Letters tO the editor

The cut which corresponds to the second integral starts at (mo + 1)2 as in the normal case. On the other hand, the first integral gives rise to an anomal branch point at ~ o = m ~ + l + 2 2mQ(1 - co~3)1/2. This is in agreement with the position of the anomal threshold, obtained by K a r p l u s , S o m m e r f i e l d and W i c h m a n n in the case of a triangle diagram [3]. Both examples of the vertex functions studied enable us to conclude that the selfconsistency condition leads in our particular cases to the usual cuts. We find this further support of the validity of the self-consistency in the teN-system. We would like to thank J. Pigfit and M. D u b e c for interesting discussions. Received 12. 5. 1965. M. PETRA~,M. NOGA Natural Science Faculty, Komensk~ University, Brat&lava*) References

[1] Balasz L. A.: Phys. Rev. 128 (1962), 1935. S i n g h V., U d g a o n k a r B. M." Phys. Rev. 130 (1963), 1177. A b e r s E., Z e m a c h , C.: Phys. Rev. 131 (i963), 2305. Ball J., W o n g D.: Phys. Rev. 133 (1964), B 179. [2] Chew G. F.: Phys. Rev. Letters 9 (1962), 233. [3] K a r p l u s R., S o r n m e r f i e l d C. M., W i c h m a n n E. H.: Phys. Rev. 111 (1958), 1187.

PLASMA INDUCTION ACCELERATOR WITH RADIAL MAGNETIC FIELD The possibility of accelerating plasmatic clusters or shock waves in plasma in a simple tube of circular cross-section by an exciting coil with a radial magnetic field as shown in Fig. 1 was investigated. It is an analogy of the acceleration of plasma between parallel conductors with an external magnetic field after [1]. A preliminary theoretical investigation was made both of pulse operation for constant B, and of continuous acceleration for high-frequency changes of the exciting current I and magnetic induction B r. The time change of the exciting current I creates a current of azimuthal direction in the plasma. This accelerates with the radial magnetic field of the plasma. If the usual stabilization conditions and certain assumptions from [1] are considered, the basic magnetohydrodynamical equations can be written in the following form for a cylindrically orthogonal system of coordinates: (1)

duz S--= dt

-- S~oBr,

J~ = aE~, + au=B r . *) Bratislava, ~meralova 2/b. 56

Czech. I. Phys. B 16

(1966)

Letterg to the editor

The quantity J~ is the current density of the azimuthal current of plasma, u z the velocity of the plasma in the direction of the tube axis, s and a the specific mass and specific electrical conductivity of the plasma. It is assumed that they are constant. For pulse operation, when the condenser discharges into the exciting coil, resultant relations were found for a finite velocity of the cluster and its path. Similarly as in [1], thes~ expressions showed that maximum velocity is obtained for a certain magnitude of B,.. This is the so-called resonance (magnetic) state. It is more difficult to calculate in this case than in [1] and it was necessary to use numerical methods for certain intervals of the coefficients of the equations. The results then showed that maximum velocity of the cluster is reached approximately at 3/4 of the period of damped oscillations of the discharging current. For the ratio ~5/~o= 0.1, where 6 is the damping coefficient and co the angular frequency of the oscillations of the discharging current, a resonance radial magnetic field is obtained having inFig. 1. I n d u c t i o n accelerator with radial duction magnetic field o f induction Br; I is curBres

rent o f e x c i t i n g coil.

/0"34

where a is a constant giving the quality of the discharge circuit. With such a field the maximum attainable cluster velocity is given by the relation (2)

Ures = fl

Uo ro2 (1 + (LolL1)) Mo 409

for a path 1"86 Zre s --

Ure s 9

Here fl is a constant that again represents the quality of the set-up, U o is the initial voltage of the discharging condenser, ro the radius of the accelerating tube, L 0 and L 1 the inductance of the discharge circuit and exciting coil, and Mo the mass of the duster. The situation is different when analyzing the equations describing the cluster acceleration for time periodical changes both of the discharging current and of the radial magnetic field. Equations (1) can again be used as the basis. It is interesting that the resultant relation for the cluster density reached could again be obtained in the closed form only for decreasing magnetic radial field. If it was required that the induction (3)

Br = _bo cos ~ot, Uz

Czech. J. Phys. B 16 (1966)

57

Letters to the editor

i.e. the amplitude Br, decreased with increasing density velocity, the resultant relation for the density velocity was (4)

2

uz - u~o

Yt

~2Iom-- bo

1

t +--sin2cot 2co

).

Here Uzo is the initial cluster velocity at a time t = 0, 71 and 72 are constants, Io the amplitude of the exciting current. The other quantities are analogous to those in relations (2) and (3). Relation (4) is very remarkable. It shows primarily that with increasing time the cluster velocity continually grows and from the condition ~(u~) - 0 dbo an analogy of the resonance (magnetic) state with single accelerations is again obtained, i.e. it must hold for maximum velocity increment that (S)

bo = 89

The induction accelerator with a radial magnetic field has maximum velocity given by relation (1) for single acceleration, and the cluster velocity continually increases according to equation (4) for continual acceleration and decreasing radial magnetic field. Received 14. 6. 1965. J. KRACiK, B. I%~OVAK,F. HANITZ, Electrotechnical Faculty, Czech Technical University, Prague*) References ll] K r a c i k J.: Czech. J. Phys. B14 (1963), 683.

C I R C U L A R AUTOSTABLE PATH OF M A G N E T O H Y D R O D Y N A M I C PLASMA ACCELERATOR The magnetic resonance states of magnetohydrodynamic accelerators of plasmatic clusters with parallel conductors, coaxial conductors or without conductors in an induction arrangement are very effective but the paths along which acceleration occurs are very long. An attempt has therefore been made to find an arrangement of an accelerator whereby a circular and, if possible, an autostable path would be possible. An autostable circular path is conceived as a path where the centrifugal force is balanced by a spontaneously produced centric magnetohydrodynamic force. *) Praha 6, Technick~l 1902. 58

Czech, J. Phys. B 16 (1966)