DOSE, DAMAGE AND RESOLUTION - FUNDAMENTAL LIMITS IN COHERENT X- RAY-DIFFRACTION IMAGING H. Chapman % R. M. Glaeser * S. Hau-Riege % H. He* M. Howells* J. Kirz* $ S. Marchesini % H. A. Padmore * J. C. H. Spence # * U. Weierstall # * Advanced Light Source, Lawrence Berkeley National Laboratory # Dept. of Physics, Arizona State University % Lawrence Livermore National Laboratory $ State University of New York at Stony Brook

SUMMARY Our recent 3D experiments and the light they shed on the time needed Dose fractionation The x-ray scattering cross section of a voxel – dose and flux calculations Issues involved in choosing the x-ray wavelength Projections of ultimate achievable resolution for – Life science experiments (dose limited) – Materials-science experiments (flux limited)

ALS BEAM LINE COHERENT OPTICS NOTES: Our experiments are done at 588 eV in undulator 3rd harmonic Be window and zone-plate monochromator both 0.8 mm in size are designed to withstand pink (once reflected) beam Diffractive elements of the zone plate mono (Charalambous) made of silicon nitride coated on both sides with aluminium for better mechanical stability and heat removal

BEAM LINE EFFICIENCY FACTORS.

SOFT-X-RAY DIFFRACTION EXPERIMENT

LATEST EXPERIMENTS We recently took our first 3D data sets over an angular range ±70° Total time for about sec views was about 10 hours Number of viewsExposure time (sec)Nominal Angle increment (°)

,000 MERGE OF 100, 3, 1 SEC EXPOSURES Spatial period=200Å Rayleigh res’n=100Å Typical CCD “counts” Statistically significant out to near the edge Not necessary according to dose fractionation theorem

SUMMARY OF 3D DATA By S. Marchesini 150 views at 100 sec exposure, 1-degree intervals

The dose fractionation theorem [Hegerl and Hoppe 1976] says that the radiation dose required to measure the strength of a voxel is the same for a multi-view 3-D experiment on a large object (with appropriate reconstruction algorithms) as it is for a single measurement of one voxel alone provided that the voxel size and the statistical accuracy are the same in both cases. Study by [McEwen 1995] extended the validity to cases with: – high absorption – signal-dependent noise – varying sample contrast – missing angular range – a need to align the recorded patterns by cross-correlation methods THE DOSE FRACTIONATION THEOREM OF STANDARD TOMOGRAPHY

The theorem is evidently based on the amount of information that can be gathered by x-ray illumination and recording We tentatively accept that the theorem is applicable to coherent x-ray diffraction imaging and we proceed to consider diffraction by an object consisting of a single cubic voxel of size Note that it is unlikely that anyone would want to do an experiment on a single voxel. Therefore this is an abstract calculation that we need to do in order to apply the theorem. It is not an attempt to model a real experiment. For this reason it is hard to compare this calculation with experiment. The only coherent diffraction experiment published on biological material (to my knowledge) is by Miao et al (PNAS 2003) and this was not on a cubic voxel. BASIS FOR CALCULATION

FOURIER-OPTICS CALCULATION OF THE VOXEL SCATTERING CROSS SECTION

VOXEL SCATTERING CROSS SECTION CONTINUED

Start with the one-electron cross section and make a coherent sum over the diffraction solid angle corresponding to resolution d Use results by earlier workers [Henke 1955, London 1989] for scattering by a spherical particle Both of these give the same result up to a constant factor of order one WAYS TO VERIFY THE CROSS SECTION RESULT

ESTIMATE OF THE DOSE AND FLUX

The dose and fluence N have the same scaling with d We can argue from the Rose criterion as follows [Glaeser 2002] – the contrast (C) between a material-filled voxel and empty space scales as the thickness d – according to Rose the number of photons per unit area N must satisfy Nd 2 >25/C 2 – therefore since C scales as d, N scales as 1/d 4 and so does the dose ANOTHER VIEWPOINT ON RESOLUTION SCALING

