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Magnetic acceleration of a superconducting solenoid to hypervelocities

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1966 J. Nucl. Energy, Part C Plasma Phys. 8 541

(http://iopscience.iop.org/0368-3281/8/5/306)

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Plasma Physics (Journal of Nuclear Energy Par1 C) 1966, Vol. 8, pp. 541 to 553. Pergamon Press Ltd. Printed in N. Ireland

MAGNETIC ACCELERATION OF A SUPERCONDUCTING SOLENOID TO HYPERVELOCITIES

F. WINTERBERG Desert Research Institute and Department of Physics,

University of Nevada, Reno, Nevada

(Received 9 February 1966)

Abstract-A new method is described which promises to accelerate large macroscopic particles up to meteoric velocities and beyond. In this method a superconducting solenoid is trapped and accelerated in front of a magnetic travelling wave. The magnetic travelling wave is generated by a lumped parameter transmission line. It is demonstrated that with relatively modest accelerator dimensions velocities in the meteoric range of lo6 “sec up to 10’ “sec may be attainable. To reach a velocity of 108cm/sec the length of the accelerator will be at least of the order of a few kilometres. At velocities of IO* cm/sec the controlled release of thermonuclear power should become feasible. The high pressure generated at impact should make it possible to study matter under conditions similar to those that exist in the interior of planets and stars. For velocities of IO8 “sec and beyond it should be possible to observe the Lorentz contraction.

1. I N T R O D U C T I O N THE acceleration of macroscopic solid particles to ultra-high velocities is of increasing scientific and technological importance. For example, one obvious goal is the gen- eration of artificial meteors, within the study of the various phenomena created by fast moving objects.

The impact of a fast moving projectile into dense matter generates a shock wave which is accompanied by very high pressures and temperatures. These high pressures which exist for a very short time are comparable to pressures in the centre of celestial bodies. It is obvious that the study of matter under these high pressures is necessary for understanding the structures of planetary bodies. Also, research in the field of ultra-high pressures is relevant to areas other than planetary physics. In chemistry for instance, there is an interest in studying reactions at very high pressures.

By the impact, very high temperatures are attainable in condensed matter. Tem- peratures in the range of lo6 OK are predicted (WINTERBERG, 1963) for projectile velocities of lo7 cmjsec. It has been previously shown that with velocities of 10s cm/ sec it should be possible to ignite a thermonuclear explosion of controllable size (WINTERBERG, 1963, 1964; and HARRISON, 1963). At these velocities and even higher, the Lorentz contraction should be within reach of experimental detection.

Because of the widespread interest in hypervelocity research, considerable efforts have already been made to accelerate macroparticles to high velocities.

One advanced chemical method to attain high velocities is the light gas gun (HENDRICKS JR., 1958). The light gas gun is essentially a two-stage gun where in the first stage, a solid chemical propellant accelerates a piston to high velocities moving into a chamber filled with hydrogen gas. As a result the hydrogen gas is heated up to high temperatures and thus propels a projectile to a velocity considerably higher than the velocity of the piston. The velocities attainable with light gas guns are in the range of a few lo5 cmisec, and therefore below meteoric velocities by one order of magnitude.

541

542 F. WINTERBERG

Higher velocities have been obtained with the electrostatic acceleration method (SHELTON er al., 1960; FRIICHTENICHT, 1962; VEDDER, 1963 ; MUKHAMEDZHANOV, 1963). In this method, small micron-size particles are charged upelectrically and then accelerated by an applied electric field. With Van de Graff accelerators micron-size particles have been brought to velocities of IO6 cmjsec which is a t the lower end of the meteoric velocity range. By technical perfection of the method and by using smaller- size particles it should be possible to reach IO7 cmjsec. The drawback of the electro- static acceleration method is that it is efficient only for very small particles in the micron-size range and below, and therefore not very useful for most applications of interest. For larger particles, the length of the accelerator becomes prohibitive.

