S C H O O L , OF P H Y S I C A L S C I E N C E S

AN EXPERIMENTAL INVESTIGATION OF CURRENT PRODUCTION BY MEANS OF ROTATING MAGNETIC FIELDS

W.N. HUGRASS, I.R. JONES AND M.G.R. PHILLIPS

FUPH-R-168 AUGUST, 1980

AN EXPERIMENTAL INVESTIGATION OF CURRENT PRODUCTION BY MEANS OF ROTATING MAGNETIC FIELDS

N.N. Hugrass, I.R. Jones and M.G.R. Phillips The School of Physical Sciences

The Flinders University of South Australia South Australia, 5042.

ABSTRACT

An investigation of current production by means of a rotating

magnetic field is made in an experiment in which the technique is

used to generate a theta-pinch like distribution of field and plasma.

Detailed measurements are made of both the generated unidirectional

azimuthal electron current and the penetration of the rotating field

into the plasma. The experimental results support the theoretical

prediction that a threshold value of the amplitude of the applied

rotating field exists for setting the electrons into rotation.

1.

1. INTRODUCTION

Significant theoretical extensions to the early work of Blevin and

Thonemann (1962) on the generation of plasma currents by means of rotating

magnetic fields are presented in the two preceding papers (Jones and

Hugrass 1980; Hugrass and Grimm 1980). These theoretical studies show

that provided the amplitude and frequency of an externally imposed

rotating field are suitably chosen, the field will penetrate a plasma

column to an extent greater than that predicted by the usual classical

skin effect, the degree of penetration depending on the amplitude of the

rotating field. This enhanced penetration is accompanied by the

generation of an azimuthal electron current. The amount of generated

current is a nonlinear function of the amplitude of the rotating field;

it saturates at a value corresponding to complete penetration of the

field and the perfect synchronous rotation of the electrons (see, for

example. Fig. 1 in Hugrass and Grimm 1980).

The rotating magnetic field technique can, in principle, be used to

generate the steady equilibrium currents appropriate to a variety of

plasma/field configurations. In two earlier publications (Hugrass,

Jones and Phillips 1979; Hugrass et al. 1980), the feasibility of using

this technique to drive plasma currents in the screw-pinch and compact

torus configuration was demonstrated. Since the purpose of these two

papers was to stress the application of the technique, little experimental

detail is to be found in them regarding the basic physics of the current

drive mechanism. We remedy this situation in the present paper where

an experiment is described in which the rotating field technique is used

to generate a theta-pinch like distribution of field and plasma.

Detailed measurements made in this experiment illustrate the most important

features of the theoretical investigations presented in the two preceding

papers. In particular, they indicate the existence of the nonlinear

2.

characteristic which links the generated plasma current to the amplitude of the rotating field.

2. THEORY

Provided the angular frequency, fa), of an applied transverse rotating magnetic field lies between the ion and electron cyclotron frequencies (u ., u ; calculated with reference to the amplitude of the rotating field) and provided the electron-ion momentum transfer collision frequency is much less than the electron cyclotron frequency, then the magnetic field can completely penetrate a cylindrical plasma column and entrain the electrons so as to produce a steady azimuthal current. At any radial position r, the electron drift velocity is no and hence the azimuthal current density is

j Q(r) = n e(r) |e| no (1)

The component, B (r), of the total axial magnetic field, B (r), which is generated solely by the rotating electrons is given by

u |e|u> N(r) B„(r) e v * 2TT

where

N(r) = 2mr n g(r) dr (3) r

is the line density of electrons lying between r and R; R is the radius of the plasma column. Note that

%l el u N0 B e(R) - 0, B e(0) = 2* ( 4 )

where N is the electron line density. The direction of B (r) depends on the sense of rotation of the

rotating field. For the experiment described in this paper, B (r) must 6

3.

be in the opposite direction to an initially applied steady axial

Magnetic field, B in order to achieve magnetic confinement. The total

axial field is thus given by

B z(r) = B a - B e(r) (5)

It should be noted that provided B (0) < B , the total axial magnetic

field is unidirectional and decreases from the discharge tube wall to

the centre.

The discharge tube which is used in the present experiment is

equipped with a set of tightly fitting and closely spaced conducting

rings which is located close to the outside surface of the tube and

coaxial with it. Each ring is electrically insulated from its neighbours.

