Magnetic nozzle

Last updated April 21, 2019

A magnetic nozzle is a convergent-divergent magnetic field that guides, expands and accelerates a plasma jet into vacuum for the purpose of space propulsion.^[1] The magnetic field in a magnetic nozzle plays a similar role to the convergent-divergent solid walls in a de Laval nozzle, wherein a hot neutral gas is expanded first subsonically and then supersonically to increase thrust. Like a de Laval nozzle, a magnetic nozzle converts the internal energy of the plasma into directed kinetic energy, but the operation is based on the interaction of the applied magnetic field with the electric charges in the plasma, rather than on pressure forces acting on solid walls.^[2] The main advantage of a magnetic nozzle over a solid one is that it can operate contactlessly, i.e. avoiding the material contact with the hot plasma, which would lead to system inefficiencies and reduced lifetime of the nozzle. Additional advantages include the capability of modifying the strength and geometry of the applied magnetic field in-flight, allowing the nozzle to adapt to different propulsive requirements and space missions. Magnetic nozzles are the fundamental acceleration stage of several next-generation plasma thrusters currently under development, such as the helicon plasma thruster, the electron-cyclotron resonance plasma thruster, the VASIMR, and the applied-field magnetoplasmadynamic thruster. Magnetic nozzles also find another field of application in advanced plasma manufacturing processes, and their physics are related to those of several magnetic confinement plasma fusion devices.

A magnetic field is a vector field that describes the magnetic influence of electric charges in relative motion and magnetized materials. Magnetic fields are observed in a wide range of size scales, from subatomic particles to galaxies. In everyday life, the effects of magnetic fields are often seen in permanent magnets, which pull on magnetic materials and attract or repel other magnets. Magnetic fields surround and are created by magnetized material and by moving electric charges such as those used in electromagnets. Magnetic fields exert forces on nearby moving electrical charges and torques on nearby magnets. In addition, a magnetic field that varies with location exerts a force on magnetic materials. Both the strength and direction of a magnetic field vary with location. As such, it is an example of a vector field.

Plasma is one of the four fundamental states of matter, and was first described by chemist Irving Langmuir in the 1920s. Plasma can be artificially generated by heating or subjecting a neutral gas to a strong electromagnetic field to the point where an ionized gaseous substance becomes increasingly electrically conductive, and long-range electromagnetic fields dominate the behaviour of the matter.

A de Laval nozzle is a tube that is pinched in the middle, making a carefully balanced, asymmetric hourglass shape. It is used to accelerate a hot, pressurized gas passing through it to a higher supersonic speed in the axial (thrust) direction, by converting the heat energy of the flow into kinetic energy. Because of this, the nozzle is widely used in some types of steam turbines and rocket engine nozzles. It also sees use in supersonic jet engines.

Basic operation of a magnetic nozzle

The expansion of a plasma in a magnetic nozzle is inherently more complex than the expansion of a gas in a solid nozzle, and is the result of several intertwined phenomena, which ultimately rely on the large mass difference between electrons and ions and the electric and magnetic interactions between them and the applied field.

The electron is a subatomic particle, symbol e⁻
or β⁻
, whose electric charge is negative one elementary charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no known components or substructure. The electron has a mass that is approximately 1/1836 that of the proton. Quantum mechanical properties of the electron include an intrinsic angular momentum (spin) of a half-integer value, expressed in units of the reduced Planck constant, ħ. As it is a fermion, no two electrons can occupy the same quantum state, in accordance with the Pauli exclusion principle. Like all elementary particles, electrons exhibit properties of both particles and waves: they can collide with other particles and can be diffracted like light. The wave properties of electrons are easier to observe with experiments than those of other particles like neutrons and protons because electrons have a lower mass and hence a longer de Broglie wavelength for a given energy.

An ion is an atom or molecule that has a net electrical charge. Since the charge of the electron is equal and opposite to that of the proton, the net charge of an ion is non-zero due to its total number of electrons being unequal to its total number of protons. A cation is a positively charged ion, with fewer electrons than protons, while an anion is negatively charged, with more electrons than protons. Because of their opposite electric charges, cations and anions attract each other and readily form ionic compounds.

If the strength of the applied magnetic field is sufficient, it magnetizes the light electrons in the plasma, which therefore describe a helicoidal motion about the magnetic lines. In practice, this is achieved with magnetic fields in the range of a few hundred Gauss. The guiding center of each electron is forced to travel along one magnetic tube.^[2] This magnetic confinement prevents the uncontrolled expansion of the electrons in the radial direction and guides them axially downstream. The heavier ions are typically unmagnetized or only partially magnetized, but are forced to expand with the electrons thanks to the electric field that is set up in the plasma to maintain quasineutrality.^[3] As a result of the ensuing electric field, the ions are accelerated downstream, while all electrons except the more energetic ones are confined upstream. In this way, the electric field helps convert the electron internal energy into directed ion kinetic energy.

