Heisenberg group

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https://en.wikipedia.org/wiki/Heisenberg_group

In mathematics, the Heisenberg group ${\displaystyle H}$, named after Werner Heisenberg, is the group of 3×3 upper triangular matrices of the form

${\displaystyle \left(\begin{array}{ccc}1& a& c\\ 0& 1& b\\ 0& 0& 1\end{array}\right)}$

under the operation of matrix multiplication. Elements a, b and c can be taken from any commutative ring with identity, often taken to be the ring of real numbers (resulting in the "continuous Heisenberg group") or the ring of integers (resulting in the "discrete Heisenberg group").

The continuous Heisenberg group arises in the description of one-dimensional quantum mechanical systems, especially in the context of the Stone–von Neumann theorem. More generally, one can consider Heisenberg groups associated to n-dimensional systems, and most generally, to any symplectic vector space.

The three-dimensional case

In the three-dimensional case, the product of two Heisenberg matrices is given by:

${\displaystyle \left(\begin{array}{ccc}1& a& c\\ 0& 1& b\\ 0& 0& 1\end{array}\right)\left(\begin{array}{ccc}1& a^{\prime }& c^{\prime }\\ 0& 1& b^{\prime }\\ 0& 0& 1\end{array}\right)=\left(\begin{array}{ccc}1& a+a^{\prime }& c+a{b}^{\prime }+c^{\prime }\\ 0& 1& b+b^{\prime }\\ 0& 0& 1\end{array}\right).}$

As one can see from the term ab', the group is non-abelian.

The neutral element of the Heisenberg group is the identity matrix, and inverses are given by

${\displaystyle {\left(\begin{array}{ccc}1& a& c\\ 0& 1& b\\ 0& 0& 1\end{array}\right)}^{-1}=\left(\begin{array}{ccc}1& -a& ab-c\\ 0& 1& -b\\ 0& 0& 1\end{array}\right).}$

The group is a subgroup of the 2-dimensional affine group Aff(2): ${\displaystyle \left(\begin{array}{ccc}1& a& c\\ 0& 1& b\\ 0& 0& 1\end{array}\right)}$ acting on ${\displaystyle (\overrightarrow{x},1)}$ corresponds to the affine transform ${\displaystyle \left(\begin{array}{cc}1& a\\ 0& 1\end{array}\right)\overrightarrow{x}+\left(\begin{array}{c}c\\ b\end{array}\right)}$.

There are several prominent examples of the three-dimensional case.

Continuous Heisenberg group

If a, b, c, are real numbers (in the ring R) then one has the continuous Heisenberg group H₃(R).

It is a nilpotent real Lie group of dimension 3.

In addition to the representation as real 3×3 matrices, the continuous Heisenberg group also has several different representations in terms of function spaces. By Stone–von Neumann theorem, there is, up to isomorphism, a unique irreducible unitary representation of H in which its centre acts by a given nontrivial character. This representation has several important realizations, or models. In the Schrödinger model, the Heisenberg group acts on the space of square integrable functions. In the theta representation, it acts on the space of holomorphic functions on the upper half-plane; it is so named for its connection with the theta functions.

Discrete Heisenberg group

A portion of the Cayley graph of the discrete Heisenberg group, with generators x,y,z as in the text. (The coloring is only for visual aid.)

If a, b, c, are integers (in the ring Z) then one has the discrete Heisenberg group H₃(Z). It is a non-abelian nilpotent group. It has two generators,

${\displaystyle x=\left(\begin{array}{ccc}1& 1& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),y=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 1\\ 0& 0& 1\end{array}\right)}$

and relations

${\displaystyle z_{}^{}=xy{x}^{-1}{y}^{-1},xz=zx,yz=zy}$,

where

${\displaystyle z=\left(\begin{array}{ccc}1& 0& 1\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)}$

is the generator of the center of H₃. (Note that the inverses of x, y, and z replace the 1 above the diagonal with −1.)

By Bass's theorem, it has a polynomial growth rate of order 4.

One can generate any element through

${\displaystyle \left(\begin{array}{ccc}1& a& c\\ 0& 1& b\\ 0& 0& 1\end{array}\right)=y^b{z}^c{x}^a.}$

Heisenberg group modulo an odd prime p

If one takes a, b, c in Z/p Z for an odd prime p, then one has the Heisenberg group modulo p. It is a group of order p³ with generators x,y and relations:

${\displaystyle z_{}^{}=xy{x}^{-1}{y}^{-1},x^p=y^p=z^p=1,xz=zx,yz=zy.}$

Analogues of Heisenberg groups over finite fields of odd prime order p are called extra special groups, or more properly, extra special groups of exponent p. More generally, if the derived subgroup of a group G is contained in the center Z of G, then the map from G/Z × G/Z → Z is a skew-symmetric bilinear operator on abelian groups.

However, requiring that G/Z to be a finite vector space requires the Frattini subgroup of G to be contained in the center, and requiring that Z be a one-dimensional vector space over Z/p Z requires that Z have order p, so if G is not abelian, then G is extra special. If G is extra special but does not have exponent p, then the general construction below applied to the symplectic vector space G/Z does not yield a group isomorphic to G.

Heisenberg group modulo 2

The Heisenberg group modulo 2 is of order 8 and is isomorphic to the dihedral group D₄ (the symmetries of a square). Observe that if

${\displaystyle x=\left(\begin{array}{ccc}1& 1& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right),y=\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 1\\ 0& 0& 1\end{array}\right)}$.

Then

${\displaystyle xy=\left(\begin{array}{ccc}1& 1& 1\\ 0& 1& 1\\ 0& 0& 1\end{array}\right),}$

and

${\displaystyle yx=\left(\begin{array}{ccc}1& 1& 0\\ 0& 1& 1\\ 0& 0& 1\end{array}\right).}$

The elements x and y correspond to reflections (with 45° between them), whereas xy and yx correspond to rotations by 90°. The other reflections are xyx and yxy, and rotation by 180° is xyxy (=yxyx).

Heisenberg algebra

The Lie algebra ${\displaystyle \mathfrak{h}}$ of the Heisenberg group ${\displaystyle H}$ (over the real numbers) is known as the Heisenberg algebra. It may be represented using the space of 3×3 matrices of the form

${\displaystyle \left(\begin{array}{ccc}0& a& c\\ 0& 0& b\\ 0& 0& 0\end{array}\right),}$

with ${\displaystyle a,b,c\in \mathbb{R}}$.

The following three elements form a basis for ${\displaystyle \mathfrak{h}}$,

${\displaystyle X=\left(\begin{array}{ccc}0& 1& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right);Y=\left(\begin{array}{ccc}0& 0& 0\\ 0& 0& 1\\ 0& 0& 0\end{array}\right);Z=\left(\begin{array}{ccc}0& 0& 1\\ 0& 0& 0\\ 0& 0& 0\end{array}\right).}$

These basis elements satisfy the commutation relations,

${\displaystyle [X,Y]=Z;[X,Z]=0;[Y,Z]=0}$.

