- Email: [email protected]
Effect of temperature on transmission of planar channeled positrons in cubic metals containing point defects
Nuclear Instruments and Methods in Physics Research B 267 (2009) 2515–2520
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Effect of temperature on transmission of planar channeled positrons in cubic metals containing point defects M.K. Abu-Assy * Physics Department, Faculty of Science, Suez-Canal University, Ismailia 41522, Egypt
a r t i c l e
i n f o
Article history: Received 2 March 2009 Received in revised form 23 May 2009 Available online 18 June 2009 PACS: 61.85.+p Keywords: Planar channeling in crystals Disorder lattice Point defects Cubic metals
a b s t r a c t Characterization the point defects in cubic metals can be attained by investigation the transmission of planar channeled positively charged particles thorough the regions containing point defects in the metal. A quantum-mechanical calculation for the transmission coefficient and the probability distribution of the transmitted positrons after passing the region containing point defect was performed in the planar channel (1 0 0). The effect of temperature was taken into consideration in the calculations which was made at metal temperatures up to 600 K by using the Debye approximation of lattice thermal vibrations. The calculations showed that the net planar potential at the channel center is slightly higher at higher temperatures and the transmission probability is decreasing with increasing the temperature. The incident energy of the channeled positron was taken in MeV region. The effect of higher anharmonic terms on the planar channeling potential was considered where new bound states was found. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction If a positively charged particle penetrates through a metal in the direction of the major axis or planes, its trajectory will be governed by a corresponding axial or planar potential of the lattice. Lindhard [1] introduced a continuum model approximation for the potential by averaging the potential over a direction parallel to an axis or a plane. The stability of the particle motion in the open channels (axial or planar) of the crystal lattice to move in channeling regime, depends mainly on the incidence angle of the particle beam with respect to the channel direction. This angle must be less than or equal to a definite angle, called critical angle for channeling, and its value depends on the particle incidence energy and on the string or planar atomic density in the case of axial or planar channeling respectively [2]. I.e. on the channeling potential, and for the same incident particle energy and crystal lattice it depends on the channel direction. Therefore, under these conditions for planar channeling, the particle can propagate through the normal lattice between two parallel planes under the influence of a net planar potential transverse to the direction of the particle propagation in the channel. However, if there exist any disorder in the lattice, as that caused by the presence of any point defects such as self interstitials or vacancies, this will alter the planar potential of the channel [3].
* Tel.: +20 12 4177447; fax: +20 64 3226307. E-mail address: [email protected] 0168-583X/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2009.05.065
Ion channeling measurements have been successfully used as a tool for determining the lattice location of dopant and impurity atoms in crystals [2] and to study crystalline defects in compound semiconductor materials [4,5], which strongly influence on the physical properties. In cubic metals there may occur three different types of point defects, namely octahedral (OH), dumb-bell configuration (DBC) and body-centered interstitial (BCI) [6]. This work concentrated on such point defects by using the energetic positrons as planar channeled particles in the metal. In the case of planar channeled positrons, the transverse motion can be described by a one-dimensional Schrödinger wave equation and by solving it one can obtain transverse eigenstates and eigenvalues by using a planar potential function in