In physics, the motion of an electrically charged particle such as an electron or ion in a plasma in a magnetic field can be treated as the superposition of a relatively fast circular motion around a point called the guiding center and a relatively slow drift of this point. The drift speeds may differ for various species depending on their charge states, masses, or temperatures, possibly resulting in electric currents or chemical separation.

A helix, plural helixes or helices, is a type of smooth space curve, i.e. a curve in three-dimensional space. It has the property that the tangent line at any point makes a constant angle with a fixed line called the axis. Examples of helices are coil springs and the handrails of spiral staircases. A "filled-in" helix – for example, a "spiral" (helical) ramp – is called a helicoid. Helices are important in biology, as the DNA molecule is formed as two intertwined helices, and many proteins have helical substructures, known as alpha helices. The word helix comes from the Greek word ἕλιξ, "twisted, curved".

An electric field surrounds an electric charge, and exerts force on other charges in the field, attracting or repelling them. Electric field is sometimes abbreviated as E-field. The electric field is defined mathematically as a vector field that associates to each point in space the force per unit of charge exerted on an infinitesimal positive test charge at rest at that point. The SI unit for electric field strength is volt per meter (V/m). Newtons per coulomb (N/C) is also used as a unit of electric field strengh. Electric fields are created by electric charges, or by time-varying magnetic fields. Electric fields are important in many areas of physics, and are exploited practically in electrical technology. On an atomic scale, the electric field is responsible for the attractive force between the atomic nucleus and electrons that holds atoms together, and the forces between atoms that cause chemical bonding. Electric fields and magnetic fields are both manifestations of the electromagnetic force, one of the four fundamental forces of nature.

In steady-state operation, the exhausted plasma jet is globally current-free, i.e., the total ion current and electron current at each section are equal. This condition prevents the continuous electrical charging of the spacecraft on which the magnetic nozzle is mounted, which would result if the amount of ions and electrons emitted per unit time differ.

The electron pressure being confined by the magnetic field gives rise to a diamagnetic drift, which is proportional to the pressure of electrons and inversely proportional to the magnetic field strength. Together with the $E\times B$ drift, the diamagnetic drift is responsible of the formation of an azimuthal electric current $j_{\theta }$ in the plasma domain. This azimuthal electric current generates an induced magnetic field which opposes the applied one, generating a repulsive magnetic force $\propto j_{\theta }B$ that pushes the plasma downstream. The reaction to this force is felt on the magnetic generator of the magnetic nozzle and is called magnetic thrust.^[3] This is the main thrust generation mechanism in a magnetic nozzle.

An electric current is the rate of flow of electric charge past a point or region. An electric current is said to exist when there is a net flow of electric charge through a region. In electric circuits this charge is often carried by electrons moving through a wire. It can also be carried by ions in an electrolyte, or by both ions and electrons such as in an ionized gas (plasma).

As described by the third of Newton's laws of motion of classical mechanics, all forces occur in pairs such that if one object exerts a force on another object, then the second object exerts an equal and opposite reaction force on the first. The third law is also more generally stated as: "To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts." The attribution of which of the two forces is the action and which is the reaction is arbitrary. Either of the two can be considered the action, while the other is its associated reaction.

Plasma detachment

The closed nature of the magnetic lines means that unless the plasma separates from the guiding magnetic field downstream, it will turn around along the field lines back to the thruster. This would defeat the propulsive purpose of the magnetic nozzle, as the returning plasma would cancel thrust and could endanger the integrity of the spacecraft and the plasma thruster. A plasma detachment mechanism is therefore necessary for the correct operation of the magnetic nozzle.^[4]

As the plasma expands in the divergent side of the magnetic nozzle, ions are gradually accelerated to hypersonic velocities thanks to the role of the internal electric field in the plasma. Eventually, the unmagnetized, massive ions are fast enough that the weak electric and magnetic forces in the downstream region become insufficient to deflect the ion trajectories except for extremely high magnetic strengths. As a natural consequence, plasma detachment starts to take place^[5] and, the amount of plasma mass flow rate that is actually deflected along the magnetic field and turns back to maintain quasineutral conditions in the plasma is negligible. In consequence, the magnetic nozzle is capable of delivering detached plasma jets usable for propulsion.

In physics and engineering, mass flow rate is the mass of a substance which passes per unit of time. Its unit is kilogram per second in SI units, and slug per second or pound per second in US customary units. The common symbol is $, although sometimes μ is used.$

The separation of ions due to their inertia leads to the formation of local longitudinal electric currents, that do not violate however the global current-free condition in the jet. The influence of the plasma-induced magnetic field, which can deform the magnetic nozzle downstream, and the formation of non-neutral regions, can further reduce the turn-back plasma losses.^[6]

Propulsive performance

The performance of a magnetic nozzle, in terms of its specific impulse, generated thrust and overall efficiency depends on the plasma thruster to which it is connected. The magnetic nozzle should be regarded as a thrust augmentation device, whose role is to convert plasma thermal energy into directed kinetic energy as discussed above. Therefore, thrust and specific impulse are strongly dependent on the electron temperature of the plasma inside the plasma source. A high electron temperature (i.e., a hot plasma) is required to have an effective plasma thruster.

