AE-395 UDC 533.9

Magnetoacoustic Waves and Instabilities

in a Hall-Effect-Dominated Plasma

S. Palmgren

This report is intended for publication in a perodical. Referen­ces may not be published prior to such publication without the consent of the author.

AKTIEBOLAGET ATOMENERGI

STUDSVIK, NYKOPING, SWEDEN 1970

AE-395

MAGNETOACOUSTIC WAVES AND INSTABILITIES IN A H A L L - E F ­

FECT-DOMINATED PLASMA

S. Pa lmgren

ABSTRACT

The dispers ion equation is studied for smal l -ampl i tude plane h a r ­

monic waves in a compress ib le p lasma moving perpendicular to a m a g ­

netic field with a constant fractional ionization. The modes of propaga­

tion are analysed mainly from a qualitative point of view and one of

them is shown to be unstable due to the Hall effect. This mode has been

previously analysed by other authors in connexion with MHD power ge ­

ne ra to r s but in a more r e s t r i c t ed and isolated sense . The p resen t work

not only genera l izes and modifies their resu l t s on this special mode,

but also makes it possible to picture the whole spect rum of propagation

modes in a simple and physically intelligible way.

Pr in ted and dis tr ibuted in May 1970.

LIST OF CONTENTS

1. INTRODUCTION

2. BASIC ASSUMPTION AND EQUATIONS

3. PROPAGATION MODES

3. 1 Velikhov's magnetoacoustic instabili ty

3.2 Alfve*n's magnetoacoustic waves

3, 3 General ized magnetoacoust ic wave

3 . 3 . 1 k / / v ' — " —e

3 .3 .2 k i y '

4. NUMERICAL EXAMPLE AND DISCUSSION

5. SUMMARY

ACKNOWLEDGEMENT

APPENDIX

REFERENCES

FIGURES

- 3 -

1. INTRODUCTION

Hydromagnetic waves in compress ib le media, magnetoacoust ic

waves (MAW), have been t rea ted by Alfven and Fa l t hammar (1963).

They studied the propagation of smal l -ampl i tude , plane harmonic wa­

ves in the low-frequency approximation and introduced l imitat ions to

infinite conductivity and to a p lasma at r e s t in a l abora tory frame of

re fe rence . We shall omit these two la t te r r es t r i c t ions and consider a

finite conducting p lasma moving perpendicular to a magnetic field and

between two electrode p la tes , connected via an external load. The s i ­

tuation now is formally that of a magnetohydrodynamic power gene ra ­

tor duct and we shall have this application in mind la ter on. In fact

one branch of the MAW in compress ib le moving media has e a r l i e r been

par t ly investigated in connexion with MHD power generat ion, or ig ina l ­

ly by Velikhov (1962). This mode has been called the magnetoacoust ic

instability mode (MAI), because it r ep resen t s a modified sound wave

and turns out to be unstable provided that the e lect ron drift velocity

measu red in the gas frame exceeds the speed of sound. This m a c r o i n -

stability is explained in t e r m s of a coupling between the p lasma p a r a ­

m e t e r s and the thermodynamic state of the gas , see McCune (1964).

Inspection shows immediately that their derivat ion cannot be used to

study the t rans i t ion between MAI and MAW when the p lasma velocity

dec rease s to ze ro . The reason is simply that the MAI is der ived for

MA smal l in some respec t , formally that n -> o, but the upper l imit of

pA has not been specified in the above-mentioned p a p e r s , n is the p e r ­

meabil i ty of free space and X is the wavelength.

The main purpose of this note is now to synthetize Alfven's m a g ­

netoacoustic waves and the magnetoacoust ic instabil i ty derived by Velik­

hov and consequently these will be called the general ized magnetoacous-

- 4 -

t i c w a v e s . F o r such a r e l a t i v e l y c o m p l e x t r e a t m e n t to be mean ing fu l

a suff ic ient condi t ion i s tha t we have p r o n o u n c e d h y d r o m a g n e t i c w a v e s ,

which m e a n s tha t the Lrundqvist n u m b e r , i . e . the m a g n e t i c R e y n o l d s

n u m b e r b a s e d on the Alfven v e l o c i t y v . , o r L = uo"£v > should e x -A u A

ceed un i ty , cr is the conduc t iv i ty and t i s a c h a r a c t e r i s t i c l eng th . H o w ­

e v e r , th i s c r i t e r i o n cannot be n e c e s s a r y when c o n s i d e r i n g the whole

s p e c t r u m of d i f fe ren t m o d e s , wh ich a l s o i nc l udes sk in e f f ec t s , b e c a u ­

se L u n d q v i s t ' s c r i t e r i o n i s a p r i o r i app l i cab l e to the m o d e s of m o d i ­

f ied sound w a v e s .

