Karl Weissenberg - The 80th Birthday
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Karl Weissenberg and the development of X-ray crystallography
PROFESSOR MARTIN J. BUERGER
Massachusetts Institute of Technology, The University of Connecticut U.S.S.
This appreciation of Karl Weissenberg’s work is concerned chiefly with the history of certain X-ray diffraction methods. While this writer has commented on some of these developments from time to time*, this material is set down here as a more connected history of an early problem of X-ray crystallography, and of Karl Weissenberg’s inspired solution and some of the developments which followed his elegant contribution.
In effect, Weissenberg made a specific and far-reaching contribution to the rotating-crystal method. This method of recording the X-ray diffraction from a single crystal was apparently first practised by de Broglie1, Wagner2 and Seeman3 in the half-dozen years immediately following the discovery of X-ray diffraction by crystals. In this method, a large single crystal was rotated about a fixed axis set normal to the X-ray beam. The X-ray diffraction by the crystal was received and recorded (in those early days) on a flat photographic plate, also placed normal to the X-ray beam. In this way, the X-ray reflections directed along the generators of the Laue cones within the range of the photographic plate were recorded as seen in the diagram of Fig. I. The earliest workers appear to have been specifically interested in the spectrographic aspects of this experiment, that is, in the wavelength distribution of the diffracted radiation, rather than the information which the spectra revealed about the crystal.
The Saxon mineralogists, Schiebold4 and Rinne,7, however, were interested in the crystallographic information obtainable by this method, and students at the University of Leipzig actually determined cell parameters, indexed reflections and (doubtless inspired by Niggli’s book**) attempted to determine the space groups of some minerals whose crystal structures were unknown. (For at least some crystals, such as tourmaline, this attempt failed because, for reasons to be noted later, indexing reflections in the rotating-crystal method is generally indeterminate. The seriousness of this problem is exacerbated for crystals with low symmetry and large cells).
About this time, several investigators5 7 10 12 began to examine natural fibres, such as cellulose, by X-ray diffraction methods. Polanyi8 was a leader in this field and was the first to obtain what is now known as a “fibre diagram”. This is the X-ray diffraction pattern produced by a stationary “fibre” which is not a single crystal, but rather an aggregate composed of many single-crystal subfibers or fibrils, all with parallel long axes. The fibrils typically have random orientation about the common axis, so that, together, they are equivalent to the many identical crystals, each rotated into a different orientation about the common axis. The X-ray diffraction was generally recorded on a flat film normal to the X-ray beam, or on a cylindrical film whose axis was concentric with the fibre axis. Weissenberg9 became interested in the study of fibres in this way, and later joined Polanyi10 12 in this research field.
As publications of these fibre studies appeared, it became clear to Schiebold11 that the ideal fibre photograph was essentially the same as the rotating-crystal photograph. Later, Polanyi, Schiebold and Weissenberg13 discussed the common features of these two kinds of research which had such different origins.
2. THE PROBLEM
In spite of the attempts of the early investigators to interpret fibre diagrams and rotating-crystal photographs, such photographs are defective in two respects. Qualitatively, they yield no direct symmetry information about the crystal whatsoever, and quantitatively they are incapable of supplying sufficient information for indexing the reflections. Both of these deficiencies are so serious that they render the rotating-crystal method a weak vehicle for crystal-structure analysis. Every rotating-crystal photograph has symmetry 2mm when the angle between the rotation axes and the X-ray beam is 900. This symmetry is not concerned with the crystal symmetry but rather with the fact that every stack of planes oblique to the rotation axis satisfies the Bragg reflection condition four times on a single rotation of the crystal; this occurs when the common normal to the planes reaches the film in the first, second, third and fourth quadrants about the X-ray beam.
Fig. 1. Recording of the Laue cones parallel to a rotation axis on a flat film. [From “X-ray Crystallography” by permission of John Wiley and Sons. New York (1942)].
The indexing of the reflections on a rotating-crystal photograph is inherently impossible because there are three unknowns h, k and I to be determined, but the photographic film on which these three variables occur has only two dimensions. When the crystal is rotated about a crystallographic axis, then the one index (corresponding to the axis about which the crystal is rotated) is fixed as a specific integer n because this is the common index for all reflections on the nth-order Laue cone. Thus, if the crystal is rotated about the c axis, all reflections with a common index l occur along generators of the same Laue cone and so record on the same layer line. If the spot occurs on the nth layer line its index is hkn. But this is all that can be learned about the indices of the reflection. The other two indices h and k are indeterminate since these two variables are functions of the one measurable parameter along the layer line.