 = the voxel intensity absorption coefficient h = the photon energy r e = the classical electron radius = the photon wave length  = the scattering strength of the voxel material in electrons per unit volume  = the density The coherent scattering cross section of a cubic voxel is r e 2 2 |  | 2 d 4 whence the dose D and the fluence N required to deliver P scattered x- rays into a detector with collection angle chosen for resolution d is DOSE SCALING WITH RESOLUTION: SUMMARY N

FLUX REQUIREMENTS Flux to detect a 10 nm voxel made of protein according to the Rose criterion. A detector collecting an angle chosen for 10 nm resolution is assumed

Below is the dose to detect a 10 nm voxel made of protein according to the Rose criterion. A detector collecting an angle chosen for 10 nm resolution is assumed DOSE REQUIREMENTS

DOSE-RESOLUTION RELATIONSHIP FOR A FROZEN-HYDRATED PROTEIN SAMPLE

DOSE-RESOLUTION RELATIONSHIP FOR A GOLD SAMPLE

The dose used in the two gold experiments by Miao et al (ABCDEF experiment) and ourselves (in 2D) was about 1000 times higher than the Rose dose calculated in the above way - Miao et al made a 2D reconstruction - our data is recent and has not yet been reconstructed To get a good image we expect to use somewhat more than the Rose- criterion dose (P = 25) which is for bare detectability against shot noise Nevertheless a factor 1000 is much more than one would expect and may mean that we are using “overkill” exposures in these experiments It may also mean that our system DQE is poor i. e. not shot-noise- limited However note that both experiments were done with an expectation to reconstruct the 2D data set - therefore if we allow for fractionation of the dose among 180 views the dose per view would be 180 times less MEANING OF THE RESULTS OF THE 2D GOLD EXPERIMENTS

w  A B P Path difference = AP – [AB + BP] = w – w cos  ≈ w  2 /2 d = 0.61 /  Require path difference < coherence length = 2 /  Conclude:  < 5.4 d 2 /w Sample size = w Resolution = d Far-field diffraction BANDWIDTH REQUIREMENT

FLUX CONSIDERATIONS: The required fluence is P/[r e 2 2 |  | 2 d 4 ] which scales like –2 (coming from the voxel cross section so this is neglecting |  | 2 dependence) (unfavorable to HXR) For a given source brightness B the coherent flux is B( /2) 2 (unfavorable to HXR) For given resolution and sample size, the bandwith  is prescribed. Thus the required fractional bandwidth varies like –1 (favorable to HXR) The flux required goes up like –3 as diminishes DOSE CONSIDERATIONS: The dose for light elements (biology say) is roughly flat with wavelength DIFFRACTION CONSIDERATIONS: Need short to approach the flux-limited resolution (maybe about 1-2 nm) for radiation-resistant i.e. materials-science samples - say <0.5  resolution RECONSTRUCTION CONSIDERATIONS Harder x-rays will make the scattering factors essentially real These are not the only considerations - ongoing discussion - my take keV but the upgraded ALS will go to 5 keV - awkward technically WHAT IS THE BEST WAVELENGTH?

Our last experiment suggests that even with our present beam line the exposure times for 3D at 10 nm resolution are reasonable (about 10 hours) and this resolution is near the dose limit for life-science Thus we conclude that 3D experiments on frozen hydrated samples are possible at 10 nm resolution and maybe 5 nm (?) with staining Based on known shortcomings of our present beam line for this experiment (it was built for chemistry) we can project an improvement factor of about 500 (due to undulator, optics, zone plate monochromator) with present ALS We can project another factor 30 due to the ALS upgrade giving an overall factor of about 10 4 This is important because we also see applications in material science where the dose limitation is much less severe. For the inverse fourth power scaling of the required flux with resolution the above factor would allow a factor ten improvement in resolution to about 1 nm We have still not taken dose fractionation into account which gives (say) another factor of 180 which may allow for wavelength reductions required in moving to 1 nm resolution PROJECTIONS INTO THE FUTURE