2 . BASIC ASSUMPTION AND EQUATIONS

We c o n s i d e r a weak ly o r p a r t i a l l y i on i zed c o m p r e s s i b l e p l a s m a

m o v i n g p e r p e n d i c u l a r to a cons t an t and h o m o g e n e o u s m a g n e t i c f ield

B with a cons t an t and h o m o g e n e o u s v e l o c i t y v. The m o m e n t equa t ions

for th i s t h r e e - c o m p o n e n t fluid a r e

| | + div P v = 0 (1 )

D d | + g r a d p = j_x B (2)

and the a d i a b a t i c expans ion law

w h e r e p is the m a s s dens i t y and p is the to ta l ion and gas p r e s s u r e .

The O h m - H a l l law, neg lec t ing ion s l i p , r e a d s

j _ + |a e j_x B_ = O - ( J E + V X . B ) (4)

w h e r e n, = e / m v is the e l e c t r o n m o b i l i t y and a = en LL is the e l e c t r o n ^e ' e e ' e ^ e

conduc t iv i ty , v is the effect ive c o l l i s i o n f r equency for m o m e n t u m t r a n s ­

f e r . F i n a l l y , M a x w e l l ' s equa t ions wi th in the l o w - f r e q u e n c y a p p r o x i m a t i o n

r e a d

c u r l it, = - ~= — at

curl B_ = M. j _ (6)

Now we i m m e r s e two electrode plates into the p lasma para l l e l to

the magnetic field and connected via an external load. Then the e l e c ­

t r i c field s t rength in the unperturbed state is uniquely defined. Next

we introduce two main assumpt ions: that of smal l magnet ic Reynolds

number , R = u, avt « 1, which means that the induced magnet ic field m b

B. can be d i s regarded in the equations above when compared to the i m ­posed field B , and fur thermore that of constant fractional ionization,

—o

or "frozen ionization" (see, for example, Witalis, 1968). This means

that the mobility is inversely proport ional to the gas density p but that

the conductivity is independent of the state of the gas .

F r o m these equations and assumptions we can derive the d i s p e r ­

sion relat ion for smal l -ampl i tude , plane harmonic wave per turbat ions

f = f exp f i (cot - k • x)}

and we r e s t r i c t ourse lves to wave vectors k perpendicular to the impo­

sed magnetic field. We consider rea l wave vectors k, and so the f r e ­

quency co may in general be complex. This a r b i t r a r y choice is of no

physical impor tance . After some lengthy algebraic manipulations in the

usual fashion we a r r ive at the resu l t 2 2

a - i k g • k = L _ ^ = 1 - , l V , (7)

w' - c ek

where a now is the Alfven velocity and c is the speed of sound. For

p rac t ica l purposes we have defined a complex phase velocity w = co/k =

= u + iv and pr ime denotes a t ransla t ional t ransformat ion to the gas

frame of re fe rence , co = u> - k v, i . e . the Doppler effect. Fu r the r no­

tations are V = k/n<j, which is the penetrat ion velocity (cf. skin effect),

- -6 - •

t and v , which is the projection on the wave vector of the electron eic

drift velocity measured in the gas frame. Finally, g = j . x S/p ^ s t n e

Lorentz force per unit mass. This equation, which describes what we

might call the generalized magnetoacoustic waves, is, apart from de­

generated cases, of third degree in w' and will then have three diffe­

rent modes of propagation. Later on we shall examine the shape, asymp­

totic behaviour and stability character of the different modes in the

complex w1-plane for different values of the electron drift velocity,

but first we shall rearrange equation (7) in such a way thai: compari­

sons with the special cases of MAI and MAW are possible..

3. PROPAGATION MODES

3. 1 Velikhov's magnetoacoustic instability

For comparisons with the MAI we rearrange equation (7) to

' 2 2 . - 1 , ? t w - c - ivk w - v , ) =

ek'

= -V (w - vek) (w - c + lk £ • k) \°>

-1 2

where v = D/O~B is the characteristic time for vorticity suppres­

sion (see, for example, Shercliff, 1965). For the RHS to be negligible

in comparison with the terms in the L.HS, we require formally that

\i -> o, which results in a quadratic equation in w and has been deri­

ved also by, e.g. , Rosa (1968, chapter 4). The two branches, which

are always numbered consecutively from left to right, are sketched

in figure 1 (a) and (b) for two values of the parameter v . The scales e K.

of the abscissa and ordinate are the same. The arrows on the differ­

ent modes indicate direction of increasing wave vector and small circ­

les denote the limit points, k = «> or 0. Due to the low-frequency app­

roximation and the length scale limitation we have to exclude a neigh-

- 7 -

bourhood of these l imit points symbolized by these c i r c l e s . The f i rs t

mode r ep resen t s vort ici ty suppress ion and the second a modified sound

wave which turns out to be unstable if the e lect ron drift velocity m e a ­

sured in the gas frame exceeds the speed of sound. In figure 1 (b) we

have inser ted the unstable mode pictured in a (u , ou .)-plane, and the

limiting value of the growth t ime for short wavelength is

i v \

T = v = 2/v ( — - 1) (9) o o ' v c

which is inverse ly proport ional to the magnetic field strenght squared.

