Calculation of the optimal specimen thickness for small-angle X-ray experiments with the use of polychromatic synchrotron radiation

Nuclear Instruments

and Methods

in Physics

Research

A 405

( 1998) 476-479

NUCLEAR INSTRUMENTS & METHODS IN PHVSbCS =zrY

ELSEVIER

Calculation of the optimal specimen thickness for small-angle X-ray experiments with the use of polychromatic synchrotron radiation A.M. Matjushin Institute of Theoretical and Experimental Biophysics. Russian Academy of Science, 142292. Pushchino. Moscow Region, Russian Federation

Abstract An evaluation method of the optimal specimen thickness for small-angle X-ray experiments using polychromatic synchrotron radiation (SR) is put forward. Curves of the dependence of the optimal specimen thickness on various SR-spectra with a one-dimensional detector and Si(Li) SSD are calculated. The curves allow one to optimize an experiment by selecting either the optimal specimen thickness for an available SR-spectrum or the short-wave spectrum limit for a given specimen thickness (fibers, tissues, etc.). Besides increasing both precision and reproducibility of the X-ray experiments, the method gives the possibility either to increase the statistical precision within a given recording time of an X-ray pattern and/or to decrease the recording time with retention of the same statistical precision. It is shown that a deviation of 15-20% in the specimen thickness from the optimal thickness leads to a decrease in the scattering intensity of less than 2%.

Synchrotron radiation (SR) emitted in accelerators and especially in storage rings has become a common research tool in the field of X-ray structural analysis. It is widely used in small-angle investigations of polymers and biopolymers where the radiation intensity problem is crucial. Monochromatized SR with Ah = 10~3-10~4 b; is used for this purpose as a rule now. The ratio of the number of photons in polychromatic and monochromatic beams ranges from 10’ up to 10’. Practically, this ratio represents the intensity losses due to the use of the monochromator in the experiments. A number of papers [l-S] examine the prospects of using polychromatic radiation for the study of the structure parameters by means of X-ray small-angle pattern analysis. This approach promises an intensity gain of 103, which is badly needed in many kinds of biological object investigations. This work is devoted to the determination of the optimal thickness of the biological samples in small-angle experiments of transmission geometry using polychromatic SR. In experiments with monochromatic radiation the maximum scattered radiation intensity occurs at the optimum thickness 1,, = l/p. Fortunately, a deviation of the sample thickness from the optimal value (or a deviation of the sample’s linear attenuation coefficient y of the order of lo-20%) causes only a small change (of the order of l-4%) [6] in the scattered radiation intensity. In the case of polychromatic radiation the question of the scattering radiation intensity dependence on the sample thickness (especially in the vicinity of lo) arises. The efficiency and 0168.9002/98/$19.00 0 1998 Elsevier Science B.V. All rights reserved PII SO168-9002(96)01053-4

in some cases even the possibility of such an approach depends on the answer to this question. Indeed, in the case of a rapidly changing intensity the reproducibility of the results can be substantially impeded. Therefore, smallangle scattering experiments involving dilute solutions of biological macromolecules become very problematic. The point is that the difference between the scattering of the solution and of the solvent is measured in these experiments. When the difference is too small the requirements for reproduction are quite strict. Consequently, repeated measurements inside the same cell are often impossible for a number of reasons. In the case of experiments with diffracting objects, for example a muscle in isotonic contraction or in the rigorisation process, or polymer or biological swelling fibers, the sample thickness can vary in quite a wide range, and to compare the intensities correctly one must allow for a correction of these changes, the determination of the thickness change being impossible or very difficult. The number of quanta of all wavelengths I(1) (between A, and AZ, i.e., the spectrum limits), scattered by an object with thickness 1 in the 8,,i, - a,,, angular interval and recorded by the detector will be

l(1) = I

AZ Hln,x B,,, f(A) exp(-y(A)I)]F(&

IIAl

A)12d0 dA ,

where f( A), p(A), and F(0, A) represent the photon spectral distribution in the original SR beam taking into account the

detector quantum efficiency and the distortions due to the X-ray beamline ~ryllium windows, the X-ray chamber and the sample cell. the linear attenuation coefficient of the sampIe, and the scattering amplitude of the object, respectively. One might obtain the optimal thickness I,, by maximizing I(f) with respect to I. However, to do this one should know F(6! A), which is an unknown quantity that is to be found from the results of the experiment. Therefore, instead of the precise value 1(i), we take an approximate value J(I ), J(I) =

