On Pierre Curie’s Works in the Field of Symmetry

A. V. Shubnikov

Pierre Curie is known in broad circles of scientific workers as the author of remarkable works in the field of radioactivity and is almost entirely unknown as the author of profound investigations in the field of symmetry and its applications in physics. Meanwhile, these investigations, had they been continued by Pierre Curie, could probably have had for the development of natural science as a whole scarcely less significance than the works on radioactivity had for the development of physics and chemistry. According to Marie Curie’s testimony, Pierre Curie himself repeatedly expressed regret that his investigations on radioactivity had drawn him away from investigations in the field of symmetry.

For Pierre Curie’s works on symmetry, as indeed for all his works, an extraordinary brevity of exposition is characteristic. The complete collection of Pierre Curie’s writings, which includes 61 of his papers and a rather extensive introductory article by Marie Curie, comprises about 610 pages. This means that, on average, each of Pierre Curie’s papers accounts for fewer than ten pages. It should be noted, however, that while the indicated conciseness of exposition does not in the least hinder the reading of most of Pierre Curie’s papers, the same cannot be said of his investigations on symmetry. It is possible that precisely this circumstance was the reason why they were not sufficiently understood and appreciated by physicists.

In one of his works on symmetry Pierre Curie writes: “I think that in the study of physical phenomena it would be of interest to introduce considerations of symmetry, so familiar to crystallographers.” “Physicists often make use of conditions following from symmetry, but usually neglect the exact determination of the symmetry of a phenomenon, because quite often these conditions prove to be simple and a priori almost obvious.”

This remark of Pierre Curie’s, made 62 years ago, fully retains its force to this day. This is evidenced if only by the fact that in almost all modern textbooks of physics we encounter the term “axial symmetry,” which is considered

requiring no explanation; meanwhile, there is not one “axial symmetry,” i.e., symmetry group containing a single axis of infinite order, but five different ones. We must dwell on this question in somewhat greater detail.

Pierre Curie was the first to single out, as especially important for physics, those symmetry groups that we now call limiting point symmetry groups. In all, there are seven such groups. They are easily remembered by the simplest model figures possessing the corresponding symmetry (Fig. 1). The first

Fig. 1. Seven point limiting symmetry groups represented by model figures.

group \((\infty)\) has no symmetry elements except an axis of infinite order; this is the group of a rotating cone. It permits the existence of enantiomorphic (right- and left-handed) forms: a cone rotating to the right and a cone rotating to the left. The second group \((\infty \cdot m)\), besides the axis of infinite order, has only an infinite set of longitudinal planes of symmetry. This is the group of a cone at rest, which does not permit the existence of enantiomorphic forms. The third group \((\infty : m)\) has only the following symmetry elements: an axis of infinite order, one transverse plane of symmetry, and a center of symmetry. This is the group of a rotating cylinder. It is important to note that this group, like the preceding one, does not permit the existence of enantiomorphic forms. This means that a cylinder rotating to the right does not differ from a cylinder rotating to the left with respect to right-handedness and left-handedness: one can be

combined (upon overturning) with another simple superposition without reflection in a plane. The fourth group \((\infty : 2)\) has only an axis of infinite order and an infinite set of transverse axes of symmetry of the second order. This is the group of the twisted cylinder. It permits the existence of enantiomorphic forms (right- and left-handed screws). The fifth group \((m \cdot \infty : m)\) contains only an axis of infinite order, an infinite set of longitudinal and one transverse plane of symmetry, as well as an infinite set of transverse axes of the second order and a center of symmetry. This is the group of the resting cylinder, which does not permit the existence of enantiomorphic forms.

