ABSTRACTS

MULTIPARAMETER PLANT

WITH

OPTIMIZATION

BASE-POINT

O F AN A T O M I C

DISTILLATION

OF

POWER

SEA WATER

Yu. D. A r s e n ' e v , S. V. R a d c h e n k o , a n d V. A. C h e r n y a e v

UDC 621.039.516.338.4

Dual-purpose atomic power plants with distillation installations are c h a r a c t e r i z e d by complicated physical interactions of the d e s i r e d p a r a m e t e r s Pl 9 9 9 Pn. To find the dimensionless complexes g e n e r a l izing all the initial costs (a I . . . . . ap) and technical indices (l t . . . . . /k) which govern the reduced costs c = co+Q + . . . +c~,

(1)

one should approximate each component Ci of the cost by an equation [1] like C~=[a~+ant~(h ... Pn)] h.

(2)

The functions describing the physical relations in the object a r e replaced by approximate functions by B r a n d o n ' s method [2]: fl (P1 -.. P a)=lcq)ii (P1) ... Win (fn) it,

(3)

where q~il (P1) is a dimensionless function of only one p a r a m e t e r P1; k is a d i m e n s i o n l e s s c o r r e c t i o n factor; and Ji indicates the dimensionality. The individual functions q~in (Pi) can be r e p r e s e n t e d as polynomials (it is sufficient to r e s t r i c t the discussion to the third power) according to a standard computer p r o g r a m [3]:

(Pii ('P1):bilo+biii Pl+ bil~ P'~+bil3P~-

(4)

The optimization is c a r r i e d out according to each p a r a m e t e r s e p a r a t e l y with subsequent refinement of the solution by iteration. We s t a r t by grouping by powers of the p a r a m e t e r ; then we go to i n c r e m e n t s about the base point [11; then, a f t e r minimization, we find the analytic solution

P op,p t

Q~+-~/~

fl~i '

(5)

where ~2~ and ~II are dimensionless complexes which are governed by the initial costs, indices, and the base point chosen. T

It is noted in the complete paper that the n u m b e r of dimensionless complexes ~ p e r p a r a m e t e r Pi is the degree of the polynomial (4), reduced by one. F o r s e c o n d - o r d e r polynomials, e.g., the final equation depends on one dimensionless complex: ~=~i.

(6)

The iteration s y s t e m is constructed in such a m a n n e r that, during the optimization of one of the p a r a f T m e t e r s P i ' the optimization popt depends on the base values P1 9 9 . P:1 - 1 , P i + l - 9 Pn, but does not depend on Pi. This method has been c o m p a r e d with the dimensional c o m p u t e r - o p t i m i z a t i o n p r o g r a m for an atomic power plant having a w a t e r - m o d e r a t e d w a t e r - c o o l e d r e a c t o r , for the optimization of three p a r a m e t e r s of the distillation installation: the number of evaporation stages (5), the t e m p e r a t u r e of the heating vapor (6) and the underheating of the sea water. T r a n s l a t e d from Atomnaya l~nergiya, Vol. 28, No. 5, p. 418, May, 1970, A b s t r a c t submitted Sept e m b e r 2, 1969.

9 Consultants Bureau~ a division of Plenum Publishin~ Corporation, 227 g/est ]7th Stree6 New YorI~, N. Y. 10011. All rights reserved. This article cannot be reproduced for an T purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.

528

This r a t h e r g e n e r a l d i s c u s s i o n can be applied to solve m u t t i p a r a m e t e r p r o b l e m s by the b a s e - p o i n t method in d i m e n s i o n l e s s o r dimensional form: p o p t / p =F1(~%, ~II), p ~ ~2II), where F 2 = F I P ' . LITERATURE i.

2. 3.

CITED

Yu. D. Arsen'ev, Similarity Theory in Problems of Engineering Shkola, Moscow (1967). D. Brandon, ISA Journal, No. 71 (1959). O. D. Kazachkovskii et al., At. Energ., 27, 183 (1969).

ACCUMULATION

OF

DEVELOPMENT O.

OF D.

PLUTONIUM FAST

FOR

THE

V.

Kirillov

Economies

[in Russian],

Vysshaya

REACTORS

Kazaehkovskii

and

E.