The name "Heisenberg group" is motivated by the preceding relations, which have the same form as the canonical commutation relations in quantum mechanics,

${\displaystyle \left[\hat{x},\hat{p}\right]=i\hslash I;\left[\hat{x},i\hslash I\right]=0;\left[\hat{p},i\hslash I\right]=0,}$

where ${\displaystyle \hat{x}}$ is the position operator, ${\displaystyle \hat{p}}$ is the momentum operator, and ${\displaystyle \hslash }$ is Planck's constant.

The Heisenberg group H has the special property that the exponential map is a one-to-one and onto map from the Lie algebra ${\displaystyle \mathfrak{h}}$ to the group H,

${\displaystyle \exp \left(\begin{array}{ccc}0& a& c\\ 0& 0& b\\ 0& 0& 0\end{array}\right)=\left(\begin{array}{ccc}1& a& c+\frac{ab}{2}\\ 0& 1& b\\ 0& 0& 1\end{array}\right).}$

In conformal field theory

In conformal field theory, the term Heisenberg algebra is used to refer to an infinite-dimensional generalization of the above algebra. It is spanned by elements ${\displaystyle a_n,n\in \mathbb{Z}}$, with commutation relations

${\displaystyle [a_n,a_m]=n{\delta }_{n+m,0}.}$

Under a rescaling, this is simply a countably-infinite number of copies of the above algebra.

Higher dimensions

More general Heisenberg groups ${\displaystyle H_{2n+1}}$ may be defined for higher dimensions in Euclidean space, and more generally on symplectic vector spaces. The simplest general case is the real Heisenberg group of dimension ${\displaystyle 2n+1}$, for any integer ${\displaystyle n\ge 1}$. As a group of matrices, ${\displaystyle H_{2n+1}}$ (or ${\displaystyle H_{2n+1}(\mathbb{R})}$ to indicate this is the Heisenberg group over the field ${\displaystyle \mathbb{R}}$ of real numbers) is defined as the group ${\displaystyle (n+2)\times (n+2)}$ matrices with entries in ${\displaystyle \mathbb{R}}$ and having the form:

${\displaystyle \left[\begin{array}{ccc}1& \mathbf{a}& c\\ \mathbf{0}& I_n& \mathbf{b}\\ 0& \mathbf{0}& 1\end{array}\right]}$

where

a is a row vector of length n,

b is a column vector of length n,

I_n is the identity matrix of size n.

Group structure

This is indeed a group, as is shown by the multiplication:

${\displaystyle \left[\begin{array}{ccc}1& \mathbf{a}& c\\ 0& I_n& \mathbf{b}\\ 0& 0& 1\end{array}\right]\cdot \left[\begin{array}{ccc}1& {\mathbf{a}}^{\prime }& c^{\prime }\\ 0& I_n& {\mathbf{b}}^{\prime }\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}1& \mathbf{a}+{\mathbf{a}}^{\prime }& c+c^{\prime }+\mathbf{a}\cdot {\mathbf{b}}^{\prime }\\ 0& I_n& \mathbf{b}+{\mathbf{b}}^{\prime }\\ 0& 0& 1\end{array}\right]}$

and

${\displaystyle \left[\begin{array}{ccc}1& \mathbf{a}& c\\ 0& I_n& \mathbf{b}\\ 0& 0& 1\end{array}\right]\cdot \left[\begin{array}{ccc}1& -\mathbf{a}& -c+\mathbf{a}\cdot \mathbf{b}\\ 0& I_n& -\mathbf{b}\\ 0& 0& 1\end{array}\right]=\left[\begin{array}{ccc}1& 0& 0\\ 0& I_n& 0\\ 0& 0& 1\end{array}\right].}$

Lie algebra

The Heisenberg group is a simply-connected Lie group whose Lie algebra consists of matrices

${\displaystyle \left[\begin{array}{ccc}0& \mathbf{a}& c\\ 0& 0_n& \mathbf{b}\\ 0& 0& 0\end{array}\right],}$

where

a is a row vector of length n,

b is a column vector of length n,

0_n is the zero matrix of size n.

By letting e₁, ..., e_n be the canonical basis of Rⁿ, and setting

${\displaystyle \begin{array}{rl}{p}_i& =\left[\begin{array}{ccc}0& {\mathrm{e}}_i^{\mathrm{T}}& 0\\ 0& 0_n& 0\\ 0& 0& 0\end{array}\right],\\ q_j& =\left[\begin{array}{ccc}0& 0& 0\\ 0& 0_n& {\mathrm{e}}_j\\ 0& 0& 0\end{array}\right],\\ z& =\left[\begin{array}{ccc}0& 0& 1\\ 0& 0_n& 0\\ 0& 0& 0\end{array}\right],\end{array}}$

the associated Lie algebra can be characterized by the canonical commutation relations,

${\displaystyle \left[\begin{array}{c}{p}_i,q_j\end{array}\right]={\delta }_{ij}z,\left[\begin{array}{c}{p}_i,z\end{array}\right]=0,\left[\begin{array}{c}{q}_j,z\end{array}\right]=0,}$

(1)

where p₁, ..., p_n, q₁, ..., q_n, z are the algebra generators.

In particular, z is a central element of the Heisenberg Lie algebra. Note that the Lie algebra of the Heisenberg group is nilpotent.

Exponential map

Let

${\displaystyle u=\left[\begin{array}{ccc}0& \mathbf{a}& c\\ 0& 0_n& \mathbf{b}\\ 0& 0& 0\end{array}\right],}$

which fulfills ${\displaystyle u^3=0_{n+2}}$. The exponential map evaluates to

${\displaystyle \exp (u)=\sum _{k=0}^{\mathrm{\infty }}\frac{1}{k!}{u}^k=I_{n+2}+u+\frac{1}{2}{u}^2=\left[\begin{array}{ccc}1& \mathbf{a}& c+\frac{1}{2}\mathbf{a}\cdot \mathbf{b}\\ 0& I_n& \mathbf{b}\\ 0& 0& 1\end{array}\right].}$

The exponential map of any nilpotent Lie algebra is a diffeomorphism between the Lie algebra and the unique associated connected, simply-connected Lie group.

This discussion (aside from statements referring to dimension and Lie group) further applies if we replace R by any commutative ring A. The corresponding group is denoted H_n(A ).

Under the additional assumption that the prime 2 is invertible in the ring A, the exponential map is also defined, since it reduces to a finite sum and has the form above (e.g. A could be a ring Z/p Z with an odd prime p or any field of characteristic 0).