The groups just enumerated exhaust precisely those symmetry groups which, without detailed distinction among them, are often called “axial symmetry.” In addition to them there are still two limiting groups—the sixth and the seventh. The sixth group \(\infty/\infty \cdot m\) is the symmetry group of the ordinary sphere, containing an infinite set of axes of infinite order, an infinite set of planes of symmetry, and a center of symmetry. This group does not permit the existence of enantiomorphic figures. The seventh group \(\infty/\infty\) is the symmetry group of a sphere without planes and a center of symmetry, but with an infinite set of axes of infinite order. This group permits the existence of enantiomorphic—right- and left-handed—spheres. One may (formally) regard that in the right-handed sphere all its diameters are twisted according to a right-handed screw, and in the left-handed one according to a left-handed screw.

Using the limiting symmetry groups, Pierre Curie for the first time established one of the most essential features distinguishing the electric field from the magnetic field, and on this basis fully explained why, in contrast to positive and negative electric charges, northern magnetism cannot be separated from southern. The point is that a cylindrical magnet together with the magnetic field surrounding it has the symmetry \((\infty : m)\) of a rotating cylinder, whereas the electrical analogue of the magnet—a voltaic pile or a cylindrical dielectric polarized along its axis—has the symmetry \((\infty \cdot m)\) of a resting cone. This means that the magnet, in accordance with the idea of the existence in it of amperian currents (rotating electrons), has a transverse plane of symmetry and has no longitudinal planes of symmetry, whereas the voltaic pile, on the contrary, has an infinite number of longitudinal planes of symmetry and no transverse one. This also means that the amperian currents at the north and south poles of a magnet flow in one direction, which we call either clockwise or counterclockwise depending on whether we look at the magnet from the side of the south or of the north pole. The impossibility of separating northern magnetism (“magnetic mass”) from southern precisely signifies the impossibility of the separate existence of right- and left-handed rotation, since every rotation is simultaneously both right-handed and left-handed.

In the doctrine of symmetry we call equal those parts of a symmetrical figure which are transformed into one another by symmetry operations. In doing so we have to distinguish two kinds of equality: congruent equality and mirror equality. It is, of course, also possible to have an equality which is simultaneously congruent and mirror. The poles of a magnet are equal to one another, since one is transformed into the other by reflection in a transverse plane of symmetry; but this equality is purely mirror equality, since the indicated transformation is effected only by an operation of the second kind, i.e., an operation containing reflection: in the present case reflection in a transverse plane of symmetry or inversion (reflection in a center). Thus the north pole differs from the south pole only with respect to right-handedness and left-handedness, i.e., no more than the right hand differs from the left. The situation is different with the electric poles of a voltaic pile. They are not transformed into one another by any symmetry operations of this object; they are not equal to one another. This is the essential difference between magnetic and electric polarity. Pierre Curie’s merit is that he was the first to recognize this difference fully and consistently.

Fig. 2. Examples of directed quantities: a—polar vector (electric-field strength); b—axial vector (magnetic-field strength); v, g—axial tensor (magnitude of left- and right-handed specific rotation of the plane of polarization); d, e—polar tensor (tensile and compressive stress).

Here it would be appropriate to make the following digression. It seems to us that one of the basic tasks of any science is, briefly speaking, to compare the incomparable and to distinguish the indistinguishable, i.e., to find essential features of similarity and difference where previously they had not been noticed. Before Pierre Curie, physicists were more interested in features of similarity than in features of difference between electric and magnetic fields, finding support in this from mathematicians. This can be seen at least from the fact that the vectors of electric- and magnetic-field strength were depicted, and, strangely enough, continue to be depicted to this day, in the same way—by a rectilinear arrow (Fig. 2, a), despite the fact that the electric vector is polar—and for it

such a representation correctly reflects its symmetry,—whereas the magnetic vector is axial—and for it such a representation incorrectly reflects its symmetry: a rectilinear arrow does not have the transverse plane of symmetry that every axial vector has. The representation of an axial vector that correctly reflects its symmetry could obviously be a segment of a straight line having a length proportional to the magnitude of the vector, and a circular arrow indicating the direction of rotation ascribed to this vector (Fig. 2, b). As we see, Pierre Curie was indeed ahead of his contemporaries, having discovered in magnetic and electric fields such an essential feature of their difference as a difference with respect to symmetry, unnoticed by others.