UDC 621.039.526:546.799.4

Interest in the secondary fuel plutonium, produced in ever increasing amounts in thermal power reactors, continues to grow. Plutonium is most effectively used in fast reactors. Even though fast power reactors still do not exist, the time which remains before a large number of them are put into operation may be used to accumulate plutonium to use in them. We present a method for establishing the maximum before using it in fast reactors. This time is determined

possible (critical) time tCR for storing plutonium from the relation

F96~ F9T (1T P)tCR, w h e r e FgF and FgT a r e r e s p e c t i v e l y the values of plutonium when used in fast and t h e r m a l r e a c t o r s [1] and p is a standard reduction factor.

This relation i m p l i e s the following: since FgF > FgT, plutonium produced

in t h e r m a l r e a c t o r s m a y be s t o r e d f o r u s e in fast r e a c t o r s f o r a t i m e tCR, a f t e r which its value FgT i n c r e a s e s to the m a x i m u m p o s s i b l e magnitude FgF. Depending on the magnitude of the ratio F g F / F g T and the reduction f a c t o r p, the c r i t i c a l t i m e is 3.57 yr. Under actual conditions the t i m e f r o m which it is p r a c t i c a l to begin to a c c u m u l a t e plutonium m a y be e a r l i e r than the t i m e fast r e a c t o r s a r e put into operation by l e s s than tCR, and is d e t e r m i n e d by the ratio of the r a t e s of d e l i v e r y of plutonium f r o m t h e r m a l r e a c t o r s and its consumption in fast r e a c t o r s . F o r s e v e r a l possible c a s e s of this ratio equations a r e obtained for d e t e r m i n i n g the t i m e f r o m which the a c c u m u l a t i o n of plutonium should begin and the t i m e at which fast r e a c t o r - c o n v e r t e r s as well as fast r e a c t o r - m u l t i p l i e r s should be put into operation to e n s u r e a given rate of i n c r e a s e of fast r e a c t o r s . Numerical calculations performed for a typical case show that the introduction of an additional amount of plutonium into a s y s t e m of fast r e a c t o r s in the initial stage of t h e i r development enables substantial amounts of U 23~ to be s a v e d l a t e r . LITERATURE

CITED J

1.

O. D. Kazaehkovskii and E. V. Kirillov, Atomnaya E n e r g i y a , 2._22, 439 (1967)o

p

T r a n s l a t e d f r o m Atomnaya E n e r g i y a , Vol. 28, No. 5, pp. 418-419, May, 1970. Original a r t i c l e s u b m i t t e d August 10, 1969; a b s t r a c t s u b m i t t e d D e c e m b e r 8, 1969.

529

ELECTROCHEMICAL SODIUM

BEHAVIOR

CHLORIDE

POTASSIUM

AND

AND

V.

THORIUM

AN EQUIMOLAR

SODIUM

M. V. Smirnov, Yu. V. Posokhin,

OF

IN MOLTEN

MIXTURE

OF

CHLORIDES

Ya. Kudyakov, and Yu. N. Krasnov

UDC 541,122,3-143

The e q u i l i b r i u m p o t e n t i a l s of t h o r i u m f o r v a r i o u s c o n c e n t r a t i o n s a n d t e m p e r a t u r e s i n m e l t s of NaCi a n d K C I - N a C 1 (50 m o l e % NaC1) w e r e m e a s u r e d r e l a t i v e to a c h l o r i d e e l e c t r o d e b y a m e t h o d d e s c r i b e d e a r l i e r [1-3]. E m p i r i c a l e q u a t i o n s w e r e o b t a i n e d f o r the i s o t h e r m s i n NaC1 m e l t : El . . . . 2.325-1--6,70.10-2. lg [Th] • 0.003 V (8t4~C); /z"2. . . . 2.291-i-7,20-t0-2. lg [Th] :i: 0.003 V (860~ C); f'~'3= -- 2.227-~ 8.27.10-2.lg [Th] • 0.003V (966~ C) a n d in K C 1 - NaC1 m e l t s ; E~ . . . . 2.473~-5A7.10-2. lg [Th] • 0,002 V(680~ C); E 2 = - - 2 . 4 2 5 q- 6,0t. 10-2.lg [Th] • 0.002 V(750~ C); E3 = -- 2.3t0 i- 7,48.10-2. Ig [Th] i- 0,002 V(900~ C). T h e y show that the t h o r i u m e l e c t r o d e is r e v e r s i b l e to a m i x t u r e of i t s ions Th 2+ and T h 4+, E x p r e s s i o n s w e r e found f o r the t e m p e r a t u r e d e p e n d e n c e s of the a p p a r e n t s t a n d a r d p o t e n t i a l s (volts) of the e l e c t r o d e s T h / T h ( I I ) a n d T h / T h ( I V ) i n K C 1 - NaCh E~h, TL (IV) =: 3,093 q- 6 . 4 2 - 1 0 - ~ . T ; J~' ~Th/Th (I[) . . . . . 2,9~6 -t- 6.39.10-'~. Y

a n d i n NaCh ~ Th (IV) -LTh, -- --2.976 ~5.87.10-~-T; /L~'h;Th (! I) ~ - - 2,832 q-- 5,84- I 0 - ~ . T.