.I> .fcA) exp(-p(A)l)(

1 At

I - exp(-cr(A)I))

dh ,

(1)

based on the estimate WIL.(H.A)/‘dHa

1 -exp(-d&l),

where rr( A) is the linear coefficient of coherent scattering on the sample. This approximation seems to be valid enough because most of the biological objects scatter the radiation to small angles. However, this estimation is somewhat overrated because in real measurements Qmln> 0; the smaller the real minimum angle, the more precise the estimate. As most of the biological objects consist for 90% of water, it will be justified to use p(A) of water instead of that of the sample; besides they are quite close to each other. To evaluate the sample a(A) we used the data of the total element composition of the averaged protein 171. The spectral distribution of the quanta of the incident SR beam was calculated taking into account all kinds of distortions. The following formula takes into account the distortions due to X-ray absorption in the Be windows of the SR beamline and in the one-dimensional detector (the total thickness is 0.08 cm) and in the mylar windows of the X-ray chamber and of the sample cell (the total thickness is 0.012 cm) and due to the 30 cm air gaps, and the quantum efficiency of the 1 cm deep detector filled with Ar or Xe plus 10% of CO, under atmospheric pressure:

- 30pu,( A))( 1 - expf-0.9ti

A))),

wheref* is the spectral photon flux at a given wavelength emitted by an electron with energy E when the critical wavelength is A,; k(h). I*,( A), ~~(4 are the total attenuation linear coefficients for Be, mylar and air, respectively; 7(A) is the linear photoabsorption coefficient for Xe or At. For the calculation of the linear coefficients /.&A) and n(A) we used tabulated data [IO] for the elements from which the coefficients in air, mylar and the studied specimens were calculated. For the calculations we used well known formulas for compound substances and mixtures [l I]. The obtained numerical data were interpolated by the least-square method involving the orthogonal Chebyshev polynomials to minimize the error and to

smooth it un~formiy over the whole ~nte~(~lation interval. The inte~olation polynomials were formed for the total integration range for all the elements, except Xe, because in xenon the K- and the L-photoabso~tion edges fall within the integration limits. In this special case there are five polynomials intended for the corresponding energy intervals between the absorption edges. At a given wavelength A the SR spectral photon flux ,f*( A) in the relative wavelength interval AA//t for the storage-ring current I and the electron energy E is given by the following formula (see Ref. 191): ,f*( A) = 2.46 X 10”’ X I(A)E(GeV)v( A/A,)

&A/A,

where v( A/A,) is the universal spectral function that has been calculated using the expansion stated in Ref. 1121. The critical wavelength AC is the parameter determining the spectral distribution of the radiation emitted by a monoenergetic electron. It is expressed in terms of the main storage-ring operating mode characteristics as follows: AC =SS!JR(m)E

‘(GeV)=

186X’CkOe)E

‘CGeV),

where H and R are the local magnetic field and the trajectory curvature radius, at the radiation point. Now the determination of I,, can be reduced to the procedure of the J(I) maximization with respect to I. However, before the determination of I,, we should elucidate the influence of 1 on J(I) in the vicinity of I,,. Fig. I presents the J(/) curves calculated for the argon-filled linear position-sensitive detector and VEPP-4 storage ring (Budker INP, Novosibirsk) with the following operating parameters: E = 3.7 GeV is the electron energy, I\< = LOIOE is the critical wavelength. In addition. the calculations involved the values of the short-wavelength spectrum r range: 0.8. I .1, 7.0 and 2.6 A. It follows from these curves 0.6

2

I”““

““““““1

$05 $

0.4

03

0.2

0.1

0.0 00

1.0

20

30 I

Fig. 1. Dependence thickness.

of the specimen

scattering

40 (mm)

intensity

on its

IV. X-RAY DIFFRACTOMETRY

478

A.M. Matjushin

I Nucl. Instr.

and Meth.

that J(I) in the vicinity of 1,) is weakly sensitive to the variation of 1 (lo-20% deviation from 1,) and it causes a l-3% change in the scattering intensity. The situation is quite similar to the case of monochromatic ~diation, and there are no additional difficulties and restrictions when using polychromatic radiation. To achieve a sharp shortwavelength cut-off one can use a total external refIection mirror with a critical reflection angle cp,. The reflectivity of such mirrors is close to unity at wavelengths .ALr and vanishes at shorter wavelengths. The following expression [8] yields the dependence of pc on A,,:

in Phys. Res. A 40,~ (1998)