In the development of various sciences we often observe a return to old ideas on a new basis and the emergence of new ideas on an old basis. This bears directly on the question we are considering of the symmetry of polar and axial vectors. The difference, discovered by Pierre Curie, between polar and axial vectors, consisting in the fact that an axial vector has, while a polar vector does not have, a transverse plane of symmetry, is at present softened by the introduction into the doctrine of symmetry of the concept of opposite equality and, correspondingly, of the concept of antisymmetric transformations and elements of antisymmetry. Using these concepts, we come to the conclusion that a polar vector, represented by a rectilinear arrow, has a transverse plane of antisymmetry, which possesses the property that the rectilinear arrow, after reflection in this plane accompanied by a change of sign of the figure (replacement of the positive sign of the end of the vector by a negative one and of the negative by a positive one), is transformed into itself (is superposed on itself).

When a physics student first becomes acquainted with limiting symmetry groups, the greatest bewilderment on his part is caused by the symmetry group of a sphere without planes of symmetry, denoted by us by the symbol $\infty/\infty$. It seems quite incredible that such spheres can occur anywhere. Our survey of the results of Pierre Curie’s work in the field of the doctrine of symmetry would not achieve its purpose if we did not take this occasion to answer this question. For this it is necessary to recall what is meant by the well-known quantity of specific rotation of the plane of polarization. First of all it should be noted that this quantity is a directed quantity; in crystals it changes its value with a change of direction. Since with this quantity we associate the idea of a certain rotation, at first the natural thought arises to assign it to axial vectors having the symmetry $(\infty : m)$ of a rotating cylinder. It is not hard to see, however, that such a solution of the question would be incorrect. This follows from the fact that the quantity of specific rotation does not change its “sign” upon

of changing the forward direction to the backward direction, i.e., right rotation remains right, and left remains left (this does not apply to the magnetic rotation of the plane of polarization). This means that specific rotation is not a vector, but a tensor, and moreover an axial tensor having the symmetry \((\infty : 2)\) of a cylinder twisted by a right- or left-handed screw. Such a quantity is conveniently represented by a segment of a straight line with two circular arrows (Fig. 2, b, c) directed to one side from the point of view of an observer looking at these arrows in turn from one and the other end of the segment, i.e., in the way the phenomenon of rotation of the plane of polarization is actually studied. In isotropic media that rotate the plane of polarization—for example, in an aqueous solution of sugar—the specific rotation is the same in all directions, i.e., the indicatrix of rotation (the gyration surface) of such media is a sphere, which, obviously, has no planes of symmetry and may be either right-handed or left-handed depending on the character of the rotation.

Pierre Curie was not the only scientist who understood that the property of symmetry may be possessed not only by crystals and other material objects, but also by physical fields, as well as by physical phenomena. What is important is that this understanding was deeper in Pierre Curie than in any of his contemporaries, and that Pierre Curie’s reflections on the symmetry of crystals and of the phenomena occurring in them led him to important generalizations, which will be discussed below. Touching on this question, Marie Curie writes in her recollections of Pierre Curie that the discovery by the Curie brothers of piezoelectric polarization “was by no means accidental. It arose as the result of reflection on the symmetry of crystallized matter, which allowed the brothers to foresee the possibility of this polarization.”

Let us proceed to the general principles of symmetry set forth by Pierre Curie in his remarkable work “On Symmetry in Physical Phenomena.”

This work begins with the following italicized lines. “The symmetry characteristic of one phenomenon or another is the maximal symmetry of the medium compatible with the existence of the phenomenon.”

“A phenomenon can exist in a medium that possesses either the characteristic symmetry, or one of its subgroups.”

“In other words, some symmetry elements of the medium may coexist with the phenomenon, but they are not obligatory. Only the absence of certain symmetry elements is obligatory. It is this—dissymmetry—that creates phenomena.”

Let us give several examples from crystal physics that clarify the meaning of these propositions of Pierre Curie.

