Neumann's principle, as one sum

Franz Neumann’s principle, from lectures in the 1830s and printed in 1885, is one sentence:

The symmetry of any physical property of a crystal must include the symmetry of the crystal’s point group.

It reads like a philosophical remark. It is a computation, and the computation is short enough to do by hand for any of the thirty-two classes.

A property is a tensor, and the group acts on it #

Almost every property that relates one physical quantity to another is a tensor, and its rank is fixed by what it relates.

A spontaneous polarisation is a vector: three numbers, rank one. Permittivity relates an applied field to a resulting polarisation, so it is a matrix taking a vector to a vector — rank two, and symmetric, so six independent numbers rather than nine. Piezoelectric moduli relate a stress to a polarisation, so they take a symmetric matrix to a vector: rank three, symmetric in the last two indices, eighteen numbers. Elastic constants relate a strain to a stress, both symmetric matrices, and with the extra symmetry that comes from energy being a state function: rank four, twenty-one numbers.

The crystal’s point group acts on each of these in the obvious way — rotate the crystal and the tensor’s components change as a tensor of its rank should — and Neumann’s principle says the tensor must come back unchanged. Rotate the crystal by one of its own symmetry operations and it is the same crystal, so its permittivity is the same permittivity.

The permitted tensors are therefore the ones the group leaves alone, and the question “how many independent components” is the question “how big is that space”.

The averaging projector #

The set of tensors fixed by every operation of a finite group G is a linear subspace, and there is a standard way to find its dimension without solving anything.

Build the averaging operator

$$P = \frac{1}{|G|}\sum_{M \in G} \rho(M)$$

where ρ(M) is however M acts on the tensors in question. P is a projector — applying it twice is applying it once, because averaging an average changes nothing — and its image is exactly the invariant subspace.

The dimension of the image of a projector is its trace. So

$$\dim(\text{invariant tensors}) = \operatorname{tr} P = \frac{1}{|G|}\sum_{M \in G} \chi(M)$$

where χ(M) = tr ρ(M) is the character. One sum, one division, and the answer is an integer.

That the answer comes out an integer is not automatic in floating point and is checked on every build: a non-integer here would mean the input was not a group.

The characters are built from two traces #

What makes this practical rather than merely correct is that χ never has to be computed by writing out ρ(M) as a large matrix. For every property the essays here use, the character is a polynomial in two numbers — tr M and tr M².

• A vector. ρ is M itself, so χ = tr M.
• A symmetric rank-2 tensor. χ = ½[(tr M)² + tr M²]. Six dimensions at the identity, as it should be.
• The piezoelectric moduli — a vector tensored with a symmetric rank-2 — is the product of the two: χ = tr M · ½[(tr M)² + tr M²]. Eighteen at the identity.
• The elastic constants are the symmetric square of the symmetric square, since the tensor is symmetric under exchanging its first pair with its second. Twenty-one at the identity.

So each property is one line of arithmetic per operation, and the whole of Neumann’s principle for one class and one property is a loop over at most forty-eight terms.

Worked, for the smallest interesting case #

Class 2, a single two-fold axis along b, two operations. Take the permittivity — symmetric rank two, six components before symmetry.

The identity has trace 3, so χ = ½[9 + 3] = 6. The two-fold about b is diag(−1, 1, −1) in a Cartesian frame with b along y, so tr M = −1, and its square is the identity with trace 3, giving χ = ½[1 + 3] = 2.

The average is (6 + 2)/2 = 4. A monoclinic crystal has four independent dielectric constants, and the table agrees.

The four are worth naming, because the picture behind them is the one the abstraction is standing in for. The two-fold about y sends x to −x and z to −z, so it leaves xx, yy, zz and xz alone and reverses xy and yz. The two reversed components must equal their own negatives and are therefore zero. Four survive: three diagonal, and the one off-diagonal component that lies in the plane perpendicular to the axis.

That is the calculation the character sum is doing, and doing it this way for six properties across thirty-two classes would be a long afternoon of sign-chasing. Doing it as a trace is 192 sums.

It is worth being clear about what the trace has given up in exchange. The hand calculation above produced two things: the number four, and the list xx, yy, zz, xz of which four. The character sum produces only the number. That is not a defect — a count of degrees of freedom is what an experimentalist planning a measurement needs, and it is the quantity that survives every change of coordinates — but it means that a second calculation has to be run whenever the question is which. There are two ways for a component to disappear and the count cannot tell them apart: a component can be forced to zero, as xy and yz are here, or it can be forced equal to another, which removes a degree of freedom without emptying a cell. The tetragonal case below is the second kind, and reading its answer of two as “two components survive and four vanish” would be wrong in a way the number itself cannot correct.

