## teaching and education

## Why

analysis works: a one-dimensional crystallography teaching tool^{a}Department of Chemistry, University of Toledo, Toledo, OH 43606, USA^{*}Correspondence e-mail: a.pinkerton@utoledo.edu

A teaching tool is proposed to help beginner students of crystallography understand how crystallographic calculations work. Examples of the most important methods taught in X-ray crystallography courses have been adapted to a one-dimensional hypothetical structure. All calculations can be carried out in class with a scientific calculator or by using a simple spreadsheet.

### 1. Introduction

One of the challenges in teaching crystallography is to provide a digestible explanation of what goes on inside the very powerful black boxes that we all currently use for structure solution and showed how the principles of this daunting mathematical landscape could be clarified by examining the calculation of structure factors and Fourier series for a hypothetical one-dimensional one-atom problem. We have taken this approach and extended it to the and least-squares By simplifying the three-dimensional expressions to analyze a one-dimensional one-atom problem, it is possible to carry out all relevant calculations with a pocket calculator or simple spreadsheet in class. In common with Stout & Jensen, the chosen one-dimensional structure has a unit-cell length of 10 Å, has an inversion center and contains two carbon atoms at *x* = ±0.1833, as shown below. The hypothetical experiment was carried out with Cu *K*α radiation, λ = 1.5418 Å.

As this is proposed as a teaching aid, minimal information is provided to introduce each section, but the transformation of three-dimensional (3D) to one-dimensional (1D) equations is given, followed by numerical or partial numerical examples. For class use, it is simple to remove calculated numbers from the tables to allow hand calculations as exercises or as homework. Details of the underlying theory may be found in a number of standard books or from web-based materials (see http://iycr2014.org/learn/educational-materials).

### 2. Bragg's law

The following table of `experimental' observations may be generated for the hypothetical structure, along with the corresponding

below.### 3. Unit-cell refinement

Given the observed Bragg angles for the series of indexed reflections shown above, we can obtain the best values for the unit-cell dimensions using a linear least-squares approach. The following treatment is numerically equivalent to minimizing .

For *n* observations, we can write

We may thus solve for **v** in the following way:

Using the values of and *h* given above we obtain the results tabulated below:

### 4. Patterson function

Taking the square of the *x*, −*x*, the Patterson vectors will be at 2*x*, −2*x*. From the observed maximum at 0.36, we obtain *x* = 0.18 for one carbon atom.

### 5. calculation

If the structure is known, then we can calculate structure factors from the atomic positions and the scattering power of the atoms. The following example uses the result of the *x*(C1) = 0.18] to calculate the corresponding structure factors and compare them with their observed values. The comparison of the two sets of values provides the traditional crystallographic *R* factor.

### 6. Fourier summation

Using appropriate phases computed as above coupled with the observed *x* = 0.18.

### 7. Direct methods

*E*), which are calculated from *F* for stationary point atoms. The following sections define the appropriate *E* values, show how the origin is defined, employ the probabilities obtained from Σ_{1} and Σ_{2} relationships to obtain probable phases for a few reflections, use symbolic addition to extend these phases, and then use this subset of phased *E* values to compute a Fourier map.

#### 7.1. *E* values

The following expressions remove the dependence of structure factors on and normalize with respect to the total number of electrons in the structure.

#### 7.2. Origin definition

As shown in the following table, the phases of reflections with all even indices are invariant with respect to the choice of origin, whereas all other parities may change phase in a predictable manner. Thus a suitable choice of a few phases may be used to define the origin of the structure.

#### 7.3. Probabilities from Σ_{1} and Σ_{2} relationships

In the following, *H* and *K* represent different values of *h* forming triples, whereas in the three-dimensional case the triples would contain *h*, *k*, *l*. The values of *P _{+}* indicate the probability that the product of the phases is positive. For Σ

_{1}(involving only pairs of reflections) the most probable result is highlighted in bold. For Σ

_{2}, only a selection of triples has been calculated. Again the most probable results have been highlighted in bold, plus two more of more modest probability in bold italics.

#### 7.4. Structure solution

Define origin – choose *h* odd with high *E*.

*h* = 5, *E* = 1.223, phase +.

Insert known phases for the high-probability triples such that the product is positive, and add symbols for unknowns:

Thus *A* = −, or else all would be + (the `uranium' disaster).

Calculate a Fourier map using only the phased *E* values –

Thus the maximum in the density is again at *x* = 0.18.

### 8. Structure refinement

When atomic positions have been determined by any of the methods described above, it is common practice to refine this structure model using nonlinear least-squares methods. Shifts in atomic coordinates are thus calculated to iteratively obtain the best fit between the model structure and the experimental data. The following equations are mathematically equivalent to minimizing . All *F*s in the following expressions represent amplitudes.

For the general case

For one parameter in one dimension

We may solve for δ*x* as follows: , thus .

### References

Stout, G. H. & Jensen, L. H. (1968). *X-ray Structure Determination, A Practical Guide*. New York: Macmillan. Google Scholar

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