3.2 Alfven^s magnetoacoust ic waves

On the other hand, for comparisons with the MAW we wri te equa­

tion (7) in the form

( w ' 2 - c2) (w* - v ^ k - iV) = (w' - v*k) (a2 - i k _ 2 £ • k) (10)

which in the case of a p lasma at r e s t reduces to

(w - c ) (w - iV) = wa (11)

and has been derived by Alfven and Fa l thammar (1963, chapter 3). Three

modes should then appear , but for symmet ry reasons only two of them

are essent ia l . The f i rs t and third ones, w. and w_, r ep resen t forward

and backward propagation of sound waves, and the last one, w ? , is pu­

rely imaginary and pic tures the skin effect. They a r e all stable and

outlined in figure 2 (a). The limiting behaviour for short wavelenghts

is found to be

lim Im (a>, o) = v and lim Im (co?) ~ k /JJ,cr -t oo K -» oo

3. 3 General ized magnetoacoustic wave

In studying the complete equation (7), we shall confine our a t ten­

tion to wave propagation ei ther para l le l or perpendicular to the e l e c -

t r o n dr i f t v e loc i t y , or equ iva l en t ly the e l e c t r o n i c c u r r e n t . In bo th c a ­

s e s we s h a l l out l ine the m o d e s of p r o p a g a t i o n in the c o m p l e x w p lane

for f u n d a m e n t a l l y d i f fe ren t c h o i c e s of the e l e c t r o n dr i f t v e l o c i t y .

3 . 3 . 1 k / / v *

If k i s p a r a l l e l to v , equa t ion (7) r e d u c e s to

^ T= 1 - i , V , (12) ,2 2 w* - v w* - c e

and we have ac tua l l y t h r e e t ypes of d i s p e r s i o n d i a g r a m s , o < v < c,

c < v < c and v > c , p lus t h r e e o r two d e g e n e r a t e d c a s e s , d e p e n d ­

ing on w h e t h e r we inc lude v = 0 as a d e g e n e r a t e d c a s e o r not . H e r e

c = (c + a ) is the a u g m e n t e d sound v e l o c i t y , n u m e r i c a l l y t a k e n

to be twice the o r d i n a r y sound v e l o c i t y , s ee the d i s c u s s i o n be low of the

n u m e r i c a l p a r a m e t e r s . In f igure 2 (a) we s t a r t wi th v = 0, a p l a s m a

at r e s t , and th i s is the MAW as m e n t i o n e d above . F o r i n c r e a s i n g dr i f t

v e l o c i t i e s , the m a i n d i f fe rence is at f i r s t only tha t the s.kin effect

m o d e has a p h a s e v e l o c i t y e s s e n t i a l l y equa l to the e l e c t r o n dr i f t v e l o c ­

i ty . Th i s is shown in f igure 2 (b) w h e r e only the p o s i t i v e a b s c i s s a has

been p lo t t ed , the b a c k w a r d mod i f i ed sound r e m a i n i n g about the s a m e

as in f igure 2 (a ) . At v ^ c the p i c t u r e changes c o m p l e t e l y but , v i e w ­

ed as a b i fu rca t i on , s t i l l in a cont inuous way . F i r s t , at v = c, f igure

2 (c) , we have a d e g e n e r a t e d c a s e w h e r e two of the b r a n c h e s p a r t l y

c a n c e l e a c h o t h e r . In the r ange c < v < c , f igure 2 (d), the f o r w a r d

p r o p a g a t i o n of sound w a v e s t u r n s out to be u n s t a b l e , and t'nis is the

MAI ana logue as d e s c r i b e d by Ve l ikhov . In fact t h e y a r e found to be

i d e n t i c a l for s h o r t w a v e - l e n g t h s , and in the r e l e v a n t i n t e r v a l of v

th i s b r a n c h a l s o has the s a m e l i m i t p o i n t s , c and v . But for dr i f t ve­

l o c i t i e s above the a u g m e n t e d sound v e l o c i t y , f igu re 2 (f), ;he l i m i t

t

e

- 9 -

points a re different from those obtained in figure 1. We may s u m m a ­

r ize the last p ic ture 2 (f) as follows. The backward modified sound wa­

ve is unchanged except that the logari thmic decrement is lowered com­

pared to a p lasma at r e s t , the second mode constitutes an unstable mo­

dified sound wave (c and c* are l imit .points) propagating along the elec­

t ron drift direct ion, and the third mode is the skin effect with a phase

velocity st i l l equal to the e lectron drift velocity.

