The long-range force that does not reach

A plasma reflects radio waves below its own frequency, which is what its mobile charges do to a wave — and the whistler that arrives sorted is the same charges acting on a wave in a magnetic field. This essay is about what the same charges do to something that is not moving at all.

The rearrangement, and the equation it satisfies #

Drop a positive test charge into a plasma. The electrons are attracted and the ions repelled, so the density of each stops being uniform: in a potential $\phi$, thermal equilibrium puts the electron density at $n e^{e\phi/kT}$ and the ion density at $n e^{-e\phi/kT}$.

That excess charge is itself a source, so Poisson’s equation has to include it:

$$\nabla^2\phi = -\frac{\rho}{\varepsilon_0} = \frac{2ne}{\varepsilon_0}\sinh\frac{e\phi}{kT}$$

which is nonlinear and, for $e\phi \ll kT$, becomes

$$\nabla^2\phi = \frac{\phi}{\lambda_D^2}, \qquad \lambda_D = \sqrt{\frac{\varepsilon_0 kT}{ne^2}}$$

The solution with a point charge at the origin is the Coulomb potential multiplied by $e^{-r/\lambda_D}$.

The unscreened law falls as one over the distance squared and therefore reaches everywhere. Screening does not change that law — it multiplies its solution by a decaying exponential, and an exponential beats every power. So the force between two charges in a plasma is the Coulomb force at short range and nothing at all beyond a few Debye lengths, with the crossover happening over a distance rather than at a boundary.

Two things about that expression are worth reading slowly. The screening length rises with temperature, because a hotter electron is harder to hold in place near the charge, and falls with density, because more electrons are available to do the holding. And it contains no reference to the test charge at all — a screening length is a property of the plasma, and the same length screens a large charge and a small one.

Screening is the Boltzmann factor applied to position rather than to state: the electrons are slightly more likely to be found where the potential is favourable, and slightly less where it is not. That bias is tiny per particle and enormous in aggregate, and the whole of Debye’s theory is that one substitution followed by the requirement that the potential be consistent with the charge it induces.

An exponential is a different kind of falloff #

The Coulomb force is called long-ranged because a power law has no scale: at ten times the distance it is a hundredth, at a thousand times a millionth, and it never becomes zero. That is what makes electrostatics in a vacuum a problem about every charge at once.

A point, a line and a plane of charge all give power laws, and that is what “long-ranged” means: their shape is the same at every distance, so there is no scale in the problem. An exponential has one, and introducing it changes the character of the force rather than its strength — which is why screening is a qualitative change and not a correction.

An exponential is not like this. At three screening lengths the potential is five per cent of the unscreened value, at ten it is four parts in a hundred thousand, and past twenty there is nothing at all in any practical sense. The force acquires a range, and a plasma can then be treated as a collection of independent regions in a way a vacuum cannot.

That is the reason the concept matters. Almost every tractable statement about plasmas — that they are quasineutral, that they support fluid-like descriptions, that a distant charge can be ignored — depends on the interaction having a finite reach.

Gauss’s law is not violated by any of this. The flux through a large surface around a screened charge really is nearly zero, because the enclosed charge really is nearly zero — the cloud has brought up almost exactly the opposite charge. Nothing has been hidden; the charge has been cancelled, and the law reports the cancellation faithfully.

Which ionised gases the theory applies to #

The derivation used the Boltzmann distribution, which is a statement about averages over many particles. So there is an implied condition: the region doing the screening must contain enough particles for an average to mean anything.

The count is called the plasma parameter, and it is the ratio that decides whether an ionised gas behaves collectively at all:

$$N_D = \frac{4}{3}\pi n\lambda_D^3 \propto \frac{T^{3/2}}{\sqrt{n}}$$

With $N_D \gg 1$ the screening is a smooth statistical response, the plasma is weakly coupled, and the standard theory applies. With $N_D$ of order one the potential energy between neighbours is comparable with their thermal energy, the gas is strongly coupled, and the picture of nearly free particles with occasional collisions is wrong.

The extremes on that chart are the interesting ones. Interstellar gas is about as weakly coupled as anything gets. The core of the Sun, at eight particles per Debye sphere, is a case where corrections matter — and they matter for a practical reason, because the screening of nuclei by electrons raises fusion rates by tens of per cent and that correction is part of every stellar model.