harmonic oscillator approximation or a planar potential function including harmonic and higher anharmonic terms [7]. Now the planar channeled positrons in the normal region of the lattice have discrete energy states dependent on the planar potential in this region, but in the region containing point defect (disordered region) the corresponding energy states of the channeled positrons will be altered according to the type of the defect. The transmission of the channeled positrons from the normal region into the disordered one depends on a transition-channeling probability for a channeled particle in the normal region to penetrate and still move in the channeling regime into the disordered region in the lattice. The wave function of the channeled positron in the normal and disordered regions respectively was used in the calculation of the transition probability for each case of lattice defect and hence one can calculate the trans-
2516
M.K. Abu-Assy / Nuclear Instruments and Methods in Physics Research B 267 (2009) 2515–2520
mission coefficient and also the probability distribution of the transmitted positrons after leaving the defected region. In this work, a quantum mechanical calculation was presented for the transmission coefficient and the probability distribution of the transmitted positrons through the disordered planar channel (1 0 0) at metal temperatures up to 600 K. Using the Debye approximation for lattice thermal vibrations, the continuum potential experienced by the particles is obtained by averaging over the thermal displacements of the atoms [2]. A planar potential function based on the Moliêre model modified by the effect of lattice thermal vibrations as given by Appleton et al. [8] was used. The effect of anharmonic terms on the planar potential function was taken into consideration by expanding the planar potential function in a series form around x ¼ 0, where x is measured from the midpoint between the two planes. The series includes x2 ; x4 ; x6 ; . . . . . .. terms, so we can consider the higher anharmonic terms like x4 ; x6 ; . . . in the potential function. Here, we considered the higher terms up to x6 term. 2. Quantum mechanical calculation of transmission coefficient In order to execute the calculations of transmission probability and transmission coefficient through the disordered region in cubic metals we assume that the incident positron moves in the xz-plane in the direction of z-axis (the planar direction (1 0 0)) as shown in Fig. 1 for crystals containing a point defect by self interstitial where there are one plane contains point defect in the disordered region. The positron comes from the left side in the normal lattice region (region I, where z < 0) and penetrates through the region contains point defect in the lattice (region II where 0 < z < a), where a, is the disordered region containing the point defect, then transmits into the normal lattice region (region III, where z > a). In this situation, the positron motion is influenced by a transverse planar potential in the direction of the x-axis. By using the appropriate planar potential function in each region we can solve the wave equation of the channeled positron in the three regions respectively to obtain an eigenfunctions describe the spatial distribution of the channeled positron in the transverse x-direction namely, ð1Þ ð2Þ ð3Þ /i ðxÞ; /j ðxÞ and /k ðxÞ together with an eigenvalues of the channeled positrons. We note that the normal lattice regions I and III have the same transverse planar potential function, denoted here by VðxÞ. In this work, an expression for the planar potential at large distance from one plane was used which based on the Moliêre ionatom potential due to one plane modified by the effect of lattice thermal vibration and given by [8]:
VðxÞ ¼
3 X
V i ðxÞ exp u21 =2a2i ;
ð1Þ
i¼1
where
V i ðxÞ ¼ 2pðN:dpÞz1 z2 e2 aTF
ai bi
ebi x=aTF ;
ð2Þ