With i n c r e a s i n g r a d i u s rR+ of the a l k a l i m e t a l c a t i o n , the v a l u e s of E T ) h / T* h ( I V a n d E T ) h / *T h ( I I a r e d i s p l a c e d i n the n e g a t i v e d i r e c t i o n a c c o r d i n g to the following e m p i r i c a l equations" ,l E~:l ~h ~Tv~- --3 729q-9 08.i0-4.T -L-( O . 7 3 7 - - 3 . J 2 . 1 0 - i . T ) .- r R + • 0,005V; _ _ I / T _ . . _

_

E~h/Th ([I)

.

'

'

'

~

3,387 :- 7.78. i0 -~- T-!- (0.53~--i 82. t0-'~'1") 9

....

1

rlt+ + 0,005 V .

E q u a t i o n s w e r e d e r i v e d f o r the t e m p e r a t u r e d e p e n d e n c e of the a p p a r e n t e q u i l i b r i u m c o n s t a n t of the 4+ c + T h s o 1 ~ 2Thde 2+ c i n the i n v e s t i g a t e d m e l t s : r e a c t i o n Thde ~. z 2966 K*~0.0a4.~--~--for KCI--NaCI

and K* = 0 . 0 6 0 - - ~

for NaCI.

E x p r e s s i o n s a r e cited f o r the d e p e n d e n c e s of the e q u i l i b r i u m p o t e n t i a l of t h o r i u m on i t s t o t a l ( a n a l y t i c a l l y d e t e r m i n e d ) c o n c e n t r a t i o n : [Th] = [Th 2+] + [Th4+]. On the b a s i s of the e x p e r i m e n t a l data we c a l c u l a t e d the l i m i t i n g m o l e h ~ c t i o n c o n c e n t r a t i o n s of p o s s i b l e i m p u r i t i e s of i r o n a n d z i r c o n i u m , a b o v e which t h e y do not c o n t a m i n a t e the t h o r i u m d e p o s i t e d on the cathode.

LITERATURE

CITED /

I.

V. Ya. Kudyakov, (1968).

M. V. Smirnov, J

N. Ya. Chukreev,

and Yu. V. Posokhin,

At. Energ.,

24, 448

T r a n s l a t e d f r o m A t o m n a y a E n e r g t y a , Vol. 28, No. 5, p. 419, May, 1970. O r i g i n a l a r t i c l e s u b m i t t e d J u l y 8, 1969; a b s t r a c t s u b m i t t e d N o v e m b e r 17, 1969.

530

M. V. S m i r n o v and V. Ya. Kudyakov, in: T r a n s a c t i o n s of the Institute of E l e c t r o c h e m i s t r y of the U r a l B r a n c h of the A c a d e m y of S c i e n c e s of the USSR [in R u s s i a n ] , No. 12, S v e r d l o v s k (1969), p. 55. M. V. S m i r n o v and N. Ya. C h u k r e e v , i b i d . , No. 3 (1962), p. 3.

2,

3.

MEASUREMENT IN A

A

OF

CELL

OF

RHODIUM

A

THE

FAST

URANIUM

NEUTRON

-- GRAPHITE

THRESHOLD

DISTRIBUTION REACTOR

WITH

DETECTOR

A. V. Bushuev, V. and V. M. Duvanov

G.