Magnetic accelerators, which use ferromagnetic projectiles, can accelerate large particles (OBERTH, 1929). I n such a n accelerator, the ferromagnetic projectile is accelerated in front of a travelling magnetic wave. Because of the relatively low saturation field strength for ferromagnetic materials, the accelerator dimensions turn out to be too prohibitive for meteoric velocities.

Another method which seems to promise very high velocities is the induction-type electromagnetic gun (Report of High Velocity Conference, 1955). In suchan induction- type gun, large eddy currents are induced in a highly conducting projectile by a travelling magnetic wave. As a consequence, the projectile acquires a large magnetic moment and is thus accelerated by the travelling magnetic wave. In such a device the field of the travelling magnetic wave has to serve two purposes. The first is to induce eddy currents in the projectile, and the second is to accelerate it. In one possible scheme, the projectile has the shape of a torus. The rapidly rising field in the magnetic travelling wave induces toroidal ring currents by which the toroidal projectile acquires a magnetic moment directed along the principal axis of the torus. One can, however, show that the generation of eddy currents in the conducting torus is accompanied by large Joule heating. The heating will be so severe that the projectile will evaporate before reaching a high velocity.

It should be possible also to accelerate a conducting particle by a rapidly collapsing magnetic field (WINTERBERG, 1965a). In tnis case, the time for the magnetic field to collapse must be short compared with the time for the field to penetrate into the conductor, which is the condition for the skin effect. As a consequence of the skin effect, the eddy currents in the particle are generated in a surface layer which evapo- rates rapidly by Joule heating, again limiting the attainable velocities.

In order to avoid this disastrous heating effect, it seems obvious to replace ordinary conductors by superconductors. However, to achieve useful velocities with induction- type guns, type I superconductors must be excluded because their critical fields do not exceed a few kilogauss. Superconductors of type 11 have much higher critical fields, but in view of their large hysteresis losses in a.c. fields, induction-type electromagnetic guns using superconductor I1 projectiles are unfeasible.

A solenoid formed by a superconductor of type 11, however, can acquire a large intrinsic magnetic moment. Consequently, such a solenoid can be accelerated in front of a travelling magnetic wave similar to the magnetostatic method described above using ferromagnetic projectiles. In contrast to the ferromagnetic case, the acceleration is much more efficient because of the high critical magnetic fields in superconductors 11. An accelerator of this type, properly called a magnetostatic accelerator, is described in the following section.

Magnetic acceleration of a superconducting solenoid to hypervelocities 543

In principle such a magnetostatic accelerator could also work with bulk objects of superconductor I. The acceleration there is limited for the same reasons as in induction- type guns using superconductors I because of their low critical fields.

2. MAGNETIC ACCELERATION OF A SUPERCONDUCTING SOLENOID Acceleration of a superconductor by a travelling magnetic wave is efficient only for

a superconducting solenoid (WINTERBERG, 1965b) and because of the demand for high magnetic fields only with type I1 superconductors.

The necessity to employ a superconducting solenoid (coil) can be seen as follows: The superconductor of type I1 has n o perfect Meissner effect. If a superconductor

I1 is topologically simply connected, an externally applied magnetic field will therefore permeate the superconductor uniformly and its magnetic moment will be small. If a toroidal superconductor I1 is placed into an external magnetic field rising in time, toroidal ring currents will be induced as a consequence of Maxwell's equations. These currents can in principle be volume currents because a superconductor I1 has no perfect Meissner effect. The Hall effect, however, limits these volume currents severely so that a superconductor I1 of toroidal shape cannot acquire a large magnetic moment. This important fact has been apparently overlooked in a similar study of the idea t o accelerate a superconductor (MAISONNIER, 1964). The upper limit of the current density resulting from the Hall effect can be substantially increased if the superconductor I1 is subdivided into many coaxial rings which are insulated from each other, or alternatively by winding a thin superconducting wire into a coil(HANAK, 1964). Such superconducting high magnetic field coils have been already operational a t magnetic field strengths of IO5 G and current densities of lo5 A/cm2. The material used for these superconducting coils has been Nb,Sn. Higher critical fields should be expected with Va,Ga. The Landau-Ginzburg theory predicts for Va,Ga a critical field of approximately 5 x lo5 G (NEWHOUSE, 1964). Because of the tensile strength of the superconducting material there is another limitation derived by equating the tensile strength r~ with the maximum possible magnetic stress HZmax/4r:

