Optimum operating conditions for a laser uranium enrichment plant

OPTIMUM

OPERATING CONDITIONS FOR A LASER URANIUM ENRICHMENT PLANT

KIMIO YAMADA and NORIHIKO OZAKI

Atomic Energy Research Laboratory, Hitachi Ltd, Ozenji, Tama-ku,Kawasaki, Kanagawa (Japan) and MANABU YAMAMOTO and KIICHI UEYANAGI

Central Research Laboratory, Hitachi Ltd, Higashi-Koigakubo, Kokubunji, Tokyo (Japan)

S UMMA R Y

Operating conditions of the laser uranium enrichment plant to obtain cheaper enriched uranium are optimised by using the standard optimisation procedure. A simple kinetic model is given to obtain the ion production rate as a function of the laser energy density, ultraviolet light energy density, atomic density and depth and height of the reaction region. The unit cost of enriched uranium is chosen as a valuefimction instead of the unit cost of the separative work. The construction cost is expressed by means of an exponential fi~nction to take the scale merit into account. Two numerical results are given. In case 1, the laser power and efficiency are subject to the restraints determined by the present technical levels and in case 2, they arej}'ee. The unit cost of the enriched uranium is higher than those of the gaseous dijfitsion and gas centrifuge methods by a factor of 2 ~ 11. Results indicate that laser uranium enrichment is probably competitive with the other uranium enrichment methods, provided that the laser efficiency is improved by up to 1% and the laser lijetime is extended several times.

1.

INTRODUCTION

The technique for laser isotope separation has been developed and recently the separation of uranium isotopes was achieved by the selective two-step photoionisation process, in the United States.1 A point common to the laser isotope separation is to use isotope shift, and excite selectively specific isotopes with the laser. Laser isotope separation involves various methods, depending on the differences in separation, such as two-step 287 Applied Energy (3) ( 1 9 7 7 ) - - © Applied Science Publishers Ltd, England, 1977 Printed in Great Britain

288

K I M I O Y A M A D A , N O R I H I K O OZAKI, M A N A B U Y A M A M O T O , KI1CHI U E Y A N A G I

photoionisation, photodeflection, photodissociation and photochemistry. The two-step photoionisation method for the uranium atom is the most promising of these methods. The reasons are: (1) this method has already achieved success on a laboratory scale; (2) the uranium a t o m has well known and resolved isotope shifts in electronic levels in comparison with uranium molecules. The purpose of this paper is to discuss the optimum conditions for the laser uranium enrichment plant. The two-step photoionisation method of the uranium atom is selected, and the Complex method 2 is employed as an optimisation procedure. The optimum conditions for two typical cases are surveyed. In case 1, the laser power and e~ciency are subject to restraints determined by the present technical levels, and in case 2 they are free from the technical requirements; i.e. no technical restrictions were put on them.

2.

LASER I S O T O P E S E P A R A T I O N PROCESS AS A P P L I E D T O U R A N I U M E N R I C H M E N T

2.1. Uranium energy levels Energy levels for the uranium atom employed in the selective two-step photoionisation process are shown in Fig. 1. The fairly strong transition (7s) 2 5L 6

1st ionization level I

6

E

UV Light

_~~

4x104

3100 In -

e--

Excited level

2x104

(Isotopeshift~ \ 0.08 A

)

0

s915'4U

_

~k

ta~ -

tO

oi

J

Laser l i g h t / / ~ 7 p

X IJ.J

4

- 2 7M 7

Gro,und level

.~X

- 0

its) 2 5L 6 Fig. 1. Uranium energy levels for the selective two-step photoionisation process. The 235U atom is selectivelyexcited to the 7s7p7Mr levelwith dye laser light of wavelength 5915-4A.The excited atom is successively ionised by the ultraviolet light of wavelength between 2000 and 3100A.

