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

© International Union of Crystallography 2006
International Tables for Crystallography (2006). Vol. C. ch. 2.6, pp. 91-93

Section 2.6.1.3.1. Parameters of a particle

O. Glattera

2.6.1.3.1. Parameters of a particle

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Total scattering length. The scattering intensity at h = 0 must be equal to the square of the number of excess electrons, as follows from equations (2.6.1.7)[link] and (2.6.1.12)[link]: [I(0)=(\Delta\rho)^2V^2=4\pi\textstyle\int\limits^\infty _0 p(r){\,{\rm d}}r.\eqno (2.6.1.15)]This value is important for the determination of the molecular weight if we perform our experiments on an absolute scale (see below).

Radius of gyration. The electronic radius of gyration of the whole particle is defined in analogy to the radius of gyration in mechanics: [R^2_g={\int\limits_V \rho(r_i)r^2_i {\,{\rm d}}V_i\over \int\limits_V \rho(r_i){\,{\rm d}}V_i}.\eqno (2.6.1.16)]It can be obtained from the PDDF by [R^2_g={\int\limits ^\infty _0 p(r)r^2{\,{\rm d}}r \over 2\int\limits^\infty _0p(r){\,{\rm d}}r}\eqno (2.6.1.17)]or from the innermost part of the scattering curve [Guinier approximation (Guinier, 1939[link])]: [I(h)=I(0)\exp (-h^2R^2_g/3).\eqno (2.6.1.18)]A plot of log[I(h)] vs h2 (Guinier plot) shows at its innermost part a linear descent with a slope tan α, where [R_g=K \surd \overline{\tan \alpha}](see Table 2.6.1.1[link]).

Table 2.6.1.1| top | pdf |
Formulae for the various parameters for h (left) and m (right) scales

[\eqalign{R&=K\surd\tan \alpha\cr \tan \alpha &={\textstyle \Delta \log I(h) \over\textstyle\Delta h^2}\cr}][\eqalign{R&=K{\textstyle\lambda a\over \textstyle2\pi}\surd\tan \alpha\cr \tan \alpha&={\textstyle\Delta \log I(m) \over \textstyle\Delta m^2}}]
[K=\surd {{3\over \log {\rm e}}} = 2.628]
[\eqalign{R_c&=K_c\surd\tan \alpha\cr \tan \alpha &={\textstyle \Delta \log [I(h)h] \over\textstyle\Delta h^2}\cr}][\eqalign{R_c&=K_c{\textstyle\lambda a\over \textstyle2\pi}\surd\tan \alpha\cr \tan \alpha&={\textstyle\Delta \log [I(m)m] \over \textstyle\Delta m^2}}]
[K_c=\surd {\textstyle 2\over \log {\rm e}}=2.146]
[\eqalign{R_t&=K_t\surd\tan \alpha\cr \tan \alpha &={\textstyle \Delta \log [I(h)h^2] \over\textstyle\Delta h^2}\cr}][\eqalign{R_t&=K_t{\textstyle\lambda a\over \textstyle2\pi}\surd\tan \alpha\cr \tan \alpha&={\textstyle\Delta \log [I(m)m^2] \over \textstyle\Delta m^2}}]
[K_t=\surd {\textstyle 1\over \log {\rm e}}=1.517]
[\eqalign{V&=2\pi^2{I(0)\over Q}\cr Q&=\int I(h)h^2\,{\rm d}h\cr}][\eqalign{V&={\textstyle \lambda^3 a^3\over 4\pi}\ {\textstyle I(0)\over Q_m}\cr Q_m&=\int I(m)m^2{\,\rm d}m}]
[A=2\pi{\textstyle [I(h)h]_0\over Q}][A={\textstyle \lambda^2a^2\over 2\pi}\ {\textstyle [I(m)m]_0\over Q_m}]
[T=\pi{\textstyle [I(h)h^2]_0\over Q}][T={\textstyle \lambda a\over 2}\ {\textstyle [I(m)m^2]_0\over Q_m}]
[M={\textstyle I(0)\over P}K{\textstyle a^2\over cd(\Delta z)^2}][K={\textstyle 1\over I_eN_L}=21.0]
[M_c={\textstyle [I(h)h]_0 \over P}\ {\textstyle K\over \pi}\ {\textstyle a^2 \over cd(\Delta z)^2}][M_c={\textstyle [I(m)m]_0\over P}\ {\textstyle 2K\over \lambda}\ {\textstyle a\over cd(\Delta z)^2}]
[M_t={\textstyle [I(h)h^2]_0 \over P}\ {\textstyle K\over 2\pi}\ {\textstyle a^2 \over cd(\Delta z)^2}][M_t={\textstyle [I(m)m^2]_0\over P}\ {\textstyle 2\pi K\over \lambda^2}\ {\textstyle 1\over cd(\Delta z)^2}]
[\overline {(\Delta\rho)^2}={\textstyle Q\over P}\ {\textstyle a^2\over 2\pi^2d}K][ \overline {(\Delta\rho)^2}={\textstyle Q_m\over P}\ {\textstyle 4\pi\over \lambda^3ad}K]
[\eqalign{K&=10^{24}/I_e\cr (10^{24}&=[{\rm cm}/{\rm \AA}]^3)}]
[\eqalign{O_s&=\pi {\textstyle K\over Q}\cr K&= \lim\limits_{h \rightarrow \infty} I(h)h^4\cr}][\eqalign{O_s&={\textstyle 2\pi ^2\over \lambda a}\ {\textstyle K_m\over Q_m}\cr K&= \lim\limits_{m \rightarrow \infty} I(m)m^4}]