Measuring it #

A screening length that could not be measured would be a modelling convenience, and it is measured in several quite different ways.

By scattering. A beam of charged particles crossing a plasma is deflected by the screened potential rather than the bare one, so the total scattering cross-section is finite where the Coulomb one diverges. The divergence of Rutherford scattering at small angles is cut off at exactly the angle corresponding to an impact parameter of λ_D, and the logarithm of the ratio of the two lengths — the Coulomb logarithm — appears in every transport coefficient a plasma has. Measuring a resistivity is therefore an indirect measurement of a screening length inside a logarithm.

By probes. A Langmuir probe inserted into a plasma collects a current that depends on the sheath around it, and the sheath is a few screening lengths thick. Fitting the current–voltage curve gives the density and the temperature, and the length follows.

In a plasma the scattering cross-section is finite only because the encounter is cut off at the screening length. Without that cut-off the integral diverges logarithmically — every distant particle contributes a little and there are unboundedly many of them — so the Coulomb logarithm that appears in every transport coefficient is the ratio of two lengths this essay is about.

And by light. Thomson scattering from a plasma has a spectrum that depends on whether the probing wavelength is longer or shorter than the screening length: shorter, and the light scatters from individual electrons; longer, and it scatters from the collective density fluctuations. The changeover in the spectrum happens at the screening length and is the standard diagnostic on large fusion machines.

What screening does not do #

It does not make the plasma inert. Fields on scales larger than the screening length are unscreened, and they are what plasma physics is about. The screening length is the boundary between the scales on which a plasma insists on neutrality and the scales on which it does not.

That relation is worth stating explicitly, because it makes the two halves of this anchor one subject. The plasma frequency is $\omega_p = \sqrt{ne^2/\varepsilon_0 m}$ and the screening length is $\lambda_D = v_{\text{th}}/\omega_p$: the screening length is how far a thermal electron gets in one plasma oscillation. Screening is the static face of the same restoring force that makes the oscillation.

And it does not prevent structure. A plasma sheath at a wall is a region of exactly a few screening lengths where quasineutrality fails, because electrons reach the wall faster than ions and leave the plasma slightly positive. Every probe, every wall, every spacecraft in the solar wind acquires such a sheath, and its thickness is the length computed here.

The field inside a conductor is zero, and screening in a metal is the same argument with degenerate electrons in place of thermal ones. The screening length is then the Thomas–Fermi one rather than the Debye one, it is set by the Fermi energy instead of the temperature, and it comes out at about an ångström — which is why a metal screens so completely that the interior field is zero for practical purposes.

How big the rearrangement actually is #

It is worth putting a number on the disturbance, because the word “screening” suggests something dramatic and the reality is a very small effect spread over a great many particles.

At one screening length from a singly charged ion in the ionosphere, the potential is of order $kT/e$ divided by the number of particles in a Debye sphere — so the fractional change in electron density there is about one part in $10^5$. The screening is done by a hundred thousand particles each being displaced by almost nothing.

That is why the derivation is allowed to linearise, and it is also the answer to the natural picture of a cloud of electrons clinging to the charge. Nothing clings. Every electron in the region is moving at several kilometres a second and spends a moment near the charge; what is steady is the average density, enhanced by one part in a hundred thousand, integrated over a volume large enough that the total excess charge is one electron’s worth.

The Debye cloud is a tiny bias applied to a huge number of particles whose sum is exactly one unit of charge. That is worth putting a number to: in a laboratory plasma there may be $10^4$ particles in a Debye sphere, each displaced by a fraction of a per cent, and the total displacement is one electron’s worth. The rearrangement is invisible per particle and complete in aggregate.

The same reading explains the plasma-parameter condition from the other direction. If there were only a few particles in the screening region, the excess charge could not be spread thinly; it would have to be one or two particles actually sitting near the charge, which is a bound state rather than a statistical cloud. That is what strong coupling means.

The same idea in four other places #

In electrolytes, where it was first written down. Debye and Hückel derived exactly this in 1923 for ions in water, and the screening length there is a nanometre or so at physiological concentration. It is why salt makes protein solutions behave differently, why colloids flocculate when salt is added, and why the electrical double layer at every electrode has the thickness it has.

