Pinch (plasma physics)

Pinched aluminium can, produced via a pulsed magnetic field created by rapidly discharging 2 kilojoules from a high voltage capacitor bank into a 3-turn coil of heavy gauge wire.

Many high-voltage electronics enthusiasts make their own crude electromagnetic forming devices.[32][33][34] They use pulsed power techniques to produce a theta pinch able to crush an aluminium soft drink can using the Lorentz forces created when large currents are induced in the can by the strong magnetic field of the primary coil.[35][36]

An electromagnetic aluminium can crusher consists of four main components: a high-voltage DC power supply, which provides a source of electrical energy, a large energy discharge capacitor to accumulate the electrical energy, a high voltage switch or spark gap, and a robust coil (capable of surviving high magnetic pressure) through which the stored electrical energy can be quickly discharged in order to generate a correspondingly strong pinching magnetic field (see diagram below).

Electromagnetic pinch "can crusher": schematic diagram

In practice, such a device is somewhat more sophisticated than the schematic diagram suggests, including electrical components that control the current in order to maximize the resulting pinch, and to ensure that the device works safely. For more details, see the notes.[37]

In plasma physics three pinch geometries are commonly studied: the θ-pinch, the Z-pinch, and the screw pinch. These are cylindrically shaped. The cylinder is symmetric in the axial (z) direction and the azimuthal (θ) directions. The one-dimensional pinches are named for the direction the current travels.

A sketch of the θ-pinch equilibrium. The   z-directed magnetic field corresponds to a   θ-directed plasma current.

The θ-pinch has a magnetic field directed in the z direction and a large diamagnetic current directed in the θ direction. Using Ampère's circuital law (discarding the displacement term)

${\displaystyle {\begin{aligned}{\vec {B}}&=B_{z}(r){\hat {z}}\\\mu _{0}{\vec {J}}&=\nabla \times {\vec {B}}\\&={\frac {1}{r}}{\frac {d}{d\theta }}B_{z}(r){\hat {r}}-{\frac {d}{dr}}B_{z}(r){\hat {\theta }}\end{aligned}}}$

Since B is only a function of r we can simplify this to

${\displaystyle \mu _{0}{\vec {J}}=-{\frac {d}{dr}}B_{z}(r){\hat {\theta }}}$

So J points in the θ direction.

Thus, the equilibrium condition (${\displaystyle \nabla p=\mathbf {j} \times \mathbf {B} }$) for the θ-pinch reads:

${\displaystyle {\frac {d}{dr}}\left(p+{\frac {B_{z}^{2}}{2\mu _{0}}}\right)=0}$

θ-pinches tend to be resistant to plasma instabilities; This is due in part to Alfvén's theorem (also known as the frozen-in flux theorem).

A sketch of the Z-pinch equilibrium. A   θ-directed magnetic field corresponds to a   z-directed plasma current.

The Z-pinch has a magnetic field in the θ direction and a current J flowing in the z direction. Again, by electrostatic Ampère's law,

${\displaystyle {\begin{aligned}{\vec {B}}&=B_{\theta }(r){\hat {\theta }}\\\mu _{0}{\vec {J}}&=\nabla \times {\vec {B}}\\&={\frac {1}{r}}{\frac {d}{dr}}\left(rB_{\theta }(r)\right){\hat {z}}-{\frac {d}{dz}}B_{\theta }(r){\hat {r}}\\&={\frac {1}{r}}{\frac {d}{dr}}\left(rB_{\theta }(r)\right){\hat {z}}\end{aligned}}}$

Thus, the equilibrium condition, ${\displaystyle \nabla p=\mathbf {j} \times \mathbf {B} }$, for the Z-pinch reads:

${\displaystyle {\frac {d}{dr}}\left(p+{\frac {B_{\theta }^{2}}{2\mu _{0}}}\right)+{\frac {B_{\theta }^{2}}{\mu _{0}r}}=0}$

Since particles in a plasma basically follow magnetic field lines, Z-pinches lead them around in circles. Therefore, they tend to have excellent confinement properties.