Representation theory

Main article: Stone–von Neumann theorem

The unitary representation theory of the Heisenberg group is fairly simple – later generalized by Mackey theory – and was the motivation for its introduction in quantum physics, as discussed below.

For each nonzero real number ${\displaystyle \hslash }$, we can define an irreducible unitary representation ${\displaystyle {\mathrm{\Pi }}_{\hslash }}$ of ${\displaystyle H_{2n+1}}$ acting on the Hilbert space ${\displaystyle L^2({\mathbb{R}}^n)}$ by the formula:

${\displaystyle \left[{\mathrm{\Pi }}_{\hslash }\left(\begin{array}{ccc}1& \mathbf{a}& c\\ 0& I_n& \mathbf{b}\\ 0& 0& 1\end{array}\right)\psi \right](x)=e^{i\hslash c}{e}^{ib\cdot x}\psi (x+\hslash a)}$

This representation is known as the Schrödinger representation. The motivation for this representation is the action of the exponentiated position and momentum operators in quantum mechanics. The parameter ${\displaystyle a}$ describes translations in position space, the parameter ${\displaystyle b}$ describes translations in momentum space, and the parameter ${\displaystyle c}$ gives an overall phase factor. The phase factor is needed to obtain a group of operators, since translations in position space and translations in momentum space do not commute.

The key result is the Stone–von Neumann theorem, which states that every (strongly continuous) irreducible unitary representation of the Heisenberg group in which the center acts nontrivially is equivalent to ${\displaystyle {\mathrm{\Pi }}_{\hslash }}$ for some ${\displaystyle \hslash }$. Alternatively, that they are all equivalent to the Weyl algebra (or CCR algebra) on a symplectic space of dimension 2n.

Since the Heisenberg group is a one-dimensional central extension of ${\displaystyle {\mathbb{R}}^{2n}}$, its irreducible unitary representations can be viewed as irreducible unitary projective representations of ${\displaystyle {\mathbb{R}}^{2n}}$. Conceptually, the representation given above constitutes the quantum mechanical counterpart to the group of translational symmetries on the classical phase space, ${\displaystyle {\mathbb{R}}^{2n}}$. The fact that the quantum version is only a projective representation of ${\displaystyle {\mathbb{R}}^{2n}}$ is suggested already at the classical level. The Hamiltonian generators of translations in phase space are the position and momentum functions. The span of these functions do not form a Lie algebra under the Poisson bracket however, because ${\displaystyle \{x_i,p_j\}={\delta }_{i,j}.}$ Rather, the span of the position and momentum functions and the constants forms a Lie algebra under the Poisson bracket. This Lie algebra is a one-dimensional central extension of the commutative Lie algebra ${\displaystyle {\mathbb{R}}^{2n}}$, isomorphic to the Lie algebra of the Heisenberg group.

On symplectic vector spaces

The general abstraction of a Heisenberg group is constructed from any symplectic vector space. For example, let (V, ω) be a finite-dimensional real symplectic vector space (so ω is a nondegenerate skew symmetric bilinear form on V). The Heisenberg group H(V) on (V, ω) (or simply V for brevity) is the set V×R endowed with the group law

${\displaystyle (v,t)\cdot \left(v^{\prime },t^{\prime }\right)=\left(v+v^{\prime },t+t^{\prime }+\frac{1}{2}\omega \left(v,v^{\prime }\right)\right).}$

The Heisenberg group is a central extension of the additive group V. Thus there is an exact sequence

${\displaystyle 0\to \mathbf{R}\to H(V)\to V\to 0.}$

Any symplectic vector space admits a Darboux basis {e_j, f^k}_{1 ≤ j,k ≤ n} satisfying ω(e_j, f^k) = δ_j^k and where 2n is the dimension of V (the dimension of V is necessarily even). In terms of this basis, every vector decomposes as

${\displaystyle v=q^a{\mathbf{e}}_a+p_a{\mathbf{f}}^a.}$

The q^a and p_a are canonically conjugate coordinates.

If {e_j, f^k}_{1 ≤ j,k ≤ n} is a Darboux basis for V, then let {E} be a basis for R, and {e_j, f^k, E}_{1 ≤ j,k ≤ n} is the corresponding basis for V×R. A vector in H(V) is then given by

${\displaystyle v=q^a{\mathbf{e}}_a+p_a{\mathbf{f}}^a+tE}$

and the group law becomes

${\displaystyle (p,q,t)\cdot \left(p^{\prime },q^{\prime },t^{\prime }\right)=\left(p+p^{\prime },q+q^{\prime },t+t^{\prime }+\frac{1}{2}(p{q}^{\prime }-p^{\prime }q)\right).}$

Because the underlying manifold of the Heisenberg group is a linear space, vectors in the Lie algebra can be canonically identified with vectors in the group. The Lie algebra of the Heisenberg group is given by the commutation relation

${\displaystyle \left[\begin{array}{c}(v_1,t_1),(v_2,t_2)\end{array}\right]=\omega (v_1,v_2)}$

or written in terms of the Darboux basis

${\displaystyle \left[{\mathbf{e}}_a,{\mathbf{f}}^b\right]={\delta }_a^b}$

and all other commutators vanish.

It is also possible to define the group law in a different way but which yields a group isomorphic to the group we have just defined. To avoid confusion, we will use u instead of t, so a vector is given by

${\displaystyle v=q^a{\mathbf{e}}_a+p_a{\mathbf{f}}^a+uE}$

and the group law is

${\displaystyle (p,q,u)\cdot \left(p^{\prime },q^{\prime },u^{\prime }\right)=\left(p+p^{\prime },q+q^{\prime },u+u^{\prime }+p{q}^{\prime }\right).}$

An element of the group

${\displaystyle v=q^a{\mathbf{e}}_a+p_a{\mathbf{f}}^a+uE}$

can then be expressed as a matrix

${\displaystyle \left[\begin{array}{ccc}1& p& u\\ 0& I_n& q\\ 0& 0& 1\end{array}\right]}$ ,

which gives a faithful matrix representation of H(V). The u in this formulation is related to t in our previous formulation by ${\displaystyle u=t+\frac{1}{2}pq}$, so that the t value for the product comes to

${\displaystyle \begin{array}{rl}& u+u^{\prime }+p{q}^{\prime }-\frac{1}{2}\left(p+p^{\prime }\right)\left(q+q^{\prime }\right)\\ =& t+\frac{1}{2}pq+t^{\prime }+\frac{1}{2}{p}^{\prime }{q}^{\prime }+p{q}^{\prime }-\frac{1}{2}\left(p+p^{\prime }\right)\left(q+q^{\prime }\right)\\ =& t+t^{\prime }+\frac{1}{2}\left(p{q}^{\prime }-p^{\prime }q\right)\end{array}}$ ,

as before.