Bortsov,

UDC 621.039.524.034.3

F a s t n e u t r o n s have an i m p o r t a n t effect on c e r t a i n p r o c e s s e s in t h e r m a l r e a c t o r s . T h r e s h o l d d e t e c t o r s can be u s e d to study the fast n e u t r o n d i s t r i b u t i o n . The m o s t s e n s i t i v e of t h e s e d e t e c t o r s is b a s e d on the r e a c t i o n Rhl~ which has the lowest t h r e s h o l d and the l a r g e s t c r o s s s e c t i o n and t h e r e f o r e such a d e t e c t o r i s suitabIe f o r m a k i n g m e a s u r e m e n t s in e x p e r i m e n t a l s y s t e m s . We d e s c r i b e the a p p a r a t u s and the method u s e d to m a k e m e a s u r e m e n t s with a r h o d i u m d e t e c t o r . The b a c k g r o u n d s o u r c e s - Rh 1~ and the i m p u r i t i e s I r 192 and I r ~4 - a r e d i s c u s s e d . It is shown that in c e r t a i n c a s e s the u s e of c a d m i u m to s u p p r e s s the b a c k g r o u n d of t h e r m a l n e u t r o n s l e a d s to e r r o r s . The e x p e r i m e n t s w e r e p e r f o r m e d in a g r a p h i t e a s s e m b l y with nine n a t u r a l u r a n i u m rods p l a c e d in a c a v i t y in the r e f l e c t o r of the F - 1 r e a c t o r at the I n s t i t u t e of A t o m i c E n e r g y . R e s u l t s a r e p r e s e n t e d of m e a s u r e m e n t s in a d r y celt and in a ceil with l a y e r s of w a t e r 2 and 11 m m thick around the fuel e l e m e n t s . Using the method of effective t h r e s h o l d c r o s s s e c t i o n s the t h r e s h o l d of the r e a c t i o n was found to be 0.72 MeV and the effective c r o s s s e c t i o n 0.68b with an u n c e r t a i n t y of ~1%. It was established that the II m m thick water layer decreased the fast neutron flux by 9.5 :~1.5% and left the spatial d i s t r i b u t i o n of f a s t n e u t r o n s in the c e l l p r a c t i c a l l y unchanged (Fig. 1). v~ ~9

&

ctors

9Moderator --20

D

29

40

EO

80

]90

Distance from surface of fuel element, mm F i g . 1. D i s t r i b u t i o n of f a s t n e u t r o n s in a cell: 9 c e l l without w a t e r ; e, A) l a y e r s of w a t e r 2 and 11 m m thick, r e s p e c t i v e l y a r o u n d fuel e l e m e n t .

C a l c u l a t i o n s w e r e p e r f o r m e d in the m u l t i g r o u p d i f f u s i o n t r a n s p o r t a p p r o x i m a t i o n and nondiffusion c o r r e c t i o n s w e r e m a d e by using a p r o g r a m given by I. S. S t e s a r e v et al. [1]. The c a l c u l a t e d ratio of the fast and t h e r m a l neutron fluxes in the c e l l with the 11 mm w a t e r l a y e r was 8.5% l e s s than the s a m e r a t i o f o r the d r y cell. The c a l c u l a t e d d i s t r i b u t i o n of the Rh l~ n') r e a c t i o n through the c e i l a g r e e d with the e x p e r i m e n t a l . LITERATURE 1.

CITED

I. S. S l e s a r e v et a l . , in: R e a c t o r T h e o r y and P h y s i c s [in R u s s i a n ] , A t o m i z d a t , Moscow (1967), p. 30.

T r a n s l a t e d f r o m A t o m n a y a E n e r g i y a ' Vol. 28, No. 5, p. 420, May, 1970. O r i g i n a l a r t i c l e s u b m i t t e d May 13, 1969; r e v i s i o n s u b m i t t e d J u l y 1, 1969; a b s t r a c t s u b m i t t e d O c t o b e r 27, 1969.

53t

OPTIMUM

PARAMETERS

OPTICAL

OF

THE

U 238

NEUTRON

POTENTIAL

G. V. A n i k i n , A . G. D o v b e n k o , L. Ya. Kazakova, V. E . K o l e s o v , V. I . P o p o v , G. N . S m i r e n k i n , a n d A. S. T i s h i n

UDC 539.125.52

The complete p a p e r d e s c r i b e s the principle features of a p r o g r a m to s e a r c h for the optical-potential p a r a m e t e r s which best describe the total c r o s s sections and angular distributions of elastically s c a t t e r e d neutrons. The potential is used in the f o r m -

-

tr = vo! (r) + ~ [W d (r) + Wag (r)] +

V~0 h (r) 75,

where / (r)= [t+exp ( ~ ) I

2

h(r)=(--~j) 7

l-i ;

d1(r)

dr I;

This form p e r m i t s one to study potentials with volume, s u r f a c e , and combined absorption. Experimental data on the s c a t t e r i n g of neutrons of 14 different energies in the selected energy range are used to fit the p a r a m e t e r s for a given nucleus. During the s e a r c h , the following fianctional is minimized: la N n

~exp(Oi En)--ocalc (01, En)

%

]

i

oealc(En) oexp

5 x [~,'

_,

~exp (En)

