# Pieter M. de Wolff

**Pieter M. de Wolff** (Pieter Maarten de Wolff, 1919–1998) was a Dutch crystallographer at Delft Institute of Technology who introduced the superspace approach for treating incommensurately modulated crystal structures, embedding a crystal whose diffraction pattern cannot be indexed in three dimensions into a periodic space of \\( 3+d \\) dimensions<sup>[1](https://www.nobelprize.org/uploads/2018/06/advanced-chemistryprize2011.pdf)</sup><sup> • </sup><sup>[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC4285887/)</sup>. The Nobel Committee's scientific background to the 2011 [Nobel Prize in Chemistry](https://www.edgepedia.org/nobel-prize-in-chemistry) credits de Wolff, together with [Aloysio Janner](https://www.edgepedia.org/aloysio-janner) and Ted Janssen, with introducing this comprehensive superspace treatment, alongside [Dan Shechtman](https://www.edgepedia.org/dan-shechtman)'s discovery of quasicrystals<sup>[1](https://www.nobelprize.org/uploads/2018/06/advanced-chemistryprize2011.pdf)</sup>.

| Key fact | Detail |
|---|---|
| Life | Born in Bandung on Java (present Indonesia); died 10 April 1998 aged 78 in Delft<sup>[3](https://doi.org/10.1107/s0021889898009182)</sup> |
| Career | Delft diploma 1941, PhD 1951 while at TNO; Professor of Applied Physics at Delft 1959–1984<sup>[3](https://doi.org/10.1107/s0021889898009182)</sup> |
| Signature idea | 1974 superspace description embedding modulated phases in \\( (3+d) \\)-dimensional space; the crystal is a 3D section of a periodic "supercrystal"<sup>[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC4285887/)</sup> |
| Triggering discovery | 1964: nonindexable weak satellite lines in the powder diagram of anhydrous sodium carbonate, indexed with nonintegral indices<sup>[3](https://doi.org/10.1107/s0021889898009182)</sup> |
| Key papers | "The pseudo-symmetry of modulated crystal structures" (1974, Acta Cryst. A30, 777–785); joint list of four-dimensional superspace groups<sup>[3](https://doi.org/10.1107/s0021889898009182)</sup><sup> • </sup><sup>[4](https://journals.iucr.org/b/issues/2014/04/00/dq5009/index.html)</sup> |
| Instrument | Guinier–de Wolff focusing camera; more than 1000 sold by Enraf-Nonius<sup>[3](https://doi.org/10.1107/s0021889898009182)</sup> |
| Honors | Gilles-Holst Medal 1976, Gotlob-Werner Medal 1986, ICDD Distinguished Fellowship 1994, Gregori Aminoff Prize 1998<sup>[3](https://doi.org/10.1107/s0021889898009182)</sup><sup> • </sup><sup>[5](https://www.crystallography.org.uk/old-bca-website/cnews/1998/jun98/gaprz.html)</sup><sup> • </sup><sup>[6](https://www.icdd.com/distinguished_fellow/)</sup> |

## Life and career

De Wolff was born in Bandung on Java, in present-day Indonesia, and died on 10 April 1998 at the age of 78<sup>[3](https://doi.org/10.1107/s0021889898009182)</sup>. He took his diploma at Delft in 1941 and his doctorate there in 1951, while working at the Dutch research organization TNO; in 1959 he became full Professor of Applied Physics at Delft, retiring in 1984<sup>[3](https://doi.org/10.1107/s0021889898009182)</sup>. His entire documented career was at Delft and TNO.

Around 1947 he improved Guinier's 1939 focusing X-ray camera, and the resulting instrument, known as the Guinier–de Wolff camera, sold more than 1000 units through the Dutch firm Enraf-Nonius<sup>[3](https://doi.org/10.1107/s0021889898009182)</sup>. He later chaired the IUCr Committee on the Nomenclature of Symmetry<sup>[3](https://doi.org/10.1107/s0021889898009182)</sup>.

## The problem of incommensurate structures

A conventional crystal is periodic in three dimensions, and every Bragg reflection can be labeled with three integers. Structures that violate this had been seen for decades: incommensurability was inferred in cold-worked metals as early as 1927, and Dehlinger's unexplained "Gittergeister" satellite spots of that year were explained by periodically arranged defects rather than by intrinsic incommensurability<sup>[1](https://www.nobelprize.org/uploads/2018/06/advanced-chemistryprize2011.pdf)</sup><sup> • </sup><sup>[4](https://journals.iucr.org/b/issues/2014/04/00/dq5009/index.html)</sup>.

