Polarizability

For electromagnetic waves, see Polarization (waves). For other uses, see Polarization (disambiguation).

Polarizability is the measure of the change in a molecule's electron distribution in response to an applied electric field, which can also be induced by electric interactions with solvents or ionic reagents. It is a property of matter. Polarizabilities determine the dynamical response of a bound system to external fields, and provide insight into a molecule's internal structure.

Electric polarizability

Definition

Electric polarizability is the relative tendency of a charge distribution, like the electron cloud of an atom or molecule, to be distorted from its normal shape by an external electric field, which is applied typically by inserting the molecule in a charged parallel-plate capacitor, but may also be caused by the presence of a nearby ion or dipole.

The electronic polarizability α is defined as the ratio of the induced dipole moment $\boldsymbol{p}$ of an atom to the electric field $\boldsymbol{E}$ that produces this dipole moment.

$\boldsymbol{p} = \alpha \boldsymbol{E}$

Polarizability has the SI units of C·m²·V⁻¹ = A²·s⁴·kg⁻¹ but is more often expressed as polarizability volume with units of cm³ or in ų = 10⁻²⁴ cm³.

$\alpha (\mathrm{cm}^3) = \frac{10^{6}}{ 4 \pi \varepsilon_0 }\alpha (\mathrm{C} \cdot \mathrm{m}^2 \cdot \mathrm{V}^{-1})$ where ε₀ is the vacuum permittivity.

The polarizability of individual particles is related to the average electric susceptibility of the medium by the Clausius-Mossotti relation.

Note that the polarizability α as defined above is a scalar quantity. This implies that the applied electric fields can only produce polarization components parallel to the field. For example, an electric field in the x-direction can only produce an x component in $\boldsymbol{p}$. However, it can happen that an electric field in the x-direction, produces a y or z component in the vector $\boldsymbol{p}$. In this case α is described as a tensor of rank 2, which is represented with respect to a given system of axes (frame of reference) by a $3 \times 3$ matrix.

Tendencies

Generally, polarizability increases as volume occupied by electrons increases. In atoms, this occurs because larger atoms have more loosely held electrons in contrast to smaller atoms with tightly bound electrons. On rows of the periodic table, polarizability therefore decreases from left to right. Polarizability increases down on columns of the periodic table. Likewise, larger molecules are generally more polarizable than smaller ones.

Though water is a very polar molecule, alkanes and other hydrophobic molecules are more polarizable. Alkanes are the most polarizable molecules. Although alkenes and arenes are expected to have larger polarizability than alkanes because of their higher reactivity compared to alkanes, alkanes are in fact more polarizable. This results because of alkene's and arene's more electronegative sp² carbons to the alkane's less electronegative sp³ carbons.

It is important to note that ground state electron configuration models are often inadequate in studying the polarizability of bonds because dramatic changes in molecular structure occur in a reaction.

Magnetic polarizability

Magnetic polarizability defined by spin interactions of nucleons is an important parameter of deuterons and hadrons. In particular, measurement of tensor polarizabilities of nucleons yields important information about spin-dependent nuclear forces.

The method of spin amplitudes uses quantum mechanics formalism to more easily describe spin dynamics. Vector and tensor polarization of particle/nuclei with spin S ≥ 1 are specified by the unit polarization vector $\boldsymbol{p}$ and the polarization tensor P_`. Additional tensors composed of products of three or more spin matrices are needed only for the exhaustive description of polarization of particles/nuclei with spin S ≥ ³⁄₂ .

See also

• Polarization density
• MOSCED, an estimation method for activity coefficients; uses polarizability as parameter

External links

• The theory of the electromagnetic field by David M. Cook
• Consistent Calculation of the Nucleon Electromagnetic Polarizabilities in Chiral Perturbation Theory Beyond Next-to-Leading Order
• Hadron polarizabilities and magnetic moments with background field methods (PDF)