The factor B takes into account the different statistical e r r o r s of the experimental data on the total c r o s s seetions and the angular distributions of the s c a t t e r e d neutrons. Energy-independent optical-potential p a r a m e t e r s have been found for the nucleus U 238 which yield a generally s a t i s f a c t o r y description of the total c r o s s sections (within ~3%) and the angular distributions of elastically s c a t t e r e d neutrons (within ~10-15%) over the energy range 0.075-15 MeV. The p a r a m e t e r s a r e shown in Table 1. The complete p a p e r contains detailed illustrative m a t e r i a l showing the a g r e e m e n t between calculated and experimental c r o s s sections. TABLE 1. Optical-Potential P a r a m e t e r s Surface

Absorption group number B Vo rt ct

WI W2 r2 b

Vco m'i g~ ( x tO-D

II

100 4t,5 1,3i 0,36 4-1,17 0,95 4,9 0,t37 0,17

4392, 1,28

:

Volume

9

4t ,9 1,3

0,29

0,25

4.7 t ,17 1,0 3,3 0,077 1,2

5,6 t ,0,5 0,95 17,8 0,054 3,6

lI

IV

III

0 43,3 i,29 0,27

106 44,3 1,25 0,6 3,6

9 43

1,26 0,56

4,5

Ill

9 39, t

IV

o

1,35 0,36

42,2 t ,3i 0,3

14,9 0,058 2,2

t7 0,04 9,7

4,2

t0

l,l 0,72 t8,5 0,037 7

0

0,t 0,79

3 ,l

0,057 t,2

Combined II

I, 40,9 9

t ,3t 0,46 2,i 5 1,06 0,72 0 0,06t 0,64

III

9

38,9 t ,35 0,3i t,5 5 t ,26 0,64 t6,4 0,05,1 t ,4

T r a n s l a t e d f r o m Atomnaya ]~nergiya, Vot. 28, No. 5, pp. 420-421, May, 1970. Original article submitted July 19, 1969; revision submitted October 1, 1969; a b s t r a c t submitted October 27, 1969.

532

TEMPERATURE

FIELD

TWO-PHASE

IN

A

NONISOTHE

R1VIAL

FLOW

~VI. Kho I b r a g i m o v , a n d V. I . S i d o r o v

O.

I. Sabelev,

UDC

621.039.534

In the operation of p o w e r installations with a liquid h e a t - t r a n s f e r agent a c e r t a i n amount of gas m a y e n t e r , which changes the conditions of heat t r a n s f e r . The p r e s e n t w o r k is devoted to the investigation of the effect of the gaseous phase on the heat t r a n s p o r t in the flow of a two-phase m i x t u r e of w a t e r and a i r . The e x p e r i m e n t s w e r e conducted in a v e r t i c a l square channel having a c r o s s section of 30 • 30 m m 2 and a length of 1800 m m with f o u r - s i d e d heating at a constant heat flux of q =1"104 k c a i / m 2 . h to the wall. A i r was injected into the bottom entrance c h a m b e r of the channel. The walI t e m p e r a t u r e was m e a s u r e d by t h r e e m i c r o - t h e r m o c o u p l e s ; the t e m p e r a t u r e in the liquid flow was m e a s u r e d by a movable thermoeouple having a ]unction d i a m e t e r of 0.2 m m . A c o m p a r i s o n of the t e m p e r a t u r e distributions in s i n g l e - p h a s e and two-phase flows shows (Fig. la) that the f o r m of the t e m p e r a t u r e profile changes s h a r p l y on i n troducing even a s m a l l amount of a i r into the w a t e r flow: in the t w o - p h a s e flow the t e m p e r a t u r e profile gets f l a t tened and b e c o m e s uniform o v e r a l a r g e p a r t of the channel c r o s s section. At the s a m e t i m e the wall t e m p e r a t u r e d e c r e a s e s , while the t e m p e r a t u r e of the liquid at the exit f r o m the channel r e m a i n s p r a c t i c a l l y constant. T h e p r e s e n c e of a i r bubbles in the liquid leads to a b e t t e r mixing of the liquid, to a m o r e intensive heat t r a n s p o r t , and to an i n c r e a s e in the heat t r a n s f e r . Thus, for the flow r e g i m e of the two-phase m i x t u r e shown in (Fig. l a , c u r v e 2), the coefficient of h e a t - t r a n s f e r i n c r e a s e s a l m o s t by a factor of t h r e e in c o m p a r i s o n with the c a s e of the flow of w a t e r (see Fig. l a , c u r v e 1).

o

~2

o,,

o,e

0,8

y/~

IRFflc 2

0,2

0,4

0,6

0,8

y/a

Fig. 1. T e m p e r a t u r e distribution (a) and the change in the intensity of the fluctuations (b): 1) s i n g l e - p h a s e flow (G=0.755 m 3 / h , q=10,000 k o a l / m 3 . h ; 2) t w o - p h a s e flow (G=0.755 m 3 / h , G ' = 0 . 0 5 5 m 3 / h , z /a =0.

