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G.Bardos, V .K.Fedyanin, G.M.Gavrilenko

RADIATION DAMAGE DEPTH

DISTRIBUTIONS IN Al AND Zn INDUCED

ВУ Ar (400 MeV), Ne (200 MeV)

AND С (112.5 MeV) ION BOMBARDMENT

Submitted to "ЖТФ"

1983

One of the most important aspects of research in the field of radiation effects is the determination of the relation ·ьet­ween the defects induced Ьу radiation and the associated changes in physical properties. These changes depend on the spatial distribution of radiation damage. In this paper we determine the depth distribution of the average number of displacements per atom induced Ьу Ar (400 MeV), Ne (200 MeV) and С (112.5 MeV ) ion beams.

The model applied is briefly described in the first parts. The results of our calculations are then presented in the se­cond part of the paper.

1. MODEL

This section deals with the construction of а model describing the average number of displacements per atom, as а function of the penetration depth. The model is based on the fact that the behaviour of the electronic stop~ing cross section differs in the high and low energy regions 11. (In the low energy region contrary to the high energy region the electronic stopping power is linear in the velocity). So, we investigate aF(~ in the two regions of electronic stopping cross section separately:

{

а~(х) at Х ~ X*,which corresponds to Е > Е * ; а F(x) =

а 11 (х) at Х ~ · Х *, which corresponds to Е < Е * energy ' F of ion.

1 . 1 . Regi on I. Е > Е*

In this high energy region the average number of displace­ments per atom is given Ьу integral

а ~(Е) у Е

f da(T,E)N d( T), Ed

(1 )

where Е is the ion energy, N d(T) is the average number of dis­placements produced Ьу а primary of energy Т , da(T,E) is t he d i f ferential cross sect i on for the energy transfer to t he pri ma-

2

У""'·

...

..

ry knock on atom, Ed is the energy required to displace а lat­tice atom, уЕ is the maximum energy which can Ье transferred in the collision, у= 4М 1М 2/ (М 1 + М 2)2 , м 1 , м 2 are the masses of the ion and target atoms, respectively.

The relationship between the incident beam energy and the path length is determined from the following. The formula con­necting the linear path length with its energy is known:

R(E) Е

dE' f--~---0 N[ Sn(E') + Se(E')]

(2)

target atoms per unit volume, Sn(E) where N is the number of and Se(E) are the nuclear respectively 121.

and electronic stopping cross sections,

It is well known that in the high energy region the electro­nic interactions dominate Ьу more than an order of magnitude, and in а first approximation the nuclear interactions can Ье neglected. As electronic straggling is also small, we neglect it too in this approximation.

Consequently, in the high energy region the mean path length is equal to the path length R(E) as well as to the penetration depth х(Е).

From equation (2) we, thus obtain the function Е(х) and further with equation (1) the Frenkel pair cross section for the given penetration distance х.

1.2. Region II. Е < Е*

The nuclear energy straggling of the ion Ъecomes more impor­tant in the low energy regime. This is due to an increase of the nuclear cross sections / 3/ and leads to а range straggling. In this energy region aF(x) is given Ьу '

а~ (х) уЕ(х-х') ,

j f dx'da(E(x'-x),'I)Nd(T) dR(x :х) f(x'), Е - - dx

х d (З)

where f(~ is the fraction of ions in the Ъеаm with projected ranges in the neighbourhood dx of х/4/. It is given Ъу a .Gaus-sian distribution: ·

1 (х- xrJ2 - ехр(- )

-'2rта2 2а2 v х х

f(x) =

In this region the electronic stopping cross section tional to the s~uare root of the eц~rgy and ~given LSS estimates 11

• i OЬ'JJeJ;:~~l"L. • .' i~l иисn..т;т ~, ~ &Aetm.ЫZ [Ю' J!~ДО!'I'ё!fШ';\ t бИБJ ivi:,./Л:::Н ,\

(4)

is propor­here Ьу

3

The mutual continuation of the two regions is determined Ьу the condition

а ~(х *) = а~ (х *) • (5)

2. CALCULATIONS AND RESULTS

The differential projectile-target cross section was chosen to Ье represented Ьу:

а) the Rutherford differential cross section

2 2 М 1 (Z 1 Z 2 е ) d т da = rr - -;:2 •

м 2 Е т

Ь) the Lindhard differential cross section

2 dt . 3/ da = тта -- f(t 2 ) 2 t 3/ 2 . •

(б)

(7)

where t = ( 2Т / у Е, ( = M2t;ai(M 1 + M2)Z 1 Z2e? а= 0.8853 a0(z ~/ З + z;/з)-I /З

Z1 , Z2 are the atomic numbers of the scattered and recoiling particles, respectively, ао is the Bohr radius, е is the electron charge С(t ~)= Лt 11б[1+(2ЛV3 )21 З]-3 / 2, Л=1.309. .