Hm,, = d(477-a). (1) Inserting (T = 1O1O dyn/cm2, (a value typical for steel) into (1) we find Hmax = 3.5 x IO5 G. In the future, it may be possible to reach the theoretical limit of 5 x lo5 G by the development of superconducting whiskers. Wires consisting of Nb,Sn have been kept superconducting in magnetic fields of 2 x lo5 G at a temperature of 8°K. Theoretical considerations indicate that Va,Ga should still be superconducting in a field of 3 x IO5 G a t approximately the same temperature (NEWHOUSE). In general higher critical fields can be expected by lowering the temperature.

In a superconductor 11, the theoretical expression for the critical current density valid for zero temperature is given by (NEWHOUSE)

j = neaA/h (2) where n is the particle number density of the conduction electrons, a thelattice distance, A the energy gap of the superconductor, and h the Planck constant. Inserting into (2) n = 1O2I ~ m - ~ , a = lo-' cm, A = 3 x loF4 eV = 5 x ergs, there results j = 2.5 x 10l6 e.s.u. = lo7 A/cmz. The highest current densities already obtained are approximately 105 A/cm2 in a transverse magnetic field of lo5 G at temperatures of

544 F. WINTERBERG

4.2 OK (NEWHOUSE). By lowering the temperature it may be possible in the not too far distant future to raise the current density to 5 x lo5 A/cm2.

The inner layers of a superconducting coil will be exposed to a higher magnetic field. In general, a magnetic field will reduce the critical current density. The coil for this reason has to be optimized so as to give each layer its maximum current density compatible with the limitations imposed by the magnetic field, which is different in different layers (NEWHOUSE). The coil optimization can be designed if the dependence of the critical current density on the magnetic field strength is known. Since this function is given only by experimentally measured values depending on wire diameters and metallurgical properties, we neglect the coil optimization by assuming a constant value for the current density in every layer of the coil.

For a coil of length I , inner radius rl and outer radius r2 for which r2 <l, one computes the magnetic moment M(rl, r2) (electrostatic c.g.s. units):

j h M(r,, rz ) = - (rZ3 - r13) .

3c

If the magnetic field along the axis of the coil is Ho one has furthermore

(3)

H, is the maximum field in the solenoid, and therefore must be kept below the critical field strength of the superconducting material. From (4) it follows that for a given maximum field strength and a given maximum current density, there results an expression for the thickness of the solenoid d = r2 - r1 which is a fixed design param- eter

d = H0c/4vj. (5) In Fig. 1, a plot of d vs. j for H,, = lo5, 2 x IO5 and 3 x IO5 G is given. Eliminating j from (3) and (5) yields

HoI M(rl,r2) = - ( r z - r:) , 12d

Depending on its magnetic history, a superconducting solenoid can acquire a magnetic moment behaving either diamagnetically or paramagnetically. Apart from the sign, the maximum magnetic moment in both cases is the same and given by equation (6).

The proposal is for the solenoid to be accelerated by a travelling magnetic wave which can be generated by a lumped parameter transmission line (CLAUSER, 1961) and shall produce the wave form drawn in Fig. 2 . After closing the switch in the trans- mission line, a travelling magnetic wave starts with a phase velocity U given by

U = l/.\/(LC). (7 )

In ( 7 ) L and C are the inductance and capacitance per unit length of the transmission line. The solenoid is placed in the reference system of the travelling magnetic wave as shown in Fig. 2 and is thereby exposed to the maximum field gradient. In the reference system of the wave, a magnetic force Fac ts on the solenoid given by

dH1 F = M - - , d x

Magnetic acceleration of a superconducting solenoid to hypervelocities 545

4 -

3 -

H,= 3x10' gauss 1 H,= 2 x IO' gauss k I I I I I b IO* 2x10' 3x10' 4x10' 5x10' j[A/cm*]