The screw pinch is an effort to combine the stability aspects of the θ-pinch and the confinement aspects of the Z-pinch. Referring once again to Ampère's law,

${\displaystyle \nabla \times {\vec {B}}=\mu _{0}{\vec {J}}}$

But this time, the B field has a θ component and a z component

${\displaystyle {\begin{aligned}{\vec {B}}&=B_{\theta }{\hat {\theta }}+B_{z}{\hat {z}}\\\mu _{0}{\vec {J}}&={\frac {1}{r}}{\frac {d}{dr}}\left(rB_{\theta }\right){\hat {z}}-{\frac {d}{dr}}B_{z}{\hat {\theta }}\end{aligned}}}$

So this time J has a component in the z direction and a component in the θ direction.

Finally, the equilibrium condition (${\displaystyle \nabla p=\mathbf {j} \times \mathbf {B} }$) for the screw pinch reads:

${\displaystyle {\frac {d}{dr}}\left(p+{\frac {B_{z}^{2}+B_{\theta }^{2}}{2\mu _{0}}}\right)+{\frac {B_{\theta }^{2}}{\mu _{0}r}}=0}$

The screw pinch might be produced in laser plasma by colliding optical vortices of ultrashort duration.[52] For this purpose optical vortices should be phase-conjugated.[53] The magnetic field distribution is given here again via Ampère's law:

${\displaystyle \nabla \times {\vec {B}}=\mu _{0}{\vec {J}}}$

A toroidal coordinate system in common use in plasma physics.

  The red arrow denotes the poloidal direction (θ)

  The blue arrow denotes the toroidal direction (φ)

A common problem with one-dimensional pinches is the end losses. Most of the motion of particles is along the magnetic field. With the θ-pinch and the screw-pinch, this leads particles out of the end of the machine very quickly, leading to a loss of mass and energy. Along with this problem, the Z-pinch has major stability problems. Though particles can be reflected to some extent with magnetic mirrors, even these allow many particles to pass. A common method of beating these end losses, is to bend the cylinder around into a torus. Unfortunately this breaks θ symmetry, as paths on the inner portion (inboard side) of the torus are shorter than similar paths on the outer portion (outboard side). Thus, a new theory is needed. This gives rise to the famous Grad–Shafranov equation. Numerical solutions to the Grad–Shafranov equation have also yielded some equilibria, most notably that of the reversed field pinch.

As of 2015, there is no coherent analytical theory for three-dimensional equilibria. The general approach to finding such equilibria is to solve the vacuum ideal MHD equations. Numerical solutions have yielded designs for stellarators. Some machines take advantage of simplification techniques such as helical symmetry (for example University of Wisconsin's Helically Symmetric eXperiment). However, for an arbitrary three-dimensional configuration, an equilibrium relation, similar to that of the 1-D configurations exists:[54]

${\displaystyle \nabla _{\perp }\left(p+{\frac {B^{2}}{2\mu _{0}}}\right)-{\frac {B^{2}}{\mu _{0}}}{\vec {\kappa }}=0}$

Where κ is the curvature vector defined as:

${\displaystyle {\vec {\kappa }}=\left({\vec {b}}\cdot \nabla \right){\vec {b}}}$

with b the unit vector tangent to B.

Consider a cylindrical column of fully ionized quasineutral plasma, with an axial electric field, producing an axial current density, j, and associated azimuthal magnetic field, B. As the current flows through its own magnetic field, a pinch is generated with an inward radial force density of j x B. In a steady state with forces balancing:

${\displaystyle \nabla p=\nabla (p_{e}+p_{i})=\mathbf {j} \times \mathbf {B} }$

where ∇p is the magnetic pressure gradient, and p_e and p_i are the electron and ion pressures, respectively. Then using Maxwell's equation ${\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {j} }$ and the ideal gas law ${\displaystyle p=NkT}$, we derive:

${\displaystyle 2Nk(T_{e}+T_{i})={\frac {\mu _{0}}{4\pi }}I^{2}}$ (the Bennett relation)

where N is the number of electrons per unit length along the axis, T_e and T_i are the electron and ion temperatures, I is the total beam current, and k is the Boltzmann constant.