OPERATING CONDITIONS FOR URANIUM ENRICHMENT

289

7s7p 7M7, 2 = 5915.4 A, is chosen for isotope specific excitation, because this level has a remarkable isotope shift, approximately 0-08 A, and a fairly strong absorption cross-section. Also the excitation energy of this level agrees with the wavelength of the maximum output power of the dye laser. The ionisation limit from this excitationlevel is3100A for a single quantum photoionisation.The lower limit of wavelength to prevent direct ionisation from the ground state is 2000A. The ultraviolet light in this wavelength range is obtainable by frequency doubling of visible laser light such as the Ar ion laser lines. The 7s7p 7M7 excited level of 235U atom is split into eight hyperfine structures due to the interaction between spins of the electrons and nucleus. In the selective two-step photoionisation process, only one component of the hyperfine structure is used.

2.2. Photoion production rate In this section, a simple kinetic model to calculate the photoion production rate is given. The thermalisation processes to be considered are illustrated in Fig. 2. Major transition processes are resonant excitation, photoionisation, induced emission and thermal relaxation. The resonant excitation and photoionisation take place when the atom absorbs a photon with energy h v= and h vi respectively. I fwe neglect charge transfer and energy transfer between 235U in the ground state,

1st ionization Charge transfer lavel //~//~,~~/~ ~ //~"~

Vi Wi Energy transfer Excited level

We

l 1 '~ Ve

Spontaneous decay Induced emission

Ground level 235U

238u

Fig. 2. A three-levelsystemillustrating a selectivetwo-stepphotoionisationprocess. Major transition processes are resonant excitation, photoionisation, induced emission and thermal relaxation.

290

KIMIO YAMADA,NOR1HIKO OZAK1, MANABUYAMAMOTO,KIICHI UEYANAGI

and 238U in the excited and ionisation states, respectively, the kinetic equations of the populations of the ground, excited and ionised 235U levels are written as below: d~

- WeN~ +

dN~=dt

W e N ~ - ( We+

dt

We + - + Ve Nt + VIN~

z

Wi+

Ve+!) N~

dN~ = WiN~_ viNi5

(1)

(2) (3)

dt

The production rate of EasU ions is obtained from the charge transfer between Easu ion and 23aU atom as dN~

dt

i -

(4)

ViN 5

where N~, N~, N~ are populations of ground level, excited level and ionisation level of the 235U atoms respectively, N~ is the population of 23s U ions, z is the lifetime of spontaneous decay and Wk and Vj are the transition probabilities from the excited level and the ionisation level so defined by

Wk = akPk/hVk Vj = AvajN i

(5) (6)

respectively. Here ~k is the absorption cross-section for a photon having frequency Vk, Pk the energy density of radiation with frequency vk, Av the difference in the thermal velocity of the uranium isotopes, aj the charge transfer or energy transfer cross-section, and N~ the population of 238U atoms. Using the initial conditions that at t = 0, N~ = N~ = 0, and N g = ~No, where a is the concentration of 235U atoms and No the atomic density, we obtain the solution of kinetic eqns. (1), (2) and (3) as a function of time:

N~ =

" N ( WeWiO~ 0 ~

1

exp(21t)

exp(~Et) "~

+ ).1(). 1 _ /-2) -{'- ~2()~2 - A1)/

( t exp()Llt)-- 1 exp(22t)-- 1) N~ : WeWiVi~No ~ + 22(21 _ )~z) + ~ - ~11)

(7) (8)

where

+

2W e + Wi + Ve + Vi+

4 We

(9)

OPERATING CONDITIONS FOR URANIUM ENRICHMENT

3.

291

OPTIMISATION PROCEDURE

3.1. Optimisation calculation The complex method 2 was used as the optimisation procedure. In order to optimise with this method, two prerequisites are in general necessary. Firstly, one has to define the independent variables of the optimisation problem. The necessary conditions for independent variables are: (1) the effect upon the value function is great and (2) they have finite optimum values. Secondly, a suitable value function has to be defined. Optimisation of the operating conditions of the laser uranium enrichment plant means the determination of the optimum compositions and the specifications which minimise the unit cost of enriched uranium. It is therefore desirable to choose the unit cost of enriched uranium instead of the unit cost of separative work as a value function. The value function is, of course, expressed as a function of the independent variables. The standard deviation of independent variables is applied to a convergence judgement in the optimisation calculation. The conventional judgement for convergence is to compare the standard deviation of the value function with a small positive number. However, it is desirable to use the standard deviation of the independent variables as a judgement condition for convergence in case one is interested in the convergence values of the independent variables.

fl S~~lon

collector Reactioregion n Slit Crucible

Fig. 3. The schematicview of the isotope separation region. The uranium atom evaporating from the high temperature crucible is irradiated by laser light and ultraviolet light in the reaction region, and ionised 235U atoms are collected by the ion collector electrodes. Distance a = 50mm.