Take a crystal of tourmaline, which, as is known, possesses the symmetry \((3 \cdot m)\) (one axis of the third order and three intersecting...

along it, longitudinal planes of symmetry). It is also known that tourmaline, when uniformly heated, becomes electrically polarized, i.e. a homogeneous electric field arises in it, directed along the axis (the pyroelectric effect). We have already seen above that a homogeneous electric field at each of its points has the symmetry \((\infty \cdot m)\). In the example under consideration, the “medium” in which the pyroeffect arises is tourmaline. However, tourmaline is not the only medium possessing this property: the pyroeffect is also possible in other media (crystals and textures), if by symmetry they belong either to the group \(\infty \cdot m\), or to one of the subgroups \((1, 2, 3 \ldots m, 2 \cdot m, 3 \cdot m, \ldots)\) of this group of “maximal symmetry.” The common property of all these groups is that certain symmetry elements are “absent” in them: a center of symmetry, a transverse plane of symmetry, and an infinite set of axes of symmetry (ordinary and mirror axes) situated perpendicularly and obliquely with respect to the existing axis. The totality of these absent symmetry elements is what Pierre Curie calls “dissymmetry.” Summing up, we can therefore say that the pyroeffect is possible in all media possessing the indicated dissymmetry. It is this that “creates the phenomenon.”

The term “dissymmetry” is widespread in the crystallographic, chemical, and physical literature. It was apparently first introduced into science by L. Pasteur, who understood by dissymmetry the property of certain figures of not being superposable by simple displacement upon their mirror image. An example of such figures may be the figure of a human hand: it is known that the figure of the right hand cannot be made to coincide by simple displacement with its mirror image, i.e. with the figure of the left hand. At present we can define L. Pasteur’s dissymmetry as the absence in a figure of symmetry elements of the second kind; these correspond to symmetry operations equivalent to an odd number of reflections in planes (simple reflection in one plane, inversion, mirror rotations, glide reflection). Pierre Curie’s concept of dissymmetry is broader. By dissymmetry he means simply the totality of all symmetry elements absent from a figure. It is very important to note the following essential distinction between symmetry (the totality of the symmetry elements present) and dissymmetry (the totality of the absent symmetry elements). It is known that the complete set of symmetry operations corresponding to all symmetry elements present in a figure forms a group in the mathematical sense. This means that the product of any two operations of the group is equivalent, in result, to some one operation of the same group. In contrast, the complete set of symmetry operations corresponding to all symmetry elements absent from a figure does not form a group in the mathematical sense. According to Pierre Curie, for the prediction of new phenomena,

dissymmetry is more essential than symmetry. Since, however, as he notes, the number of absent elements of symmetry is always infinitely large, it is simpler to enumerate the elements of symmetry (the present ones) than the elements of dissymmetry (the absent elements of symmetry).

In broad scientific circles dissymmetry—whether the dissymmetry of L. Pasteur or the dissymmetry of Pierre Curie—is often confused with asymmetry, i.e. the complete absence of symmetry. Asymmetry, obviously, is only a special case of dissymmetry. Dissymmetry is also sometimes confused with antisymmetry—opposite symmetry, described by special groups of four-dimensional symmetry.

Developing his fundamental propositions, cited by us above, Pierre Curie arrives at the following extremely important conclusion:

“When several different natural phenomena are superimposed on one another, forming a single system, their dissymmetries are compounded. As a result, only those elements of symmetry remain which are common to each phenomenon taken separately.”

This proposition of Pierre Curie is far from a trivial extension to physical phenomena of the truth, trivial for geometrical figures, which consists in the fact that when two (or many) unequal symmetrical constituent figures are combined into one composite figure, in the latter only those elements of symmetry remain which are common to all the constituent figures under a given method of arranging them in space. Suppose, for example, that we are given a cube and a cone, arranged relative to one another so that the axis of the cone coincides with a diagonal of the cube (Fig. 3). It is immediately evident that the composite figure has the symmetry \((3 \cdot m)\) (one axis of the third order and three longitudinal planes of symmetry). It is easy to see that these elements of symmetry are contained both in the cube and in the cone. It is also easy to see that the elements of symmetry absent in both the cube and the cone will also be absent in the composite figure: the dissymmetry of the composite figure is higher; it is made up of the dissymmetries of the component figures.