This system of rings acts as a conducting shell conducting only in the

azimuthal direction; it conserves the axial magnetic flux. The presence

of these rings modifies (5) in the following manner. During the period

when azimuthal electron current is driven in the plasma, current must

flow in the system of conducting rings in order to conserve the initial

axial magnetic flux, tfR*B (R is the radius of the conducting rings).

The flux due to the electron current in the plasma is R

l*e| - " 0|e|w | N(r) dr (6) o

Hence the current in the conducting rings must produce an uniform

magnetic field

1 B C < c l<»el (7)

in the opposite direction to B g(r) to cancel this flux. The total axial

magnetic field is now given by

B z(r) - B a • B c - B e(r) (8)

The various components of B_(r) are shown in Pig. 1. Note that

B z(r) - Bfl • B c (9)

4.

f or R < r < R and it is clear that the effect of the conducting rings c

is simply to increase the effective value of the initial bias field, B g.

Fig. 1 shows that the axial field configuration obtained in this

manner resembles that of a theta pinch but with a broader field

distribution. The on-axis value of 0 is 2

0(0) - 1 - 1 -B e(0)

(B • B ) v a c1

= 1 1 - 2ir (K * B ) *• a c

(10)

This quantity can therefore be controlled at will by varying the values

of to. B and N . a o

APPARATUS

Fig. 2 is a schematic diagram of the apparatus used in this

experiment. The discharge vessel consisted of a 50 mm internal diameter

Pyrex tube. This tube was equipped with two orthogonal dipole coils

which were located on the outer wall of the discharge tube at a radius

of 30 mm. The dipole coils, which were 0.75 m in length, were made

of 10 mm wide, 1.2 mm thick copper strips covered with two layers of

heat shrinkable plastic tubing for insulation purposes. The discharge

tube was also equipped with a set of closely spaced conducting rings

which were insulated from each other and which were wound closely over

the dipole coils. Each ring was made of PVC insulated, 0.74 mm diameter,

single strand copper wire. The time constant of this set of rings

was long enough to conserve the axial magnetic flux during the period

of an experimental discharge and yet was short enough to allow an

initially applied, quasi-steady, bias magnetic field, B . (produced by

an external solenoid not shown in Fig. 2) to penetrate the discharge

region.

5.

The discharge tube was connected to a conventional vacuus systea;

the base pressure was 2 x 10~ Torr. Prior to an experimental shot,

the discharge tube was isolated fro* the vacuus system by aeans of a gate

valve and a controllable amount of gas was admitted through a fast

piezoelectric valve. In the experiments reports here, argon was used

as the filling gas both to ensure that the necessary inequality u > tn.

was well satisfied and to take advantage of its relative ease of ionization

at the low filling pressures (1 - 10 mTorr) which were used. No pre-

ionization arrangement was provided; the breakdown of the neutral gas was

initiated by the electric field associated with the rotating magnetic

field. Satisfactory ionization could be achieved for filling pressures

greater than about 2.5 mTorr. Experience showed that it was possible

to achieve acceptable shot-to-shot reproducibility provided that the

filling pressure was kept constant to within ±0.05 mTorr.

A transverse magnetic field which rotated about the axis of the

discharge tube was produced by passing r.f. currents of the same frequency

and amplitude, but dephased by 90 , through the two orthogonal dipole

coils. In past experiments (Blevin and Thonemann 1962; Davenport et

al. 1966) the r.f. currents have beeT< generated by means of conventional

ringing capacitor discharges. This technique has the major disadvantage

that the envelope of the r.f. current waveform is sensitive to the load.

Therefore, when one is studying rotating field discharges of different

initial conditions, the envelope of the r.f. pulse will, of necessity,

vary. It becomes difficult to separate effects which are due to

different plasma parameters from those which are solely due to different

r.f, pulse shapes. Ideally, experiments should be conducted using

continuous (CW) r.f. generators operating in a constant current mode

(i.e. generator impedance much larger than load impedance). Since the

cost and scale of such r.f. generators are, however, well beyond the

6.

scope of the present work, a compromise solution was adopted wherein

the r.f. currents were generated by means of pulsed, high-power r.f.

line generators of the type pioneered by the Swiss plasma physics group

at Lausanne (Weibel 1964, Lietti 1969 and Keller 1965). In the

experiments reported here, the required r.f. currents were obtained

from Neibel type r.f. line generators which were modified in such a

manner as to reduce the necessary number of spark gaps. This modification

eased considerably the construction and operation of the line generators.