3. KARL WEISSENBERG’S SOLUTION TO INDEXING REFLECTIONS
The problem of indexing rotating-crystal photographs was completely solved by Karl Weissenberg17. Judging by the brief description in his paper, Weissenberg obtained his solution by intuition. Once the solution is presented, it is easy to rationalize it. The essence of Weissenberg’s solution was to spread the two unknown variables (h and k when rotating a crystal about its c axis) of the spots of one specific layer line over the two dimensions of the film. To accomplish this, Weissenberg proceeded as follows: First, he took a classical rotating-crystal photograph to find out where each layer line was located. Then he arranged to screen out all but one layer line, as in Fig. 2a by letting the diffracted rays corresponding to this layer reach the film through an annular slit in a cylinder placed coaxial with the film. This amounts to recording only one Laue cone, say the nth. The directions of the reflections hkn occur along different generators of this cone, but generally each different reflection occurs as the crystal is in a different specific orientation φ about the rotation axis, such that the Bragg reflection condition is satisfied for that dhkn. Weissenberg preserved these different orientations by coupling the rotation of the crystal to a translation motion of the cylindrical film in the direction of the cylinder axis, as seen in Fig. 2b. Thus, the various reflections hkn were spread over the two dimensions of the photographic film. The only remaining problem was to determine the transformation from the two variables h and k to the two variables on the film, say the linear coordinates x and y along the two orthogonal edges of the film. Here x is proportional to the projection ү of the Bragg deviation angle 2θ on a plane normal to the rotation axis, and y is proportional to the crystal-orientation angle φ.
Fig. 2a. Elimination of all Laue cones except one, by means of a layer-line screen. [From “X-ray Crystallography” by permission of John Wiley and Sons. New York (1942)].
4. FURTHER DEVELOPMENTS OF WEISSENBERG’S METHOD
It is a curious fact that Weissenberg’s advance, published in 1924, was slow to be adopted. In fact, two years later, Bernal15 published his famous paper “On the interpretation of X-ray, single-crystal, rotation photographs”. Because of the earlier publication of Weissenberg’s paper, the oscillating-crystal method, which Bernal’s publication taught, was already an anachronism, yet it dominated the teaching and practice of X-ray crystallography, especially in England, until after the second world war. Meanwhile, the Weissenberg camera began to be adopted in Germany. Böhm rearranged the parts of Weissenberg’s original apparatus so that the cylindrical camera was translated horizontally, an improvement in design which was retained in all later versions. Schneider19 showed how the indexing of Weissenberg-type photographs could be graphically solved. At the same time, the American mineralogists, Tunell19 and Barth20 began using the Weissenberg method in studying the cells and space groups of minerals, thus introducing the technique into the United States, especially among mineralogists.
In these early days, Weissenberg photographs had customarily been taken with the X-ray beam normal to the rotation axis of the crystal. The Wooster and Wooster22 publication of 1933 provided a new graphical method of transforming the points of a Weissenberg photograph, made with this “normal-beam” technique, to a map of the reciprocal lattice.
Fig. 2b. Spreading the reflections of a selected layer line, n, over the two-dimensional surface of a film. [From “X-ray Crystallography” by permission of John Wiley and Sons. New York (1942)].
A year later, Buerger23 pointed out the great advantages of inclining the primary X-ray beam to the rotation axis at the same angle that the generators of the Laue cone of the photographed level are inclined to that axis. This technique, called “equi-inclination”, made the interpretation of Weissenberg photographs very easy: this is because this inclination provides a similar geometry of diffraction for all levels of the reciprocal lattice, thus giving rise to a great invariance in the appearance of photographs of all levels. This invariance not only permitted easy transfer of spots on the Weissenberg photographs to the reciprocal lattice, but even made easy the indexing of spots on the Weissenberg photograph itself. The basis of this is that all reciprocal-lattice lines mapped on the Weissenberg photograph have the same shape in the same region of the photograph of any level. Equi-inclination also eliminates the otherwise “blind regions” of the reciprocal lattice which had hitherto been unrecorded on rotating-crystal and Weissenberg photographs. It also permits an easy interpretation of the Friedel symmetry25 of crystals by a mere inspection of Weissenberg photographs, and makes possible a quick and certain determination of the space group by the use of diffraction symbols26. With the design of a convenient apparatus27 (Figs. 3 and 4) for making equi-inclination Weissenberg photographs, the Weissenberg method became the most powerful and convenient method of recording X-ray diffraction data from single crystals and nearly every laboratory in which X-ray diffraction by single crystals was studied adopted the Weissenberg method.
