The accuracy of atomic co-ordinates derived from Fourier series in X-ray structure analysis

Abstract

A brief introduction to the subject of X-ray structure analysis is followed by a discussion of various conjectures regarding the accuracy of derived atomic co-ordinates and the importance of the latter in the derivation of molecular theory. Representing the synthesis in the form <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>D</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace" /> <mml:mi>y</mml:mi> <mml:mo>,</mml:mo> <mml:mi>z</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mstyle> <mml:mrow> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mi>V</mml:mi> </mml:mfrac> </mml:mrow> <mml:munderover> <mml:mrow> <mml:mo>∑</mml:mo> <mml:mo>∑</mml:mo> <mml:mo>∑</mml:mo> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>H</mml:mi> <mml:mi>K</mml:mi> <mml:mi>L</mml:mi> </mml:mrow> <mml:mrow> <mml:mo>+</mml:mo> <mml:mi>H</mml:mi> <mml:mi>K</mml:mi> <mml:mi>L</mml:mi> </mml:mrow> </mml:munderover> <mml:mo>⁡</mml:mo> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>F</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>h</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace" /> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace" /> <mml:mi>l</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mo stretchy="false">|</mml:mo> </mml:mrow> <mml:mi>cos</mml:mi> <mml:mfenced close="]" open="["> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>π</mml:mi> <mml:mfenced close=")" open="("> <mml:mrow> <mml:mi>h</mml:mi> <mml:mstyle> <mml:mrow> <mml:mfrac> <mml:mi>x</mml:mi> <mml:mi>a</mml:mi> </mml:mfrac> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>k</mml:mi> <mml:mstyle> <mml:mrow> <mml:mfrac> <mml:mi>y</mml:mi> <mml:mi>b</mml:mi> </mml:mfrac> </mml:mrow> <mml:mo>+</mml:mo> <mml:mi>l</mml:mi> <mml:mstyle> <mml:mrow> <mml:mfrac> <mml:mi>z</mml:mi> <mml:mi>c</mml:mi> </mml:mfrac> </mml:mrow> </mml:mstyle> </mml:mstyle> </mml:mstyle> </mml:mrow> </mml:mfenced> <mml:mo>−</mml:mo> <mml:mi>x</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>h</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace" /> <mml:mi>k</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace" /> <mml:mi>l</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mfenced> <mml:mo>,</mml:mo> </mml:mstyle> </mml:math> , (1) and neglecting errors of computation, two sources of inaccuracy are shown to occur: ( a ) experimental errors in the | F | values; ( b ) errors due to ( H, K, L ) being finite. Information as to the magnitude of the errors in the | F | values has recently become available, and a comparison of two sets of experimentally determined | F | values shows that the errors are independent of the magnitude of the structure factors and have a most probable value <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="normal">Δ</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>e</mml:mi> </mml:mrow> <mml:mspace width="thinmathspace" /> <mml:mrow> <mml:mo>=</mml:mo> </mml:mrow> <mml:mspace width="thinmathspace" /> <mml:mo>±</mml:mo> <mml:mrow> <mml:mn>0</mml:mn> </mml:mrow> <mml:mrow> <mml:mn>.6</mml:mn> </mml:mrow> </mml:math> . (2) Examination of the shape of the atomic peaks derived from a number of Fourier syntheses shows that the radial density distribution can be closely represented by the function <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>d</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>r</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mi>A</mml:mi> <mml:msup> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>−</mml:mo> <mml:mi>p</mml:mi> <mml:msup> <mml:mi>r</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:msup> </mml:math> , (3) where A depends on the atomic number of the particular atom and p appears to be fairly constant over a number of atoms from carbon to sulphur, a mean value being p =4.69. (4) A combined analytical-statistical analysis leads to the relation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>ε</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mn>90.8</mml:mn> <mml:mi mathvariant="normal">Δ</mml:mi> <mml:mi>e</mml:mi> <mml:mrow> <mml:mo>/</mml:mo> </mml:mrow> <mml:mi>N</mml:mi> <mml:mspace width="thinmathspace" /> <mml:msqrt> <mml:mi>V</mml:mi> </mml:msqrt> <mml:mo stretchy="false">(</mml:mo> <mml:mi>λ</mml:mi> <mml:mi>p</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow> <mml:mstyle displaystyle="true"> <mml:mrow> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mrow> </mml:mstyle> </mml:mrow> </mml:msup> </mml:math> , (5) where ε is the most probable error in the co-ordinate, N is the atomic number of the particular atom, V is the volume of the unit cell in A 3 , and λ is the wave-length corresponding to the smallest spacings observed. Taking the values of Δe and p given in (2) and (4) and considering a carbon atom in a unit cell of volume V = 583A 3 , (6) equation (5) leads, when all the information obtainable with copper K α radiation is used, to the value, ε &lt; 0.0027A. A formula is also given for the case in which errors are proportional to the order of their parent reflexions. The problem of finite limits of summation is dealt with in Part 2. For a simple system containing only two carbon atoms the errors, calculated as upper limits, are: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mtable columnspacing="1em" rowspacing="4pt"> <mml:mtr> <mml:mtd> <mml:mi>ρ</mml:mi> </mml:mtd> <mml:mtd>

Keywords

Affiliated Institutions

• Birkbeck, University of London GB

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