476-479

6

6

4

pc = 2.33 x IO-‘A&Z/A)“”

>

where p, Z, and A represent the density, the atomic number, and the atomic weight of the mirror material, respectively. Thus, one can vary the spectrum short-wavelength cut-off by choosing an appropriate reflection angle. Fig. 2 depicts the dependence of I, on the short-wavelength spectrum cutoff A, for three operating modes of the VEPP-4 storage ring (for AC= 2.010, 1.409 and 0.879E, curves 1.4, 2.5 and 3,6 respectively) and on filling the detector with Ar or Xe (curves 1, 2, 3 correspond to xenon, curves 5, 6. 7 pertain to argon). The plateau observed in the short wavelength region is due to both the low efficiency of the detector (especially for Ar) and the low intensity of the SR in this spectral region for A, = 2.010 and 1.409 A. For the A, 2 I .2 A short-wavelength cut-off, I,, is more sensitive to the detector filling than in the storage ring o~rating mode (i.e.. with the SR spectrum). The xenon K-edge is responsible for the slight inflections of curves 1 and 2 within the 0.3-0.5 A range. Fig. 3 shows a similar dependence of the SifLi) SSD with 5 and 3 mm thickness (curves 1-3 and 4-6).

5.0 ,

I

Fig. 2. The specimen optimai thickness dependence on the shortwavelength spectrnm limit for one-dimensional detectors.

2

0 0.0

10

2.0

30 h, (A)

Fig. 3. The specimen optimal thickness dependence on the shortwavelength spectrum limit for Si(Li) SSD.

The knowledge of the [,(A, ) dependence allows one either to set up an optimal sample thickness for the given SR spectrum, or to optimize the experiments with specimens of fixed thickness (e.g., fibers, tissues, organels, etc.). The latter includes the shop-wavelength spectrum cut-off matching the specimen thickness, making the fixed thickness optimal or at least quasi-optimal. This provides useful oppo~unities to improve the precision of the X-ray pattern; to enhance the reproducibility of the experiment; either to reduce the exposure time necessary for the given quality of the scattering record or to increase the data statistical accuracy at a fixed exposure time. There is a circumstance that deserves to be outlined. Although the progressive elimination of the SR shortwavelength component reduces the number of X-ray photons in the main beam, the signal to noise ratio increases, in spite of a seeming statistic deterioration, Due to this fact, with the increase of the wavelength coherent scattering increases and Compton (incoherent) scattering decreases. Therefore, the quality of X-ray pattern records is ‘improved. Fig. 4 shows the relative deviation of the sample thickness from its optimal value (1,, - 1)/1,, as a function of A,, which produced a decrease of the scattering intensity of 1, 2 and 5%. It can be seen that the admissible margins of the relative thickness deviation from the optimal value are quite wide. The thickness excess is somewhat more detrimental than the thickness deficit. It is advisable to choose the initial specimen thickness a Iittle different from the optimum: the difference should be compensated by the thickness change during the expe~ment. The method described above of the sample thickness optimization in small-angle scattering can be easily extend-

A.M.

Matjushin

I Nucl. Instr.

and Meth.

in Ph.vs. Rrs. A 40-T f/99&‘)

476-479

37”)

Acknowledgement This work was supported by the Russian Fundamental Research Foundation, grant 94-1)4- 13343.

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ed to the general case of any angle range. To do this, we should use. instead of formula (I), the following expression:

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cow, 19.57). in Russian.

J(I) =

A? “,“LJX ,J1A) exp(-fi( .I/AI “mm

- exp(-a(

h)llcos 28)( I

h)i)) d8 dA

Handbook of Spectroscopy. vol. I, ed. Robinson. Cleveland, Ohio

( 1974).

V.N. Korchuganov et al., Preprint 77-Y. Novosibirsk, INP USSR

Budker

( 1977).

In addition, exp(-0.9fi A)/cos 28) should be substituted for exp(--0.9?f A)) in the expression ,fr h) pe~aining to the detector efficiency.

IV. X-RAY

DIFFRACTOMETRY