At short wave-lengths the growth t ime of the unstable mode, de ­

noted by subscr ipt ' 2 ' , is s t i l l of the same order as that der ived by

Velikhov, eq. (9). If higher o rder t e r m s are included, obtained from

eq. (12) by an i terat ion p r o c e s s , we get after s e r i e s expansion in

(v ' - c)/V

t V

I m w 2 = " 2 ¥ ("f " J) i - {(v;-c)2v-2](i+(i4(|)V) (13)

The expansion p rocess itself is valid only if the square root is

comparable to unity, which also means that the s ta tements of eq. (9)

are justified. Finally, in case v » c we get a growth t ime

T = ' - ^ - ( 1 - a R 2) (14) 1 vv me '

e where Rrvi„ = ^cAv is the magnet ic Reynolds number based on the e l e c -

1 l a 2 2

t ron drift velocity and a = —=- (1 + (1 + •=- (—) ) ). This means that the 8ff L C

instabili ty growth is somewhat decreased in compar ison with Velikhov "s

r e s u l t s .

3. 3.2 k , v* — x —e

Final ly, we consider waves perpendicular both to the imposed

magnetic field and the e lect ron drift in such a way that (k, v , B) con-— ,i C —~"

st i tutes a r ight-handed car tes ian coordinate sys tem. Then

- 10 -

k • £ = k (I x B)/p = -en p " 1 k (v' x B) = - a2k2v 7fCV

*- — e — —e — e

where VC = JJ, B is the Hall coefficient. Then equation (7) reads e

2

w

(i + i «) = i . i j ; (i5) v K.V w ' 72 2 v x XKV

which is shown very schematically in figure 3 for not too high electron

drift velocities. The first mode is a modified sound wave provided that

we restrict ourselves to short wave-lengths. For increasing wave­

lengths this mode is unstable and the critical wave-length is given by

the intersection with the abscissa, V . = v c /K. This corresponds cr it e

in general to wave-lengths which are very long when compared to a

laboratory length scale. The asymptotic behaviour of the growth time

when k -* o

rr~' lim T2 = k a V - ^ - . O (k1/2) k - o

which shows that the instability growth has one or more extrema in the

interval o < k < k . . The physical interpretation and consequences of

this long wave-length instability are still an open question. The second

mode constitutes essentially the skin effect and the third represents,

at least for short wave-lengths, a stable modified sound wave.

4. NUMERICAL EXAMPLE AND DISCUSSION

Concerning the application to MHD generator plasmas we shall

consider a potassium-seeded argon plasma. For a detailed discussion

of the plasma parameters, see for example Kerrebrock (] 964) or Sa-

kao et al. (1969). First of all, from the gas parameters p = 1 atm

T = 1500 K and the adiabatic index a = 1. 668, and an imposed mag-

netic field strength B = 0. 8 T, we obtain a sound velocity c = 720 m/s

- 11 -

and an Alfven velocity a = 1270 m / s , so that the augmented sound v e ­

locity is about twice the ordinary sound velocity. F r o m the p la sma pa­

r a m e t e r s n = 10 m , T = 2500 K and m / m = 1. 36 • 10 we

e e e' a

have an e lec t ron-neu t ra l collision frequency v = 2 . 3 * 1 0 s and e a 8 -1 for e l ec t ron-e lec t ron collisions v = 2.7 • 10 s , so that the p l a s -ee 2

n e Q

ma is par t ia l ly ionized. F u r t h e r m o r e the conductivity a = — 12 . e ea

A/Vm. An obtainable value of the cur ren t is up to 2 • 10 A / m , which 4

means that the e lectron drift velocity is below 10 m / s (the t he rma l 5 .

velocity of the e lect rons is about 2 • 10 m / s ) . The cha rac t e r i s t i c t i --1 -2

me for vort ici ty suppresion is v = 4 * 1 0 s and the c r i t e r ion for

M.X to be smal l in eq. (8) (only concerning the unstable mode) is numer i -

2 2

cally X « 5 * 10 , i . e . wave-lengths smal le r than about 7 m e t r e s .

We may note that, although this is l a rge r than any genera tor size up

to date, it may be important in other situations where , for example ,

the conductivity is inc reased . The growth t ime of the instabil i ty is he ­

re cor rec t ly expressed by eq. (9) T ^=6 m s . Finally, the instabil i ty perpendicular to v occurs if

X > I n / _ _ _ = 400 m u, a V . --^ v ' c e

and is of no in te res t in this context.

5. SUMMARY

The propagation of magnetoacoust ic waves in a Hall effect p l a s ­

ma has been t rea ted in the present paper . The evolution of the differ­

ent modes with increasing p lasma velocity was followed, most ly d e ­

scr ibed in a qualitative way. It was found that the skin effect mode

could with advantage be studied in the e lectron frame of re fe rence ,

whereas hydromagnetic effects, including vort ici ty suppress ion, occu-

- 12 -

red in the gas f rame. One of these modes , called MAI mode, has p a r t ­

ly been analyzed by Velikhov and many others in a simplified way. The

two descr ipt ions of this instabil i ty, propagating preferent ia l ly in a di­

rect ion para l le l to the e lectron drift, are sti l l the same with r ega rd

to the instabili ty cr i ter ion and within the limit of smal l wave- lengths .