Two further developments related to the Weissenberg method may be briefly noted. The first is concerned with the precision with which the cell geometry can be determined. Before the advent of the Weissenberg method, precise measurements of cell geometry could only be made by the powder method. The development of the back-reflection Weissenberg instrument27 made possible the extension of the well-developed extrapolation techniques of the powder method to reflection data taken from single crystals. A second development is concerned with the measurement of reflection intensities with a diffractometer. Powder diffractometers had been developed during World War II. In 1952, a diffractometer for single crystals31 was designed which made use of the same geometry as the Weissenberg camera. A reflection hkl on a Weissenberg film has orthogonal film coordinates φhkl (the angle through which the crystal must be turned to provide the reflection) and үhkI (the projection of the Bragg deviation angle 2θhkl on the plane normal to the rotation axis).
Fig. 3. Weissenberg apparatus designed so that the angle μ, between the plane normal to the rotation axis and the x-ray beam, can be varied without disturbing the adjustment of the instrument to the x-ray beam.
Fig. 4a. First model of the Weissenberg apparatus conforming to the design of Fig. 3 [From M.J. Buerger Z. Krist. 94 (1936) 87-99]
Fig. 4b. Apparatus of Fig. 4a, with camera and layer-line screen removed, and shifted out of the X-ray beam.
The single-crystal diffractometers, shown in Fig. 5, contained provision for turning the crystal through angle φhkl and then turning a photon counter (originally a Geiger counter) through an angle үhkI to record the reflection at the same angular position at which the Weissenberg film would have received it. The single-crystal diffractometer was first set manually, but later automated31, so that the instrument sets the crystal and counter automatically, and, allowing for background, records the integrated intensities automatically.
A curious development in Weissenberg recording took place in 1952. The appearance of the points of the reciprocal lattice on the Weissenberg photograph may be described as due to a transformation of reciprocal space onto Weissenberg space. This transformation transforms a central line of the reciprocal lattice into a straight line on the Weissenberg photograph whose slope is ү/ω; for standardized coupling constants this slope is 2. This caused a plane of symmetry, for example, to have a slope of 2 with respect to the centre line of the film. But in transferring from ordinary polar coordinates r, θ to a linear representation of r and an orthogonal θ, a symmetry plane through the centre of the polar coordinate system ordinarily transforms to a line orthogonal to the axis of the linear representation of θ. Huerta29, 30 thought that a modified Weissenberg motion should be designed to provide this geometry. He accomplished this by adding a rotational motion to the translation motion of the Weissenberg cylindrical camera. Thus, the motion of the camera was a screw motion, which gave rise to the appelation “the helicoidal method”. This was designed so that the screw had a pitch ω′ such that ω-ω′ = 0. Thus, central reciprocal -lattice lines had slopes ү/(ω-ω′)= ∞, so that they appeared in the Weissenberg photograph at right angles to the centre line. Unfortunately, this addition to the Weissenberg motion was devised subsequent to the Sauter, Schiebold, de Jong-Bouman, and precession methods, each of which already provided photographs which approximated better (in the first two cases) or actually attained (in the last two) the arrangement in the reciprocal lattice itself.
Fig. 5a. Single-crystal diffractometer based upon the Weissenberg geometry. Original Instrument constructed in 1952.
Fig. 5b. Improved manually set model.
Fig. 5c. Commercial, automated model.
5. THE SAUTER AND SCHIEBOLD METHODS
Weissenberg may be regarded as the father of moving-film methods. When it was realized that, by spreading the records of reflections with only two variable indices over the two-dimensional film, other methods arose to take advantage of this. The first of these was the Sauter32 34 method, shown schematically in Fig. 6. By this method, the reciprocal lattice was projected on a flat film, but with moderate radial and angular distortions, because үhkI ≠ ﻉhkI and φfilm hkl ≠ φreciprocal lattice hkI. But because of the obvious resemblance of the Sauter photographs to the reciprocal lattice, the method came into limited use.