In metals, as the Thomas–Fermi length, with the electrons’ Fermi energy in place of $kT$. It is under an ångström, which is why a charged impurity in a metal is invisible a few atoms away and why metals can be treated as neutral with a surface charge.

In the strong interaction, where colour charge is screened over a fermi. The analogy has real limits — the QCD case also confines, which has no counterpart here — but the language of a screening length carries over directly.

And in gravity, where it does not happen at all. Gravitational charge has one sign, so nothing can gather around a mass to cancel its field. That single fact is why gravity dominates on large scales despite being forty orders of magnitude weaker: electric forces screen themselves and gravity cannot. What decides a falloff is usually geometry; here it is the availability of the opposite sign.

Screening is the arrangement of two opposite charges produced spontaneously and continuously: the test charge polarises everything around it, and what a distant probe sees is the pair rather than the charge. The same idea runs through electrolytes, semiconductors, nuclear matter and the vacuum itself — in each case a bare quantity is dressed by what it induces, and only the dressed one is measurable.

The consequence for fusion, which is worth a number #

The correction the solar core needs is small, well understood and load-bearing, and it makes a good example of a screening length being used rather than described.

Two nuclei approaching each other in a plasma do not see each other’s bare Coulomb repulsion; they see it screened by the electrons between them. The barrier is therefore lower than the vacuum calculation says, and since the tunnelling probability is exponentially sensitive to the barrier, a small reduction is a large rate change.

For the Sun’s core the enhancement is around ten to thirty per cent depending on the reaction — small enough to be a correction and large enough that solar models cannot be compared with neutrino measurements without it. Salpeter worked it out in 1954 using exactly the Debye–Hückel expression derived here, and the size of the correction is set by the ratio of the screening energy to the thermal energy, which is the plasma parameter again.

That is the general shape of the thing. A screening length is rarely interesting on its own; it becomes interesting when it modifies something that depends on it exponentially.

The same length in a glass of salt water #

The electrolyte case was given a sentence above and deserves more, because it is where the calculation was first done and where it is used most often.

For ions in water at room temperature the same expression, with the water’s dielectric constant in place of the vacuum’s, gives a screening length of about 0.3 nanometres divided by the square root of the concentration in moles per litre. Tenth-molar salt water screens over a nanometre; a millimolar solution over ten. Every charged surface in contact with water carries a diffuse layer of counterions of that thickness, and every measurement of such a surface is a measurement through it.

The first consequence is thermodynamic. An ion in solution is stabilised by its own screening cloud — the counterions gathered around it lower its energy — so its effective concentration is less than its actual one. Working the energy out from the screened potential gives the Debye–Hückel limiting law: the logarithm of the activity coefficient falls as the square root of the ionic strength, with a coefficient that contains no property of the particular ion beyond its charge.

The square root is the screening length showing through. Nothing in solution chemistry has an obvious reason to depend on the square root of a concentration, and it does because the stabilisation goes as one over the screening length and the screening length goes as one over the square root of the concentration. The law was the first successful theory of electrolyte solutions and it is still the standard starting point.

The second consequence is mechanical and decides whether suspensions stay suspended. Two like-charged colloidal particles repel through their screened potentials and attract through the van der Waals force, which is not screened. Adding salt shortens the screening length without touching the attraction, so the repulsive barrier between two particles shrinks, and past some concentration it disappears entirely and the particles stick on contact.

That is the DLVO picture of colloid stability, and it explains a rule found empirically long before: the concentration of salt needed to coagulate a suspension falls extremely steeply with the counterion’s charge — roughly as its sixth power, so aluminium is hundreds of times more effective than sodium. Water treatment exploits it directly, dosing with aluminium or iron salts to make suspended clay flocculate and settle.

And it explains a landform. River water carries clay in suspension because the particles are charged and screened over tens of nanometres; where the river meets the sea, the screening length collapses to under a nanometre, the barrier vanishes, and the clay flocculates and drops. Every delta is a Debye length changing by a factor of ten.

The screening that cannot happen #

The essay noted that gravity has only one sign and therefore cannot screen. It is worth doing the arithmetic, because the failure is not a missing effect but a sign change, and the sign change is the whole of a different subject.

Run the identical derivation with mass in place of charge. A gravitational potential $\phi$ gives a density $n_0 e^{-m\phi/kT}$ — more particles where the potential is lower, exactly as before. Poisson’s equation for gravity carries the opposite sign from the electrostatic one, because masses attract. Putting the two together and linearising gives

$$\nabla^2\delta\phi = -\frac{\delta\phi}{\lambda^2},$$

with a minus where the plasma case had a plus.