A cr i te r ion for the applicability of their simplified t rea tment was de ­

rived and we found that R = uoAv should be smal l . We may first me ^ e }

note that this cr i ter ion concerns only one special mode and not all of

them, and further that, even if the magnetic Reynolds number R = ° m

= (icrA-v and/or the Lundqvist number L = ^ cr^-vA a r e s r n a ^ > this might

not be t rue for R because of a strong Hall effect. me °

From the descript ion of the general ized magnetoacoust ic waves

we have also found that a new instabili ty might occur for long wave-

lenths, k < k = u a v v c /K, but due to the infinitesimal pe r tu rba-cr e

tion method used cautiousness is demanded on this point (for various

aspec ts , see , for example, Grad, 1965). Anyhow this instabil i ty is of

no in te res t in a laboratory scale , e . g . in a MHD generator p lasma.

- 13 -

ACKNOWLEDGEMENT

The author is much indebted to Dr. E . Dahlberg for st imulating

discussions and valuable c r i t i c i sm.

This work has been financially supported by the Swedish Board

of Technical Development, contract 69-187/U115.

- 14 -

APPENDIX

A.s soon as v > c, it i s usefu l to w r i t e eq . (12) in the f o r m

r a 2 v e - w 2 n 1 / 2

~ C L1 + Q v' - wl + iV e 2

w h e r e s u b s c r i p t s denote the m o d e s u n d e r c o n s i d e r a t i o n . F o r s h o r t

w a v e l e n g t h s , V-» » , we obta in by an i t e r a t i o n p r o c e s s

,.(0) w 2 ' = C

(o = c r 1 + ( a ) 2 _ 4 _ L L . . ! 1 / 2

\ ^ Kc> v ' - c + iVj

or in t e r m s of p = (v ' - c ) / V

2 / c> ' \ 2 . 2,a>

w. ( i ) _ c r ] + p ( ~ } - i p (c) -i

2 1/2

w h e r e s u p e r s c r i p t ( ) deno te s the n u m b e r of i t e r a t i o n s t e p s . Concern ing

the i m a g i n a r y p a r t we obta in

T . 0 ) Im w ' v ' = - c

r 2 a 4 2 c* 2 V / 2

J 2 c* 2 1 / 2

2(1 + p ' ) 'J

and by s e r i e s expans ion in p , for s m a l l p , we ge t

I m w z i c p'_ _p o + o +-£y ) )j o r

/ n (V - c) ,_ v ' - c , 2 2 . 1 / 2

• , ( U - - - - - - ^ L1 " ( - £ T - ) 0 + 0 + ? ( | ) ) ) j ^i2 - " 2

Highe r o r d e r i t e r a t i o n s a r e neg l ig ib l e if |w~^ ' - w r ' | i s of the o r d e r

p o r s m a l l e r .

- 15 -

R E F E R E N C E S

A L F V E N H. and F A L T H A M M A R C. - G . , C o s m i c a l e l e c t r o d y n a m i c s . The C l a r e n d o n P r e s s , Oxford . 1963.

GRAD H. , V a r i a t i o n a l p r i n c i p l e for a g u i d i n g - c e n t e r p l a s m a . P h y s . F l u i d s 9 (1966) p . 225 .

KERREBROCK J.L. , N o n e q u i l i b r i u m ion i za t i on due to e l e c t r o n h e a t i n g . L T h e o r y . AIAA J . 2 (1964) p . 1072.

McCUNE J . E . , Wave g r o w t h an i n s t a b i l i t y in p a r t i a l l y - i o n i z e d g a s e s . Int . s y m p . on m a g n e t o h y d r o d y n a m i c e l e c t r i c a l p o w e r g e n e r a t i o n . P a r i s , 6-11 Ju ly 1964. ( O E C D / E N E A ) P a r i s 1964. Vo l . 2, p . 5 2 3 .

ROSA R . J . , M a g n e t o h y d r o d y n a m i c e n e r g y c o n v e r s i o n . M c G r a w - H i l l , New Y o r k . 1968.

SAKAO F . and SATO H. , N o n e q u i l i b r i u m e l e c t r i c a l conduc t iv i ty of a p o t a s s i u m - s e e d e d a r g o n p l a s m a . P h y s . F l u i d s 12 (1969) p . 2063 .

S H E R C L I F F J . A. , A tex tbook of m a g n e t o h y d r o d y n a m i c s . P e r g a m o n P r e s s , Oxford . 1965.

VELIKHOV E . P . , Ha l l i n s t a b i l i t y of c u r r e n t - c a r r y i n g s l i g h t l y - i o n i z e d p l a s m a s . M a g n e t o p l a s m a d y n a m i c e l e c t r i c a l power g e n e r a t i o n . R e p . of a s y m p . N e w c a s t l e upon T y n e , 6 -8 th S e p t e m b e r 1962. ( IEE) London 1963, p . 135.

WITALIS E . A. , R o t a t i o n a l m o t i o n of m a g n e t i z e d p l a s m a s . P l a s m a P h y s . 10 (1968) p . 747 .