The Schiebold method35, Fig. 7, was an improvement on the Sauter method in that, by bending a flexible photographic film into a cylinder coaxial with the rotation axis of the crystal, the range of recordable reflections was extended so that it was equal to that of the Weissenberg film. Unfortunately, the film had to be continually rotated about its centre while the cylinder axis remained parallel with the rotation axis. This impractical mechanical arrangement, while providing greater reflection range, was difficult to practise and maintain. The method never came into general use.
Fig. 6. Diagram of the instrument used in the sauter method, showing the coupling of rotations of film and crystal. [From “X-ray Crystallography” by permission of John Wiley and Sons. New York (1942)].
Fig. 7. Relation between the film motion in the Sauter method (a) and the Schiebold method (b). [From “X-ray Crystallography” by permission of John Wiley and Sons. New York (1942)].
6. THE METHOD OF DE JONG AND BOUMAN
Shortly after the invention of the Sauter and Schiebold methods, de Jong and Bouman began to experiment with other rotation axes for the photographic film; this resulted in their discovering how to photograph the reciprocal lattice without distortion36. A diagram of the geometrical arrangement in their method is shown in Fig. 8. The general key to this is that, when the crystal, along with its reciprocal lattice, is rotated about a rotation axis, then, in order to photograph a level of the reciprocal lattice without distortion, the film and its rotation axis must be parallel projections from the centre of the crystal, of the reciprocal-lattice level and its rotation axis, and these two axes must be coupled to rotate in the same sense at the same rate. This writer followed with interest the de Jong and Bouman publications as they appeared, and devised his own version37 of their apparatus, shown in Fig. 9. This version, like the equi-inclination Weissenberg instrument, is designed to permit any inclination of the rotation axis to the X-ray beam, and also to operate with any separation of the rotation axes of the film and crystal.
Fig. 8. Diagram showing the geometry of recording the reciprocal lattice by the method of de Jong and Bouman. [From “X-ray Crystallography” by permission of John Wiley and Sons. New York (1942)].
Rimsky38 noted that the method of de Jong and Bourman, when used with anti-equi-inclination, could record the reciprocal lattice without a central blind region. Unfortunately, anti-equi-inclination requires a collection of circular casettes of different radii in order to avoid letting too large a casette get in the path of the primary X-ray beam. The Rimsky version, called the “retigraph” has achieved a limited popularity in France.
Fig. 9a. Improved model of apparatus for the de Jong and Bouman method.
Fig. 9b. Improved model of apparatus for the de Jong and Bouman method.
7. THE PRECESSION METHOD
The precession method is not a direct descendant of the Weissenberg method, but is included here because revision of the first primitive version required the addition of a moving film. The original version of the precession method was the consequence of an attempt to correct the lack of information about the symmetry of the crystal revealed by a rotating-crystal photograph. The symmetry which such a photograph shows is that of a projection of the rotation motion on the crystal, which is universally 2mm. To correct this required, basically, a motion whose projection has the same symmetry as a circle, which would not, therefore, degrade the crystal’s symmetry. For this purpose, a precessing motion for a rational axis of the crystal was selected.
In the first version of the instrument39, seen in Fig. 10, the crystal alone was caused to undergo a precessing motion, yet prevented from rotating by being attached to a gimbal. This gave the desired symmetry results, but the photographs displayed a doubling of the diffraction spots except along a line parallel to the horizontal axis of the gimbal. When the de Jong and Bouman paper appeared, it was evident that, by applying an extension of the principle used by them for a rotating crystal, this defect could be corrected. Specifically, it was necessary to cause the film normally to undergo a precessing motion parallel to that of the rotational axis of the crystal40. The coupling arrangement of the final instrument is shown in Fig. 11.
Both the method of de Jong and Bouman and the precession method produce photographs which are undistorted images of the reciprocal lattice. From these photographs the unit cell can be determined with ease and considerable accuracy41. The symmetry and missing reflections of such photographs yield the diffraction symbol, which is the maximum qualitative information which can be determined from a set of X-ray photographs.
Fig. 10. Early version of the precession instrument.
Fig. 11. Diagram showing the coupling of the precession motions of the film and crystal. [From “X-ray Crystallography” by permission of John Wiley and Sons. New York (1942)].