The intensity distribution of the t e m p e r a t u r e fluctuations in the w a t e r flow (G=0.755 m3/h) is p r e s e n t e d in {Fig. l b , curve 1). On injecting a s m a l l quantity of a i r into the w a t e r flow (G' = 0.55 mS/h) the i n t e n s i t y of the t e m p e r a t u r e fluctuations d e c r e a s e s s h a r p l y (see Fig. l b , c u r v e 2), which is caused by the flattening of the t e m p e r a t u r e p r o file and the d e c r e a s e in the t e m p e r a t u r e head. When the flow rate of a i r a c r o s s the channel is c o m p a r a b l e to that of w a t e r , the nature of the fluctuations changes. F o r e x a m p l e , at a volume flow rate of G = 0.5 m 3 / h f o r w a t e r and G ' = 0.5 m S / h f o r a i r infrequent intensive fluctuations w e r e o b s e r v e d ; these w e r e a p p a r e n t l y caused by the c o r k r e g i m e of the flow. At a fixed point in the flow the t e m p e r a t u r e of the l i q uid f i r s t i n c r e a s e s s h a r p l y and then d e c r e a s e s s h a r p l y with r e f e r e n c e to some m e a n value. The initial i n c r e a s e of the t e m p e r a t u r e c a n b e explained by the i n c r e a s e d liquid t e m p e r a ture ahead of the front p a r t of the bubble; the d e c r e a s e of the t e m p e r a t u r e below the m e a n value m a y be explained by the i n c r e a s e d turbulence in the s t e r n zone of the bubble.

T r a n s l a t e d f r o m Atomnaya t~nergiya, Vol. 28, No. 5, pp. 421-422, May, 1970. Original a r t i c l e submitted May 19, 1969.

533

ROLE OF

OF

ENERGY

TRANSPORT Yu.

DEPENDENCE

IN

PROBLEMS

THEORY* I. E rshov

and

S.

B.

UDC 517.9:533.9

Shikhov

We consider the integral equation of neutron t r a n s p o r t with an isotropic s c a t t e r i n g indicatrix in a homogeneous nonconcave volume: E~

%(r, E):= .t- dE'g(E,

E')• ~ drY%(r% t'7')

E1

e--(~) Ir--r'[

I r__r, lU ~-f(r,E),

(I)

V

The solution of this kind of p r o b l e m is obtained in [1, 2] with the use of Keyes method only for a plane g e ometry: however, the solution is not r i g o r o u s l y substantiated and is valid only when certain r e s t r i c t i o n s a r e imposed on the function g(E, E'). The investigation of the m o r e general case with the use of Keyes m e t h od is perhaps not possible, since s e r i o u s difficulties appear in proving the completeness of the s y s t e m of eigenfunctions of the t r a n s p o r t operator. The use of the method of integral t r a n s f o r m s made it possible to eliminate such difficulties and obtain m o r e general results not only in the plane g e o m e t r y , but also for an a r b i t r a r y nonconcave volume. F o r an infinite plane l a y e r the problem was reduced to the investigation of a certain integral o p e r a t o r of F r e d h o l m type with the use of F o u r i e r t r a n s f o r m . It could be shown that for sufficiently general r e p r e sentations for the function g(E, E') the general solution of Eq. (1) for the plane l a y e r can contain, besides the exponential t e r m s , t e r m s that are products of exponent and power of the space variable. In the case of an a r b i t r a r y nonconcave volume V Eq. (1) was reduced with the use of Radon's t r a n s form to a form which formalty coincides with the equation for an infinite plane layer; therefore the results obtained for the plane g e o m e t r y can be c a r r i e d o v e r to the a r b i t r a r y geometry. F u r t h e r m o r e , a simple approximation of the isotropic scattering indicatrix is proposed for which the theoretical possibility of solving the p r o b l e m s of t r a n s p o r t theory, taking account of the continuous dependence of the neutron c r o s s s e c tions on the energies, is demonstrated on a simple example. LITERATURE 1.