Fig. 3. A complex figure composed of a cube and a cone.

The possibility, asserted by Pierre Curie, of transferring this geometrical truth to physical phenomena means the possibility of representing physical phenomena by figures and, in particular, by such material figures to which, besides purely geometrical properties, we also ascribe certain physical properties. We have already encountered examples of such figures; this is a rotating cyli-

...a cylinder representing a magnetic field or the magnetic polarization of a substance, a twisted cylinder representing the rotation of the plane of polarization in crystals, etc.

Let us give several simple examples of the application of Pierre Curie’s proposition under consideration.

Let us imagine a stream of water flowing along a channel in the direction indicated in Fig. 4 by the rectilinear arrow. A cylinder is immersed in the water, rotating to the right (clockwise) about a fixed axis normal to the surface of the water. It is easy to see that under these conditions one half of the lateral surface of the cylinder moves with the flow of the water, accelerating it; the other half moves against the flow, retarding it. As a result, the water level rises at one bank (++), and falls at the other (—). This dissymmetry arises as a result of the superposition of the dissymmetries of the rectilinear and circular arrows, as a consequence of which there remains only one element of symmetry common to them—a plane of symmetry parallel to the surface of the water. In figures possessing only one plane of symmetry (such as the figure of a human being), all directions parallel to the plane of symmetry (such as up and down, forward and backward, but not right and left in the human figure) are polar, i.e., their ends are not identical with one another. The mechanical phenomenon described corresponds exactly to the Hall effect in electrodynamics. If a constant electric current is passed through a thin metallic plate placed between the poles of a magnet, then a potential difference arises at the edges of the plate. The magnetic field has the symmetry of a rotating cylinder, the electric field the symmetry of a rectilinear arrow (a cone). The combination of the two figures has only a single plane of symmetry, common to the circular and rectilinear arrows.

Fig. 4. Drawing for explaining the Hall phenomenon.

In scientific circles, greater renown than the principle of superposition of symmetries just considered is enjoyed (without proper understanding of them) by three principles of symmetry due to Pierre Curie, by which a connection is established between the symmetry of cause and effect. They obtained this fame, it seems to us, because in Marie Curie’s published memoirs about her husband they are set in italics and numbered as the most important. Pierre Curie himself, as was already mentioned above, does not set them in italics and does not number them.

Here are these principles:

1. “When definite causes produce known effects, the elements of symmetry of the cause must be contained in the effects produced.”

2. “When known effects exhibit a certain dissymmetry, this latter must also be contained in the causes that produced these effects.”

3. “Propositions converse to the two preceding ones are incorrect, at least in practice, i.e. the effects may be more symmetrical than the causes that give rise to them.”

The difficulty in understanding these principles lies in the lack of clarity as to what, in concrete cases, should be understood by “cause” and “effect,” and what should be meant by their “symmetry” and “dissymmetry.”

We can give here the following interpretation of these principles, reducing them to a single principle of superposition of symmetries.

Let us suppose that a compressive stress \(t_{33}\), directed along one of the fourth-order axes of this crystal, is applied to a cube of rock salt (Fig. 5). The question is: what symmetry must the crystal acquire under these conditions?

Fig. 5. Compression of a rock-salt cube.

Let us agree to take as the “cause” all the initial data, i.e. the prescribed but as yet nowhere applied stress \(t_{33}\), and the prescribed but as yet unstressed and undeformed crystal; and as the “effect,” that which is being asked about, i.e. the deformed and stressed crystal. We are accustomed to represent the tensor of compressive stress by oppositely directed rectilinear arrows (Fig. 2, e). Such a pair of arrows has the symmetry of a stationary cylinder \((m \cdot \infty : m)\) (one axis of infinite order, one transverse plane, and a set of longitudinal planes of symmetry, a set of transverse axes of the second order, and a center of symmetry). It can be proved that the same symmetry is possessed also by the magnitude of the tensor

\[ \begin{pmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & t_{33} \end{pmatrix}, \]

by which the indicated stress is described. This follows from the fact that all nine components of the tensor under consideration transform into themselves under all operations of symmetry of the indicated group and under no others. From this we conclude that the symmetry of the stress and the symmetry of the stress tensor are one and the same. Thus, in applying the principles of Pierre Curie, it is possible and necessary to take as the ...

