractional coords):
| -1.027e-02 6.622e-20 4.542e-19 |
| 6.622e-20 2.367e-03 -1.264e-19 |
| 4.542e-19 -1.264e-19 1.234e-03 |
Minimum resolution = 26.263 A
Maximum resolution = 2.013 A
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thii_pk example P3221
Cell: 81.40 81.40 140.47 90.00 90.00 120.00
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Refmac:
Overall : scale = 0.674, B = 2.478
Partial structure 1: scale = 0.253, B = 36.212
Overall anisotropic scale factors
B11 = 1.74 B22 = 1.74 B33 = -2.61 B12 = 0.87 B13 = 0.00 B23 = 0.00
-----------------------------------------------------------------------------
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SFCHECK
Ratio of Eigen values : 1.0000 1.0000 0.6070
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Phaser:
Principal components of anisotropic part of B affecting observed amplitudes:
eigenB (A^2) direction cosines (orthogonal coordinates)
8.591 0.8023 -0.5969 0.0000
8.591 0.5969 0.8023 -0.0000
-17.182 -0.0000 0.0000 1.0000
Anisotropic deltaB (i.e. range of principal components): 25.773
--------------
ABSOLUTE SCALE
--------------
Scale factor to put input Fs on absolute scale
Wilson Scale: 1.32506
Wilson B-factor: 66.4817
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Ctruncate ANISOTROPY CORRECTION:
ANISOTROPY CORRECTION:
Anisotropic scaling (orthogonal coords):
| -0.2516 -0.0000 -0.0000 |
| -0.0000 -0.2516 0.0000 |
| -0.0000 0.0000 0.5032 |
Anisotropic U scaling (fractional coords):
| -5.063e-05 -2.532e-05 -2.288e-20 |
| -2.532e-05 -5.063e-05 -1.336e-21 |
| -2.288e-20 -1.336e-21 2.550e-05 |
Anisotropic B scaling (fractional coords):
| -3.998e-03 -1.999e-03 -1.807e-18 |
| -1.999e-03 -3.998e-03 -1.055e-19 |
| -1.807e-18 -1.055e-19 2.014e-03 |
Minimum resolution = 28.161 A
Maximum resolution = 3.000 A
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3) am782.mtz P1
P1 Cell: 51.312 62.722 66.795 77.082 81.135 89.685
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REFMAC
-----------------------------------------------------------------------------
Overall : scale = 0.398, B = -6.531
Partial structure 1: scale = 0.401, B = -5.462
Overall anisotropic scale factors
B11 = -0.15 B22 = -0.32 B33 = 0.78 B12 = 0.22 B13 = -0.49 B23 = -0.35
-----------------------------------------------------------------------------
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SFCHECK
Anisotropic distribution of Structure Factors:
Ratio of Eigen values : 0.8224 0.7447 1.0000
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Phaser:
Refined Anisotropy Parameters
-----------------------------
Principal components of anisotropic part of B affecting observed amplitudes:
eigenB (A^2) direction cosines (orthogonal coordinates)
1.173 0.1725 -0.5952 0.7849
0.960 -0.6181 0.5550 0.5567
-2.133 0.7670 0.5811 0.2722
Anisotropic deltaB (i.e. range of principal components): 3.306
--------------
ABSOLUTE SCALE
--------------
Scale factor to put input Fs on absolute scale
Wilson Scale: 1.80035
Wilson B-factor: 19.4283
++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
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Ctruncate:
ANISOTROPY CORRECTION:
Anisotropic scaling (orthogonal coords):
| -0.0043 -0.0410 0.0426 |
| -0.0410 0.1191 0.0136 |
| 0.0426 0.0136 -0.1148 |
Anisotropic U scaling (fractional coords):
| -7.689e-06 -1.799e-05 1.842e-05 |
| -1.799e-05 2.710e-05 9.959e-06 |
| 1.842e-05 9.959e-06 -2.776e-05 |
Anisotropic B scaling (fractional coords):
| -6.071e-04 -1.420e-03 1.454e-03 |
| -1.420e-03 2.140e-03 7.864e-04 |
| 1.454e-03 7.864e-04 -2.192e-03 |
B = 18.806 intercept = 6.446 siga = 0.218 sigb = 0.040
scale factor on intensity = 630.0471
results from fitting Truncate style Wilson plot
B = 22.621 intercept = 6.194 siga = 2.556 sigb = 0.437
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You wrote:
Clipper_reci_orth => hkl.coord_reci_orth(cell)
Then n = [hkl.cord_reci_orth} [U-ortho_aniso] [hkl.cord_reci_orth]transpose
gives temp coeff exp(0.5n)
I copied this down from you - but I realise now that I dont know whether this
is to be applied to intensities or amplitudes. - amplitudes I guess from the
0.5 division..
It would also be nice to have the coefficients used in the more traditional
U_aniso form where u_ij = betaij/8PI**2 for comparison to other software.
It would mean
1) changing signs of your n_ij
2) getting the matrix product of [reci_orth] [U-ortho_aniso] [reci_orth]
transpose
E
exp(-0.25( h**2 * (a*)**2 * beta11 + k**2 * (b*)**2 * beta22 + l**2 *
(c*)**2 * beta33
+ 2*k*l*(b*)*(c*)*beta23 + 2*l*h*(c*)*(a*) *beta31 + 2*h*k*(a*)*(b*)
*beta12))