**The sodium carbonate anomaly.** In 1964 the group of Pim de Wolff in Delft found an anomaly in the diffraction pattern of dehydrated sodium carbonate: peaks that could only be indexed using three irrational indices or four integer indices, so the phase has no lattice periodicity and is aperiodic<sup>[4](https://journals.iucr.org/b/issues/2014/04/00/dq5009/index.html)</sup>. The obituary account places these nonindexable weak satellite lines in the powder diagram of anhydrous sodium carbonate<sup>[3](https://doi.org/10.1107/s0021889898009182)</sup>, while the IUCr historical review identifies the phase as the incommensurately modulated γ-Na₂CO₃<sup>[4](https://journals.iucr.org/b/issues/2014/04/00/dq5009/index.html)</sup>. De Wolff indexed the lines with nonintegral indices and showed that anhydrous soda is modulated and not lattice periodic<sup>[3](https://doi.org/10.1107/s0021889898009182)</sup>.

The structural picture is a periodic distortion whose period is incompatible with that of the underlying parent lattice<sup>[1](https://www.nobelprize.org/uploads/2018/06/advanced-chemistryprize2011.pdf)</sup>. In the modulated γ-phase of Na₂CO₃, the main reflections belong to a monoclinic lattice and the satellites correspond to a one-dimensional modulation with wavevector \\( q = \\alpha a^{*} + \\gamma c^{*} \\) (unique axis b)<sup>[7](https://docenten.science.ru.nl/Janner/pa/ITCCtot1.pdf)</sup>. Because \\( \\alpha \\) and \\( \\gamma \\) depend on temperature and are generally irrational, no three-integer indexing is possible<sup>[4](https://journals.iucr.org/b/issues/2014/04/00/dq5009/index.html)</sup>.

## The superspace approach

In 1974 de Wolff presented a superspace description for modulated phases, embedding the structures into \\( (3+d) \\)-dimensional space, where 3 is physical space and \\( d \\) the additional internal dimensions<sup>[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC4285887/)</sup>. The real crystal is regarded as a three-dimensional section through the \\( (3+d) \\)-dimensional periodic "supercrystal", and the diffraction pattern of the modulated crystal as the projection of its \\( (3+d) \\)-dimensional reciprocal lattice<sup>[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC4285887/)</sup>. [Translational symmetry](https://www.edgepedia.org/translational-symmetry) lost by an otherwise periodic three-dimensional structure that experiences \\( d \\) integrally independent incommensurate modulation waves is restored upon embedding in the higher-dimensional space<sup>[8](https://iso.byu.edu/2011%20Stokes.pdf)</sup>.

**Indexing in four integers.** In an incommensurately modulated crystal, main Bragg reflections lie on the reciprocal lattice of the 3D-periodic basic structure, and first-order satellite reflections appear at positions \\( \\pm q \\) from the main reflections; the vector \\( q \\) is the wave vector of the modulation wave, and its length is the reciprocal of the modulation wavelength<sup>[9](https://link.springer.com/article/10.1007/s12210-023-01167-z)</sup>. In a one-dimensionally modulated crystal, any reflection can be uniquely indexed by four integers \\( (h, k, l, m) \\), with \\( m = 0 \\) for main reflections and \\( m \\neq 0 \\) for satellites; spots with \\( h_4 = 0 \\) are the main reflections, the others satellites<sup>[9](https://link.springer.com/article/10.1007/s12210-023-01167-z)</sup><sup> • </sup><sup>[4](https://journals.iucr.org/b/issues/2014/04/00/dq5009/index.html)</sup>. De Wolff, Janner, and Janssen observed in 1981 that a set of points described by \\( (3+d) \\) integer indices forms a reciprocal lattice in \\( (3+d) \\) space, and accordingly defined superspace as a space of dimension \\( (3+d) \\)<sup>[9](https://link.springer.com/article/10.1007/s12210-023-01167-z)</sup>.

The formalism of de Wolff (1974) and Janner and Janssen (1977) has become the accepted standard for describing incommensurate structures involving displacive and compositional waves<sup>[8](https://iso.byu.edu/2011%20Stokes.pdf)</sup>. Incommensurability as an intrinsic crystallographic property was first proposed by de Wolff and co-workers in 1974 (Acta Cryst. A30, 777–785) and 1977, followed by Janner and Janssen in 1977 (Phys. Rev. B, 15, 643–658)<sup>[4](https://journals.iucr.org/b/issues/2014/04/00/dq5009/index.html)</sup>.

## Reception and later development

In 1972 de Wolff presented his way of describing the symmetry of anhydrous soda using a four-dimensional space group at the International Congress of the IUCr in Kyoto; his 1974 paper was titled "The pseudo-symmetry of modulated crystal structures"<sup>[3](https://doi.org/10.1107/s0021889898009182)</sup>. He cooperated in determining the list of all four-dimensional superspace groups for modulated crystals and studied modulated structures such as Rb₂ZnBr₄<sup>[3](https://doi.org/10.1107/s0021889898009182)</sup>. The \\( (3+d) \\)-dimensional superspace Bravais classes for \\( d = 1 \\), \\( d = 2 \\) and \\( d = 3 \\) were determined and classified by Janner and colleagues<sup>[8](https://iso.byu.edu/2011%20Stokes.pdf)</sup>.