A detailed description of these modified Weibel line generators can be

found in Hugrass, Jones and Phillips 1980.

Each of the two line generators could produce a short r.f. current

pulse consisting of eight periods at a frequency of 0.67 MHz. The

required 90 phase difference was achieved by triggering one line

generator one quarter of a cycle after the other. The equivalent

generator impedance and root-mean-square open circuit voltage of each

generator was 9.70 and 14.1 kV, respectively. No attempt was made to

match the generators to the loads. Measurements verified that the

load impedance was at all times a small fraction of the generator

impedance. The generators therefore behaved as constant current sources

and the applied rotating magnetic field was independent of the plasma

discharge parameters.

Experience showed that it was not convenient to check the phase

difference between the two line generators by direct measurements from

an oscillogram of the two r.f. generator currents, I«(t) and I 2(t).

However, the phase difference, $, between two oscillatory currents,

I. and l~, of equal amplitude and frequency, is related to the amplitudes

of their sum and difference through the equation

*s 2 tan w^ (11)

7.

If the phase angle is 90 , the signals I~ + I. and I. - I, have equal

amplitudes and hence phase quadrature can be monitored by comparing the

magnitudes of these two signals. Fig. 3 shows the measured line

generator currents, I. and I.; the measured sum and difference of the

r.f. currents, I. • I_ and I. - l~, are also shown in this figure.

Note the steady decrease in amplitude during the r.f. pulse followed by

one cycle of increased amplitude. The occurrence of current increase

at the end of the r.f. pulse is a characteristic of this type of line

generator.

A calibrated magnetic probe located at various radial positions

and with appropriate orientations was used to measure the time-varying

axial (B ) and azimuthal (B Q) components of the magnetic field across

a horizontal diameter in the central plane of the discharge tube.

A ceramic tube of 2 mm outer diameter served as a guide for the magnetic

probe. The probe signal was filtered by a balanced low pass RC filter

and integrated with a passive RC integrator. The filter was used to

eliminate high frequency interference caused by the spark gaps of the

r.f. line generators.

4. RESULTS

(i) The dependence of AB on the filling pressure

From an experimental viewpoint, the quantity AB , defined by if

AB Z = B2(R) - B 2(0) (12)

is easily measured. Using (8) and (4),

AB Z = Be(0) (13)

If all the electrons rotate synchronously with the field, then

M wJe|w N

8.

(The superscript M is used to emphasize the fact that the above value

represents the maximum) attainable value of AB ) . Any 'slip' between

electrons and field leads to a saaller value of AB .

With the Magnetic probe orientated to Measure the B component it

of the magnetic field, the following quantities

[B (0) - BJ(probe located at r = 0) c c

and [-B ](probe located r = R), c

were measured for different values of the argon filling pressure.

Note that the measured signals do not include a contribution due to the

bias magnetic field, B (- 280 gauss in these experiments), because the

time constant (~ 100 usee) of the integrator was much smaller than the

period of the bias field (8 msec). The results of these measurements

are shown in Fig. 4.

The peak values of the signals obtained at r = 0 and r = R are

plotted against the filling pressure in Fig. 5(a). The difference between

these two curves, AB , is plotted in Fig. 5(b). The ideal linear relation M between AB and the filling pressure (obtained from (14) assuming that

each argon atom contributes one electron to the plasma) is also plotted

in Fig. 5(b). It appears that for filling pressures in the range M 3 - 5 mTorr, AB = AB , and the electrons are 'tied' to the rotating

field.

A quantitative comparison between the experimental results and the

predictions of the theoretical studies (Jones and Hugrass, 1980; Hugrass

and Grimm 1980) is not possible for the following reasons:

(a) The number density and collision frequency were assumed uniform

and constant in the theoretical model.

(b) In the theoretical model, the ions were assumed to form a fixed

neutralizing background of positive charge. In the experiment,

however, dynamical effects occur during the formation of the

9.

magnetically confined plasma column,

(c) The amplitude of the rotating field was not constant during the

experiment (see, for example, the current waveforms shown in

Fig. 3).

(dj In the theoretical model it was assumed that the rotating field

was applied to a completely ionized plasma column. In practice,

the initially neutral filling gas was ionized by the electric

field associated with the rotating magnetic field.

