@text
MAD.kin
 "L1" measurement near adsorption close to minimum of f'
|Fph-| = 122.396      |Fph|av = 120.902  take average as Fph
|Fph+| = 119.408
del anom = 2.99 with - > +

 "L2" measurement near adsorption close to maximum of f''
|Fph-| = 125.658      |Fph|av = 122.893
|Fph+| = 120.127
del anom = 5.531 with - > +

 "L3" measurement far from adsorption edge
|Fph-| = 128.965      |Fph|av = 127.407  take average as "Native"== Fp
|Fph+| = 125.849
del anom = 3.116 with - > +


MAD:  Multiple wavelength Anomalous Dispersion

     There are two approaches to abstracting information from a MAD experiment.  One is to use all information in a set of simultaneous equations, mathematically satisfying perhaps, but using a different intuition then our geometrical approach.  Another is to treat the experiment as a combination of isomorphous changes in the scattering power of the "heavy" atoms along with associated anomalous scattering differences.  This uses the same conceptual machinery already developed for the SIRAS method, and, indeed, one of the commonly used computer programs for MAD phasing was originally developed for SIRAS.

     Theoretically, all that is really needed is data at two wavelengths.  Recall that in SIRAS a difference in isomorphous scattering along with one pair of anomalous scattering sets is sufficient.  However, not only does redundancy of taking data at more than 2 wavelengths help in this experiment where one is interested in small differences between large numbers, three wavelengths can be selected to optimize the phasing information (and often data at 4 wavelengths is taken for further assurance).
     Three wavelengths can be chosen, in terms of the SIRAS analysis, such that one pair  has maximal "isomorphous" differences, and an intermediate wavelength gives maximal anomalous differences.  At the adsorption edge of an element, inner electrons come into resonance with the incident x-ray energy, these electrons scatter more strongly and with a different phase shift (i.e. more like 90 phase lag) to the incident wave.  There also are slightly fewer electrons scattering with the usual 180 phase lag, so the usual scattering falls off while the anomalous scattering increases as the adsorption edge is crossed.  The effect is specific to the element but is subject to slight shifts because of interactions with surrounding atoms, so good practice is to run a scattering scan across the wavelength region and determine exactly the scattering profiles for the crystal under study.
     Scattering from an element can be expressed as  f(total) = fo + f' + f"  where fo is the expected scattering for that element at the angle of the measurement, f' is the change in the usual 180 deg. phase lagged scatter, and f' is the amount of scatter with a 90 deg. phase lag.  (2 componets at 90 deg. of course, can represent all scattering.)
f' hits a minimum just at the absorptin edge where f'" is rapidly changing, and f" has a sharp maximum at the high energy side of the edge.  Let L1 be the point of minimum usual scattering, L2 the point of maximal anomalous scattering, and L3 be remote from the edge where the usual scattering is no longer depressed.  Then most of the phasing power can be extracted from the experiment just by taking the difference between L1 and L3 as the "isomorphous" part (averaging the Bijvoet pairs taken at those wavelengths) and the Bijvoet pair at L2 for the anomalous signel.  Of course, with 3 sets of Bijvoet pairs there are other ways to combine the data sets to enhance the accuracy, but most of the power is gained from those optimal combinations.  
     Modern "area" detectors and high intensity x-rays from a synchrotron source on frozen crystals, allow for efficient collection of the Bijvoet pairs and redundancy to provide accurate intensity measurements.


     The diagram shown in the MAD kinemage mimics the phase determination for a particular hkl reflection using the SIRAS paradigm, from a crystal containing Selenium.  The heavy atom derivative is data taken at L1 where the usual scattering from the crystal is most different, the "native" data is from L3, far from the edge, and the anomalous pair is taken at L2.
     Thus  Fph - Fp ~= Fh becomes  L1av - L3av ~= fh , the putative "heavy atom".  This is a perfect isomorphous change of scattering power at the Se position.  It is a small change, only a fraction of the scattering power of that atom, but all measurements are done on the same crystal at close to the same time.
     The convention of considering the L3 data to be the "native" set makes a certain kind of sense, but the "derivative" has less scattering, L1 on average is smaller than L3,  so the resultant vectors are backwards with respect to the actual Se atom scattering factor.  However, the actual physics is the same no matter how things are defined, so the effective +,- sides still have the same relationship to the actual Se scattering factor and have to be assigned on the diagram accordingly.  In the kinemage, all the +anomalous effect is on the right of the Se vector and all the -anomalous effect is on the left of the Se vector  (Recall that the diagram is of the +h+k+l side of the crystal with the -h-k-l side information reflected onto the +h+k+l side diagram.



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@master {MAD anom}
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@master {MAD common}
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