Contrast variation

May, R.

doi:10.1107/97809553602060000581

International
Tables for
Crystallography
Volume C
Mathematical, physical and chemical tables
Edited by E. Prince

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International Tables for Crystallography (2006). Vol. C. ch. 2.6, p. 107

Section 2.6.2.2.1. Contrast variation

R. May^b

2.6.2.2.1. Contrast variation

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The easiest way of using the scattering-amplitude difference between ¹H and ²H is the so-called contrast variation. It was introduced into SANS by Ibel & Stuhrmann (1975) on the basis of X-ray crystallographic (Bragg & Perutz, 1952), SAXS (Stuhrmann & Kirste, 1965), and light-scattering (Benoit & Wippler, 1960) work. Most frequently, contrast variation is carried out with mixtures of light (¹H₂O) and heavy water (²H₂O), but also with other solvents available in protonated and deuterated form (ethanol, cyclohexane, etc.). The scattering-length density of H₂O varies between −0.562 × 10¹⁰ cm⁻² for normal water, which is nearly pure ¹H₂O, and 6.404 × 10¹⁰ cm⁻² for pure heavy water.

The scattering-length densities of other molecules, in general, are different from each other and from pure protonated and deuterated solvents and can be matched by ¹H/²H mixture ratios characteristic for their chemical compositions. This mixture ratio (or the corresponding absolute scattering-length density) is called the scattering-length-density match point , or, semantically incorrect, contrast match point . If a molecule contains non-covalently bound hydrogens, they can be exchanged for solvent hydrogens. This exchange is proportional to the ratio of all labile ¹H and ²H present; in dilute aqueous solutions, it is dominated by the solvent hydrogens. A plot of the scattering-length density versus the ²H/(²H + ¹H) ratio in the solvent shows a linear increase if there is exchange; the value of the match point also depends on solvent exchange. The fact that many particles have high contrast with respect to ²H₂O makes neutrons superior to X-rays for studying small particles at low concentrations.

The scattered neutron intensity from N identical particles without long-range interactions in a (very) dilute solution with solvent scattering density $[\rho _s]$ can be written as $[I(Q)=[{{\rm d}}\sigma(Q)/{{\rm d}} \Omega]NTAI_0\Delta\Omega, \eqno (2.6.2.1)]$ with the scattering cross section per particle and unit solid angle $[{{\rm d}}\sigma (Q)/{{\rm d}} \Omega=\left\langle\left\vert\textstyle\int[\rho({\bf r})-{\rho}_s]\exp (i{\bf Q\cdot r}){\,{\rm d}}{\bf r}\right\vert^2\right\rangle. \eqno(2.6.2.1a)]$

The angle brackets indicate averaging over all particle orientations. With $[\rho(r)=\sum b_i/V_p]$ and $[I(0)={\rm constant}\times \langle\vert\int[{\rho}({\bf r}) -{\rho}_s]{\,{\rm d}}{\bf r}\vert {^2}\rangle]$ , we find that the scattering intensity at zero angle is proportional to $[\Delta\rho =\textstyle\sum b_i/V_p -\rho_s, \eqno (2.6.2.2)]$ which is called the contrast. The exact meaning of [V_p] is discussed, for example, by Zaccai & Jacrot (1983), and for X-rays by Luzzati, Tardieu, Mateu & Stuhrmann (1976).

The scattering-length density ρ(r) can be written as a sum $[{\rho}({\bf r})={\rho}_0+{\rho}_F({\bf r}),\eqno (2.6.2.3)]$ where $[\rho _0]$ is the average scattering-length density of the particle at zero contrast, Δρ = 0, and $[{\rho}_F({\bf r})]$ describes the fluctuations about this mean. I(Q) can then be written $[I(Q)=(\rho _0 - \rho _2)^2I_c(Q)+(\rho _0-\rho _s)I_{cs}(Q)+I_s(Q). \eqno (2.6.2.4)]$

[I_s] is the scattering intensity due to the fluctuations at zero contrast. The cross term $[I_{cs}(Q)]$ also has to take account of solvent-exchange phenomena in the widest sense (including solvent water molecules bound to the particle surface, which can have a density different from that of bulk water). This extension is mathematically correct, since one can assume that solvent exchange is proportional to Δρ. The term [I_c] is due to the invariant volume inside which the scattering density is independent of the solvent (Luzzati, Tardieu, Mateu & Stuhrmann, 1976). This is usually not the scattering of a homogeneous particle at infinite contrast, if the exchange is not uniform over the whole particle volume, as is often the case, or if the particle can be imaged as a sponge (see Witz, 1983).

The method is still very valuable, since it allows calculation of the scattering at any given contrast on the basis of at least three measurements at well chosen ¹H/²H ratios (including data near, but preferentially not exactly at, the lowest contrasts). It is sometimes limited by ²H-dependent aggregation effects.

References

Benoit, H. & Wippler, C. (1960). Répartition angulaire de la lumière diffusée par une solution de copolymères. J. Chim. Phys. 57, 524–527.Google Scholar

Bragg, W. L. & Perutz, M. F. (1952). The external form of the haemoglobin molecule. I. Acta Cryst. 5, 277–283.Google Scholar

Ibel, K. & Stuhrmann, H. B. (1975). Comparison of neutron and X-ray scattering of dilute myoglobin solutions. J. Mol. Biol. 93, 255–265.Google Scholar

Luzzati, V., Tardieu, A., Mateu, L. & Stuhrmann, H. B. (1976). Structure of human serum lipoprotein in solution. I. Theory and techniques of an X-ray scattering approach using solvents of variable density. J. Mol. Biol. 101, 115–127.Google Scholar

Stuhrmann, H. B. & Kirste, R. G. (1965). Elimination der intrapartikulären Untergrundstreuung bei der Röntgenkleinwinkelstreuung an kompakten Teilchen (Proteinen). Z. Phys. Chem. Neue Folge, 46, 247–250.Google Scholar

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International Tables for Crystallography (2006). Vol. C. ch. 2.6, p. 107