Despite the above comments, a qualitative explanation of the

experimental results, especially regarding the time-dependence of

AB , can be given in terms of the following physical picture. ?ig. 6(a) M

shows a sketch of three curves of AB /AB as a function of the amplitude

of the rotating field, B , for three different values of v .. Theory

indicates that these curves have the general shape shown in this

diagram (see Fig. 1, Hugrass and Grimm 1980). Below this sketch is

another which shows the time variation of B during the course of one u

r.f. pulse. If one makes the assumption that time variations in AB follow instantaneously any variation in B , then the curves of Fig. 6(a)

M can be used to construct the AB (t)/AB curves shown in Fig. 6(b).

Three types of behaviour can be recognised:

(a) For curve I, the value of v . is such that the threshold value

of B is less than the amplitude of the rotating field at all

time during the r.f. pulse. The corresponding AB (t) curve is it M therefore flat with a magnitude which corresponds to AB .

(b) For curve II, the range of the rotating field amplitudes in the

r.f. pulse lies within the transition region. The peak of the M corresponding AB (t) curve is less than AB . A noteworthy

' Z

feature of this case is the increase of AB which accompanies

the momentary rise in B at the end of the r.f. pulse.

10.

(c) For curve III, the rotating field amplitude is always audi less

than the thieshold value. Consequently, the m i — value of N

AB-(t) is only a saell fraction of AB

The three types of behaviour outlined above can be identified in

the experimental results. In the pressure range 3 - 5 •Torr oscillogram

of the type shown in Fig. 7(a) were obtained. It is seen that, after

an initial period during which the gas is ionized and the electrons are

brought into rotation, AB reaains at an essentially constant value until

the end of the r.f. pulse. As seen froa Fig. 5(b) this constant value M of AB corresponds to AB The behaviour in this present range

corresponds to that described in (a) above.

In the pressure range 5 - 9 aTorr, oscillograms of the type shown

in Fig. 7(b) were obtained. It is seen that the aaxiaua value of AB M is considerably less than AB . Note also the momentary increase in AB

towards the end of the r.f. pulse. In this pressure range, the situation

is as described in (b) above.

Finally, for filling pressure higher than about 9 aTorr, behaviour

of the type outlined in (c) was observed (see Fig. 7(c)).

This correspondence between the experimental results and the simple

physical picture presented in Fig. o is consistent with the reasonable

assunption that the effective vulue of v . increases with increasing

filling pressure.

(ii) The radial dependence of B

The magnetic probe, orientated to measure the axial coaponent of

the magnetic field, was used to measure the quantity [B (r) - B ] at

various radial positions. The data required to plot a radial profile

of the magnetic field was accumulated from a large number of discharges

(typically 20), the probe being relocated at a different radial position

for each shot. The total axial aagnetic field was obtained using the

11.

equation

B z(r) = B a - [B e(r) - B c] (8)

The measurements and the radial profile of B_(r) shown in Fig. 8 were

obtained with B = 280 gauss and p = 3.7 raTorr. The conditions for

this experiment have been so chosen that B + B„ and B (0) nearly cancel

on the axis i.e. the on-axis value of 8 for this configuration is ~ 1.0.

An estimate of the total azimuthal electron current generated in

this experiment can be made using the following equation which relates

I„, the total current/unit length, to AB„: o z

h - r - A B z < 1 5 )

H o

For AB Z = 320 gauss and a discharge length of 0.75 m, the total azimuthal

current is calculated to be 19.1 kA.

(iii) The penetration of the rotating magnetic field into the plasma

Fig. 9(a) shows a plot of the computed vacuum magnetic field lines

of the rotating field generated by the dipole coils. The lines are

plotted at an instant of time when the 0-component of the field, as

measured by a suitably orientated probe located in the indicated probe

guide, is a maximum. The corresponding computed radial profile, B 0(r),

is shown in Fig. 9(b).

The penetration of the rotating magnetic field into the plasma column

was investigated by measuring the 8-component of the field across the

diameter indicated in Fig. 9(a). Fig. 10 shows the waveforms of B Q(t)

at various radial positions for filling pressures of 3.7 and 6.0 mTorr

argon and for an experimental run made with the discharge tube evacuated.