The isomorphism to the group using upper triangular matrices relies on the decomposition of V into a Darboux basis, which amounts to a choice of isomorphism V ≅ U ⊕ U*. Although the new group law yields a group isomorphic to the one given higher up, the group with this law is sometimes referred to as the polarized Heisenberg group as a reminder that this group law relies on a choice of basis (a choice of a Lagrangian subspace of V is a polarization).

To any Lie algebra, there is a unique connected, simply connected Lie group G. All other connected Lie groups with the same Lie algebra as G are of the form G/N where N is a central discrete group in G. In this case, the center of H(V) is R and the only discrete subgroups are isomorphic to Z. Thus H(V)/Z is another Lie group which shares this Lie algebra. Of note about this Lie group is that it admits no faithful finite-dimensional representations; it is not isomorphic to any matrix group. It does however have a well-known family of infinite-dimensional unitary representations.

The connection with the Weyl algebra

Main article: Weyl algebra

The Lie algebra ${\displaystyle {\mathfrak{h}}_n}$ of the Heisenberg group was described above, (1), as a Lie algebra of matrices. The Poincaré–Birkhoff–Witt theorem applies to determine the universal enveloping algebra ${\displaystyle U({\mathfrak{h}}_n)}$. Among other properties, the universal enveloping algebra is an associative algebra into which ${\displaystyle {\mathfrak{h}}_n}$ injectively imbeds.

By the Poincaré–Birkhoff–Witt theorem, it is thus the free vector space generated by the monomials

${\displaystyle z^j{p}_1^{k_1}{p}_2^{k_2}\cdots p_n^{k_n}{q}_1^{{\ell }_1}{q}_2^{{\ell }_2}\cdots q_n^{{\ell }_n},}$

where the exponents are all non-negative.

Consequently, ${\displaystyle U({\mathfrak{h}}_n)}$ consists of real polynomials

${\displaystyle \sum _{j,\overrightarrow{k},\overrightarrow{\ell }}{c}_{j\overrightarrow{k}\overrightarrow{\ell }}{z}^j{p}_1^{k_1}{p}_2^{k_2}\cdots p_n^{k_n}{q}_1^{{\ell }_1}{q}_2^{{\ell }_2}\cdots q_n^{{\ell }_n},}$

with the commutation relations

${\displaystyle p_k{p}_{\ell }=p_{\ell }{p}_k,q_k{q}_{\ell }=q_{\ell }{q}_k,p_k{q}_{\ell }-q_{\ell }{p}_k={\delta }_{k\ell }z,z{p}_k-p_{k}z=0,z{q}_k-q_{k}z=0.}$

The algebra ${\displaystyle U({\mathfrak{h}}_n)}$ is closely related to the algebra of differential operators on ℝⁿ with polynomial coefficients, since any such operator has a unique representation in the form

${\displaystyle P=\sum _{\overrightarrow{k},\overrightarrow{\ell }}{c}_{\overrightarrow{k}\overrightarrow{\ell }}{\mathrm{\partial }}_{x_1}^{k_1}{\mathrm{\partial }}_{x_2}^{k_2}\cdots {\mathrm{\partial }}_{x_n}^{k_n}{x}_1^{{\ell }_1}{x}_2^{{\ell }_2}\cdots x_n^{{\ell }_n}.}$

This algebra is called the Weyl algebra. It follows from abstract nonsense that the Weyl algebra W_n is a quotient of ${\displaystyle U({\mathfrak{h}}_n)}$. However, this is also easy to see directly from the above representations; viz. by the mapping

${\displaystyle z^j{p}_1^{k_1}{p}_2^{k_2}\cdots p_n^{k_n}{q}_1^{{\ell }_1}{q}_2^{{\ell }_2}\cdots q_n^{{\ell }_n}\mapsto {\mathrm{\partial }}_{x_1}^{k_1}{\mathrm{\partial }}_{x_2}^{k_2}\cdots {\mathrm{\partial }}_{x_n}^{k_n}{x}_1^{{\ell }_1}{x}_2^{{\ell }_2}\cdots x_n^{{\ell }_n}.}$

Applications

Weyl's parameterization of quantum mechanics

Main article: Wigner–Weyl transform

The application that led Hermann Weyl to an explicit realization of the Heisenberg group was the question of why the Schrödinger picture and Heisenberg picture are physically equivalent. Abstractly, the reason is the Stone–von Neumann theorem: there is a unique unitary representation with given action of the central Lie algebra element z, up to a unitary equivalence: the nontrivial elements of the algebra are all equivalent to the usual position and momentum operators.

Thus, the Schrödinger picture and Heisenberg picture are equivalent – they are just different ways of realizing this essentially unique representation.

Theta representation

Main article: theta representation

The same uniqueness result was used by David Mumford for discrete Heisenberg groups, in his theory of equations defining abelian varieties. This is a large generalization of the approach used in Jacobi's elliptic functions, which is the case of the modulo 2 Heisenberg group, of order 8. The simplest case is the theta representation of the Heisenberg group, of which the discrete case gives the theta function.

Fourier analysis

The Heisenberg group also occurs in Fourier analysis, where it is used in some formulations of the Stone–von Neumann theorem. In this case, the Heisenberg group can be understood to act on the space of square integrable functions; the result is a representation of the Heisenberg groups sometimes called the Weyl representation.

As a sub-Riemannian manifold

An animation of a geodesic in the Heisenberg group.

The three-dimensional Heisenberg group H₃(R) on the reals can also be understood to be a smooth manifold, and specifically, a simple example of a sub-Riemannian manifold. Given a point p=(x,y,z) in R³, define a differential 1-form Θ at this point as

${\displaystyle {\mathrm{\Theta }}_p=dz-\frac{1}{2}\left(xdy-ydx\right).}$

This one-form belongs to the cotangent bundle of R³; that is,

${\displaystyle {\mathrm{\Theta }}_p:T_p{\mathbf{R}}^3\to \mathbf{R}}$

is a map on the tangent bundle. Let

${\displaystyle H_p=\left\{v\in T_p{\mathbf{R}}^3\mid {\mathrm{\Theta }}_p(v)=0\right\}.}$

It can be seen that H is a subbundle of the tangent bundle TR³. A cometric on H is given by projecting vectors to the two-dimensional space spanned by vectors in the x and y direction. That is, given vectors ${\displaystyle v=(v_1,v_2,v_3)}$ and ${\displaystyle w=(w_1,w_2,w_3)}$ in TR³, the inner product is given by

${\displaystyle ⟨v,w⟩=v_1{w}_1+v_2{w}_2.}$

The resulting structure turns H into the manifold of the Heisenberg group. An orthonormal frame on the manifold is given by the Lie vector fields

${\displaystyle \begin{array}{rl}X& =\frac{\mathrm{\partial }}{\mathrm{\partial }x}-\frac{1}{2}y\frac{\mathrm{\partial }}{\mathrm{\partial }z},\\ Y& =\frac{\mathrm{\partial }}{\mathrm{\partial }y}+\frac{1}{2}x\frac{\mathrm{\partial }}{\mathrm{\partial }z},\\ Z& =\frac{\mathrm{\partial }}{\mathrm{\partial }z},\end{array}}$

which obey the relations [X, Y] = Z and [X, Z] = [Y, Z] = 0. Being Lie vector fields, these form a left-invariant basis for the group action. The geodesics on the manifold are spirals, projecting down to circles in two dimensions. That is, if

${\displaystyle \gamma (t)=(x(t),y(t),z(t))}$

is a geodesic curve, then the curve ${\displaystyle c(t)=(x(t),y(t))}$ is an arc of a circle, and

${\displaystyle z(t)=\frac{1}{2}{\int }_{c}xdy-ydx}$

with the integral limited to the two-dimensional plane. That is, the height of the curve is proportional to the area of the circle subtended by the circular arc, which follows by Stokes' theorem.