Two things the sum does not need #

It does not need the tensor written out. The character is a trace, and a trace is a single number; nothing here ever constructs the 3⁴ = 81 components of an elastic tensor or the 6 × 6 matrix crystallographers write it as. That is why the computation is instantaneous for all thirty-two classes and all six properties at once.

It does not need a Cartesian frame, which is the surprising half and is worth its own essay. The matrices this site holds are integer matrices in the lattice basis, which are not orthogonal, and a physical tensor lives in an orthonormal frame. The two descriptions are related by conjugation, a character is a trace, and a trace is unchanged by conjugation — so the integer matrices give the right answer for the Cartesian question, and no real number appears anywhere in the calculation.

Axial properties, and where the determinant comes in #

Not every property transforms as a plain tensor. Some carry an extra factor of det M — they are axial, or pseudotensors — and the physical meaning is that they have a sense of rotation rather than a direction.

The example the field needs is optical activity, whose gyration tensor is an axial symmetric rank-2. Its character is det M times the ordinary one. In a centrosymmetric group the inversion contributes −χ where the identity contributes +χ, the operations pair off, and the pairs annihilate: the sum is exactly zero and the property is forbidden.

That is the symmetry reason a centrosymmetric crystal cannot be optically active, stated without any physics. And it is the same reason a centrosymmetric crystal cannot be piezoelectric, which is an odd-rank polar tensor and gets its sign change from the rank rather than from an axial factor.

The exact statement is worth having, because the version usually repeated — axial properties are forbidden by a centre of symmetry — is false, and the arithmetic that says so is one exponent. Replacing M by −M multiplies a polar character of rank r by (−1)ʳ, and multiplies an axial one by (−1)ʳ⁺¹, because the extra det M changes sign as well. So the pairs annihilate for an odd-rank polar property and for an even-rank axial one — the polar vector, the piezoelectric moduli, the gyration tensor — and for nothing else. An axial vector is odd rank and axial, the two sign changes cancel, and it comes through an inversion untouched. That is why a magnetic moment is not forbidden by a centre, and it is the one line that keeps this rule from being a slogan.

The same answer, twice, by different routes #

The count above never chooses axes. A picture of which component vanishes cannot avoid it — “the xy component is zero” is a statement about named directions — so the shape figures work in a Cartesian frame, build the projector explicitly on the six components, and read off which survive and which are tied together.

Those two calculations share almost nothing. One is a sum of traces of integer matrices; the other is a sum of outer products in floating point, in a frame chosen for the picture. They must agree, and the build fails if they do not.

That is the field’s version of the site’s standing arrangement: a pattern figure asserts a group from a point set, and a diffraction figure asserts the same group’s consequences from a sum in reciprocal space, and the two share the operation list and nothing else. Here the shared input is the group and the two routes are the character and the projector.

The whole table, and what it is good for #

Six characters times thirty-two classes is 192 integers, and printing them together makes visible something no single row does.

The rows do not fall together. A crystal in class 4̅ has seven elastic constants — more than the six of 422, which has the same number of operations — and four piezoelectric moduli where 422 has one. Symmetry is not a single quantity that a class has more or less of; it is a group, and different properties are sensitive to different parts of it.

The clearest case is the pair 432 and 4̅3m, both cubic, both of order 24, both non-centrosymmetric, both with three elastic constants and one dielectric constant. One of them can be piezoelectric and the other can be optically active, and it is not the same one. That is the pair the properties ladder is built around.

Where the exactness stops #

This is the caution that matters most in the whole field, and it is stated in every caption that quotes a number.

The count says which components are permitted. It says nothing about their values. Those are a measurement. A class permitting a non-zero piezoelectric modulus does not make a material piezoelectric; it fails to forbid it, and the modulus may be too small to use.

The principle is a one-way implication. A property’s symmetry must include the crystal’s. It may be higher — and often is, which is why an isotropic-looking property can appear in a low-symmetry crystal. The permittivity of a triclinic crystal has six independent components and nothing stops five of them being nearly zero.

And the tensor’s intrinsic symmetry is an input, not an output. That elasticity is symmetric under exchanging its index pairs is thermodynamics, not group theory; the calculation is handed that symmetry and counts what survives. A property with a different intrinsic symmetry needs a different character and is a different sum.

Who said it, and what Curie added #

Neumann taught the principle in Königsberg from the 1830s and never published it; it appears in the 1885 edition of his lectures, edited by his student Woldemar Voigt, and Voigt is the reason it is a working tool rather than a remark. Voigt’s Lehrbuch der Kristallphysik of 1910 worked out the tensor forms class by class, by hand, and the tables in it are still the ones reproduced in textbooks a century later.