The formalism solved old puzzles. Attempts to index the crystal faces of calaverite (AuTe₂) by simple rational indices had failed as far back as 1902; in 1988 Schutte and de Boer solved a natural single crystal of calaverite using the superspace formalism, with an incommensurate displacement modulation of the Te atoms associated with an occupational modulation of the Au atoms<sup>[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC4285887/)</sup>.

## Modulated structures versus quasicrystals

Incommensurately modulated structures and quasicrystals both lack 3D translational symmetry, but they differ in kind. Modulated structures are periodic distortions with a period incompatible with the parent lattice; in contrast to quasicrystals, they may be regarded as distortions of periodic structures, and their point-group symmetries allow 3-dimensional periodicity<sup>[1](https://www.nobelprize.org/uploads/2018/06/advanced-chemistryprize2011.pdf)</sup>. In icosahedral and decagonal quasicrystals, by contrast, self-similarity is related to the scaling properties of the golden ratio \\( \\tau = (\\sqrt{5} + 1)/2 \\)<sup>[1](https://www.nobelprize.org/uploads/2018/06/advanced-chemistryprize2011.pdf)</sup>.

Three modes of long-range order without 3D translational symmetry have been found: incommensurately modulated crystals, incommensurate composite crystals, and quasicrystals, all described by the superspace theory developed by de Wolff, Janner, and Janssen; modulated crystals have been found with \\( d = 1 \\), \\( 2 \\) or \\( 3 \\) modulation wave vectors<sup>[9](https://link.springer.com/article/10.1007/s12210-023-01167-z)</sup>.

## By the numbers

- Dimensions: superspace embeddings use \\( d = 1 \\), \\( 2 \\) or \\( 3 \\) internal dimensions beyond physical space<sup>[9](https://link.springer.com/article/10.1007/s12210-023-01167-z)</sup>.
- Na₂CO₃ transitions: about 753 K from the hexagonal to the monoclinic phase; at about 633 K one vibration mode becomes unstable, giving the modulated γ-phase below \\( T_i = 633 \\) K; at low temperature (128 K) a transition to a commensurate phase has been reported<sup>[7](https://docenten.science.ru.nl/Janner/pa/ITCCtot1.pdf)</sup>.
- Modulation vector in γ-Na₂CO₃: \\( q = \\alpha a^{*} + \\gamma c^{*} \\) (unique axis b), a one-dimensional modulation<sup>[7](https://docenten.science.ru.nl/Janner/pa/ITCCtot1.pdf)</sup>.
- Guinier–de Wolff cameras sold: more than 1000<sup>[3](https://doi.org/10.1107/s0021889898009182)</sup>.
- Calaverite's unindexable faces resisted rational indexing from 1902 until the 1988 superspace solution<sup>[2](https://pmc.ncbi.nlm.nih.gov/articles/PMC4285887/)</sup>.

## References

1. [Scientific Background on the Nobel Prize in Chemistry 2011, Nobel Committee](https://www.nobelprize.org/uploads/2018/06/advanced-chemistryprize2011.pdf)
2. [Superspace crystallography: a key to the chemistry and properties of materials (2015). Acta Cryst. B](https://pmc.ncbi.nlm.nih.gov/articles/PMC4285887/)
3. [Pieter Maarten de Wolff 1919–1998 (obituary)](https://doi.org/10.1107/s0021889898009182)
4. [Janssen & Janner (2014). Aperiodic crystals and superspace concepts. Acta Cryst. B](https://journals.iucr.org/b/issues/2014/04/00/dq5009/index.html)
5. [Gregori Aminoff Prize 98, British Crystallographic Association](https://www.crystallography.org.uk/old-bca-website/cnews/1998/jun98/gaprz.html)
6. [ICDD Distinguished Fellow Award](https://www.icdd.com/distinguished_fellow/)
7. [Superspace symmetry for aperiodic crystals, International Tables for Crystallography chapter (Janner)](https://docenten.science.ru.nl/Janner/pa/ITCCtot1.pdf)
8. [Generation of (3+d)-dimensional superspace groups for describing the symmetry of modulated crystalline structures](https://iso.byu.edu/2011%20Stokes.pdf)
9. [Aperiodic crystals and their atomic structures in superspace: an introduction (2023). Rendiconti Lincei](https://link.springer.com/article/10.1007/s12210-023-01167-z)

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*Topic: Encyclopedia › Physical world and mathematics › Physical and mathematical scientists › Physicists and astronomers › Researchers in condensed matter physics and quantum materials › Crystallography and diffraction pioneers*

*Initially written Oct 10, 2026 · Edited: Oct 11, 2026; Oct 15, 2026; Oct 21, 2026*

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