292

KIMIO YAMADA, NORIHIKO OZAKI, MANABU YAMAMOTO, KIICHI UEYANAGI

3.2. Independent variables Figure 3 shows the schematic view of the isotope separation region. The arrangement of the laser, ultraviolet light and atomic beam and the related independent variables are shown in Fig. 4. The isotope separation region is composed of a crucible for making the uranium vapour, a slit for the atomic beam collimation and ion collector. The uranium metal in the crucible is heated using the electron gun. The uranium atoms evaporating from it enter the reaction region with a cross-section ofSuv cm in width and $1 cm in depth, and are irradiated by laser and ultraviolet beams. The laser light, being perpendicular to the atomic beam, is Suv cm in width and Sa cm in height. The ultraviolet light with cross-section of $1 cm in depth and S a cm in height meets orthogonally both the atomic beam and the laser light. The 235Uions are selectively produced in the reaction region and are collected by the ion collector located downstream of the reaction region. Five independent variables are selected according to the reasons described in section 3.1 ; these are laser energy density, ultraviolet light energy density, atomic density, depth $1 and height S, of the reaction region. The important variables which might have influence on the unit cost of enriched uranium besides the independent variables are distance between the crucible and the reaction region, gap of the ion collector electrodes and width Su~ of the reaction region. It is, of course, desirable for

Reaction region Width

ty = Pi )

Laser light

( En,r. d.,.,=p, ) Atomic beam

( nensity=No ) Fig. 4. The independent variables for optimisation of laser uranium enrichment plant. The uranium atom is irradiated by laser light and ultraviolet ]ight with energy densities Pc and P~, respectively.

OPERATING CONDITIONS FOR URANIUM ENRICHMENT

293

atomic density to reduce the distance between the crucible and the reaction region. However, it is necessary for the thermal shield and collimating slit to separate them by at least 5 cm. The density of 235U ions decreases by charge transfer as they go across the atomic beam to the ion collector. Therefore, the smaller gap of the ion collector electrodes is desirable. The width Suv of the reaction region is 1 cm. The ultraviolet light energy density differs on both sides of the reaction region because of the absorption of the excited uranium atoms. However, the amount of attenuation loss will scarcely be noticed since the photoionisation cross-section of ultraviolet light is extremely small. As long as the amount of attenuation loss can be neglected, ultraviolet light may be used. As a result, it is possible to use ultraviolet light at least 500 times.

Valuefunction (1) Production rate of enricheduranium. The concentration of 235U ions decreases

3.3.

by charge transfer collisions between z35U ions and 238U atoms when ions are separated from the neutral beam by the static electric field. The number of z 35U ions collected by the ion collector per unit time is given by the following equation (see Appendix): n

1

(10)

n

As=

~ nS,. S..V(N~5 + N~) - A,

(11)

1

where n is the number of divisions of depth S Zin the reaction region, N d and dare the number and the gap of the ion collector electrodes, respectively, v is the average thermal velocity of the atomic beam, and ar~ is the charge transfer cross-section. Though high enrichment is obtained at the ion collector, the enrichment required for the fuel used in a nuclear power plant is about 3 %. Therefore, it is necessary that highly enriched uranium is blended with natural uranium to form a nuclear fuel. The quantities of 3 % product and natural uranium for blending can be written as: w. =

MsMsAs(I - ~e) - ~e(3MsAs + 'M~A8) Na[MsMs(~ e - ce.) - 9a.ae]

We = Wn +

MsA5 + MsAs N,

(12)

(13)

where W. and /,Ize are the quantities of natural uranium and 3% product respectively, Ms and Ms the mass numbers of 23sU and z3su respectively, ae the enrichment of the uranium obtained, ~, the natural abundance ratio of uranium and N a Avogadro's number.