The value of B a was 280 gauss for this set of experimental data. ft

The penetration can be assessed by comparing the measured Bfl(r)

profiles obtained with plasma in the discharge tube to those obtained with

12.

the discharge tube evacuated. For each oscillogram, successive peak-

to-peak values of the signal were measured. Examples of the radial

profiles of the amplitude of B Q (defined as half the measured peak-to-peak

value) for the 3.7 mTorr and the vacuum case are shown in Fig. 11. The

peak-to-peak values corresponding to these profiles are idem., fied in

Fig. 10.

For the 3.7 mTorr case, the rotating field penetrates into the

plasma (curves a, b, and c in Fig. 11) during the r.f. pulse. The

azimuthal current, and the associated axial magnetic field, produced by

the rotating electrons consequently remain fairly constant for the

duration of the r.f. pulse. When the applied rotating field begins to

decay at the end of the r.f. pulse, it fails to sustain the azimuthal

electron current and the axial magnetic field pulse also decays. At

this time, the rotating field begins to be excluded from the plasma

(curve d, Fig. 11) by the action of an induced axial screening current.

This screening current manifests itself through an enhancement of the

value of BQ in the outer regions of the plasma column.

For the 6.0 mTorr argon case, the rotating field initially penetrates

the plasma (curves e and f, Fig. 11). However, for these conditions,

the amplitude of the applied rotating field becomes smaller than the

threshold value during the r.f. pulse itself. When this happens, both

the rotating field is excluded (curves g and h, Fig. 11) and the axial

field pulse decays.

5. DISCUSSION

Theoretical studies predict a threshold condition for setting plasma

electrons into rotation with a transverse rotating field. If one wishes

to generate substantial azimuthal electron current by this technique then

the existence of a threshold value for the amplitude of the applied

13.

rotating field assumes great practical importance; it is this threshold

value which determines the parameters of the costly r.f. generators

used to produce the rotating field.

The experimental results presented here provide convincing

qualitative evidence for the existence of a threshold value for the

amplitude of the applied rotating field. They are consistent with the

following simple physical picture. As long as the rotating field

amplitude is above the threshold value, the electrons are in essence

tied to the field lines and they rotate nearly synchronously producing

a considerable azimuthal current. The effective frequency of the

rotating field in a frame of reference rotating with the electrons is

essentially zero; the corresponding skin depth is, in this case, much

larger than the plasma radius. On the other hand, when the rotating

field amplitude becomes less than the threshold value, the azimuthai

electron motion decreases and the effective frequency of the rotating

field, in a frame of reference rotating with the electrons, approaches

that measured in the laboratory frame of reference. The effective skin

depth in this case can be much smaller than the radius of the plasma

column and the rotating field amplitude inside the plasma is attenuated

accordingly.

We wish to acknowledge financial support for this work from the

Australian Institute of Nuclear Science and Engineering and the Australian

Research Grants Committee.

14.

REFERENCES

BLEVIN, H.A. and THON! ANN, P.C. 1962, Nucl. Fusion, 1962 Suppl.,

Part I, p.55.

DAVENPORT, P.A., FRANCIS, G., MILLAR, W. and TAYLOR, A.G. 1966

U.K.A.E.A. Culham Report CLM-R65.

HUGRASS, W.N. and GRIMM, R.C. 1980.

HUGRASS, W.N., JONES, I.R., McKENNA, K.F., PHILLIPS, M.G.R.,

STORER, R.G. and TUCZEK, H., 1980 Phys.Rev.Lett. 4£, 1676.

HUGRASS, W.N., JONES, I.R. and PHILLIPS, M.G.R. 1979 Nucl. Fusion 19̂ , 1546.

HUGRASS, W.N., JONES, I.R. and PHILLIPS, M.G.R. 1980

J.Phys.E:Sci. Instrum. 13, 276.

JONES, I.R. and HUGRASS, W.N. 1980.

KELLER, R. 1965 Helv.Phys.Acta 38_, 328.

LIETTI, P. 1969 Rev.Sci.Instrum. 40, 473.

WEIBEL, E.S. 1964 Rev.Sci.Instrum. 35, 173.

15.

FIGURE CAPTIONS

Fig. 1. The axial magnetic field distribution for a 6-pinch-like

rotating field discharge

R is the inner radius of the discharge tube

R is the effective radius of the flux conserving rings

B is the initial bias axial magnetic field

B is the axial magnetic field produced by the current

induced in the flux conserving rings

B (r) is the axial magnetic field produced by the

azimuthal electron current

Fig. 2. Schematic diagram of the apparatus

Fig. 3. r.f. current waveforms

(a) Line generator current waveforms

(b) Sum and difference of the line generator currents

Horizontal scale 2 usec/div; Vertical scale 2.9 kA/div

Fig. 4. The axial magnetic field on the axis (r = 0) and at the

wall (r = 25 mm) for different values of the argon filling

pressure. The two bottom traces show the r.f. current pulse.

