Rubidium-87 is a long-lived radioactive isotope with thirty-seven protons and fifty neutrons. Its beta decay to strontium-87 supports rubidium-strontium geochronology.

It is also a common isotope in laser-cooled atomic-gas experiments. Its nuclear spin and hyperfine states provide experimentally accessible internal structure, while the whole atom can behave as a boson. These atomic roles are distinct from the geological use of its slow nuclear decay.

A chain of $N$ up/down segments of length $a$ has extension $L=Nax$ and binary configurational entropy. An external force $f(L)=f_{0}L/a$ has potential $-f_0{L}^2/(2a)$, obtained by integrating the force. Its mean-field free-energy minima obey the displayed equation. Expansion gives $F-F(0)=N{k}_B(T-T_c)x^2/2+N{k}_{B}T{x}^4/12+\cdots$, so the ordered extension emerges continuously, $|x|\sim \sqrt{3(T_c-T)/T}$, rather than jumping. The equation matches zero-field mean-field theory of the Ising model; finite systems round the ideal singular behavior.

The observed collection of lines or bands from vibration-rotation transitions. In the usual parallel diatomic $\Sigma \to \Sigma$ infrared band, P and R branches have $\Delta J=-1,+1$ and the Q branch is absent.

The rovibrational spectroscopy branch in which absorption lowers the total rotational angular momentum quantum number by one. A label $P(J)$ conventionally specifies the lower-state value $J$.

The rovibrational spectroscopy branch with unchanged rotational angular momentum quantum number. Differences between upper and lower rotational constants can resolve its transitions into closely spaced lines. Allowed lower-state values depend on the angular momenta of both states.

The rovibrational spectroscopy branch in which absorption raises the total rotational angular momentum quantum number by one. A label $R(J)$ conventionally specifies the lower-state value $J$.

A symmetric-top vibrational band whose transition dipole is parallel to the symmetry axis, with $\Delta K=0$ in the elementary model.

A symmetric-top vibrational band whose transition dipole is perpendicular to the symmetry axis, with $\Delta K=\pm 1$. Unresolved Q subbranches can create a regular sequence controlled mainly by $A-B$.

With additive drag from $N$ identical monomers, ${\zeta }_N=N{\zeta }_0$ and centre-of-mass diffusion is $D_N=k_{B}T/(N{\zeta }_0)$. For monomer Stokes law ${\zeta }_0=6\pi \eta a$, this gives $D_N\propto N^{-1}$. The model neglects long-range hydrodynamic coupling and does not alone describe the long-time spatial diffusion of a strongly entangled chain.

The dipole coupling $-pE\cos \theta$ connects only $J\pm 1$ at fixed $m$. For $J\ge 1$, second order gives $(pE{)}^2[J(J+1)-3m^2]/[2BJ(J+1)(2J-1)(2J+3)]$; the ground-state result is separately $-(pE{)}^2/(6B)$. Here $B$ has energy units and $\theta$ is the polar angle to the field.

For a polar rigid symmetric top with its permanent dipole on the symmetry axis, electric-dipole pure rotation has $\Delta J=\pm 1$, $\Delta K=0$. The $A-B$ term cancels within each transition; centrifugal distortion can lift the rigid-model coincidence of different K components.

A set of spectral lines associated with transitions between molecular rotational energy levels. A polar linear rigid rotor has electric-dipole transitions with $\Delta J=\pm 1$; in the ideal model adjacent absorption frequencies are separated by twice the rotational constant in frequency units.

A rotating ideal half-wave plate flips circular polarization. The spin change has magnitude $2\hslash$ per photon, so its mechanical energy exchange produces a frequency shift of magnitude $2|\Omega |$. For Jones convention $H(\theta )=R(\theta )\mathrm{diag}(1,-1)R(-\theta )$ and $Q=\mathrm{diag}(1,i)$, the combination $QH(\theta )Q=e^{i2\theta {\sigma }_x}$. Acting on $(1,1{)}^T/\sqrt{2}$ gives $e^{i2\theta }$; with time dependence $e^{-i\omega t}$ and $\theta =\Omega t$, the output frequency is $\omega -2\Omega$. The opposite input eigenpolarization reverses the shift.

At radius $a$ in a ring rotating at rate $\omega$, a walker with signed tangential speed $u$ relative to the ring has inertial speed $a\omega +u$. The normal reaction per mass is $(a\omega +u{)}^2/a$; the curvature of the relative path matters as well as the centrifugal force and the Coriolis force.

A planar quadratic saddle $m{\omega }^2(x^2-y^2)/2$ rotated at angular speed $\Omega$ becomes linearly confining for the displayed strict inequality. Its rotating-frame mode frequencies obey ${ϵ}_{\pm }^2={\Omega }^2\pm {\omega }^2$. Equality has a zero-frequency secular mode and is not robust bounded-motion stability. This is a rotating potential model; a rolling ball on an actual surface has additional constraints.

Let $\omega =e^{2\pi i/m}$ be a primitive $m$th root of unity. The finite geometric series givesConsequently, for a polynomial $F(z)={\sum }_n{a}_n{z}^n$, the expression $m^{-1}{\sum }_j{\omega }^{-jr}F({\omega }^{j}z)$ keeps exactly the terms whose exponents are congruent to $r$ modulo $m$. This is orthogonality of characters for a cyclic group in concrete form.

For $m=2$, the even and odd parts are $(F(z)+F(-z))/2$ and $(F(z)-F(-z))/2$. Taking $F(z)=(1+z{)}^N$ extracts sums of binomial coefficients in chosen residue classes. The same calculation works for formal power series coefficient by coefficient over a coefficient field containing $\omega$ and $1/m$.

Formation of root nodules. More nodules do not necessarily imply more effective nitrogen fixation or greater host benefit.

The raw root mean square of a random variable is $\sqrt{E[X^2]}$, whereas its r.m.s. fluctuation about its mean is the standard deviation $\sqrt{E[(X-\mu {)}^2]}$. The former cannot be smaller than $|\mu |$. Experimental phrases such as r.m.s. noise or temperature spread usually refer to the latter and should state the reference level.

A sphere of radius $a$ touching both faces of a symmetric groove of opening angle $\varphi$ rotates instantaneously about the line joining its contacts. The centre is a distance $h=a\sin (\varphi /2)$ from this line, so rolling without slipping means $v=h\omega$. For a homogeneous sphere, $K=m{v}^2/2+I{\omega }^2/2=m{v}^2[1+2/(5{\sin }^2(\varphi /2))]/2$. Differentiating this kinetic energy with respect to distance down an incline gives the displayed acceleration. Both contact forces do no work for fixed faces and no slip.