The efficiency of the magnetic nozzle has to be discussed in terms of divergence or radial losses. As a byproduct of the expansion in the divergent magnetic nozzle, part of the kinetic energy of ions is directed in the radial and azimuthal directions. This energy is useless for thrust generation, and therefore accounts as losses. An efficient magnetic nozzle is sufficiently long to minimize the amount of energy wasted in the radial and azimuthal directions.^[3] Additionally, an excessively weak magnetic field would fail to confine radially and guide axially the plasma, incurring in large radial losses.

Other figures of merit of the system are the electric power, mass and volume of the required magnetic field generator (magnetic coils and/or permanent magnets). A low electric power consumption, mass and volume are desirable for space propulsion applications.

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In spacecraft propulsion, a Hall-effect thruster (HET) is a type of ion thruster in which the propellant is accelerated by an electric field. Hall-effect thrusters trap electrons in a magnetic field and then use the electrons to ionize propellant, efficiently accelerate the ions to produce thrust, and neutralize the ions in the plume. Hall-effect thrusters are sometimes referred to as Hall thrusters or Hall-current thrusters. Hall thrusters are often regarded as a moderate specific impulse space propulsion technology. The Hall-effect thruster has benefited from considerable theoretical and experimental research since the 1960s.

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A nozzle is a device designed to control the direction or characteristics of a fluid flow as it exits an enclosed chamber or pipe.

Electron cyclotron resonance is a phenomenon observed in plasma physics, condensed matter physics, and accelerator physics. An electron in a static and uniform magnetic field will move in a circle due to the Lorentz force. The circular motion may be superimposed with a uniform axial motion, resulting in a helix, or with a uniform motion perpendicular to the field resulting in a cycloid. The angular frequency of this cyclotron motion for a given magnetic field strength B is given by

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In plasma physics, waves in plasmas are an interconnected set of particles and fields which propagate in a periodically repeating fashion. A plasma is a quasineutral, electrically conductive fluid. In the simplest case, it is composed of electrons and a single species of positive ions, but it may also contain multiple ion species including negative ions as well as neutral particles. Due to its electrical conductivity, a plasma couples to electric and magnetic fields. This complex of particles and fields supports a wide variety of wave phenomena.

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References

↑ Andersen et al. Physics of Fluids 12, 557 (1969)
1 2 R.A. Gerwin, G.J. Marklin, A.G. Sgro, A.H. Glasser, Characterization of plasma flow through magnetic nozzles, LANL report AL-TR-89-092 (1990)
1 2 3 E. Ahedo, M. Merino, Two-dimensional supersonic plasma acceleration in a magnetic nozzle, Physics of Plasmas 17, 073501 (2010)
↑ Ahedo, E., Merino, M., On plasma detachment in propulsive magnetic nozzles, Physics of Plasmas, Vol. 18, No. 5, 2011, pp. 053504
↑ Merino, M., Ahedo, E., Plasma detachment in a propulsive magnetic nozzle via ion demagnetization, Plasma Sources Science and Technology, Vol. 23, No. 3, 2014, pp. 032001.
↑ Merino, M., Ahedo, E., Effect of the plasma-induced magnetic field on a magnetic nozzle, Plasma Sources Science and Technology, Vol. 25, No. 4, 2016, pp. 045012.

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[1] Andersen et al. Physics of Fluids 12, 557 (1969)

[gerw90-2] 1 2 R.A. Gerwin, G.J. Marklin, A.G. Sgro, A.H. Glasser, Characterization of plasma flow through magnetic nozzles, LANL report AL-TR-89-092 (1990)

[ahed10f-3] 1 2 3 E. Ahedo, M. Merino, Two-dimensional supersonic plasma acceleration in a magnetic nozzle, Physics of Plasmas 17, 073501 (2010)

[4] Ahedo, E., Merino, M., On plasma detachment in propulsive magnetic nozzles, Physics of Plasmas, Vol. 18, No. 5, 2011, pp. 053504

[5] Merino, M., Ahedo, E., Plasma detachment in a propulsive magnetic nozzle via ion demagnetization, Plasma Sources Science and Technology, Vol. 23, No. 3, 2014, pp. 032001.

[6] Merino, M., Ahedo, E., Effect of the plasma-induced magnetic field on a magnetic nozzle, Plasma Sources Science and Technology, Vol. 25, No. 4, 2016, pp. 045012.