2.

CITED

R . Zelazny, Nucleonika, 9, Nos. 7-8, 563 (1964). K. Fuehs and S. Collatz, Kernenergie, 6 . ~ , 386 (1964).

SOURCE

OF

ATOMIZATION

MULTIPLY OF

THE

CHARGED OPERATING

IONS WITH

CATHODE

SUBSTANCE

Yu. P. Trt'yakov, A . S. P a s y u k , L. P. Kul'kina, a n d V. I . K u z n e t s o v

UDC 621.384.6

A cyclotron s o u r c e of multiply charged ions (m.e.i.), in which the operating substance is fed into the discharge impulsively by the method of cathode atomization, is developed based on a g a s - d i s c h a r g e s o u r c e of m.c.i, with preheated cathode [1]. An a r c discharge with electron oscillations in the magnetic field of the cyclotron is used in the source. In the region of the e m i s s i o n silt of the s o u r c e an additional electrode * T r a n s l a t e d f r o m Atomnaya Energiya, Vol. 28, No. 5, p. 422, May, 1970' Original a r t i c l e submitred July 24, 1969. ST ranslated f r o m Atomnaya E nergiya, Vol. 28, No. 5, p. 423, May, 1970. Original article submitred June 12, 1969.

534

TABLE

I. Current of Multiply Charged

Ions of Different Elements

o~

Ion current of the operating substance from ions with different charges in a pulse, mA ~o

o

Ca Ti Cu Zn Mo Ta W

i+

2+

3+

4+

5+

i8,4 22,0 3,0 17,7

84.9

3i,8 20,0 22 26,6 31,7 54,7 24,6

3,0 3,4 t4 12,5 35,3 29~3 23,3 tl,4 17,i

0,4 0,25 4,5 264,7 9,49 16,8 18,9 13,1

59,1 23 32 29,6 69,6 24

2O

6+

0,014 0,04

1,0 1,6 6,6 3,95 9,4 12,5 6,8

7+

8+

9+

0,005

o,7

0,25 1,9 0,76 2,0 8,4 3,3

0,035 0,04

o[~o 0,4 3,0 0,7

0,024

o~2

77 -- 100 60 60 70 93 90 65 65

of the a t o m i z a b l e o p e r a t i n g s u b s t a n c e i s i n t r o d u c e d into the d i s c h a r g e c h a m b e r . The e l e c t r o d e is fixed in a w a t e r - c o o l e d c o p p e r c o n t a i n e r . A s m o o t h r e g u l a t e d p o t e n t i a l , which is n e g a t i v e with r e s p e c t to the anode (0-2 kV), is a p p l i e d to the c o n t a i n e r f r o m a s e p a r a t e s o u r c e . In the p r e s e n c e of the d i s c h a r g e in the c h a m b e r p o s i t i v e ions of the p l a s m a b o m b a r d the s u r f a c e of the e l e c t r o d e . The e j e c t e d a t o m s a r e ionized in the d i s c h a r g e , a f t e r which the ions of the o p e r a t i n g s u b s t a n c e a r e e x t r a c t e d through the e m i s s i o n slit. The amount of the s u b s t a n c e e n t e r i n g into the d i s c h a r g e is r e g u l a t e d by v a r y i n g the p o t e n t i a l on the a t o m i z a b l e e l e c t r o d e . A d i s p l a c e m e n t of the e l e c t r o d e into the r e g i o n of the e m i s s i o n s l i t r e s u l t s in a c o n c e n t r a t i o n of the o p e r a t i n g s u b s t a n c e (in the f o r m of n e u t r a l p a r t i c l e s and ions) at the point of e x t r a c t i o n of the ions. The ions of the a u x i l i a r y gas r e m a i n p r e d o m i n a n t l y outside the r e g i o n w h e r e the e l e c t r o d e is p l a c e d . Due to this the obtained ion c u r r e n t of the o p e r a t i n g s u b s t a n c e e x c e e d s the ion c u r r e n t of the a u x i l i a r y gas (see Table 1). The m . c . i , s o u r c e was t e s t e d on bench [2] and in the c y c l o t r o n U-300 of the l a b o r a t o r y of n u c l e a r r e a c t i o n s of the Joint I n s t i t u t e for N u c l e a r R e s e a r c h . The v o l t - a m p e r e c h a r a c t e r i s t i c s of the a t o m i z a b l e e l e c t r o d e - a n o d e gap, the dependence of the r e l a t i v e i n t e n s i t y of the s p e c t r a l l i n e s of a t o m s and ions of the a t o m i z a b l e s u b s t a n c e on the e l e c t r o d e p o t e n t i a l , and the d i s t r i b u t i o n of the r e l a t i v e i n t e n s i t y of the spec~ ~ral l i n e s of a t o m s and ions of c a l c i u m o v e r the t r a n s v e r s e c r o s s s e c t i o n of the d i s c h a r g e c h a m b e r w e r e m e a s u r e d on the bench. Yield c u r v e s of m . e . i , w e r e a l s o obtained as functions of the d i s c h a r g e p a r a m e t e r s in the s o u r c e . Multiply c h a r g e d ions of m a g n e s i u m , a l u m i n u m , c a l c i u m , t i t a n i u m , c o p p e r , zinc, m o l y b d e n u m , t a n t a l u m , and t u n g s t e n w e r e obtained on the bench (see Table 1). In the c y c l o t r o n U-300 c a l c i u m ions with s e v e n and eight c h a r g e s w e r e a c c e l e r a t e d . The m e a n ion c u r r e n t of C a ~ in a r a d i u s of 100 cm c o m p r i s e d 3 /~A; f o r C a ~ the c o r r e s p o n d i n g c u r r e n t was 0.4 #A.