Heisenberg group of a locally compact abelian group

It is more generally possible to define the Heisenberg group of a locally compact abelian group K, equipped with a Haar measure. Such a group has a Pontrjagin dual ${\displaystyle \hat{K}}$, consisting of all continuous ${\displaystyle U(1)}$-valued characters on K, which is also a locally compact abelian group if endowed with the compact-open topology. The Heisenberg group associated with the locally compact abelian group K is the subgroup of the unitary group of ${\displaystyle L^2(K)}$ generated by translations from K and multiplications by elements of ${\displaystyle \hat{K}}$.

In more detail, the Hilbert space ${\displaystyle L^2(K)}$ consists of square-integrable complex-valued functions ${\displaystyle f}$ on K. The translations in K form a unitary representation of K as operators on ${\displaystyle L^2(K)}$:

${\displaystyle (T_{x}f)(y)=f(x+y)}$

for ${\displaystyle x,y\in K}$. So too do the multiplications by characters:

${\displaystyle (M_{\chi }f)(y)=\chi (y)f(y)}$

for ${\displaystyle \chi \in \hat{K}}$. These operators do not commute, and instead satisfy

${\displaystyle \left(T_x{M}_{\chi }{T}_x^{-1}{M}_{\chi }^{-1}f\right)(y)=\overline{\chi (x)}f(y)}$

multiplication by a fixed unit modulus complex number.

So the Heisenberg group ${\displaystyle H(K)}$ associated with K is a type of central extension of ${\displaystyle K\times \hat{K}}$, via an exact sequence of groups:

${\displaystyle 1\to U(1)\to H(K)\to K\times \hat{K}\to 0.}$

More general Heisenberg groups are described by 2-cocyles in the cohomology group ${\displaystyle H^2(K,U(1))}$. The existence of a duality between ${\displaystyle K}$ and ${\displaystyle \hat{K}}$ gives rise to a canonical cocycle, but there are generally others.

The Heisenberg group acts irreducibly on ${\displaystyle L^2(K)}$. Indeed, the continuous characters separate points so any unitary operator of ${\displaystyle L^2(K)}$ that commutes with them is an ${\displaystyle L^{\mathrm{\infty }}}$ multiplier. But commuting with translations implies that the multiplier is constant.

A version of the Stone–von Neumann theorem, proved by George Mackey, holds for the Heisenberg group ${\displaystyle H(K)}$. The Fourier transform is the unique intertwiner between the representations of ${\displaystyle L^2(K)}$ and ${\displaystyle L^2\left(\hat{K}\right)}$. See the discussion at Stone–von Neumann theorem#Relation to the Fourier transform for details.

Period 5 element

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Period_5_element 

A period 5 element is one of the chemical elements in the fifth row (or period) of the periodic table of the chemical elements. The periodic table is laid out in rows to illustrate recurring (periodic) trends in the chemical behaviour of the elements as their atomic number increases: a new row is begun when chemical behaviour begins to repeat, meaning that elements with similar behaviour fall into the same vertical columns. The fifth period contains 18 elements, beginning with rubidium and ending with xenon. As a rule, period 5 elements fill their 5s shells first, then their 4d, and 5p shells, in that order; however, there are exceptions, such as rhodium.

Physical properties

This period contains technetium, one of the two elements until lead that has no stable isotopes (along with promethium), as well as molybdenum and iodine, two of the heaviest elements with a known biological role, and Niobium has the largest magnetic known penetration depth of all the elements. Zirconium is one of the main components of zircon crystals, currently the oldest known minerals in the earth's crust. Many later transition metals, such as rhodium, are very commonly used in jewelry due to the fact that they are incredibly shiny.

This period is known to have a large number of exceptions to the Madelung rule.

Elements and their properties

Chemical element			Block	Electron configuration
37	Rb	Rubidium	s-block	[Kr] 5s1
38	Sr	Strontium	s-block	[Kr] 5s2
39	Y	Yttrium	d-block	[Kr] 4d1 5s2
40	Zr	Zirconium	d-block	[Kr] 4d2 5s2
41	Nb	Niobium	d-block	[Kr] 4d4 5s1 (*)
42	Mo	Molybdenum	d-block	[Kr] 4d5 5s1 (*)
43	Tc	Technetium	d-block	[Kr] 4d5 5s2
44	Ru	Ruthenium	d-block	[Kr] 4d7 5s1 (*)
45	Rh	Rhodium	d-block	[Kr] 4d8 5s1 (*)
46	Pd	Palladium	d-block	[Kr] 4d10 (*)
47	Ag	Silver	d-block	[Kr] 4d10 5s1 (*)
48	Cd	Cadmium	d-block	[Kr] 4d10 5s2
49	In	Indium	p-block	[Kr] 4d10 5s2 5p1
50	Sn	Tin	p-block	[Kr] 4d10 5s2 5p2
51	Sb	Antimony	p-block	[Kr] 4d10 5s2 5p3
52	Te	Tellurium	p-block	[Kr] 4d10 5s2 5p4
53	I	Iodine	p-block	[Kr] 4d10 5s2 5p5
54	Xe	Xenon	p-block	[Kr] 4d10 5s2 5p6

(*) Exception to the Madelung rule

s-block elements

Rubidium

Main article: Rubidium

Rubidium is the first element placed in period 5. It is an alkali metal, the most reactive group in the periodic table, having properties and similarities with both other alkali metals and other period 5 elements. For example, rubidium has 5 electron shells, a property found in all other period 5 elements, whereas its electron configuration's ending is similar to all other alkali metals: s¹. Rubidium also follows the trend of increasing reactivity as the atomic number increases in the alkali metals, for it is more reactive than potassium, but less so than caesium. In addition, both potassium and rubidium yield almost the same hue when ignited, so researchers must use different methods to differentiate between these two 1st group elements. Rubidium is very susceptible to oxidation in air, similar to most of the other alkali metals, so it readily transforms into rubidium oxide, a yellow solid with the chemical formula Rb₂O.