That single sign is decisive. The plasma equation has exponentially decaying solutions, which is screening. The gravitational equation has oscillating solutions — and, once the dynamics are put back in, growing ones. The length $\lambda$ that appears is not a screening length at all; it is the Jeans length, the scale above which a self-gravitating gas collapses rather than settling down.

So the two cases are the same calculation with one sign reversed, and they could hardly be more different in what they describe. Electric charge gathers around a disturbance and cancels it; mass gathers around a disturbance and enlarges it.

Three consequences follow and all three are structural rather than incidental.

There is no gravitational shielding and there cannot be. A Faraday cage works because a conductor can supply charge of either sign to its surface; nothing can supply negative mass, so no arrangement of matter screens a gravitational field. Every proposal to the contrary founders on this sentence.

Gravity therefore dominates at large scales despite being enormously the weaker force. Electric forces cancel themselves out over a few Debye lengths and gravitational ones accumulate over megaparsecs, which is why the structure of the universe is gravitational and the structure of a molecule is electrical.

And a self-gravitating system has no thermodynamic limit. Screening is what makes energy extensive — doubling the volume of a plasma doubles its energy, because each region only interacts with its neighbours. Without it the energy of a self-gravitating system depends on its shape and its size, standard statistical mechanics does not apply, and quantities that cannot be negative elsewhere — a heat capacity, for one — routinely are.

What the pictures cannot show #

The linearisation fails close in. The derivation assumed $e\phi \ll kT$, which breaks down within a distance where the potential energy reaches the thermal energy — the Landau length, about 10⁻⁹ metres in the ionosphere. Inside that radius the full nonlinear equation is needed and the exponential form is wrong; outside it, which is where nearly all the volume is, the linear result is excellent.

The plasma is unmagnetised and in equilibrium. A magnetic field makes the screening anisotropic, and a plasma with drifting populations screens differently along and across the drift. Both are ordinary situations and neither is drawn here.

The ions are treated as mobile and thermal. In many laboratory plasmas the ions are cold and essentially fixed on the timescale that matters, which changes the screening length by a factor of √2 — and much of the literature quietly uses the electron-only version.

And the test charge is static. A charge moving faster than the thermal speed drags its screening cloud behind it and leaves a wake, which is a real and measurable effect and the reason a fast ion in a plasma loses energy at the rate it does.

Why a plasma is a state of matter #

The last thing worth saying is why this quantity is treated as the definition rather than as a property.

An ionised gas has three lengths in it: the distance between particles, the screening length, and the size of the container. The ordering $n^{-1/3} \ll \lambda_D \ll L$ is what makes the collective description work — many particles per screening sphere, and many screening spheres in the system. When it holds, the gas supports waves, sustains currents, screens fields and behaves like a medium; when it fails, it is a collection of charged particles that happen to be in the same place.

That is a stronger statement than it sounds. A gas becomes a plasma not when its atoms ionise but when the ordering above is satisfied, and the two are not the same condition: a very dilute ionised gas in a small container fails it, and a partially ionised one in a large volume can satisfy it with a fraction of a per cent of its atoms ionised. The ionosphere is ionised to about one part in a thousand and is unambiguously a plasma; a spark in air is ionised far more heavily and is a plasma only in the region where the ordering holds.

A plasma has two lengths — the distance between neighbours and the distance over which the interaction reaches — and it is a state of matter because the second is much larger than the first. Many particles inside a Debye sphere is what makes collective behaviour possible; when the number approaches one the plasma becomes strongly coupled and behaves like a liquid instead.

The ladder from here #

Later rungs on this anchor: the sheath at a wall, and the Bohm criterion that decides how fast ions must enter it; Landau damping, which is the collisionless mechanism by which a plasma wave loses energy and has no analogue in any fluid; the dielectric function of a plasma, of which both the screening and the oscillation are limits; and strongly coupled plasmas, where the ordering that makes all of this work is reversed.

The neighbouring ladders are the plasma frequency, which is the same restoring force made dynamic, the inside of a conductor, which is screening taken to its limit, and the shape decides the falloff, which is the other way a force law’s reach can be changed.