294

KIMIO YAMADA, NORIHIKO OZAKI, MANABU YAMAMOTO, KIlCHI UEYANAGI

(2) Operating cost. This includes costs for maintenance, employment, materials and electric power. The equipment to be maintained includes the dye laser, ultraviolet laser and evacuation systems. The maintenance cost can be written as

(mr = stCl/tl + ~;,vCuv/tuv + ~vCv/tv

(14)

where C, 5, t are respectively unit cost, the ratio of maintenance cost to initial cost and the lifetime of the equipment: the subscripts 1, uv, v represent the dye laser, ultraviolet laser and evacuation systems, respectively. It is assumed that the employment cost is proportional to the number of lasers. As a laser irradiates the area of S a x S a at the reaction region, the number of lasers is given as

N, = [S,/(&. ap)] + [Suv/Sa]

(15)

I[ 1]: gauss symbol where AD is the number of re-utilisations of the ultraviolet laser, With the number of lasers, the employment cost can be written as: Gem : A e m N l + Bern

(16)

where Aem and Bern are constant. A material cost is given by natural uranium quantities passing through the reaction region and that for blending as: Cma = U m " ( W n -'~

MsNoVSr/N~)

(17)

where IV, is the natural uranium quantity for blending calculated from eqn. (12), Um the unit cost of natural uranium and S r the cross-section of the atomic beam. The electric power cost is

Ep : Up(PeSe/~e + PiSi/(t]iAp) + Eg + Or)

(18)

where P and S are energy density and irradiation area respectively, q the laser efficiency, subscripts e and i represent the dye laser and ultraviolet laser respectively, Up the unit cost of electricity, Eg the power consumed by the electron gun and O t the electric power cost including that for evacuation air conditioner and cooling water systems. (3) Plant construction cost andcapital cost. The construction cost has to be defined as a function of the independent variables. For this purpose, we express the construction cost by means of an exponential function

Cj = A j (DPj)kJ

(19)

In this expression, the cost exponent kj is a dimensionless quantity between zero and unity which determines the variations in the cost of construction with the size of the component. Both Aj and kj must be determined empirically by means of cost

OPERATING CONDITIONS FOR URANIUM ENRICHMENT

295

evaluation studies. The design parameters, DPj are quantities characterising the cost of the individual components and must themselves be the independent variables or functions of them. Figure 5 shows the empirical cost functions which are used to describe the construction costs of the various plant components. The abscissae indicate the design parameter of the construction cost functioh; the ordinates show the cost of components. Ten different cost functions are contained in Fig. 5, each of them derived by the use of goods on the market. However, the exponent kj of the electric power source and cooling water system was quoted from a paper by M. M~trtensson. 3 Assuming the plant lifetime and annual capital charge, we can deduce the capital cost from the construction cost. (4) The unit cost of enriched uranium. The unit cost of enriched uranium is obtained from the cost described above as follows: U = (Cop + Cp)/We

(20)

where Cop is the operating cost and C v the capital cost.

4.

CASESTUDY

Table 1 shows the parameters used to optimise the operating conditions of the laser uranium enrichment plant. The most favourable ones at the present time are employed in this calculation, but some of them possess the possibility of changing to TABLE 1 PARAMETERS USED IN THE COMPUTATION

Notation Ap

Up

Um i T TI Tuv ?v

Explanation The number of re-utilisations of the ultraviolet laser Unit cost of electricity, S/kWh Unit cost of natural uranium, $/kg Fixed charge rate, Lifetime, years Dye laser Ultraviolet laser Evacuation systems

Value 500 0.017 33 10 0.5 0-5 2

more favourable values in the future. For example, it is possible to extend the lifetime of a laser by a factor of 5 if the laser tube is improved. However, the unit costs of electricity and natural uranium rise gradually, and the rises in these prices have minor influences upon the unit cost of enriched uranium.