LITERATURE

2.

CITED

P. M. Morozov, B. N. Makov, and M. S. Ioffe, At. ~nerg., 2, 272 (1957); A. S. Pasy-ak, Yu. P. Tret'yakov, and S. K. Gorbachev, At. r 24, 21 (1968). A. S. Pasyuk, Kuo Ch~i-ch'ien, and Yu. P. Tret'yakov, Preprint Joint. Inst. for Nucl. IRes., 1523, Dubna (1963).

535

CONDITIONS

FOR

EQUILIBRIUM

THE

EXISTENCE

PHASES

I. D.

Dreval'

and

IN

LINEAR

V.

V.

OF

TWO

STABLE

ACCELERATORS L~DC 621.384.64

Kushin

It is shown that in l i n e a r a c c e l e r a t o r s of charged p a r t i c l e s higher (or inverse) h a r m o n i c s of the a c c e l e r a t i n g field m a y lead to a unique phenomenon by causing a l t e r n a t i n g gradients of longitudinal f o r c e s , i.e., to two stable equilibrium p h a s e s in a single p e r i o d of the h i g h - f r e q u e n c y oscillations. The p o s s i b i l i t y of the a p p e a r a n c e of such p h a s e s is d i s c u s s e d on the example of an a c c e l e r a t o r with p h a s e - c h a n g i n g f o c u s sing, in which alternating gradients of longitudinal f o r c e s a r e c r e a t e d due to the periodic change of the sign and the magnitude of the equilibrium phase. +-Yc-u(~) .............:::::::: ..........................

....::,....+,

;27};iri =:--: .....b ........................... "-.......:;.=::::-~,;:. ~ !o\

................ ..................2:./:....:"..:::,--.:::':::: ;:...::::...................... ::::::::::": .............. '......................... [l/i~

iil/iT

....................

...............................

s \

~o:,~ --

-~"

%,\

f;I;i/

~"

IZ.. x',,i\ I~ !i#

l i p ~'~

I~"

!

~Tr

-= ,.,,,

;/ ',,', ~

'~#dti~

Fig. 1. Potential functions for q~= 88 ~

It was found that the stability of the two equilibrium p h a s e s is e n s u r e d in the c a s e when the p h a s e of the highfrequency accelerating field in adjacent segments of the accelerator differs by 170-180 ~ It is true that the used accelerating field is very weak, since the main part of the energy, acquired by the particles in one segment of the accelerator, is lost in the neighboring segmentso A decrease in the above-mentioned phase shift leads to a loss of the stability of the second equilibrium phase. Representing the periodic law of variation of one of the equilibrium phases along the accelerator [n the form. ~s = ~P0=~ ~Pl, it is possible to d e t e r m i n e the effective potential e n e r g y of the phase oscillations as a function of the p a r a m e t e r s q~0 and ~i. Typical potential functions with corresponding s e p a r a t r i c e s a r e p r e s e n t e d in Fig. 1. The mutual position of both stable equilibrium p h a s e s can be found f r o m a third o r d e r a l g e b r a i c equation in t h e p h a s e deviations &~.

Translated from Atomnaya Energiya, Vol. 28, No. 5, pp. 423-424, submitted March 12, 1969; abstract submitted July 15, 1969.

536

May,

1970.

Original article