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Working with a single crystal, what is the first thing the diffractometer has access to? What is the output I see and what are the steps from this two levels of information?

If I'm correct, the output might be (hkl) indexes and their intensities, but I don't know how the instrument can assign the coordinates in reciprocal space to the spots.

Is necessary for it to know the orientation of the crystallographic axes respect to the goniometer?

How does it calculate it?

It's not tightly necessary a too much detailed answer, but I need an ensemble vision of the situation.

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    $\begingroup$ The manual, both operations and software, should cover this. $\endgroup$
    – Jon Custer
    Commented Aug 15 at 18:13
  • $\begingroup$ @JonCuster what makes you think that 1) OP is located near an instrument, and 2) that the manual is nearby? For a 30 year old instrument in a building full of people familiar with the technology, it could be that the manual is long-lost or in some random person's desk drawer who has since moved away. $\endgroup$
    – uhoh
    Commented Aug 18 at 5:23

2 Answers 2

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The crystal on the pin (or the loop) of the goniometer head, the diffraction experiment is the gradual movement around multiple axes (or circles) to access a reasonably large portion of the reciprocal space around the crystal. One of the more frequent setups is a 4-circle diffractometer

enter image description here

(credit UPenn)

With the crystal symbolized by the cube in the center you have circles $\phi$, $\omega$, $2\theta$ and $\chi$. It is highly important to keep track of these angles varied during the experiment -- just as well as the distance between the center of these circles and the detector (which is adjustable, but typically remains fix during a collection of data) while recording reflection intensity in $(x,y)$ coordinates of the detector's sensor.

For each position* an individual file (frame) is saved. For efficiency of file I/O and storage, most of it is in a binary format to be read by the specialized software you purchase with the diffractometer. However the header line needn't be. For an easier read, I wrapped it and added leading line numbers to get this (partly redacted) frame recorded on an old Bruker SMART 1K 3-circle diffractometer:

1   FORMAT :        86
2   VERSION:         9
3   HDRBLKS:        15
4   TYPE   :Scan frame
5   SITE   :UNIVERSITY OF
6   MODEL  :PLATFORM
7   USER   :UNIVERSITY OF
8   SAMPLE :sucrose
9   SETNAME:sucrose
10  RUN    :         2
11  SAMPNUM:         1
12  TITLE  :sucrose from
13  TITLE  :
14  TITLE  :
15  TITLE  :
16  TITLE  :
17  TITLE  :
18  TITLE  :
19  TITLE  :
20  NCOUNTS:     7267500           0
21  NOVERFL:        19
22  MINIMUM:         0
23  MAXIMUM:      1912
24  NONTIME: 221796409
25  NLATE  :         0
26  FILENAM:
27  CREATED:DD/MM/YY   HH:MM:SS
28  CUMULAT:    64.5835953
29  ELAPSDR:    30.0000000
30  ELAPSDA:    32.2956886
31  OSCILLA:         0
32  NSTEPS :         1
33  RANGE  :     0.3000000
34  START  :   -91.9000015
35  INCREME:    -0.3000183
36  NUMBER :       224
37  NFRAMES:       606
38  ANGLES :   332.0000000   268.1000061    90.0000000    54.7200012
39  NOVER64:         0
40  NPIXELB:         1
41  NROWS  :       512
42  NCOLS  :       512
43  WORDORD:         0
44  LONGORD:         0
45  TARGET :Mo
46  SOURCEK:    45.0000000
47  SOURCEM:    40.0000000
48  FILTER :Parallel,graphite
49  CELL   :    15.7668018    15.8232164     6.9688878    90.1392212    90.0406418
50  CELL   :   119.5063553
51  MATRIX :    -0.0514976     0.0165236    -0.0360825     0.0513622     0.0687769
52  MATRIX :    -0.0235442  0.457335E-02    -0.0164295    -0.1368749
53  LOWTEMP:           0           0       -5290
54  ZOOM   :     0.5000000     0.5000000     1.0000000
55  CENTER :   253.6660004   261.7380066
56  DISTANC:     5.4480000
57  TRAILER:        -1
58  COMPRES:NONE
59  LINEAR :     1.0000000     0.0000000
60  PHD    :     0.0000000     0.0000000
61  PREAMP :     4.0000000
62  CORRECT:K426L170
63  WARPFIL:K426L170._ix
64  WAVELEN:     0.7107300     0.0000000     0.0000000
65  MAXXY  :   331.0000000   392.0000000
66  AXIS   :         2
67  ENDING :   332.0000000   267.7999878    90.0000000    54.7200012
68  DETPAR :     0.0292703     0.0700109  0.896212E-02    -0.3303430     0.2778771
69  DETPAR :     0.1510672
70  LUT    :BB
71  DISPLIM:     0.0000000     0.0000000
72  PROGRAM:SMART V5.059
73  ROTATE :         0
74  BITMASK:$NULL
75  OCTMASK:          15          30          15         481         497         993
76  OCTMASK:         497         481
77  ESDCELL:     0.0443505     0.0362424     0.0149620     0.2186361     0.2676024
78  ESDCELL:     0.1994543
79  DETTYPE:CCD-PXL
80  NEXP   :           2         410          32           2           0
81  CCDPARM:    13.6999998     2.2000000    28.0000000   410.0000000 65535.0000000
82  CHEM   :C6 H12 O6
83  MORPH  :plate like
84  CCOLOR :yellow
85  CSIZE  :0.50 mm|0.20 mm|0.13 mm|not determined|room temperature
86  DNSMET :x
87  DARK   :dark060s._dk
88  AUTORNG:     4.0000000    13.6999998     2.2000000     0.0000000 65535.0000000
89  ZEROADJ:     0.0000000     0.0000000     0.0000000     0.0000000
90  XTRANS :     0.0000000     0.0000000     0.0000000
91  HKL&XY :     0.0000000     0.0000000     0.0000000     0.0000000     0.0000000
92  AXES2  :     0.0000000     0.0000000     0.0000000     0.0000000
93  ENDING2:     0.0000000     0.0000000     0.0000000     0.0000000

  • characteristics of the sample (e.g., 8-19, 82-85)

  • lattice constants and orientation matrix as assigned at this stage of the experiment (49-52)

  • distance center of the circles to detector (56, in centimeters, evidently to characterize a crystal of a small molecule compound), and current angle of the frame (38); the trailing entry about 54 degrees is characteristic for the fixed $\chi$ of this setup).

  • details about the radiation (45, 48, 64)

  • correction data and files about dead pixels and dark image (e.g., 63 or 87) for background correction

  • book keeping where the current frame (36) is within the current run (10) and the experiment as a whole set of frames recorded in total (37). If you look closely, one typical experiment initiates with is a quick (adjusting) large move of $\phi$, $\omega$, $2\theta$, $\chi$ (if available). This is followed by small slow moves of the camera ($2\theta$ angle), each of them to record a single frame. When meeting a geometrical constraint, these small moves stop (and close the run); the diffractometer moves to new «starting position» with the same initial $2\theta$ angle but very different $\phi$ angle to begin a new run.

    To illustrate this a little bit, I «greped» a couple of measurements of one analysis to yield a table about the file name of the individual frame recorded, the run and frame number, and the angles $\omega, 2\theta, \phi, \chi$.