V

u'

^ k " u '

Fig. l ( - Vclikhov's nugnetoacoustic inacability.

v #2 2 . v , t » v w - c - i r (* - ve k)

( . ) v ; k < c . (b) v<;k c

v (a)

Fig, 2, - Generalized magnetoacoustic waves, kll V'I . —H —e

a , . v - 2 j - i - t _ W - e u - « '

w

(a) vj - 0, Alfven waves (b) 0 < v' < c. (c) v* c.

(d) c < vj < c*. (e) v' ,f) v > c; e

Fig, 3, - Generalized magnetoacoustic waves, k J_v'•

w' - c w

LIST OF PUBLISHED AE-REPORTS

1-320 (See back cover earlier reports.)

321. Stability of a steam cooled fast power reactor, its transients due to mode­rate perturbations and accidents. By H. Vollmer. 1968. 36 p. Sw. cr. 10:-.

322. Progress report 1967. Nuclear chemistry. 1968. 30 p. Sw. cr. 10:-. 323 Noise in the measurement of light with photomultipliers. By F. Robben.

1968. 74 p. Sw. cr. 10:- . 324. Theoretical investigation of an electrogasdynamic generator. By S. Palm-

gren. 1968. 36 p. Sw. cr. 10:-.

325. Some comparisons of measured and predicted primary radiation levels in the Agesta power plant. By E. Aalto, R Sandlin and A. Krell. 1968. 44 p. Sw. cr. 10:- .

326. An investigation of an irradiated fuel pin by measurement of the production of fast neutrons in a thermal column and by pile oscillation technique. By Veine Gustavsson. 1968 24 p. Sw. cr. 10:- .

327. Phytoplankton from Tvaren, a bay of the Baltic, 1961-1963. By Torbi5rn Willen. 1968. 76 p. Sw. 10:- .

328. Electronic contributions to the phonon damping in metals. By Rune Jonson. 1968. 38 p. Sw cr. 10:-.

329. Calculation of resonance interaction effects using a rational approximation to the symmetric resonance line shape function. By H. Haggblom. 1968. 48 p. Sw. cr 10:-.

330. Studies of the effect of heavy water in the fast reactor FRO. By L. I. Tiren, R. Hakansson and B. Karmhag. 1968. 26 p. Sw. cr. 10:-.

331. A comparison of theoretical and experimental values of the activation Dop-pier effect in some fast reactor spectra By H. Haggblom and L. I. Tiren. 1968. 28 p. Sw. cr. 10:- .

332. Aspects of low temperature irradiation in neutron activation analysis. By 0 Brune 1968. 12 p. Sw. cr. 10:- .

333. Application of a betatron in photonuclear activation analysis. By D. Brune, S. Mattsson and K. Liden. 1968. 13 p. Sw. cr. 10:- .

334. Computation of resonance-screened cross section by the Dorlx-Speng system. By H. Haggblom. 1968. 34 p. Sw. cr. 10:-.

335. Solution o! large systems of linear equations in the presence of errors. A constructive criticism of the least squares method. By K. Nygaard. 1968. 28 p. Sw cr. 10:-.

336 Calculation of void volume fraction in the subcooled and quality boiling regions. By S. Z. Rouhani and E. Axelsson. 1968. 26 p. Sw. cr. 10:- .

337. Neutron elastic scattering cross sections of iron and zinc in the energy region 2.5 to 8.1 MeV. By B. Holmqvist, S. G. Johansson, A. Kiss, G. Lo-din and T. Wiedling. 1968. 30 p. Sw. cr. 10:—.

338. Calibration experiments with a DISA hot-wire anemometer. By B. Kjell-strom and S. Hedberg. 1968. 112 p. Sw. cr. 10:-.

339. Silicon diode dosimeter for fast neutrons. By L. Svansson, P. Swedberg, C-O. Widell and M. Wik. 1968. 42 p. Sw. cr. 10:- .

340. Phase diagrams of some sodium and potassium salts in light and heavy water. By K. E. Holmberg. 1968 48 p. Sw. cr. 10:-.

341. Nonl inear dynamic model of power plants with single-phase coolant reac­tors. By H. Vollmer. 1968. 26 p. Sw. cr. 10:-.

342. Report on the personnel dosimetry at AB Atomenergi during 1967. By J Carlsson and T. Wahlberg. 1968. 10 p. Sw. cr. 10:- .

343. Friction factors in rough rod bundles estimated from experiments in parti­ally rough annuli — effects of dissimilarities in the shear stress and tur­bulence distributions. By B. Kjellstrom. 1968. 22 p. Sw. cr. 10:-.

344. A study of the resonance interaction effect between >"U and u 'Pu in the lower energy region. By H. Haggblom. 1968. 48 p Sw. cr. 10:-.

345. Application of the microwave discharge modification of the Wilzbach tech­nique for the tritium labelling of some organics of biological interest. By T. Gosztonyi. 1968. 12 p. Sw. cr. 10:-.