Strontium

Main article: Strontium

Strontium is the second element placed in the 5th period. It is an alkaline earth metal, a relatively reactive group, although not nearly as reactive as the alkali metals. Like rubidium, it has 5 electron shells or energy levels, and in accordance with the Madelung rule it has two electrons in its 5s subshell. Strontium is a soft metal and is extremely reactive upon contact with water. If it does come in contact with water, though, it will combine with the atoms of both oxygen and hydrogen to form strontium hydroxide and pure hydrogen gas which quickly diffuses in the air. In addition, strontium, like rubidium, oxidizes in air and turns a yellow color. When ignited, it will burn with a strong red flame.

d-block elements

Yttrium

Main article: Yttrium

Yttrium is a chemical element with symbol Y and atomic number 39. It is a silvery-metallic transition metal chemically similar to the lanthanides and it has often been classified as a "rare earth element". Yttrium is almost always found combined with the lanthanides in rare earth minerals and is never found in nature as a free element. Its only stable isotope, ⁸⁹Y, is also its only naturally occurring isotope.

In 1787, Carl Axel Arrhenius found a new mineral near Ytterby in Sweden and named it ytterbite, after the village. Johan Gadolin discovered yttrium's oxide in Arrhenius' sample in 1789, and Anders Gustaf Ekeberg named the new oxide yttria. Elemental yttrium was first isolated in 1828 by Friedrich Wöhler.

The most important use of yttrium is in making phosphors, such as the red ones used in television set cathode ray tube (CRT) displays and in LEDs. Other uses include the production of electrodes, electrolytes, electronic filters, lasers and superconductors; various medical applications; and as traces in various materials to enhance their properties. Yttrium has no known biological role, and exposure to yttrium compounds can cause lung disease in humans.

Zirconium

Main article: Zirconium

Zirconium is a chemical element with the symbol Zr and atomic number 40. The name of zirconium is taken from the mineral zircon. Its atomic mass is 91.224. It is a lustrous, gray-white, strong transition metal that resembles titanium. Zirconium is mainly used as a refractory and opacifier, although minor amounts are used as alloying agent for its strong resistance to corrosion. Zirconium is obtained mainly from the mineral zircon, which is the most important form of zirconium in use.

Zirconium forms a variety of inorganic and organometallic compounds such as zirconium dioxide and zirconocene dichloride, respectively. Five isotopes occur naturally, three of which are stable. Zirconium compounds have no biological role.

Niobium

Main article: Niobium

Niobium, or columbium, is a chemical element with the symbol Nb and atomic number 41. It is a soft, grey, ductile transition metal, which is often found in the pyrochlore mineral, the main commercial source for niobium, and columbite. The name comes from Greek mythology: Niobe, daughter of Tantalus.

Niobium has physical and chemical properties similar to those of the element tantalum, and the two are therefore difficult to distinguish. The English chemistCharles Hatchett reported a new element similar to tantalum in 1801, and named it columbium. In 1809, the English chemist William Hyde Wollaston wrongly concluded that tantalum and columbium were identical. The German chemist Heinrich Rose determined in 1846 that tantalum ores contain a second element, which he named niobium. In 1864 and 1865, a series of scientific findings clarified that niobium and columbium were the same element (as distinguished from tantalum), and for a century both names were used interchangeably. The name of the element was officially adopted as niobium in 1949.

It was not until the early 20th century that niobium was first used commercially. Brazil is the leading producer of niobium and ferroniobium, an alloy of niobium and iron. Niobium is used mostly in alloys, the largest part in special steel such as that used in gas pipelines. Although alloys contain only a maximum of 0.1%, that small percentage of niobium improves the strength of the steel. The temperature stability of niobium-containing superalloys is important for its use in jet and rocket engines. Niobium is used in various superconducting materials. These superconducting alloys, also containing titanium and tin, are widely used in the superconducting magnets of MRI scanners. Other applications of niobium include its use in welding, nuclear industries, electronics, optics, numismatics and jewelry. In the last two applications, niobium's low toxicity and ability to be colored by anodization are particular advantages.

Molybdenum

Main article: Molybdenum

Molybdenum is a Group 6 chemical element with the symbol Mo and atomic number 42. The name is from Neo-Latin Molybdaenum, from Ancient GreekΜόλυβδος molybdos, meaning lead, itself proposed as a loanword from Anatolian Luvian and Lydian languages, since its ores were confused with lead ores. The free element, which is a silvery metal, has the sixth-highest melting point of any element. It readily forms hard, stable carbides, and for this reason it is often used in high-strength steel alloys. Molybdenum does not occur as a free metal on Earth, but rather in various oxidation states in minerals. Industrially, molybdenum compounds are used in high-pressure and high-temperature applications, as pigments and catalysts.

Molybdenum minerals have long been known, but the element was "discovered" (in the sense of differentiating it as a new entity from the mineral salts of other metals) in 1778 by Carl Wilhelm Scheele. The metal was first isolated in 1781 by Peter Jacob Hjelm.

Most molybdenum compounds have low solubility in water, but the molybdate ion MoO₄²⁻ is soluble and forms when molybdenum-containing minerals are in contact with oxygen and water.

Technetium

Main article: Technetium

Technetium is the chemical element with atomic number 43 and symbol Tc. It is the lowest atomic number element without any stable isotopes; every form of it is radioactive. Nearly all technetium is produced synthetically and only minute amounts are found in nature. Naturally occurring technetium occurs as a spontaneous fission product in uranium ore or by neutron capture in molybdenum ores. The chemical properties of this silvery gray, crystalline transition metal are intermediate between rhenium and manganese.

Many of technetium's properties were predicted by Dmitri Mendeleev before the element was discovered. Mendeleev noted a gap in his periodic table and gave the undiscovered element the provisional name ekamanganese (Em). In 1937 technetium (specifically the technetium-97 isotope) became the first predominantly artificial element to be produced, hence its name (from the Greek τεχνητός, meaning "artificial").

Its short-lived gamma ray-emitting nuclear isomer—technetium-99m—is used in nuclear medicine for a wide variety of diagnostic tests. Technetium-99 is used as a gamma ray-free source of beta particles. Long-lived technetium isotopes produced commercially are by-products of fission of uranium-235 in nuclear reactors and are extracted from nuclear fuel rods. Because no isotope of technetium has a half-life longer than 4.2 million years (technetium-98), its detection in red giants in 1952, which are billions of years old, helped bolster the theory that stars can produce heavier elements.