ON PIERRE CURIE’S WORKS IN THE FIELD OF SYMMETRY

symmetry of properties and phenomena to mean the symmetry of those quantities (tensors of various rank) or of those figures by which they (the properties and phenomena) are described.

In our case the symmetry of the cause is composed of the symmetry \((m \cdot \infty : m)\) of the stress and the symmetry \((\bar{6}/4)\) of rock salt (the symmetry of a simple cube). The highest subgroup of both these groups (under the assumption that all elements of their symmetry intersect at one point) is the group \((m \cdot 4 : m)\) (one axis of the fourth order, one transverse and four longitudinal planes of symmetry, four twofold axes, and a center of symmetry), characteristic of tetragonal crystals. The deformed crystal must evidently acquire the same symmetry, which is indeed what is observed.

Fig. 6. Diagram for explaining the principle of superposition of symmetries: the symmetry groups \(\Pi_1\) and \(\Pi_2\) generate the group \(C\).

The foregoing can be given the following visual interpretation (Fig. 6). Let us denote one of the symmetry groups of the cause, i.e., more precisely, the totality of all its symmetry operations, by the figure \(\Pi_1 = pqrsp\), and the other symmetry group of the cause by the figure \(\Pi_2 = tsuqt\) (Fig. 6). Then the symmetry of the effect will be represented unambiguously by the shaded figure \(C\), which may be regarded as a kind of “product” \(\Pi_1 \Pi_2 = C\) of the groups \(\Pi_1\) and \(\Pi_2\) for a given disposition of their symmetry elements. It is clear that at the same time the dissymmetry of the effect \(C\) with respect to \(\Pi_1\) is also uniquely determined (those symmetry elements present in \(\Pi_1\) which are absent from \(C\)), i.e. \(D_1 = \Pi_1 - C = suqrs\), and the dissymmetry of \(C\) with respect to \(\Pi_2\), i.e. \(D_2 = \Pi_2 - C = tspqt\).

Fig. 7. Second diagram for explaining the principle of superposition of symmetries: the groups \(\Pi_1\) and \(C\) generate some one of many groups \(\Pi_2, \Pi_3 \ldots\).

If we imagine that the given groups, i.e., the “cause,” are two groups \(\Pi_1\) and \(C\) (Fig. 7), then the third group \(\Pi_2\), as is seen from the figure, cannot be uniquely determined from them. As applied to the example considered by us, this means the following: if we know the symmetry of the deformed crystal and the symmetry of the deforming stress, then from these data we cannot decide what symmetry the crystal had before deformation.

From what has been set forth it is clear that all three of Pierre Curie’s principles, which relate the symmetry of the cause and the symmetry of the effect, can ultimately be effectively reduced to the principle of superposition of symmetries formulated by him.

Pierre Curie concludes his reflections on symmetry with a remark stating that conclusions drawn from the consideration of symmetry may be of two kinds: 1) indisputable, but negative, conclusions, which correspond to the proposition—there are no effects without a cause; and 2) positive conclusions, which, however, do not provide the same certainty as negative ones; they correspond to the proposition—there is no cause without an effect.

Among indisputable negative conclusions belongs, for example, the assertion that piezoelectric phenomena are impossible in crystals possessing a center of symmetry. Among uncertain positive conclusions one may include the converse proposition: piezoelectricity is possible only in crystals without a center of symmetry, but it is not obligatory.

In conclusion, let us note that Pierre Curie’s ideas in the field of the theory of symmetry cannot be regarded as fully formalized. This will be done by future generations.