B = 280 gauss

Fig. 5. (a) The measured axial magnetic field on the axis (r = 0)

and at the wall (r = 25 mm) plotted against the filling

pressure M

(b) AB and AB plotted against the filling pressure

Fig. 6. AB z(t) waveforms for different values of v . AD

(a) Sketches of the ( Z/AB-M) and (B ,t) relationships. AB

(b) Sketch of derived ( Z/AB ZM. t) relationship

16.

The measured axial magnetic field on the discharge tube

axis for three different values of the filling pressure.

Upper trace [Be(0) - B j 175 gauss/div

Lower trace r.f. current 2.9 kA/div

Horizontal scale 5 usec/div

(a) The measured axial field signal at various radial

positions for B = 280 gauss and filling pressure -

3.7 mTorr.

(b) The radial profile of the axial magnetic field measured

in the interval t : 9-li usee (t = 0 corresponds to

the start of the r.f. current).

(a) The magnetic field lines generated by the two orthogonal

dipole coils

(b) The computed radial variation of B 0 along the indicated

diameter.

The rotating field component BQ(r,t) for

(a) evacuated discharge tube

(b) 3.7 mTorr filling pressure

(c) 6.0 mTorr filling pressure

The top traces show B on-axis; the bottom traces show K + I 2 <

B a =280 gauss; Horizontal scale 2 ysec/div

B e(r] 1 for:

(a) 1 (e) 1 (b) (c) (d) J

Vacuum and

3.7 mTorr

O

•

(f) (g) (h) J

Vacuum and

6.0 mTorr •

j R c ^ B j

1 R ••»

\ I u

BQ \ * \ B 0

\

\ 1 / BO

ea

I

Fig. 1

Steady bias field, B a

Flux conserving rings

r.f. coils

Fig. 2

(a)

(b)

CM ( - 1 +

CM

I

Fig. 3

r = 0 (mTorr)

2.2

r = 25 nan

2.7

3.0

3.3

3.8

4.5

5.2

19

^W

Gauss/div

35

35

35

35

35

35

35

35

''///yw\A

5 psec/div 5 psec/div

Fig. 4

Gauss 400 r

3001-

200

100

CBe(0) - B C]

H h 8 12

+ H 16 20 mTorr • •

C-Bc]

400 h

20 mTorr

Fig. 5

on <

N <

0)

(a)

1.0

2 N

09 <J

N 0.5 -

3

(b)

Fig. 6

"V^ (a)

Cb)

(c)

3.5 mTorr

6.2 mTorr

13.0 mTorr

Fig. 7

• \ 300 - Jm

1 3 700 -\ ^̂

\ B 0

V 100 . •/

jS*

I 1 1 1 1 1 __i .. • 25 20 IS 10 5 0 5

rO-0

(b)

10 IS 20 25

r.f. current pulse

(a)

Fig. 8

Probe guide

(a)

(b)

Fig. 9

Gauss/div Tube evacuated 3.7 mTorr 6.0 mTorr

165

74

186

186

186

186

186

186

186

186

b J T e f g h

Fig. 10

• i

(a)

• • • • • • • •

• • 200 ° o ( l o o 0 • •

• 0 • 0 ° • o o

(A o 100 _ 0

• aus 0

o* DO o

CQ I

0 1 1 1 1 1 1 1 1

(b)

20 10 0 10 20 r(nun)

»• • 8°

I I I I

• •

°o,» • °8 I |

o • O *

o 8|

I I I 1

(e)

(f)

to

* i ° o8* »• ••° t»« o

I

i i i i I I I I

{ •••! I * 8o 0 o 1

1° • • 8 • 8 "8 t(

o» o <

• 1 l 1 1 1 1 1 1

(c)

(d)

I f O . n O

. o ! • 0°°B»

I I I I

8 «8 • 5 . 8o

I I I I

oo 0

,6 f • I

« 0 0 o f l

8 , #

• • - • • • , 0

f t • ••• ° i i ' i I i i i i

(g)

(h)

igi / . • •

i i i i

0 f i t

I I I I

rig. n