Ruthenium

Main article: Ruthenium

Ruthenium is a chemical element with symbol Ru and atomic number 44. It is a rare transition metal belonging to the platinum group of the periodic table. Like the other metals of the platinum group, ruthenium is inert to most chemicals. The Russian scientist Karl Ernst Claus discovered the element in 1844 and named it after Ruthenia, the Latin word for Rus'. Ruthenium usually occurs as a minor component of platinum ores and its annual production is only about 12 tonnes worldwide. Most ruthenium is used for wear-resistant electrical contacts and the production of thick-film resistors. A minor application of ruthenium is its use in some platinum alloys.

Rhodium

Main article: Rhodium

Rhodium is a chemical element that is a rare, silvery-white, hard, and chemically inert transition metal and a member of the platinum group. It has the chemical symbol Rh and atomic number 45. It is composed of only one isotope, ¹⁰³Rh. Naturally occurring rhodium is found as the free metal, alloyed with similar metals, and never as a chemical compound. It is one of the rarest precious metals and one of the most costly (gold has since taken over the top spot of cost per ounce).

Rhodium is a so-called noble metal, resistant to corrosion, found in platinum- or nickel ores together with the other members of the platinum group metals. It was discovered in 1803 by William Hyde Wollaston in one such ore, and named for the rose color of one of its chlorine compounds, produced after it reacted with the powerful acid mixture aqua regia.

The element's major use (about 80% of world rhodium production) is as one of the catalysts in the three-way catalytic converters of automobiles. Because rhodium metal is inert against corrosion and most aggressive chemicals, and because of its rarity, rhodium is usually alloyed with platinum or palladium and applied in high-temperature and corrosion-resistive coatings. White gold is often plated with a thin rhodium layer to improve its optical impression while sterling silver is often rhodium plated for tarnish resistance.

Rhodium detectors are used in nuclear reactors to measure the neutron flux level.

Palladium

Main article: Palladium

Palladium is a chemical element with the chemical symbol Pd and an atomic number of 46. It is a rare and lustrous silvery-white metal discovered in 1803 by William Hyde Wollaston. He named it after the asteroid Pallas, which was itself named after the epithet of the Greek goddess Athena, acquired by her when she slew Pallas. Palladium, platinum, rhodium, ruthenium, iridium and osmium form a group of elements referred to as the platinum group metals (PGMs). These have similar chemical properties, but palladium has the lowest melting point and is the least dense of them.

The unique properties of palladium and other platinum group metals account for their widespread use. A quarter of all goods manufactured today either contain PGMs or have a significant part in their manufacturing process played by PGMs. Over half of the supply of palladium and its congener platinum goes into catalytic converters, which convert up to 90% of harmful gases from auto exhaust (hydrocarbons, carbon monoxide, and nitrogen dioxide) into less-harmful substances (nitrogen, carbon dioxide and water vapor). Palladium is also used in electronics, dentistry, medicine, hydrogen purification, chemical applications, and groundwater treatment. Palladium plays a key role in the technology used for fuel cells, which combine hydrogen and oxygen to produce electricity, heat, and water.

Ore deposits of palladium and other PGMs are rare, and the most extensive deposits have been found in the norite belt of the Bushveld Igneous Complex covering the Transvaal Basin in South Africa, the Stillwater Complex in Montana, United States, the Thunder Bay District of Ontario, Canada, and the Norilsk Complex in Russia. Recycling is also a source of palladium, mostly from scrapped catalytic converters. The numerous applications and limited supply sources of palladium result in the metal attracting considerable investment interest.

Silver

Main article: Silver

Silver is a metallic chemical element with the chemical symbol Ag (Latin: argentum, from the Indo-European root *arg- for "grey" or "shining") and atomic number 47. A soft, white, lustrous transition metal, it has the highest electrical conductivity of any element and the highest thermal conductivity of any metal. The metal occurs naturally in its pure, free form (native silver), as an alloy with gold and other metals, and in minerals such as argentite and chlorargyrite. Most silver is produced as a byproduct of copper, gold, lead, and zinc refining.

Silver has long been valued as a precious metal, and it is used to make ornaments, jewelry, high-value tableware, utensils (hence the term silverware), and currency coins. Today, silver metal is also used in electrical contacts and conductors, in mirrors and in catalysis of chemical reactions. Its compounds are used in photographic film, and dilute silver nitrate solutions and other silver compounds are used as disinfectants and microbiocides. While many medical antimicrobial uses of silver have been supplanted by antibiotics, further research into clinical potential continues.

Cadmium

Main article: Cadmium

Cadmium is a chemical element with the symbol Cd and atomic number 48. This soft, bluish-white metal is chemically similar to the two other stable metals in group 12, zinc and mercury. Like zinc, it prefers oxidation state +2 in most of its compounds and like mercury it shows a low melting point compared to transition metals. Cadmium and its congeners are not always considered transition metals, in that they do not have partly filled d or f electron shells in the elemental or common oxidation states. The average concentration of cadmium in the Earth's crust is between 0.1 and 0.5 parts per million (ppm). It was discovered in 1817 simultaneously by Stromeyer and Hermann, both in Germany, as an impurity in zinc carbonate.

Cadmium occurs as a minor component in most zinc ores and therefore is a byproduct of zinc production. It was used for a long time as a pigment and for corrosion resistant plating on steel while cadmium compounds were used to stabilize plastic. With the exception of its use in nickel–cadmium batteries and cadmium telluride solar panels, the use of cadmium is generally decreasing. These declines have been due to competing technologies, cadmium's toxicity in certain forms and concentration and resulting regulations.

p-block elements

Indium

Main article: Indium

Indium is a chemical element with the symbol In and atomic number 49. This rare, very soft, malleable and easily fusible other metal is chemically similar to gallium and thallium, and shows the intermediate properties between these two. Indium was discovered in 1863 and named for the indigo blue line in its spectrum that was the first indication of its existence in zinc ores, as a new and unknown element. The metal was first isolated in the following year. Zinc ores continue to be the primary source of indium, where it is found in compound form. Very rarely the element can be found as grains of native (free) metal, but these are not of commercial importance.

Indium's current primary application is to form transparent electrodes from indium tin oxide in liquid crystal displays and touchscreens, and this use largely determines its global mining production. It is widely used in thin-films to form lubricated layers (during World War II it was widely used to coat bearings in high-performance aircraft). It is also used for making particularly low melting point alloys, and is a component in some lead-free solders.

Indium is not known to be used by any organism. In a similar way to aluminium salts, indium(III) ions can be toxic to the kidney when given by injection, but oral indium compounds do not have the chronic toxicity of salts of heavy metals, probably due to poor absorption in basic conditions. Radioactive indium-111 (in very small amounts on a chemical basis) is used in nuclear medicine tests, as a radiotracer to follow the movement of labeled proteins and white blood cells in the body.