Weissenberg’s short excursion into the field of X-ray crystallography was illuminated by a flash of genius for which many scientists are indebted to him. The word “breakthrough” is commonly used for an important discovery, but rarely has it been easy to trace its many consequences. In this particular case, however, this author, who appreciated what Weissenberg had accomplished shortly after Weissenberg’s work had been published, took no little pleasure in following the more obvious, direct results of Weissenberg’s inspiration. Less obvious results are the mass of new data about crystal structures which have accumulated in the last half century since Weissenberg’s contribution. Karl Weissenberg’s indirect impact on sciences such as biology, ceramics, chemistry, metallurgy, mineralogy and solid-state physics, through aiding the study of the arrangements of atoms in crystals, is very large.
*M. J. Buerger. The development of moving-film methods. Paper read at the New York Academy of Sciences, Conference on X-ray Diffraction at the American Museum of Natural History, New York, January 10 and 11, 1941.
M. J. Buerger. X-ray Crystallography (Wiley, New York, 1942) 106, 214-229, 295, 312-346, 435-465, 517.
M. J. Buerger. The development of methods and instrumentation for crystal-structure analysis. Z. Kristallogr. 120 (1964) 3-18.
M. J. Buerger. Contemporary Crystallography (McGraw-Hill, New York, 1970) 119-126-148, 185.
**PauI Niggli. Geometrische Kristallographie des Diskontinuums (Gebrader Borntraeger, Leipzig, 1919) 463-503.
THE ROTATING-CRYSTAL METHOD
1. M. de Broglie. Sur Ia spectroscopie de rayons de Röntgen. Compt. Rend. 158 (1914) 177-180.
2. E. Wagner. Spektraluntersuchungen an Röntgenstrehlen. 1. Ann. de Phys. 49 (1915) 868-892, especially 874; Fig. 4, Table V; Figs. 51-53.
3. H. Seemann. Vollständige Spektraldiagramme von Kristallen. Physik. Z.,20 (1919), 169-175.
4. E. Schiebold, in F. Rinne. Einführung in die kristallographische Formenlehre u. elem. Anl. zu kristallo-opt. sowie röntgenogr. Untersuchung (Leipzig, 1919) especially 198-200.
5. R. O. Herzog, Willi Jancke and M. Polanyi. Röntgenspektrographische Beobachtungen an Zellulose. II. Z. Physik, 3 (1920) 343-348.
6. F. Rinne. Röntgenographische Feinbaustudien. Abh. d. math.-phys. Kiasse der Sächs. Akad. der Wiss. 38 (Nr. 3)1921.
7. M. Polanyi. Faserstruktur im Röntgenlichte. Naturwissenschaften, 9 (1921) 337-349.
8. M. Polanyi. Das Röntgen-Faserdiagramm. Z. Physik, 7 (1921), 149-180.
9. K. Weissenberg. “Spiralfaser” und “Ringfaser” im Röntgendiagram. Z. Physik, 8 (1922), 20-31.
10. M. Polanyi and K. Weissenberg. Das Röntgen-Faserdiagramm. Z. Physik, 9 (1922), 123-130.
11. E. Schiebold. Bemerkungen zur Arbeit: Das Röntgenfaserdiagramm von P. Polanyi. Z. Physik, 9 (1922), 180-183.
12. M. Polanyi and K. Weissenberg. Das Röntgen-Faserdiagramm. Z. Physik, 10 (1922) 44-53.
13. M. Polanyi, E. Schiebold and K. Weissenberg. über die Entwicklung des Drehkristallverfahrens. Z. Physik, 23 (1924) 337-340.
14. E. Schiebold. Über graphische Auswertung von Röntgenphotogrammen. Z. Physik, 23 (1924), especially 360-364.
15. J. D. Bernal. On the interpretation of x-ray, single-crystal, rotation photographs. Proc. Roy. Soc. London (A), 113 (1926), especially 117-160.
16. E. Schiebold. Die Drehkristallmethode. Fortschr. Mineral., etc., 11 (1927), 111-280.
THE WELSSENBERG METHOD
17. K. Weissenberg. Ein neues Röntgengoniometer. Z. Physik, 23 (1924),229-238.
18. J. Bohm. Das Weissenbergsche Röngtengoniometer. Z. Physik, 39 (1926), 557-561.
19. W. Schneider. Über die graphische Auswertung von Aufnahmen mit dem Weissenbergschen Röntgengoniometer. Z. Krist., 69 (1928), 41-48.
20. George Tunell. Determination of the space-lattice of a triclinic mineral by means of the Weissenberg x-ray goniometer. Am. Mineral., 18 (1933),181-186.