346 A comparison between effective cross section calculations using the inter­mediate resonance approximation and more exact methods. By H. Hagg­blom. 1969. 64 p. Sw. cr. 10:- .

347. A parameter study of large fast reactor nuclear explosion accidents. By J. R. Wiesel. 1969. 34 p. Sw. cr. 10:- .

348. Computer program for inelastic neutron scattering by an anharmonic crystal. By L. Bohlin, I. Ebbsjo and T. Hogberg. 1969. 52 p. Sw. cr. 10:- .

349. On low energy levels in <»W. By S G. Malmskog, M. Hojeberg and V. Berg. 1969. 18 p. Sw. cr. 10:-.

350. Formation of negative metal ions in a field-free plasma. By E. Lanson. 1969. 32 p. Sw. cr 10:-.

351. A determination of the 2 200 m/s absorption cross section and resonance integral of arsenic by pile oscillator technique. By E. K. Sokolowski and R. Bladh. 1969. 14 p. Sw. cr. 10:-.

352. The decay of " 'Os . By S. G Malmskog and A. Backlin. 1969. 24 p. Sw cr 10:-.

353. Diffusion from a ground level point source experiment with thermolumine-sconce dosimeters and Kr 85 as tracer substance. By Ch. Gyllander, S Hollman and U Widemo. 1969. 23 p. Sw. cr. 10:-.

354 Progress report, FFN, October 1, 1967 - September 30, 1968. By T. Wied­ling. 1969. 35 p. Sw. cr. 10:-.

355. Thermodynamic analysis of a supercritical mercury power cycle. By A. S. Roberts, Jr., 1969. 25 p. Sw. cr. 10:-.

356. On the theory of compensation in lithium drifted semiconductor detectors. By A. Lauber. 1969. 45 p. Sw cr 10:-.

357 Half-life measurements of levels in "As. By M. Hojeberg and S. G. Malmskog. 1969. 14 p. Sw. cr. 10:-.

358. A non-linear digital computer model requiring short computation time for studies concerning the hydrodynamics of the BWR. By F. Reisch and G Vayssier. 1969. 38 p. Sw. cr. 1 0 : -

359. Vanadium beta emission detectors for reactor in-core neutron monitoring. I. O. Andersson and B. Soderlund. 1969. 26 p. Sw. cr. 10:-.

360. Progress report 1968 nuclear chemistry. 1969. 38 p. Sw. cr. 10:- . 361. A half-life measurement of the 343.4 keV level in " 'Lu. By M. H3jeberg

and S. G Malmskog. 1969. 10 p. Sw. cr. 10:-. 362. The application of thermoluminescence dosimeters to studies of released

activity distributions. By B-l. Ruden. 1969. 36 p. Sw. cr. 10:-. 363. Transition rates in '"Dy. By V Berg and S. G. Malmskog. 1969. 32 p.

Sw. cr. 10:- . 364. Control rod reactivity measurements in the Agesta reactor with the pulsed

neutron method. By K. Bjoreus 1969. 44 p. Sw. cr. 10:- .

365. On phonons in simple metals I I . Calculated dispersion curves in aluminium. By R. Johnson and A. Westin. 19(i9. 124 p. Sw. cr. 10:- .

366. Neutron elastic scattering cross sections. Experimental data and optical model cross section calculations A compilation of neutron data from the Studsvik neutron physics laboratory. By B. Holmqvist and T. Wiedling. 1969. 212 p. Sw. cr 10:- .

367 Gamma radiation from fission fragments. Experimental apparatus — mass spectrum resolution. By J. Higbie. 1969. 50 p. Sw. cr. 10:- .

368. Scandinavian radiation chemistry meeting Studsvik and Stockholm, Sep­tember 17-19, 1969. By H. Christensen. 1969. 34 p. Sw. cr. 10:-.

369. Report on the personnel dosimetry at AB Atomenergi during 1968. By J. Carlsson and T Wahlberg. 1969. 10 p. Sw. cr 10:-.

370. Absolute transition rates in "Mr. By S G. Malmskog and V. Berg. 1969. 16 p. Sw. cr. 10:-.

371. Transition probabilities in the 1/2h(631) Band in " ! U . By M. Hoieberg and S. G Malmskog. 1969. 18 p. Sw. =r. 10:-.

372. E2 and M1 transition probabilities in odd mass Hg nuclei. By V. Berg, A. Backlin, B. Fogelberg and S. G Malmskog. 1969. 19 p. Sw. cr. 10: - .

373. An experimental study of the accuracy of compensation In lithium drifted germanium detectors. By A. Lauber and B. Malmsten. 1969. 25 p. Sw. cr. 10:- .

374. Gamma radiation from fission fragments. By J. Higbie. 1969. 22 p. Sw. cr. 10:-.

375 Fast Neutron Elastic and Inelastic Scattering of Vanadium. By B. Holm­qvist, S. G. Johansson, G. Lodin and T. Wiedling. 1969. 48 p. Sw. cr. 10:- .

