Background

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In scattering, background refers to the unwanted scattering that arises from sources other than the sample of interest. It thus underlies the signal of interest, decreasing the signal-to-noise ratio, and making analysis more complicated.

Sources

  1. Detector: Every detector has some background signal. The detector background may also have multiple components: a component that is present in every exposure (e.g. readout noise), as well as a component that scales with the exposure time (e.g. dark current). Detectors may also exhibit signals from other sources: e.g. cosmic rays, or even ambient light.
  2. Air scattering: The incident beam, and scattered rays, will be scattered by ambient air that they travel through. This tends to broaden the beams (and thus peaks), and introduces diffuse background into the measurement. This source of background can be minimized by flushing the beam path with helium gas (which has very weak scattering), or by pumping-down to near vacuum. Air scattering is most pronounced at lower x-ray energies; it is nearly invisible for high-energy x-rays.
  3. Instrumental: Most x-ray instruments will have windows that isolate the x-ray source (which is under vacuum) from the sample chamber. Even if the sample chamber is evacuated, x-ray transparent windows will likely remain in place. These windows (although nominally x-ray transparent) will give rise to a scattering signal. This scattering can be partially blocked using guard slits downstream of the window (ideally placed close to the sample). Even so, the low-q scattering cannot be eliminated. Moreover, the slits will introduce some signal of their own (weak scattering; or bright streaks if they cut deeply into the incident beam). Windows placed after the sample (e.g. when the sample is in air but the downstream path is evacuated) will lead to scattering that cannot be eliminated. Kapton is frequently used; this material introduces diffuse low-q scattering, as well as some weak halos at intermediate-q.
  4. Sample holder: Especially in transmission-scattering experiments, the sample will typically be contained in a holder (e.g. a capillary, or between two Kapton sheets). This holder will of course introduce scattering.
  5. Matrix: For materials that are dispersed (e.g. particles in solution or dispersed in a polymer), the matrix itself will lead to scattering.
  6. Diffuse scattering: Confusingly, sometimes the diffuse scattering arising from the sample may also be referred to as a kind of background. The diffuse scattering generally arises from disorder: it may be considered an unwanted background when analyzing a structural peak; but it may be the signal of interest when analyzing heterogeneous ordering.

Measuring

One can attempt to measure the background, in preparation for subtracting it from the experimental data. A variety of measurements can be combined to assess the various sources of background.

  • Dark signal: By performing an exposure with the x-ray beam blocked, one can independently measure the detector component (#1) of the background.
  • Direct beam: By performing an exposure with the x-ray beam turned on, but without any sample (or even sample cell), one can measure the contributions from detector, air scattering, and instrumental (#1-3). The air+instrumental component can then be obtained by subtracting the dark signal from this direct beam measurement.
  • Empty cell: By measuring the empty sample cell, one additionally includes the sample holder; i.e. one measures #1-4.
  • Empty cell (w/ matrix): One can instead measure an 'empty cell' where the matrix (e.g. solvent) is present; i.e. one measures #1-5.

Subtraction

Full background subtraction

In order to remove the effect of the background, the simplest solution is to simply measure it, and subtract it from the experimental data. However, there are a few issues to consider:

  • Exposure time: Most of the sources of background scale with exposure time. So a valid subtraction will require using the same exposure time for the background and sample measurements. In principle, one can do a more general background subtraction by rescaling the background and sample measurements by the exposure time; however if the detector has readout noise (which doesn't scale with exposure time), then this procedure is not valid. In such a case, one should get a separate measure of the readout noise (dark signal), and first subtract this from both images.

I_{\mathrm{true}} = \frac{ (I_{\mathrm{sample}} - I_{\mathrm{readout}})/t_{\mathrm{sample}} }{ (I_{\mathrm{background}} - I_{\mathrm{readout}})/t_{\mathrm{background}} }
  • Flux: In fact, the exposure time is not the metric that matters: the total photon flux (over the course of the exposure) is what matters. I.e.: since a real-world x-ray beam does not have perfectly stable flux, it is better to normalize by the total photon flux during an exposure, rather than the total measurement time. This can be done if the beamline/instrument has a direct-beam monitor. (On some instruments, this is a non-blocking detector upstream of the sample; on others, the beamstop itself may be a photo-diode.)

I_{\mathrm{true}} = \frac{ (I_{\mathrm{sample}} - I_{\mathrm{readout}})/\int \mathrm{flux}_{\mathrm{sample}} \mathrm{d}t }{ (I_{\mathrm{background}} - I_{\mathrm{readout}})/\int \mathrm{flux}_{\mathrm{background}} \mathrm{d}t }
  • Shot noise: As is true for any scattering measurement, the measurement of the background will have noise arising from finite counting statistics. This complicates background subtraction, since the noise is itself noisy. This can be minimized by using a very long exposure time for the background measurement (possibly split into multiple frames to avoid detector saturation). Doing so will mean the background has good signal-to-noise. However, one must then be careful in rescaling the background to compare it to the measurement of interest.

Local background

Although a full (2D image) background subtraction works quite well for transmission-SAXS, it in general does not work for GISAXS or GIWAXS. This is because it is not possible to measure the 'empty cell' in a meaningful way. One might be tempted to do a GISAXS measurement on the bare substrate, and subtract this from the signal coming from the thin film. However, this will not work for a variety of reasons:

  1. The size of the bare substrate and the sample of interest are unlikely to be exactly matched (hence the total scattering will not be identical).
  2. The scattering from the substrate is modified by the presence of a sample layer on top of it: the reflection geometry modifies the intensity as well as the spatial distribution of scattering (e.g. refraction distortion). E.g. consider an extreme case where one is measuring below the critical angle of the sample film: the scattering of the substrate will be essentially absent.
  3. The sample film may also attenuate the substrate scattering due to absorption (the grazing-incidence geometry means that substrate scattering must travel a long path through the film; i.e. even relatively weak absorption will measurably affect the signal).
  4. The distinct dynamical scattering features of GISAXS (Yoneda streak, specular rod, reflectivity oscillations, etc.) are all influenced by the complete multi-layer stack (by the film/substrate density profile in the normal direction). Since these features are different in the background and sample measurements, a direct subtraction is not meaningful.
  5. The low-q diffuse scattering is influenced by the roughness of interfaces (and scaled by the electron-density contrast across said interfaces). This is another example wherein the scattering of the substrate will be strongly modified by the presence of the sample film on top.

Thus, although one can subtract the detector and direct-beam backgrounds, one cannot hope to subtract the 'empty cell' (substrate) background; this latter background is likely to be dominant. An alternative strategy is to instead subtract a 'local background' when extracting a linecut. For instance, if assessing a peak position/width, one can fit the local data to a 'Gaussian + linear baseline' (or 'Gaussian + power-law baseline'), where the baseline is an (ad-hoc) accounting of the background. This inherently includes all the sources of background noted above.

In the case of an arc linecut (intensity along an arc at a constant q), one can assess the 'local background' by taking a similar integration just outside the peak region, and subtracting the data from each corresponding angle. Alternatively, one can take q-linecuts at each angle along the arc, and subtract an ad-hoc background as noted above.

Subtracting a local background inherently includes all sources of background scattering noted above. This works well for well-defined structural peaks, where the contributions from the peak and the baseline can be easily identified. For diffuse scattering or even broad halos, this may not be possible.

See Also