The radius of gyration is related to the geometrical parameters of simple homogeneous triaxial bodies as follows (Mittelbach, 1964[link]):[\halign{#\hfil&\quad #\hfil\cr\hbox{{\rm sphere\ (radius}\ $R$) }& $R^2_g=(3/5)R^2$\cr \hbox{{\rm hollow \ sphere\ (radii}\ $R_1$}& $R^2_g=(3/5) {{\textstyle R^5_2-R^5_1}\over\textstyle{R^3_2-R^3_1}}$\cr\raise4ex\hbox{{\rm \quad and} \ $R_2$)}\cr \hbox{{\rm ellipsoid\ (semi}\hbox{-}{\rm axes}\ $a, b, c$)}& $R^2_g=(1/5)(a^2+b^2+c^2)$\cr\cr{\rm parallelepiped\ (edge\ lengths}&$R^2_g=(1/12)(A^2+B^2+C^2)$\cr \quad\hbox{$ A, B, C$) }\cr\cr {\rm elliptic\ cylinder}& $R^2_g={{\textstyle a^2+b^2}\over {\textstyle 4}}+{{\textstyle h^2}\over {\textstyle 12}}=R^2_c+{{\textstyle h^2}\over{\textstyle 12}}$\cr \hbox{{\rm \quad(semi}\hbox{-}{\rm axes}\ $a$,\ $b$\semi \ {\rm height} \ $h$)}&\cr\cr{\rm hollow\ cylinder}&$R^2_g={{\textstyle r^2_1+r^2_2}\over {\textstyle 2}}+{{\textstyle h^2}\over{\textstyle 12}}$.\cr \hbox{{\rm \quad(height}\ $h$\ {\rm and\ radii}\ $r_1, r_2$)}&\cr}]

Radius of gyration of the cross section. In the special case of rod-like particles, the two-dimensional analogue of [R_g] is called radius of gyration of the cross section [R_c]. It can be obtained from [R^2_c={\textstyle \int\limits^\infty_0 p_c(r)r^2{\,{\rm d}}r \over 2\int\limits ^\infty _0 p_c(r){\,{\rm d}}r},\eqno (2.6.1.19)]where [p_c(r)] is the PDDF of the cross section or it can be calculated from the innermost part of the scattering intensity of the cross section [I_c(h)]: [I_c(h)=I_c(0)\exp (-h^2R^2_c/2),\eqno (2.6.1.20)]with [I_c(h)=I(h)h] (see Table 2.6.1.1[link]).

Radius of gyration of the thickness. A similar definition exists for lamellar particles. The one-dimensional radius of gyration of the thickness [R_t] can be calculated from [R^2_t={\textstyle \int\limits^\infty _0 p_t(r)r^2{\,{\rm d}}r \over 2\int\limits^\infty _0p_t(r){\,{\rm d}}r},\eqno (2.6.1.21)]or from the innermost part of the scattered intensity of thickness [I_t(h)]: [I_t(h)=I_t(0)\exp (-h^2R^2_t),\eqno (2.6.1.22)]with [I_t(h)=I(h)h^2] (see Table 2.6.1.1[link] and §2.6.1.3.2.1[link]).

Volume. The volume of a homogeneous particle is given by [V=2\pi ^2{\textstyle I(0)\over Q}.\eqno (2.6.1.23)]This equation follows from equations (2.6.1.12)[link][link]–(2.6.1.14)[link]. Such volume determinations are subject to errors as they rely on the validity of an extrapolation to zero angle [to obtain [I(0)]] and to larger angles ([h^{-4}] extrapolation for Q). Scattering functions cannot be measured from h equal to zero to h equal to infinity.