Polarizability of the nucleon

Preface

The polarizabilities belong to the fundamental structure constants of the nucleon, in addition to the mass, the electric charge, the spin and the magnetic moment. The proposal to measure the polarizabilities dates back to the 1950th. Two experimental options were considered (i) Compton scattering by the proton and (ii) the scattering of slow neutrons in the Coulomb field of heavy nuclei. The idea was that the nucleon with its pion cloud obtains an electric dipole moment under the action of an electric field vector which is proportional to the electric polarizability. After the discovery of the photoexcitation of the Δ resonance it became obvious that the nucleon also should have a strong paramagnetic polarizability, because of a virtual spin-flip transition of one of the constituent quarks due to the magnetic field vector provided by a real photon in a Compton scattering experiment. However, experiments showed that this expected strong paramagnetism is not observed. Apparently a strong diamagnetism exists which compensates the expected strong paramagnetism. Though this explanation is straightforward, it remained unknown how it may be understood in terms of the structure of the nucleon. A solution of this problem was found very recently when it was shown that the diamagnetism is a property of the structure of the constituent quarks. In retrospect this is not a surprise, because constituent quarks generate their mass mainly through interactions with the QCD vacuum via the exchange of a σ meson. This mechanims is predicted by the linear σ model on the quark level (QLLσM) which also predicts the mass of the σ meson to be mσ=666 MeV. The σ meson has the capability of interacting with two photons being in parallel planes of linear polarization. We will show in the following that the σ meson as part of the constituent quark structure, therefore, provides the largest part of the electric polarizability and the total diamagnetic polarizability.

Definition of electromagnetic polarizabilities

A nucleon in an electric field E and a magnetic field H obtains an electric dipole moment d and magnetic dipole moment m given by.

${\mathbf d}\,\,=4\pi\,\alpha \,{\mathbf E}$

${\mathbf m}=4\pi\,\beta \,{\mathbf H}$

in a unit system where the electric charge e is given by e² / 4π = α_e = 1 / 137.04. The proportionality constants α and β are denoted as the electric and magnetic polarizabilities, respectively. These polarizabilities may be understood as a measure of the response of the nucleon structure to the fields provided by a real or virtual photon and it is evident that we need a second photon to measure the polarizabilities. This may be expressed through the relations

$\delta W=-\frac12 \,4\pi\,\alpha\,{\mathbf E}^2-\frac12 \,4\pi\,\beta\,{\mathbf H}^2$

where δW is the energy change in the electromagnetic field due to the presence of the nucleon in the field. The definition implies that the polarizabilities are measured in units of a volume, i.e. in units of fm³ (1 fm=10 ^{− 15} m).

Modes of two-photon reactions and experimental methods

Static electric fields of sufficient strength are provided by the Coulomb field of heavy nuclei. Therefore, the electric polarizability of the neutron can be measured by scattering slow neutrons in the electric field E of a Pb nucleus. The neutron has no electric charge. Therefore, two simultaneously interacting electric field vectors (two virtual photons) are required to produce a deflection of the neutron. Then the electric polarizability can be obtained from the differential cross section measured at a small deflection angle. A further possibility is provided by Compton scattering of real photons by the nucleon, where during the scattering process two electric and two magnetic field vectors simultaneously interact with the nucleon.

In the following we discuss the experimental options we have to measure the polarizabilities of the nucleon. As outlined above two photons are needed which simultaneously interact with the electrically charged parts of the nucleon. These photons may be in parallel or perpendicular planes of linear polarization and in these two modes measure the polarizabilities α, β or spinpolarizabilities γ, respectively. The spinpolarizability is nonzero only for particles having a spin.

In total the experimental options discussed above provide us with 6 combinations of two electric and magnetic field vectors. These are described in the following two equations:

• Photons in parallel planes of linear polarization

$(\text{case}\, 1)\,\, \alpha: \,\,\,\,{\mathbf E}\uparrow\uparrow {\mathbf E}'\quad\quad (\text{case}\, 2)\,\,\beta: \, {\mathbf H} \rightarrow\rightarrow {\mathbf H}'\quad\, (\text{case}\,\, 3)\,\, -\beta: \,{\mathbf H}\rightarrow\leftarrow {\mathbf H}'$

• Photons in perpendicular planes of linear polarization

$(\text{case}\,\,4)\,\, \gamma_E: {\mathbf E}\uparrow\rightarrow {\mathbf E}'\quad \,\,(\text{case}\,\,5)\,\,\gamma_H: {\mathbf H} \rightarrow\downarrow {\mathbf H}'\quad \,\,(\text{case}\,\,6)\,\, -\gamma_H: {\mathbf H}\rightarrow\uparrow {\mathbf H}'$

Case (1) corresponds to the measurement of the electric polarizability α via two parallel electric field vectors E. These parallel electric field vectors may either be provided as longitudinal photons by the Coulomb field of a heavy nucleus, or by Compton scattering in the forward direction or by reflecting the photon by 180°. Real photons simultaneously provide transvers electric E and magnetic H field vectors. This means that in a Compton scattering experiment linear combinations of electric and magnetic polarizabilities and linear combinations of electric and magnetic spinpolarizabilities are measured.