    YLID1.001  RUN: 1 NUMBER:  1 ANGLES: 332.0000000 335.0000000   0.0000000 54.7200012
YLID1.002  RUN: 1 NUMBER:  2 ANGLES: 332.0000000 334.7000122   0.0000000 54.7200012
YLID1.003  RUN: 1 NUMBER:  3 ANGLES: 332.0000000 334.3999939   0.0000000 54.7200012
YLID1.004  RUN: 1 NUMBER:  4 ANGLES: 332.0000000 334.1000061   0.0000000 54.7200012
YLID1.005  RUN: 1 NUMBER:  5 ANGLES: 332.0000000 333.7999878   0.0000000 54.7200012
YLID1.006  RUN: 1 NUMBER:  6 ANGLES: 332.0000000 333.5000000   0.0000000 54.7200012
YLID1.007  RUN: 1 NUMBER:  7 ANGLES: 332.0000000 333.2000122   0.0000000 54.7200012
YLID1.008  RUN: 1 NUMBER:  8 ANGLES: 332.0000000 332.8999939   0.0000000 54.7200012
YLID1.009  RUN: 1 NUMBER:  9 ANGLES: 332.0000000 332.6000061   0.0000000 54.7200012
YLID1.010  RUN: 1 NUMBER: 10 ANGLES: 332.0000000 332.2999878   0.0000000 54.7200012
[...]
YLID2.001  RUN: 2 NUMBER:  1 ANGLES: 332.0000000 335.0000000  90.0000000 54.7200012
YLID2.002  RUN: 2 NUMBER:  2 ANGLES: 332.0000000 334.7000122  90.0000000 54.7200012
YLID2.003  RUN: 2 NUMBER:  3 ANGLES: 332.0000000 334.3999939  90.0000000 54.7200012
YLID2.004  RUN: 2 NUMBER:  4 ANGLES: 332.0000000 334.1000061  90.0000000 54.7200012
YLID2.005  RUN: 2 NUMBER:  5 ANGLES: 332.0000000 333.7999878  90.0000000 54.7200012
[...]
YLID3.001  RUN: 3 NUMBER:  1 ANGLES: 332.0000000 335.0000000 180.0000000 54.7200012
YLID3.002  RUN: 3 NUMBER:  2 ANGLES: 332.0000000 334.7000122 180.0000000 54.7200012
YLID3.003  RUN: 3 NUMBER:  3 ANGLES: 332.0000000 334.3999939 180.0000000 54.7200012
YLID3.004  RUN: 3 NUMBER:  4 ANGLES: 332.0000000 334.1000061 180.0000000 54.7200012
YLID3.005  RUN: 3 NUMBER:  5 ANGLES: 332.0000000 333.7999878 180.0000000 54.7200012
[...]

    How many individual frames per run, how many runs in total, their start and end parameter are needed to collect data good enough in quality and quantity depends on the sample, the (presumed) symmetry of the sample and its orientation in the coordinates of the diffractometer, if accurately recording many Friedel pairs for an absolute structure determination is requested. There are programs are around to assist the operator in the decision to define the experiment accordingly.

Depending on the sample's quality, internal symmetry and the crystals's relative orientation vs. the coordinate system of the diffractometer, modern diffractometers equipped with an area detector can suggest an (initial) set of lattice vectors and an orientation matrix after recording only a few dozen diffraction frames with a couple of well resolved diffraction peaks.** With contemporary diffractometers equipped with area detectors, it usually is not problematic to collect a couple of new/additional diffraction frames which then are processed altogether with the previous data to assign a new set of lattice vectors and orientation matrix.*** If this fails (e.g., suggesting a unit cell obviously too small to accommodate the molecules in question, of wrong handiness) one can provide a transformation matrix to get this one right, and which often simultaneously affects the orientation matrix.

Addition: In a comment, you ask why we need an orientation matrix at all, given Bragg's law of $n\lambda = 2d \sin\theta$. Its purpose is to translate from coordinate system of the reciprocal space $(a^*,b^*,c^*)$ into the cartesien $(x,y,z)$ of the diffractometer. The former may be oblique because $(a,b,c)$ in the rhombohedral, hexagonal, monoclinic, triclinic crystal system enclose at least one angle which is not constrained to $90^\circ$, while all planes of the later are orthogonal to each other. (The software equally takes care for the translation of diffraction intensities on the Ewald sphere projected onto the shape of the detector area; a bit related to mapping 3D earth on a 2D sheet of paper.)