Surface. The surface S of one particle is correlated with the scattering intensity [I_1(h)] of this particle by [I_1(h)\vert _{h\to\infty}=(\Delta\rho)^2{\textstyle 2\pi\over h^4}S.\eqno (2.6.1.24)]Determination of the absolute intensity can be avoided if we calculate the specific surface [O_s] (Mittelbach & Porod, 1965[link])[O_s=S/V=\pi\, {\displaystyle \lim_{h\to \infty}[I(h)h^4]\over Q}.\eqno (2.6.1.25)]

Cross section, thickness, and correlation length . By similar equations, we can find the area A of the cross section of a rod-like particle [A=2\pi {\textstyle [I(h)h]_{h\to 0} \over Q}\eqno (2.6.1.26)]and the thickness T of lamellar particles by [T=\pi {\textstyle [I(h)h^2]_{h\to 0}\over Q}\eqno (2.6.1.27)]but the experimental accuracy of the limiting values [[I(h)h]_{h\rightarrow 0}] and [[I(h)h^2]_{h\rightarrow 0}] is usually not very high.

The correlation length [l_c] is the mean width of the correlation function γ(r) (Porod, 1982[link]) and is given by [l_c={\textstyle \pi\over Q} \int\limits^\infty _0 I(h)h{\,{\rm d}}h. \eqno (2.6.1.28)]

The maximum dimension D of a particle would be another important particle parameter, but it cannot be calculated directly from the scattering function and will be discussed later.

Persistence length [a_p]. An important model for polymers in solution is the so-called worm-like chain (Porod, 1949[link]; Kratky & Porod, 1949[link]). The degree of coiling can be characterized by the persistence length [a_p] (Kratky, 1982b[link]). Under the assumption that the persistence length is much larger than the cross section of the polymer, it is possible to find a transition point h+ in an I(h)h2 vs h plot where the function starts to be proportional to h. There is an approximation [h^+a_p\simeq2.3,\eqno (2.6.1.29)]depending on the length of the chain (Heine, Kratky & Roppert, 1962[link]). For further details, see Kratky (1982b[link]).

Molecular weight. Particles of arbitrary shape. The particle is measured at high dilution in a homogeneous solution and has an isopotential specific volume [v'_2] and [z_2] mol. electrons per gram, i.e. the molecule contains [z_2M] electrons if M is the molecular weight. The number of effective mol. electrons per gram is given by [\Delta z_2=(z_2-v'_2\rho_0),\eqno (2.6.1.30)]where [\rho _0] is the mean electron density of the solvent. The molecular weight can be determined from the intensity at zero angle I(0):[\eqalignno{M&={\textstyle I(0)\over P}\ {\textstyle a^2\over \Delta z^2dcI_eN_L}\cr &={I(0)\over P}\ {21.0a^2\over \Delta z^2dc} &(2.6.1.31)}](Kratky, Porod & Kahovec, 1951[link]), where P is the total intensity per unit time irradiating the sample, a [cm] is the distance between the sample and the plane of registration, d [cm] is the thickness of the sample, c [g cm−3] is the concentration, and NL is Loschmidt's (Avogadro's) number.

Rod-like particles. The mass per unit length Mc = M/L, i.e. the mass related to the cross section of a rod-like particle with length L, is given by a similar equation (Kratky & Porod, 1953[link]): [\eqalignno{M_c&={\textstyle [I(h)h]_{h\to 0} \over P}\ {a^2 \over \pi \Delta z^2dcI_eN_L}\cr &= {[I(h)h]_{h\to 0} \over P}\ { 6.68a^2 \over \Delta z^2dc}. &(2.6.1.32)}]

Flat particles . A similar equation holds for the mass per unit area Mt = M/A: [\eqalignno{M_t&={[I(h)h^2]_{h\to 0} \over P}\ {a^2 \over 2\pi \Delta z^2dcI_eN_L}\cr &= {[I(h)h^2]_{h\to 0} \over P}\ {3.34a^2 \over \Delta z^2dc}.& (2.6.1.33)}]

Abscissa scaling . The various molecular parameters can be evaluated from scattered intensities with different abscissa scaling. The abscissa used in theoretical work is [h=(4\pi/\lambda)\sin\theta]. The most important experimental scale is m [cm], the distance of the detector from the centre of the primary beam with the distance a [cm] between the sample and the detector plane. [h[{\rm nm}^{-1}]=T_{hm}[{\rm cm}^{-1}{\rm nm}^{-1}]m[{\rm cm}],\eqno (2.6.1.34)]with[T_{hm}=2\pi /\lambda a.\eqno (2.6.1.35)]The angular scale 2θ with [2\theta\simeq m/a=(\lambda/2\pi)h\eqno (2.6.1.36)]was used in the early years of small-angle X-ray scattering experiments. The formulae for the various parameters for m and the h scale can be found in Table 2.6.1.1[link], the formulae for the 2θ scale can be found in Glatter & Kratky (1982[link], p. 158).

References

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