Pierre Curie generalised it in 1894, and the generalisation is the more quotable half: c’est la dissymétrie qui crée le phénomène — it is the asymmetry that creates the phenomenon. Curie’s version is about the symmetry of causes and effects rather than of a crystal and its properties, and it says that the symmetry of an effect must include the symmetry of its cause, so an effect can be less symmetric than its cause only if something else broke the symmetry.

The practical difference is what happens when a field is applied. Neumann’s principle constrains the crystal’s own properties; Curie’s says what to do when a crystal is put in an electric field, which lowers the effective symmetry to the operations the crystal and the field share. That is why an applied field can produce effects a crystal’s own class forbids, and it is not a violation of anything.

This site computes Neumann’s version, which is the one that is a finite calculation over a finite group. Curie’s is a statement about symmetry groups of arbitrary configurations and is not enumerable.

Why this is not the representation theory it resembles #

Anybody who has met character tables will recognise the machinery, and the resemblance is close enough that it is worth saying what is not being done here.

Representation theory decomposes a reducible representation into irreducibles, and the standard tool is the inner product of characters: the multiplicity of an irreducible χᵢ in a representation χ is (1/|G|) Σ χ(M) χᵢ(M)*. That machinery answers questions about which transitions are allowed, which vibrational modes are infrared-active, and how a degenerate level splits in a crystal field.

What this field computes is the same formula with χᵢ set to the trivial representation, whose character is 1 everywhere. So the sum reduces to the average of χ, and the answer is one number rather than a decomposition.

That is the whole of the reduction, and it is why no character table appears here. A table is needed to identify which irreducibles are present; counting the invariant subspace needs only the character of the representation being asked about, and that is built from tr M and tr M² directly.

The larger machinery is a deliberate omission rather than an oversight. Selection rules for the crystallographic groups, irreducible representations of a space group, little groups and Brillouin-zone symmetry are the standard next step, and each is a body of work that this site has not taken and says so. molecular-geometry.com owns the vanishing-integral theorem applied to a molecular point group; the crystallographic version remains unclaimed.

Why the sum runs over every operation #

The formula averages over the whole group, and it is worth saying what goes wrong if it does not, because the shortcut is tempting and the failure is silent.

A group of order forty-eight is described by two or three generators, and it is natural to think that averaging over the generators would do — after all, a tensor invariant under the generators is invariant under everything they produce. That is true of the tensors and false of the projector. What makes P a projector is that multiplying it by any ρ(M) permutes the terms of the sum and returns P itself, so P² = P and its trace is a dimension. Average over three generators and the result is some operator with no such property, whose trace is a number with no meaning.

The consequence is that the character sum is an assertion about closure as much as about tensors. If the operation list is not a group, the average is not an integer, because it is not the trace of a projector and has no reason to be one. So the integrality check on every build is doing two jobs at once: it catches arithmetic error, and it catches an operation list that has lost a member or gained a spurious one. A class enumerated with forty-seven of its forty-eight operations gives fractional counts across the whole table, which is a far louder failure than a slightly wrong integer would be.

That is the same arrangement as elsewhere on this site — an enumeration checked by a closed form it cannot influence — with the useful difference that here the closed form is a consequence of the enumeration being right rather than an independent count of it.

An even-rank property cannot see an inversion, and an axial one sees nothing else #

One line of the same arithmetic explains a whole pattern in the table, and it is the pattern a reader notices first.

Adding an inversion to a group G gives G together with −G, twice the order. For an even-rank polar property the character is a polynomial in traces of even total degree, so χ(−M) = χ(M), the two halves of the sum are identical, and the average is unchanged. A class and the centrosymmetric class above it therefore permit exactly the same number of components of any even-rank polar property — which is why the elastic and dielectric columns take only eleven distinct patterns across the thirty-two classes, one for each Laue class, and why a measurement of either can never distinguish a class from its Laue class.

For an axial even-rank property the same substitution gives χ(−M) = −χ(M), since the extra factor of det M changes sign in three dimensions. The two halves cancel term by term and the average is exactly zero. That is the general statement behind a centrosymmetric crystal is not optically active: not a fact about light, and not a case analysis over eleven classes, but a cancellation that happens whatever the group was before the inversion was added.

And for an odd-rank polar property, χ(−M) = −χ(M) again, for the same reason with a different exponent — which is the piezoelectric cancellation and the polar vector’s at once. Three of the table’s most-quoted rules are one substitution, made three times.

Where this goes #

One sum, six properties, thirty-two classes: a table of 192 integers, none of them quoted. The three that are worth an essay each are the number of elastic constants, which runs 21, 13, 9, 7, 6, 5, 3 down the systems and is the oldest of these results; the ten polar classes, which are where ferroelectricity has to live; and the twenty of twenty-one that may be piezoelectric, whose single exception is the most interesting entry in the table.