296

KIMIO YAMADA, NORIHIKO OZAK1, MANABU YAMAMOTO, KIICH1 UEYANAGI

$

Dye laser k= 0.67

I

J

106

/

J

J

10s 101

102

103

Dye laser power ( w ) i

$

Ultraviolet I ~ k-0.67

'~

105 Jf 10 4 10 2

10 3

10 4

Ultraviolet laser power ( w ) Fig. 5. The empirical construction cost functions. In the figures, the abscissa indicates the design parameter of the construction cost function. The ordinate shows the cost of components in dollars.

OPERATING CONDITIONS FOR URANIUM ENRICHMENT

S

Electric power source k=0.6

10 5

/

/

104

/

/

/

I

w

10 6

10 7

Total electric power

10 8 (w)

/ $

Building k=0.8

10 3

/

/

f

J

J 10 2

101

10 0 Floor space

( m2 )

Fig. 5--contd.

10 2

297

7

z

2,

o

3

e-

~

\

•

,,

\

II

\

~ °

i

o

I

,

v

A

3

i

I °

oe-

-7"

3

I

0

~

\ II .E:)

-7" ~.

x

m > z > 0

=

o

0

>

>.

> z >

0 0

z 0

>

,.<

O0

OPERATING CONDITIONS FOR URANIUM ENRICHMENT

$

~P

Vacuum vessel k=0.4

f

J

104

103 100

101

10 2

Volume in vacuum vessel (m 3)

Evacuation s y s t ~

104 J

103 10 0

101

102

Volume in vacuum vessel ( m3 ) Fig. 5--contd.

299

300

KIMIO YAMADA, NORIHIKO OZAKI, MANABU YAMAMOTO, KIICHI UEYANAGI

$

;Cooling w a t e j system k=0.78 J

104

J i f

103 105

106

107

Power of cooling water pump(w)

I

$

Electron g u n j k=0.68

105

/ f

J

./

f

104

105 10 6 Electric gun power ( w Fig. 5--contd.

107

301

OPERATING CONDITIONS FOR URANIUM ENRICHMENT

Table 2 shows the physical quantities used in the optimisation programme. The excitation cross-section of uranium is from the paper by Tvccio e t al. 1 The photoionisation cross-section is the one approximated to theoretically. The charge transfer cross-section is estimated from the experimental values of the other elements. 4 The energy transfer cross-section is the calculated value from those of inert gases. These values are possibly overestimated, and this has a minor influence on the unit cost of enriched uranium. TABLE 2 PHYSICAL QUANTITIES

Input data

Numerical value

Cross-section, cmz Excitation lonisation Charge transfer Energy transfer

1.3 x 10 ~4 1.5 x 1 0 - 1 7 1.3 x 10 13 1.0 x 10 13

Wave length, A Dye laser Ultraviolet laser

5915.4 2600

2500 E

.,m

. m

e,,-=

e-

e-'

2300

O

"F.

2100, T

10 -2 10 -1 Efficiency of dye laser ~ x )

100

Fig. 6. The unit cost o f enriched u r a n i u m plotted as a function of dye laser et~ciency. The unit cost of enriched u r a n i u m is optimised via the five independent variables. The ultraviolet laser et~ciency is taken as 5 x 10-2%.

302

KIMIO YAMADA, NORIHIKO OZAKI, MANABU YAMAMOTO, KIICHI UEYANAGI

Figure 6 shows the unit cost of enriched uranium as a function of the dye laser efficiency, and Fig. 7 shows the unit cost of enriched uranium versus ultraviolet laser efficiency. On this calculation, any technical restrictions are not imposed on the independent variables. The unit cost of enriched uranium decreases exponentially as laser efficiency increases to 1 ~o where it takes the asymptotic value. It might be said that the efforts to improve the laser etficiency above 1 ~o are rather fruitless for isotope separation. However, technical restrictions exist with respect to laser energy density and efficiency. For example, the ultimate value of the dye laser efficiency is about 2 x 10-3~o, and the ultraviolet laser efficiency is about 3 x 10-2~o for the present technical levels. It is highly favourable for economic laser uranium enrichment to improve the laser efficiency up to one percent.