FIG. I .-Dependence of d upon , j and Hc.

t X

H, = -H --------- m

FIG. 2.--Schematic diagram showing the shape of the magnetic field and position of the solenoid in it (paramagnetic case),

where dH,/dx is the gradient of the magnetic field in front of the travelling magnetic wave, and x is directed along the transmission line accelerator. If the travelling magnetic wave is uniformly accelerated so that the velocity z' of the solenoid is equal to the phase velocity U in each segment of the transmission line accelerator, the inertial force -)1?0 acting on the solenoid will be in static equilibrium with the magnetic force F exerted by the travelling magnetic wave. If the solenoid behaves diamag- netically and is to be accelerated, it can be exposed to a field gradient extending only over a quarter wave length of the travelling magnetic wave drawn in Fig. 2. In the paramagnetic case, the solenoid can be exposed to the entire gradient extending over

546 F. WINTERBERG

a half wavelength. Since uniform acceleration acts like a gravitational force, the stability problem of the solenoid placed in front of a travelling magnetic wave is analogous to the stability problem of a magnetic body suspended in a magnetic field against gravity. From this analogy, one should expect stability for the diamagnetic case. In the paramagnetic case, the stability must be accomplished by feedback control of the field as being used in the suspension of ultra-centrifuges in magnetic fields.

If the magnetic field rises from -H, to +H, over the length of the solenoid (paramagnetic case), we have

dHl 2Hm dx I ' -=-

and therefore F = 2MHJI.

(9 )

The mass m = m(rl,r2) of the solenoid with a density p is given by

m(rl,rz) = pnl(r,2 - r?). (1 1) From (6) , (10) and (1 1) one obtains the acceleration, a, of the solenoid

H,H, r: - r? 6npld r t - r I 2 '

- --

For the special case of a solid coil, we have r, = 0, and r2 = d, and an acceleration a = a,:

We furthermore introduce the parameter p so that r2 = pd, p 2 1, and thus

r1 = d(p - 1). (14)

a = aof(p) (1 5 )

The acceleration given by equation (12) is then expressed by

where 3 3 1 1

f ( p , = jp - - + -- 4 4 2 p - 1 '

In Fig. 3, a plot off@) is given. From equation ( 1 5 ) it follows that the acceleration is independent of the thickness d and thereby independent of the critical current density j . A large value of j , however, will permit a small thickness d and thus reduce the mass of the solenoid for the same value of the acceleration.

The length L of the accelerator for a given final velocity 1: is determined by

L = v2/2a. (17)

Magnetic acceleration of a superconducting solenoid to hypervelocities 547

- I I 2 3 P

I I I

FIG. 3.-The auxiliary functionf(p).

Lo = u2/2a,, We introduce

and have J L = JLolf(p).

In order to keep the solenoid superconducting in the travelling magnetic wave the maximum field H,,, is not permitted to exceed the critical field of the superconducting material, It is to be remembered that Ho which is the maximum field strength within the coil has to be kept just below the critical field strength. Therefore, putting in (13) H , = H, leads to the following value of a,:

and therefore

In Fig. 4 the dependence of L as a function of L' is given for the values of H, = lo5 G, 2 x IO5 G and 3 x IO5 G. For the parameter p we assume the values 1 < p 5 2.5, furthermore 1 = 2 cm and p = 5 g/cm3.

a, = Ho216npl, (20)

Lo = 37rpl(~/H,)~. (21)

3. T H E T R A N S M I S S I O N L I N E A C C E L E R A T O R The accelerator consists of many coaxial coils charged by capacitors as indicated

in Fig. 5. The solenoid is accelerated in the centre along the axis of these coaxial coils. After closing the switch at D a magnetic travelling wave starts propagating down the transmission line. The phase velocity of the wave is given by (7)"

For a uniform acceleration, the phase velocity U must be made equal to the velocity U of the solenoid in each segment of the transmission line. This requirement leads to

LC = 1/2ax. (22) * Note added in proof: If the electric current in the coils of the transmission line flows in one

direction only the resulting wave form would be like a shock wave. In order to produce the wave form in the centre of the accelerator as shown in Fig. 2 it is necessary to construct the coils in several concentric layers in which the electric current flows n different directions.