The combination of case (1) and case (2) measures α + β and is observed in forward-direction Compton scattering. The combination of case (1) and case (3) measures α − β and is observed in backward-direction Compton scattering.The combination of case (4) and case (5) measures $\gamma_0\equiv \gamma_E+\gamma_H$ and is observed in forward-direction Compton scattering. The combination of case (4) and case (6) measures $\gamma_\pi\equiv \gamma_E-\gamma_H$ and is observed in backward-direction Compton scattering.

Compton scattering experiments exactly in the forward direction and exactly in the backward direction are not possible from a technical point of view. Therefore, the respective quantities have to be extracted from Compton scattering experiments carried out at intermediate angles.

Experimental results

The experimental polarizabilities of the proton (p) and the neutron (n) may be summarized as follows

$\alpha_p=12.0\pm 0.6,\quad \beta_p=1.9\mp 0.6, \quad \alpha_n=12.5\pm 1.7, \quad \beta_n=2.7\mp 1.8\,\text{ in units of}\,\, 10^{-4}\,{\rm fm}^3$.

The experimental spinpolarizabilities of the proton (p) and neutron (n) are

$\gamma^{(p)}_\pi=-36.4\pm 1.5, \quad \gamma^{(n)}_\pi=58.6\pm 4.0\,\text{ in units of}\,\, 10^{-4}\,{\rm fm}^4$.

The experimental polarizabilities of the proton have been obtained as an average from a larger number of Compton scattering experiments. The experimental electric polarizability of the neutron is the average of an experiment on electromagnetic scattering of a neutron in the Coulomb field of a Pb nucleus and a Compton scattering experiment on a quasifree neutron, i.e. a neutron separated from a deuteron during the scattering process. The two results are (see )

$\alpha_n=12.6\pm 2.5$ from electromagnetic scattering of a slow neutron in the electric field of a Pb nucleus, and

$\alpha_n=12.5\pm 2.3$ from quasifree Compton scattering by a neutron initially bound in the deuteron.

The avarage given above is obtained from these two numbers.

Furthermore, there are ongoing experiments at the University of Lund (Sweden) where the electric polarizability of the neutron is determined through Compton scattering by the deuteron.

Calculation of polarizabilities

Recently great progress has been made in disentangling the total photoabsorption cross section into parts separated by the spin, the isospin and the parity of the intermediate state, using the meson photoproduction amplitudes of Drechsel et al. The spin of the intermediate state may be s = 1 / 2 or s = 3 / 2 depending on the spin directions of the photon and the nucleon in the initial state. The parity change during the transion from the ground state to the intermediate state is ΔP = yes for the multipoles $E1,\,M2,\cdots$ and ΔP = no for the multipoles $M1,\,E2,\cdots$. Calculating the respective partial cross sections from photo-meson data, the following sum rules can be evaluated:

$\alpha+\beta=\frac{1}{2\pi^2} \int^\infty_{\omega_0}\frac{\sigma_{\rm tot}(\omega)}{\omega^2}d\omega$,

$\alpha-\beta=\frac{1}{2\pi^2}\int^\infty_{\omega_0}\sqrt{1+\frac{2\omega}{m}}\left[\sigma(\omega,E1,M2,\cdots)-\sigma(\omega,M1,E2,\cdots)\right]\frac{d\omega}{\omega^2} +(\alpha-\beta)^t$,

$\gamma_0=-\frac{1}{4\pi^2}\int^\infty_{\omega_0}\frac{\sigma_{3/2}(\omega)-\sigma_{1/2}(\omega)}{\omega^3}d\omega$,

$\gamma_\pi=\frac{1}{4\pi^2}\int^\infty_{\omega_0}\sqrt{1+\frac{2\omega}{m}}\left(1+\frac{\omega}{m}\right)\sum_n P_n[\sigma^n_{3/2}(\omega)-\sigma^n_{1/2}(\omega)]\frac{d\omega}{\omega^3}+\gamma^t_\pi$,

$P_n=-1\,\text{for}\,E1,M2,\cdots\,\text{multipoles and}\, P_n=+1\,\text{for} \,M1,E2,\cdots\,\text{multipoles}$.