For context, in good cases,

  • a couple dozen of frames suffice for an initial set of lattice vectors and orientation matrix
  • you continue to collect many more frames according to the presumed symmetry and hence completeness of the data; perhaps refine the orientation matrix now while indexing faces and recording a manual absorption correction with the crystal still mounted on the goniometer head
  • you perform data reduction. In this step, one often obtains an even better orientation matrix and lattice constants which then are used to extract the diffraction intensities. Alternatively to the above, it may be now the time for an absorption correction. Instead of hkl expressed in direction cosines tied on a particular setup, you obtain the device independent handy hkl file.
  • start with structure solution (determine at least all non-H atoms)
  • complete with structure refinement

* Actually it is a small and slow sweep of increments like $0.3^\circ{}$ along $2\theta$ while the other circles are kept fix while recording a frame.

** Diffractometers with the elder detection technique of a spot detector/counter tube like the famous Enraf Nonius CAD 4 in $\kappa$ geometry (example, the counting tube with pencil narrow opening just across the X-ray tube/collimator) have their own very elaborate algorithm to orbit around the crystal to determine an initial cell and orientation matrix to identify about 25 good diffraction peaks (ref) for the initial guess. Before recording the big data set eventually used for integration, absorption correction, crystal structure solution and refinement.

*** The possibility to rerun from scratch the data reduction of diffraction data collected with area detectors is a huge advantage compared to those collected with point detectors. Aside from shorter acquisition times.

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  • $\begingroup$ excellent! Last point: why in single crystal diffraction you don't use the angle of diffraction to calculate the distance d using Bragg's law and then have access to (hkl) indexes through some calulation? Why an orientation matrix? Maybe this questions are trivial, but I didn't have a practial view of the experiment before your answer $\endgroup$
    – Rif
    Commented Aug 16 at 9:11
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    $\begingroup$ @Rif The answer was edited to highlight the relationship of individual frames and «their position» on the circles for a diffraction experiment of multiple runs. The need for an orientation matrix was explained (basically, in the cubic crystal system $(a,b,c)$ in direct and thus $(a^*, b^*, c^*)$ in reciprocal space are cartesian coordinates like the $(x,y,z)$ of the diffractometer. But $(a^*, b^*, c^*)$ of the triclinic crystal system $(a,b,c)$ is oblique. Intentional pick of the two as extrema. $\endgroup$
    – Buttonwood
    Commented Aug 16 at 19:25
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This

Single crystal diffraction picture

is what your primary data looks like. I mean, you have a lot of different pictures like this one, all from one crystal.

(The picture is not mine, and my knowledge is not very recent, but I believe the basics haven't changed much).

You deduce the crystal orientation and cell dimensions; then you know which (h,k,l) each spot has. Then you integrate each spot, and then (and not before) you have the intensities.

One might argue that a diffractometer does all this (almost) automatically. I say: no, the computer does. Then the same computer, with little or not-so-little human help, does the rest and gives you the structure with atomic coordinates and stuff. As for the diffractometer alone, its output is what I said earlier.

So it goes.

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  • $\begingroup$ Thanks! The most difficult point for me is how deduce the crystal orientation...is there a procedure? $\endgroup$
    – Rif
    Commented Aug 15 at 19:36
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    $\begingroup$ My late father-in-law was the computer for his x-ray data back in the day. One gets used to seeing the pattern, mentally rotating the Ewald sphere, and determining the symmetry and potential crystal structures, then calculating things by hand. Computers? Bah! $\endgroup$
    – Jon Custer
    Commented Aug 15 at 19:51
  • $\begingroup$ The picture (source?) might show the reconstruction of a zone. Twofold rotation/mirror plane seems possible, maybe even a fourfold axis though a bit tilted. $\endgroup$
    – Buttonwood
    Commented Aug 15 at 23:44

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