The average number of displacements, Nd was calculated а) Ьу the Kinchin-Pease damage function /б /

Nd

О O< E < Ed

1 Ed < Е < 2Е d

Е/2Е 2Е d < Е < Е 1

Е 1 / 2E;d Е 1 < Е < оо

(8)

i where Е1 =M2fo/l6m, M1,mare the masses of primary knock-on atom 1 and electron, respectively, and fo is the Fermi energy,

Ь) Ьу the modified Kinchin-Pease expression / 7/

к Е l 4 N, = ""2E;J • (9)

~

3

~ -~ о.".'\ ' Ф" .-;:,-":') ,, <~

А' -1 1 б 3 / 4 where к= 0.8, ® = JYI 1 + kg(f) ], g(f) = 3.4008( 1 + 0.40244( + ( ,

1/ 6 1/ 2 2 k = 0.1337 ~1 (Z 1/ M1) , f=M 2~~ /(M 1 +M2,)\z ,Z 2e .

- \ ~ '-J..-

-15' Gr ю

1016

10·17

Al

20 50

~ ..

100

1' v'

150 200 xlpml

Fig.l. Depth distributions of displacements per atom induced Ьу Ar (400 MeV), Ne (200 HeV), С (1 12.5 MeV) ions bombard­

ment in target Al. . The calcu­lations were made Ьу model LМКР.

'1'\

·17 10

бF

Al

1 л !f

1 1

./ · .r"'

.. ~ Q/ ,?~ _."..., '·

с.. ''' _.о_,Р ~· ___ .. -" _ . .....-:. о~ .~ -·-- о RKP

• - RMKP • - LMKP

1 • ..---...",.

L. . 10 50 150 200 l pml х 100

Fig.2. Depth distributions of displacements per atom induced Ьу Ar (400 MeV), Ne (200 MeV), С (1 12.5 ИеV) ion bombardment in target Al.The calculations wer e made Ьу models RКР, RМКР, LМКР.

In calculating the integral (2) the handbook / B/ has been used. FiПures 1-4 show the results of the calculations. In the figures aFKP is the displacements per atom using the Ruther­ford differential cross section (б) and the Kinchin-Pease damage function(8).In the calculations of a~MKPand ajMKPthe modified Kinchin-Pease damage function (9) was used with Rutherford#s and Lindhard#s differential scattering cross sections, respecti­vely.

5

•

..

6F

·15

Zn

10·16

о 10 20 эо 'о 5О 60 Xlpm

Fig.3. Depth distributions of displacement per atom in­duced Ьу Ar (400 MeV), Ne (200 MeV), С (112.5 MeV) ion bombardment in target Zn. The calculations were made Ьу model LМКР.

REFERENCES

1017

GF Zn

о - А КР

" - RMKP •- LMKP

f . . . . . . 10 20 30 LO 50 60 Х lpml

Fig.4. Depth distributions of displacement per atom in­duced Ьу Ar (400 MeV), Ne (200 MeV), С (112.5 MeV) ion bombardment in target Zn. The calculations were made Ьу models RКР, RМКР, LМКР.

1. Lindhard J. et al. Мat.Fys.Medd.Dan.Vid.Selsk., 1963, 33, No.IO.

2. Лейман К. Взаимодействие излучения с твердь~ телом и образо­

вание элементарнъ~ дефектов. Атомиздат, М., 1979. 3. Бор Н. Прохождение атом~ частиц через вещество. ИИЛ, М.,

1950. 4. Мanning I., Mueller G.P. Comp.Phys.Comm., 1974, 7, р.85. 5. Lindhard J., Nielsen V., Scharff М. Mat.Fys.Medd.Dan.

Vid.Selsk., 1968, 36, No.IO. б. Kinchin G.H., Pease R.S. Rep.Progr.Phys., 1955, 18, p.l. 7. Norgett N.·J., Robinson М.Т., Torrens I.M. Nucl.Engin. and

Design, 1975, 33, р.50. 8. Ziegler J.F. Handbook of Stopping Cross Sections of

Energetic Ions in All . Elements. Pergamon Press Inc., 1980, vol.5.

6

Received Ьу PuЬlishing Department on Мау 1 О , 1 983.

•

Бардош д., Федянин В.К., Гавриленка Г.М. El7-83-303 О распределении радиационных дефектов в Af и Zn, образованных при облучении ионами Ar /400 МэБ/, Ne /200 МэВ/, С / 112,5 МэВ/

Рассчитано число выбитых ионов материала Af и Zn,при­ходящихся на единичный поток ионов Ar 1400 МэВ/ , Ne 1200 МэБ/, С / 112,5 МэВ/ в зависимости от глубины их проник­новения в материал.

Работа вьmолнена в Лаборатории теоретической физики ОИЯИ.

Препринт Объединенного института ядерных исследований. Дубна 1983

Bardos G., Fedyanin V.K., Gavrilenko G.H. El7-83-303 Radiation Damage Depth Distributions in Af and Zn Induced Ьу Ar /400 MeV /, Ne /200 MeV / and С /112.5 MeV 1 Ion Bombardment

А method for the approximate calculation of damage depth distributions i nduced Ьу ion bombardment has been de~cribed and results are presented for Ar /400 MeV/, Ne /200 MeV/ and С /112.5 MeV/ ion projects in Af and Zn subs trates.

The inves tigation has been perfor med at the Laboratory of Theoretical Physics, JINR. ,

- ·--·- -- ---

Рг ~rc h. Dubna 1983

1.