376. Experimental and Theoretical Dynamic Study of the Agesta Nuclear Power Station. By P.-A. Bliselius, H. Vcillmer and F. Akerhielm. 1969. 39 p. Sw. cr. 1 0 : -

377. Studies of Redox Equilibria at Ellevated Temperatures 1. The Estimation of Equilibrium Constants and Standard Potentials for Aqueous Systems up to 374°C. By Derek Lewis. Sw. cr. 10:-

378. The Whole Body Monitor HUGO II at Studsvik. Design and Operation. By L. Devell, I. Nilsson and L. Venner. 1970. 26 p. Sw. cr. 10:-.

379. ATOMSPHERIC DIFFUSION Investigations at Studsvik and Agesta 1960-1963. By L-E Hseggblom, Ch Gyllsnder and U Widemo. 1969. 91 p. Sw. cr.

380. An expansion method to unfold pioton recoil spectra. By J. Kockum. 1970. 20 p. Sw. cr. 10: - .

381. The 93.54 keV level in "Sr, and evidence for 3-neutron states above N=50. By S. G. Malmskog and J. Mc Donald. 1970. 24 p. Sw. cr. 10:-.

382. The low energy level structure cf 1"lr. By S. G. Malmskog, V. Berg, A. Backlin and G. Hedin. 1970. 21 p. Sw. cr. 10:-.

383. The drinking rate of fish in the Skagerack and the Baltic. J. E. Larsson. 16 p. Sw. cr. 10: - .

384. Lattice dynamics of NaCI, KCI, RbCI and RbF. G. Raunio and S. Rolandson 26 p. Sw. cr. 10: - .

385 A neutron elastic scattering study of chromium, iron and nickel in the energy region 1.77 to 2.76 MeV. 26 p. By B. Holmqvist, S .G Johansson, G. Lodin, M. Salama and T. Wiedling. 1970. Sw. cr. 10: -

386. The decay of bound isobaric analogue states in '"Si and 31S using (d. ny) reactions. By L. Nilsson, A. Nilsson and I. Bergqvist. 1970. 34 p. Sw. cr. 10:- .

337. Transition probabilities in "<Os. By S. G. Malmskog, V. Berg and A. Back­lin. 1970. 40 p. Sw. cr. 10: - .

388. Cross sections for high-energy gamma transitions from MeV neutron cap­ture in 20*Pb. By I. Bergqvist, B. Lundberg and L. Nilsson. 1970. 16 p. Sw. cr. 10: - .

389. High-speed, automatic radiochemical separations for activation analysis in the biological and medical research laboratory. By K. Samsahl. 1970. 18 p. Sw. cr. 10: - .

390. Use of fission product Ru-106 gamma activity as a method for estimating the relative number of fission events in U-235 and Pu-239 in low-enriched fuel elements. By R. S. Forsyth and W. H. Blackadder. 1970. 26 p. Sw. cr. 10: - .

391. Half-life measurements in ' " I . By V. Berg and A. Hoglund. 1970. 16 p. Sw. cr. 10: - .

392. Measurement of the neutron spectra in FRO cores 5, 9 and PuB-5 using resonance sandwich detectors. Bf T. L. Andersson and M. N. Qazi. 1970. 30 p. Sw. cr. 10: - .

393. A gamma scanner using a Ge(Li) semi-conductor detector with the possi­bility of operation in anti-coincidence mode. By R. S. Forsyth and W. H. Blackadder. 1970. 22 p. Sw. cr. 10: - .

394. A study of the 190 keV transition in '"La. By V. Berg, A. Hoglund and B. Fogelberg. 1970. 22 p. Sw. cr. 10:--.

395. Magnetoacoustic waves and instabilities in a Hall-effect-dominated plasma. By S. Palmgren. 1970. 20 p. Sw. cr. 10:- .

List of published AES-reports (In Swedish)

1. Analysis be means of gamma spectrometry. By D. Brune. 1961. 10 p. Sw. cr. 6: -

2. Irradiation changes and neutron atmosphere in reactor pressure vessels-some points of view. By M. Grounes. 1962. 33 p. Sw cr. 6 : - .

3. Study of the elongation limit in mild steel. By G. Dstbarg and R. Atter-mo. 1963 17 p. Sw. cr. 6:- .

(. Technical purchasing in the reactor field. By Erik Jonson. 1963. 64 p. Sw. er. 8 : - .

5. Agesta nuclear power station. Summary of technical data, descriptions, etc. for the reactor. By B. Lilliehcok. 1964. 336 p. Sw. cr. 15:- .

6. Atom Day 1965. Summary of lectures and discussions. By S. SandstrBm. 1966. 321 p. Sw. cr. 15:-.

7. Building materials containing radium considered from the radiation pro­tection point of view. By Stig O W. Bergstrom and Tor Wahlberg. 1967. 26 p. Sw. cr. 10:-.

Additional copies available from the library of AB Atomenergi, Fack, S-611 01 Nykoping, Sweden.

EOS-tryckenerna, Stockholm 1970