548 F. WINTERBERG

p = I

p = I IO' p = I

p = 2.5

p = 2.5 p ='2.5

10.

rd

IO'

IO'

IO

thermo- meteor nuclear

velocity velocity +- range * +range-- v[cm/secj

IO5 I os IOT IO8

FIG. 4.-Accelerator length L as a function of U, Ho andp.

I , IO5 I os IOT IO8

FIG. 4.-Accelerator length L as a function of U, Ho andp.

If R is the radius of the field coils in the transmission line, then apparently the stored energy E per unit length of the transmission line must be approximately

E = +(RH,#. (23) This energy is constant for all segments of the accelerator and must be supplied by the capacitors of the transmission line. If all capacitors are charged by the same voltage then the capacitance per unit length along the transmission line is constant. To place the solenoid in the centre of the transmission line requires that R 2 r2, therefore

Taking H,, = 3 x lo5 G which according to (1) is the highest value for the magnetic field, further rz = 1 cm one has E > 1-12 x 1O1O ergs/cm = 1.12 x lo3 J/cm. Energy of this amount can be stored in high-voltage capacitors of the type used in &pinch experiments.

Magnetic acceleration of a superconducting solenoid to hypervelocities 549

I V X

B A

FIG. fi.-Transmission line accelerator. A : accelerated solenoid; B: field coils; C : capacitors; D : switch to be closed to start travelling magnetic wave. Hl(x): magnetic field on transmission line relative to the frame of the solenoid. n(x): windings per

unit length in field coils (arbitrary units).

If the capacitance per unit length C is constant, then according to equation (22) L is inversely proportional to x. The inductance L of a coil with a radius R and n windings per unit length is given by

L = (2~rRn/c)~ . (25)

Inserting (25) into (22) and solving for n = n ( x ) yields

(26) C

2xR.\/(2aCx) ’ n(x) =

therefore n(x) a ~ - 1 / 2 . It thus follows that a uniformly accelerated magnetic travelling wave can be generated with field coils decreasing in number of turns down the trans- mission line.

550 F. WINTERBERG

U I I 11 o .-

d :lo

: 2 I

0 I I1

.- I f

1

I \ \ \ I \

.- \

4

\ \ \

I 0

I I I I

I I I 1 I I I

I

o I

0

-+------- W ----

0 I

U I I 1 .- I

O

Magnetic acceleration of a superconducting solenoid to hypervelocities

4. MAGNETIC PREPARATION OF SOLENOID PRIOR

551

TO INJECTION I N ACCELERATOR

Prior to the moment a t which the solenoid is injected into the accelerator it has to be magnetized. As pointed out previously the solenoid can either behave diamag- netically or paramagnetically.

This property is most clearly demonstrated by a diagram showing the accessible regions in a Hi - Ha -plane, where Hi is the magnetic field in the centre of the solenoid and the Ha the externally applied magnetic field. The region magnetically accessible to the solenoid has to be within the boundary lines given by

where H, is the critical magnetic field.

FIG. 7.-Magnetization curve for a superconducting solenoid in the M - &plane (schematic).

In the presence of an external magnetic field, Maxwell’s equations applied to the solenoid yield

4ad Hi - Ha = - j . (28) C

If j , is the critical current density then two other boundary lines in the Hi - Ha- plane are given by

4ad If, - Ha = 3 -j, = +If,. (29)

C

552 F. WINTERBERG

The boundary lines given by equations (27) and (29) are drawn as dotted lines in Fig. 6.