=

6000

E

4000 C

2000 0

"E

10 -2

10 -1

100

Efficiency of ultraviolet light source I ~ Fig. 7. The unit cost of enriched uranium plotted as a function of ultraviolet laser efficiency.A unit cost of enriched uranium is optimisedby the fiveindependent variables. The dye laser efficiencyis taken as 2-5 ×10 2~,. Optimisation of the operating conditions is performed for two cases: case 1, the laser energy density and efficiency are subject to restraints depending on the present technical levels; where in case 2 there is no restriction. In case 2, the efficiencies of the dye laser and the ultraviolet laser are chosen as 1 ~,,, according to the results

OPERATING CONDITIONS FOR URANIUM ENRICHMENT

303

TABLE 3 TECHNICALRESTRICTIONS Assumption

Case 1

Case 2

Power density, W/cm 2 Dye laser Ultraviolet laser

200 4000

free free

2 × 10-3 3 x 10-2

1 1

Efficiency, % Dye laser Ultraviolet laser

TABLE 4 OPTIMUMOPERATINGCONDITIONSANDCONSEQUENTCOSTS Results

Case 1

Case 2

Optimum values Dye laser power density, W/cm 2 Ultraviolet laser power density, kW/cm 2 Atomic density, cm -3 Depth Sj, cm Height S a, cm 3 % uranium production, kg/year Total electricity, MW Construction cost, $/kg Electric power cost, $/kg Maintenance cost, $/kg Material cost, $/kg Unit cost of enriched uranium, $/kg

200 4.00 7-83 x 1013 511 1.02 998 26.0 197 3830 462 888 5380

314 16.7 8.23 x 1013 510 0-998 2470 3.06 132 182 359 395 1070

m e n t i o n e d above. T h e technical restrictions i m p o s e d on the laser energy density a n d efficiency are s u m m a r i s e d in T a b l e 3. T a b l e 4 shows the o p t i m i s e d o p e r a t i n g c o n d i t i o n s o f the laser u r a n i u m e n r i c h m e n t plant. T h e a t o m i c density a n d d e p t h S l a n d height Sa o f the r e a c t i o n region are h a r d l y affected by c h a n g i n g the laser energy density a n d efficiency, a n d the electric p o w e r cost decreases as a result o f the i m p r o v e m e n t o f laser efficiency. T h e o t h e r r e m a r k a b l e result is t h a t the energy density required by the dye laser a n d ultraviolet laser with a n d w i t h o u t the restrictions are n o t a p p r e c i a b l y different. T h e laser energy density shown in T a b l e 4 is p r o b a b l y realised even at the present technical levels, whereas a t t a i n m e n t o f a laser efficiency o f 1 ~o i s v e r y difficult. A high m a i n t e n a n c e cost ensues in case 2 because the lifetime o f the laser is relatively short. Hence, it is i m p o r t a n t to extend the lifetime o f the laser for the r e d u c t i o n o f the unit cost o f enriched u r a n i u m . T h e o p t i m i s e d unit costs o f enriched u r a n i u m are 53805/kg in case 1 a n d 1070 $/kg in case 2. The unit cost o f separative w o r k d o n e by the gaseous diffusion a n d gas centrifugal m e t h o d s is e s t i m a t e d at a b o u t 100 $/kg S W U with a possible cost escalation. S u p p o s i n g t h a t the assay is 0.3 ~o, we need separative w o r k 3.5 kg S W U

304

KIMIO YAMADA, NORIHIKO OZAKI, MANABU YAMAMOTO, KI1CHI UEYANAGI

and 10 kg of feed material for the production of 1.0 kg of 3.0 ~o product. Then the unit cost of enriched uranium including the material cost is approximately 500 $/kg. Compared with the costs of the other uranium enrichment methods, the obtained optimum values are greater by a factor of 11 in case 1 and by a factor of 2 in case 2. However, the optimum value of the unit cost of enriched uranium depends largely on the unit cost of electricity Up, the lifetime z of the laser and the efficiencies of the lasers. The electricity cost of 0.017 S/kWh is probably high and the improvement of the lifetime and efficiency of the laser can reasonably be expected in the near future. It can be concluded from the results of case 2 that laser uranium enrichment is able to compete with the other methods.

5.