$(\alpha-\beta)^t=\frac{1}{2 \pi}\left[\frac{g_{\sigma NN}{ M}(\sigma\to \gamma\gamma)}{ m^2_\sigma} +\frac{g_{f_0 NN}{ M}(f_0\to \gamma\gamma)}{ m^2_{f_0}} +\frac{g_{a_0 NN}{ M}(a_0\to \gamma\gamma)}{ m^2_{a_0}}\tau_3\right]$,

$\gamma^t_\pi=\frac{1}{2\pi m}\left[ \frac{g_{\pi NN}{ M}(\pi^0\to \gamma\gamma)}{ m^2_{\pi^0}}\tau_3 +\frac{g_{\eta NN}{ M}(f_0\to \gamma\gamma)}{m^2_\eta} +\frac{g_{\eta' NN}{ M}(a_0\to \gamma\gamma)}{m^2_{\eta'}}\right]$.

where ω is the photon energy in the lab frame. The sum rules for α + β and γ₀ depend on nucleon-structure degrees of freedom only, whereas the sum rules for α − β and γ_π have to be supplemented by the quantities (α − β)^t and $\gamma^t_\pi$, respectively. These are t-channel contributions which may be interpreted as contributions of scalar and pseudoscalar mesons being parts of the constituent-quark structure. The sum rule for α + β depends on the total photoabsorption cross section and , therefore, does not require a disentangling with respect to quantum numbers. The sum rule for α − β requires a disentangling with respect to the parity change of the transition. The sum rule for γ₀ requires a disentangling with respect to the spin of the intermediate state. The sum rule for γ_π requires a disentangling with respect to spin and parity change.

The t-channel contributions depend on those scalar and pseudoscalar mesons which (i) are part of the structure of the constituent quarks and (ii) are capable of coupling to two photons. These are the mesons σ(600), f₀(980) and a₀(980) in case of (α − β)^t, and the mesons π⁰, η and η' in case of $\gamma^t_\pi$. The contribtions are dominated by the σ and the π⁰ whereas the other mesons only lead to small corrections

Results of calculation

The results of the calculation are summarized in the following eight equations :

$\alpha_p=\,\,\,+4.5\,\text{(nucleon)}+7.5\,\text{(const. quark)}=+12.0$

$\beta_p=\,\,\,+9.4\,\text{(nucleon)}-7.5\,\text{(const. quark)}=\,\,\,+1.9$

$\alpha_n=\,\,\,+5.1\,\text{(nucleon)}+8.3\,\text{(const. quark)}=+13.4$

$\beta_n=+10.1\,\text{(nucleon)}-8.3\,\text{(const. quark)}=\,\,\,+1.8\,\,\text{in units of}\,\,10^{-4}{\rm fm}^3$

$\gamma^{(p)}_0=\,\,\,-0.58\pm 0.20\,\text{(nucleon)}$

$\gamma^{(n)}_0=\,\,\,+0.38\pm 0.22\,\text{(nucleon)}$

$\gamma^{(p)}_\pi=\,\,\,+8.5\,\text{(nucleon)}-45.1\,\,\text{(const. quark)}=-36.6$

$\gamma^{(n)}_\pi=+10.0\,\text{(nucleon)}+48.3\,\,\text{(const. quark)}=+58.3\,\,\text{in units of}\,\,10^{-4}{\rm fm}^4$

The electric polarizabilities α_p and α_n are dominated by a smaller component due to the pion cloud (nucleon) and a larger component due to the σ meson as part of the constituent-quark structure (const. quark). The magnetic polarizabilities β_p and β_n have a large paramagnetic part due to the spin structure of the nucleon (nucleon) and an only slightly smaller diamagnetic part due to the σ meson as part of the constituent-quark structure (const. quark). The contributions of the σ meson are supplemented by small corrections due to f₀(980) and a₀(980) mesons .

The spinpolarizabilities $\gamma_0^{(p)}$ and $\gamma_0^{(n)}$ are dominated by destructively interfering components from the pion cloud and the spin structure of the nucleon. The different signs obtained for the proton and the neutron are due to this destructive interference. The spinpolarizabilities $\gamma^{(p)}_\pi$ and $\gamma^{(n)}_\pi$ have a minor component due to the structure of the nucleon (nucleon) and a major component due to the pseudoscalar mesons π⁰, η and η' as structure components of the constituent quarks (const. quark).

The agreement with the experimental data is excellent in all eight cases.

Summary

In the foregoing we have shown that the polarizabilities of the nucleon are well understood. Differing from previous belief the mesonic structure of the constituent quark is essential for the sizes and the general properties of the polarizabilities.