Tin

Main article: Tin

Tin is a chemical element with the symbol Sn (for Latin: stannum) and atomic number 50. It is a main-group metal in group 14 of the periodic table. Tin shows chemical similarity to both neighboring group 14 elements, germanium and lead and has two possible oxidation states, +2 and the slightly more stable +4. Tin is the 49th most abundant element and has, with 10 stable isotopes, the largest number of stable isotopes in the periodic table. Tin is obtained chiefly from the mineral cassiterite, where it occurs as tin dioxide, SnO₂.

This silvery, malleable other metal is not easily oxidized in air and is used to coat other metals to prevent corrosion. The first alloy, used in large scale since 3000 BC, was bronze, an alloy of tin and copper. After 600 BC pure metallic tin was produced. Pewter, which is an alloy of 85–90% tin with the remainder commonly consisting of copper, antimony and lead, was used for tableware from the Bronze Age until the 20th century. In modern times tin is used in many alloys, most notably tin/lead soft solders, typically containing 60% or more of tin. Another large application for tin is corrosion-resistant tin plating of steel. Because of its low toxicity, tin-plated metal is also used for food packaging, giving the name to tin cans, which are made mostly of steel.

Antimony

Main article: Antimony

Antimony (Latin: stibium) is a toxic chemical element with the symbol Sb and an atomic number of 51. A lustrous grey metalloid, it is found in nature mainly as the sulfide mineral stibnite (Sb₂S₃). Antimony compounds have been known since ancient times and were used for cosmetics, metallic antimony was also known but mostly identified as lead.

For some time China has been the largest producer of antimony and its compounds, with most production coming from the Xikuangshan Mine in Hunan. Antimony compounds are prominent additives for chlorine and bromine containing fire retardants found in many commercial and domestic products. The largest application for metallic antimony is as alloying material for lead and tin. It improves the properties of the alloys which are used as in solders, bullets and ball bearings. An emerging application is the use of antimony in microelectronics.

Tellurium

Main article: Tellurium

Tellurium is a chemical element that has the symbol Te and atomic number 52. A brittle, mildly toxic, rare, silver-white metalloid which looks similar to tin, tellurium is chemically related to selenium and sulfur. It is occasionally found in native form, as elemental crystals. Tellurium is far more common in the universe than on Earth. Its extreme rarity in the Earth's crust, comparable to that of platinum, is partly due to its high atomic number, but also due to its formation of a volatile hydride which caused the element to be lost to space as a gas during the hot nebular formation of the planet.

Tellurium was discovered in Transylvania (today part of Romania) in 1782 by Franz-Joseph Müller von Reichenstein in a mineral containing tellurium and gold. Martin Heinrich Klaproth named the new element in 1798 after the Latin word for "earth", tellus. Gold telluride minerals (responsible for the name of Telluride, Colorado) are the most notable natural gold compounds. However, they are not a commercially significant source of tellurium itself, which is normally extracted as by-product of copper and lead production.

Tellurium is commercially primarily used in alloys, foremost in steel and copper to improve machinability. Applications in solar panels and as a semiconductor material also consume a considerable fraction of tellurium production.

Iodine

Main article: Iodine

Iodine is a chemical element with the symbol I and atomic number 53. The name is from Greek ἰοειδής ioeidēs, meaning violet or purple, due to the color of elemental iodine vapor.

Iodine and its compounds are primarily used in nutrition, and industrially in the production of acetic acid and certain polymers. Iodine's relatively high atomic number, low toxicity, and ease of attachment to organic compounds have made it a part of many X-ray contrast materials in modern medicine. Iodine has only one stable isotope. A number of iodine radioisotopes are also used in medical applications.

Iodine is found on Earth mainly as the highly water-soluble iodide I⁻, which concentrates it in oceans and brine pools. Like the other halogens, free iodine occurs mainly as a diatomic molecule I₂, and then only momentarily after being oxidized from iodide by an oxidant like free oxygen. In the universe and on Earth, iodine's high atomic number makes it a relatively rare element. However, its presence in ocean water has given it a role in biology (see below).

Xenon

Main article: Xenon

Xenon is a chemical element with the symbol Xe and atomic number 54. A colorless, heavy, odorless noble gas, xenon occurs in the Earth's atmosphere in trace amounts. Although generally unreactive, xenon can undergo a few chemical reactions such as the formation of xenon hexafluoroplatinate, the first noble gas compound to be synthesized.

Naturally occurring xenon consists of nine stable isotopes. There are also over 40 unstable isotopes that undergo radioactive decay. The isotope ratios of xenon are an important tool for studying the early history of the Solar System. Radioactive xenon-135 is produced from iodine-135 as a result of nuclear fission, and it acts as the most significant neutron absorber in nuclear reactors.

Xenon is used in flash lamps and arc lamps, and as a general anesthetic. The first excimer laser design used a xenon dimer molecule (Xe₂) as its lasing medium, and the earliest laser designs used xenon flash lamps as pumps. Xenon is also being used to search for hypothetical weakly interacting massive particles and as the propellant for ion thrusters in spacecraft.

Biological role

Rubidium, strontium, yttrium, zirconium, and niobium have no biological role. Yttrium can cause lung disease in humans.

Molybdenum-containing enzymes are used as catalysts by some bacteria to break the chemical bond in atmospheric molecular nitrogen, allowing biological nitrogen fixation. At least 50 molybdenum-containing enzymes are now known in bacteria and animals, though only the bacterial and cyanobacterial enzymes are involved in nitrogen fixation. Owing to the diverse functions of the remainder of the enzymes, molybdenum is a required element for life in higher organisms (eukaryotes), though not in all bacteria.

Technetium, ruthenium, rhodium, palladium, silver, tin, and antimony have no biological role. Although cadmium has no known biological role in higher organisms, a cadmium-dependent carbonic anhydrase has been found in marine diatoms. Indium has no biological role and can be toxic as well as Antimony.

Tellurium has no biological role, although fungi can incorporate it in place of sulfur and selenium into amino acids such as tellurocysteine and telluromethionine. In humans, tellurium is partly metabolized into dimethyl telluride, (CH₃)₂Te, a gas with a garlic-like odor which is exhaled in the breath of victims of tellurium toxicity or exposure.

Iodine is the heaviest essential element utilized widely by life in biological functions (only tungsten, employed in enzymes by a few species of bacteria, is heavier). Iodine's rarity in many soils, due to initial low abundance as a crust-element, and also leaching of soluble iodide by rainwater, has led to many deficiency problems in land animals and inland human populations. Iodine deficiency affects about two billion people and is the leading preventable cause of intellectual disabilities. Iodine is required by higher animals, which use it to synthesize thyroid hormones, which contain the element. Because of this function, radioisotopes of iodine are concentrated in the thyroid gland along with nonradioactive iodine. The radioisotope iodine-131, which has a high fission product yield, concentrates in the thyroid, and is one of the most carcinogenic of nuclear fission products.

Xenon has no biological role, and is used as a general anaesthetic.