CONCLUSION

Optimum operating conditions of the laser uranium enrichment plant, in which the selective two-step photoionisation process of atomic uranium is employed, has been studied. The unit cost of enriched uranium at present is fairly expensive in comparison with those by the gaseous diffusion and gas centrifuge methods. However, we can reasonably say that laser uranium enrichment is competitive with the other uranium enrichment methods provided that the following requirements are satisfied: (1) (2)

the laser efficiency is improved to about one percent. the laser lifetime is extended to about one year.

ACKN OWLEDGEM ENTS

The authors would like to thank Drs K. Taniguchi, S. Yamada and A. Doi of the Atomic Energy Research Laboratory for their constant encouragement throughout the study.

REFERENCES

1. S. A. T v c c I o et al., Two-step selective photoionization of 235U in uranium vapor, 8th International ConJerence on Quantum Electronics, San Francisco, USA (1974). 2. M. J. Box, A new method of constrained optimization and a comparison with other methods, Computer J., 8, 42 (1965). 3. M. MARTEr~SSON,Swedish studies on the economics of uranium enrichment, J. of the British Nuclear Energy Society, 10, 191 (1971), 4. B. M. SMIRNOV and M. 1. CHIBISOV,Resonance charge transfer in inert gases, Soviet Phys.-Tech. Phys., 10, No. 1, 88 (1965).

305

O P E R A T I N G C O N D I T I O N S FOR U R A N I U M E N R I C H M E N T

APPENDIX

The number of 235U ions collected at the ion collectors is given by considering the charge transfer collisions which take place when the ion is extracted from the neutral beam. As there is no excited atom in the collection region, the equations describing the populations of the ionisation and ground states of uranium isotopes as a function of the distance between the ion and ion collector can be written as: dA~ _

ar~(A'sAg5 - AgsAis)

(A.I)

dA~ g i d x -- ar~(AsA5 - AisA~)

(A.2)

dA ~ d x = ar'(A~A~ - AgsAis)

(A.3)

dAg8 d x - av~(A~A~ - AisA~)

(A.4)

dx

where A is the population (subscripts 5 and 8 indicate that the quantity belongs to 235U and 23Su respectively, superscripts g and i indicate ground and ionisation states respectively), and x is the distance between the ion and ion collector. From eqns. (A.I) to (A.4), we have dA~ dA~ dA~ dA~ + -- - - + . dx dx dx dx

dA~ . dx

.

dA~ dA~ . dx dx

dA~ dx

0

(A.5)

The solutions of eqn. (A.5) is given as:

~1~ = A~ + fl

A~ = 6 - A~

(A.6)

where 7, fl, 7 and 6 are constant. Then, we obtain the following equation:

dA~ dx

+ ar~(cc + fl + 27)A~ - crrY(~ + 7) = 0

(A.7)

Eliminating the constants a, fl and 7 from eqn. (A.7) by using the solutions (A.6) and the initial condition that at x = 0, the populations A~ = N~, A~ = N~a,A~ = N~ and A~ = N~, we obtain the solution of the linear differential equation as follows: A~5(x) = N~ol [(N~N~ - N ~ N ~ ) e x p ( - a r ~ N o x

) + (Ug5 + Ns)(N 8 ~ i + Nis)] (A.8)

where N~, N~, N~ and N~ are the ion and atom densities after passing through the reaction region. Assuming that the density distribution of the incident ion beam to the ion

306

KIM10 YAMADA, NORIHIKO OZAKI, MANABU YAMAMOTO, KI1CH1 UEYANAGI

collector electrodes is uniform and the ion velocity occurring from the electric field of the ion collector electrodes is much faster than the thermal velocity of atomic beam, we have the total number of 23su ions: n

.,t5 =

2s,-~VNd f 1

A~(x)dx 0

n

=Y/ ~SvlNdFNNsiN~O-~NgLsNsin

Noar~

( 1 - exp(-or~Nod))+

(Ngs + Nis)(Ni8 + Nis)d]

1

(A.9) where v is the average thermalvelocity of the atomic beam, Nd the number of the ion collector electrodes, d the electrode gap, the summation is based on considering the decrease of laser energy density in the reaction region and n is the number of divisions of the depth S, of the reaction region.