The generalized Bennett relation considers a current-carrying magnetic-field-aligned cylindrical plasma pinch undergoing rotation at angular frequency ω

The generalized Bennett relation considers a current-carrying magnetic-field-aligned cylindrical plasma pinch undergoing rotation at angular frequency ω. Along the axis of the plasma cylinder flows a current density j_z, resulting in an azimuthal magnetic field Β_φ. Originally derived by Witalis,[57] the generalized Bennett relation results in:[58]

${\displaystyle {\begin{aligned}{\frac {1}{4}}{\frac {\partial ^{2}J_{0}}{\partial t^{2}}}={}&W_{\perp {\text{kin}}}+\Delta W_{E_{z}}+\Delta W_{B_{z}}+\Delta W_{k}-{\frac {\mu _{0}}{8\pi }}I^{2}(a)\\[8pt]&{}-{\frac {1}{2}}G{\overline {m}}^{2}N^{2}(a)+{\frac {1}{2}}\pi a^{2}\epsilon _{0}\left(E_{r}^{2}(a)-E_{\phi }^{2}(a)\right)\\\end{aligned}}}$

• where a current-carrying, magnetic-field-aligned cylindrical plasma has a radius a,
• J₀ is the total moment of inertia with respect to the z axis,
• W_⊥kin is the kinetic energy per unit length due to beam motion transverse to the beam axis
• W_Bz is the self-consistent B_z energy per unit length
• W_Ez is the self-consistent E_z energy per unit length
• W_k is thermokinetic energy per unit length
• I(a) is the axial current inside the radius a (r in diagram)
• N(a) is the total number of particles per unit length
• E_r is the radial electric field
• E_φ is the rotational electric field

The positive terms in the equation are expansional forces while the negative terms represent beam compressional forces.

The Carlqvist relation, published by Per Carlqvist in 1988,[12] is a specialization of the generalized Bennett relation (above), for the case that the kinetic pressure is much smaller at the border of the pinch than in the inner parts. It takes the form

${\displaystyle {\frac {\mu _{0}}{8\pi }}I^{2}(a)+{\frac {1}{2}}G{\overline {m}}^{2}N^{2}(a)=\Delta W_{B_{z}}+\Delta W_{k}}$

and is applicable to many space plasmas.

The Bennett pinch showing the total current (I) versus the number of particles per unit length (N). The chart illustrates four physically distinct regions. The plasma temperature is 20 K, the mean particle mass 3×10⁻²⁷ kg, and ΔW_Bz is the excess magnetic energy per unit length due to the axial magnetic field B_z. The plasma is assumed to be non-rotational, and the kinetic pressure at the edges is much smaller than inside.

The Carlqvist relation can be illustrated (see right), showing the total current (I) versus the number of particles per unit length (N) in a Bennett pinch. The chart illustrates four physically distinct regions. The plasma temperature is quite cold (T_i = T_e = T_n = 20 K), containing mainly hydrogen with a mean particle mass 3×10⁻²⁷ kg. The thermokinetic energy W_k >> πa² p_k(a). The curves, ΔW_Bz show different amounts of excess magnetic energy per unit length due to the axial magnetic field B_z. The plasma is assumed to be non-rotational, and the kinetic pressure at the edges is much smaller than inside.

Chart regions: (a) In the top-left region, the pinching force dominates. (b) Towards the bottom, outward kinetic pressures balance inwards magnetic pressure, and the total pressure is constant. (c) To the right of the vertical line ΔW_Bz = 0, the magnetic pressures balances the gravitational pressure, and the pinching force is negligible. (d) To the left of the sloping curve ΔW_Bz = 0, the gravitational force is negligible. Note that the chart shows a special case of the Carlqvist relation, and if it is replaced by the more general Bennett relation, then the designated regions of the chart are not valid.

Carlqvist further notes that by using the relations above, and a derivative, it is possible to describe the Bennett pinch, the Jeans criterion (for gravitational instability,[59] in one and two dimensions), force-free magnetic fields, gravitationally balanced magnetic pressures, and continuous transitions between these states.