The magnetic moment of the solenoid expressed in terms of the internal and external magnetic fields is obtained from equation (3) after inserting (28):

In Fig. 7, we have plotted M i n arbitrary units as a function of Ha. ' T h e points indicated by the letters A-G on Figs. 6 and 7 correspond to each other.

If we insert a superconducting solenoid with no previous magnetic history in a magnetic field slowly rising in time, the solenoid will go from the initial point A to the point B in the Hi - Ha -plane and in the M - Ha -plane. In going from A to B, induced currents in the solenoid will shield its interior against the applied external magnetic field thereby making Hi = 0. From A to B, the solenoid will behave like a perfect diamagnetic body and will acquire the maximum magnetic moment a t point B.

Further increasing the external magnetic field, Ha, will lead to a flux jump from point B to point C where the magnetie field penetrates into the solenoid until the whole volume is filled uniformly. This flux jump is accompanied by the release of heat. By decreasing Ha in going from C to D the magnetic moment becomes positive, and the solenoid acts like a paramagnetic body reaching its maximum magnetic moment a t point D. If Ha, at position D , is decreased to negative values, the solenoid will move under the release of heat along the curve D E in order not to exceed the critical current density. The behaviour along the lines FG and GB is quite analogous to the behaviour along CD and DE. The closed loop in the Hi - Ha-, and M - Ha -plane represents the hysteresis loop characteristic for a particular solenoid.

For the maximum possible force acting on the solenoid by a n external inhomo- geneous magnetic field, the magnetic moment must acquire its maximum value. The solenoid should therefore preferably operate on the line D E for the paramagnetic case and on the line BG for the diamagnetic case.

Since the magnetic states on these lines can be reached only be slowly changing the externally applied magnetic field and cooling at the same time, this must be done in an auxiliary field before the instant when the solenoid is injected into the accelerator.

5. C O N C L U S I O N

The described method of accelerating a superconducting solenoid seems to promise velocities by a t least one order of magnitude beyond what is attainable with light gas guns, thus making the generation of artificial meteors possible. Further development of the process should ultimately lead to the controlled release of thermonuclear power.

CLAUSER M. U. (1961) in Advanced Propulsion Systems, p. 138, Pergamon Press, Oxford. FRIICHTENICHT J. F. (1962) Rev. Scient. Instrum. 33, 209. HANAK T. T. (1964) RCA Rev. 25, 551. HARRISON E. R. (1963) Phys. Rev. Lett. 11, 535. HENDRICKS C. D. JR. (1958) Technical Report ERL-LM-154, Electronics Research Laboratory,

MAISONNIER C. H. (1964) Report 64/21 Laboratorio Gas Ionizzati, Euratom. MUKHAMEDZHANOV A. K. (1963) Planet. Space Sci. 11, 1485. NEWHOUSE V. L. (1964) Applied Superco/ldrrctiui/.y, Wiley, New York. OBERTH H. (1929) Wege zur Raumsch/fluhrf, R. Oldenbourg, Munich.

R E F E R E N C E S

The Ramo-Wooldrige Corporation.

Magnetic acceleration of a superconducting solenoid to hypervelocities 553

Report of High Velocity Conference at the Rand Corporation, (1955) [Several unclassified reports

SHELTON H. et al. (1960)J. uppl. Phys. 31, 1243. VEDDER J. F. (1963) Rev. sci. Instrum. 34, 1175. WINTERBERG F. (1963) Case Institute of Technology, Plasma Research Program, Technical Report

WINTERBERG F. (1964) 2. Nuturfi 19a, 231. WINTERBERG F. (1965a) Bull. Am. phys. Soc. 10, 239. WINTERBERG F. (1965b) University of Nevada, Desert Research Institute Preprint No. 11 ; (1966)

on induction-type guns].

No. A21, June 1963; (1964) Bull. Amphys. Soc. 9,309.

Nucl. Fusion. To be published.

J