21. Tom. F. W. Barth and George Tunell. The space-lattice and optical orientation of chalcanthite (CuSO45H2O): an illustration of the use of the Weissenberg x-ray goniometer in the triclinic system. Ajm. Mineral., 18 (1933),187-194.
22. W. A. Wooster and Nora Wooster. A graphical method of interpreting Weissenberg photographs. Z. Krist. (A), 84 (1933), 327-331.
23. M. J. Buerger. The Weissenberg reciprocal lattice projection and the technique of interpreting Weissenberg photographs. Z. Krist. (A), 88 (1934), 356-380.
24. D. Crowfoot. The interpretation of Weissenberg photographs in relation to crystal symmetry. Z. Krist. (A), 90 (1935), 215-236.
25. M. J. Buerger. The application of plane groups to the interpretation of Weissenberg photographs. Z. Krist. (A), 91 (1925), 255-289.
26. M. J. Buerger. An apparatus for conveniently taking equi-inclination Weissenberg photographs. Z. Krist. (A), 94 (1936), 87-99.
27. M. J. Buerger. The precision determination of the linear and angular lattice constants of single crystals. Z. Krist. (A) 97 (1937) 433-468.
28. M. J. Buerger and William Parrish. The unit cell and space group of tourmaline (an example of the inspective equi-inclination treatment of trigonal crystals). Am. Mineral., 22 (1937), 1139-1150.
29. Fernando Huerta. Los metodos del cristal giratorio. (C. Bermejo, Madrid, 1952) 109 pages, especially 83-94.
30. Fernando Huerta. Teoria de los metodos roentgenograficos del cristal giratorio. Consejo Superior de Investigaciones Cientificas. Monografias de Cienica Moderna 48 (1955) 7-135.
31. M. J. Buerger. The development of methods and instrumentation for crystal-structure analyses. Z. Kristallogr. 120 (1964) 3-18.
THE SAUTER METHOD
32. E. Schiebold. Ergebnisse der technische Röntgenkunde. IL. (Leipzig, 1931),86-87.
33. Erwin Sauter. Zur Kenntnis des Rotations-Röntgengoniometer-diagrams. Z. Krist. (A), 84 (1933), 461-467.
34. E. Schiebold. Über ein neues Röntgengoniometer. Gleichzeitig Bemerkung zu der Arbeit von E. Sauter; “Eine einfache Universalkamera für Röntgen-kristallstrukturanalysen”. Z. Krist. (A), 86 (1933), 370-377.
35. E. Schiebold. Über ein neues Röntgengoniometer. Z. Krist. (A) 86 (1933), especially 377-383.
36. W. F. de Jong and J. Bouman. Das Photographieren von reziproken Kristallnetzen mittels Röntgenstrahlen. Z. Krist. (A) 98 (1938). 456-459.
37. M. J. Buerger. X-Ray Crystallography (John Wiley, New York, 1942) 331-346, especially pages 332-333.
38. A. Rimsky. Appareil permettant la photographie directe de l’espace réciproque. Bull. soc. fr. Min. Cristal. 75 (1952) 500.
THE PRECESSION METHOD
39. M. J. Buerger. X-Ray Crystallography (John Wiley, New York, 1942)206-211.
40. M. J. Buerger. The photography of the reciprocal lattice. Am. Soc. X-ray and Electr. Diff. Monograph 1 (1944) 37 pages.
41. Martin J. Buerger. The precession method in x-ray crystallography. (John Wiley, New York, 1964) 276 pages.
Preface / Acknowledgements / Biographical Notes
Weissenberg’s Influence on Crystallography
Karl Weissenberg and the Development of X-Ray Crystallography
The Isolation of, and the Initial Measurements of the Weissenberg Effect
The Role of Similitude in Continuum Mechanics
The Effect of Molecular Weight and Concentration of Polymers in Solutions on the Normal Stress Coefficient
Elasticity in Incompressible Liquids
The Physical Meaning of Weissenberg's Hypothesis with Regard to the Second Normal-Stress Difference
A Study of Weissenberg's Holistic Approach to Biorheology
The Weissenberg Rheogoniometer Adapted for Biorheological Studies
Dr. Karl Weissenberg, 1922-28
Weissenberg’s Contributions to Rheology
The Early Development of the Rheogoniometer
Some of Weissenberg's More Important Contributions to Rheology: An Appreciation
Publications of Karl Weissenberg and Collaborators / List of Contributors
© Copyright John Harris