where N is the bulk density, dp is the interplanar distance, z1 and z2 are the charge numbers of the channeled positron and the crystal 1=2 2=3 2=3 is the Thoatoms respectively, and aTF ¼ 0:8853a0 = z1 þ z2 mas–Fermi screening radius and a0 ¼ 0:529 Å is the Bohr radius and fai g ¼ f0:1; 0:55; 0:35g and fbi g ¼ f6:0; 1:2; 0:3g. Also, u1 is the thermal vibration amplitude and may be estimated from the Debye approximation, indicating that VðxÞ increasing slightly with increasing temperature at large x (at x dp =2). In this work, the transmission of positrons channeled through (1 0 0) direction and around the midpoint between the two planes was considered i.e. at large distance from the plane. Therefore, we used in the calculations the planar potential function as given in (1). Hence, the net planar potential function due to two planes is given as:
VðxÞ ¼ 2pðN:dpÞz1 z2 e2 aTF
3 X ai bi d2p x=aTF bi d2p þx=aTF u1 =2a2i e e þe : b i¼1 i ð3Þ
In previous work [9], the anharmonic effects has been considered in the calculations, by expanding VðxÞ as given in (3) around x ¼ 0, where x is measured from the midpoint between the two planes. We get
1 1 1 1 VðxÞ ¼ V 0 þ k1 x2 þ k2 x4 þ k3 x6 þ k4 x8 þ ; 2 4 6 8
ð4Þ
where 3 X ai bi dp =2aTF u1 =2a2i e e ; bi i¼1 3 X ai bi 2 bi dp =2aTF u1 =2a2i ¼ 2pðNdp Þz1 z2 e2 aTF e e ; bi aTF i¼1 3 X 1 ai bi 4 bi dp =2aTF u1 =2a2i ¼ pðNdp Þz1 z2 e2 aTF e e ; 3 bi aTF i¼1 3 X 1 ai bi 6 bi dp =2aTF u1 =2a2i ¼ pðNdp Þz1 z2 e2 aTF e e ; 60 bi aTF i¼1 3 X 1 ai bi 2n bi dp =2aTF u1 =2a2i ¼ e e ; 2pðNdp Þz1 z2 e2 aTF ð2n 1Þ! bi aTF i¼1
V 0 ¼ 4pðNdp Þz1 z2 e2 aTF k1 k2 k3 kn
where k1 is the force constant of the harmonic oscillator and k2 ; k3 ; . . . represents the force constants of the higher anharmonic oscillators. The eigenfunctions of the channeled positron in harmonic approximation in the three regions are given respectively as [10]: ð1Þ
/i ðxÞ ¼ ð2Þ /j ðxÞ
a
pffiffiffiffi i p2 i!
!1=2 Hi ðaxÞ expða2 x2 =2Þ;
ð5Þ
Hj ða0 xÞ expða02 x2 =2Þ;
ð6Þ
!1=2
a0 ¼ pffiffiffiffi j p2 j!
Table 1 Maximum number of bound states, nmax , for positrons with incident energy 50 MeV move in (1 0 0) direction in normal static lattice and for normal lattice at 600 K of the given cubic metals.
Fig. 1. The transmission of positrons in the planar direction (1 0 0) passing through the region of one plane containing point defect.
Cubic metal
Ni
Cu
Ag
V
Cr
Fe
Static lattice 600 K
11 12
12 12
15 16
7 7
7 7
7 7
2517
M.K. Abu-Assy / Nuclear Instruments and Methods in Physics Research B 267 (2009) 2515–2520
e ¼ 3hk2 =4cm0 x0 k1 and e1 ¼ k3 h 2 =k21 cm0 :
ð11Þ
The calculated values of nmax are given in Table 1 for the normal lattice of the given cubic metals. It was found that the maximum number of bound states nmax , of the channeled positron is increasing as the planar atomic density ðN:dp Þ is increasing and also is slightly increasing with increasing the temperature as shown in Table 1. In addition, we found that the higher anharmonic terms give a new bound states [7]. 2.1. Transition channeling probability between two states in two regions
Fig. 2. Transmission probability as a function of bound state number n for a positron incident energy 50 MeV in the planar direction (1 0 0) of Ag crystal containing point defect by octahedral, dumb-bell or by body centered interstitial at 600 K.
!1=2
ð3Þ
/k ðxÞ ¼
a pffiffiffiffi k p2 k!
Hk ðaxÞ expða2 x2 =2Þ
ð7Þ
For the channeled particle in region I to transmit into region II and still move in channeling regime D E is governed by the value of ð1Þ ð2Þ the overlap integral /i ðxÞj/j ðxÞ and for the channeled particle in region II to transmit into region III and still D move in channeling E ð2Þ ð3Þ regime is governed by the overlap integral /j ðxÞj/k ðxÞ . In this respect the transition channeling probability of the particle with an initial state jii to cross the first interface and to be in the state jji in a region containing point defect can be defined as
pi!j ¼ jhjjiij2
Table 3 Transmission probabilities for bound states n ¼ 0; 1; 2; 3 of static lattice and for lattices at 600 K for different planar atomic densities of the disordered plane ðN:dp Þd .
and the eigenvalues are given by:
1 En ¼ V 0 þ n þ hx0 ; 2
Ag
by substituting k1 by its numerical value in the normal region and 1=2 . in the region containing point defects respectively, x0 ¼ ckm10 The maximum number of bound states ðnmax Þ in harmonic approximation can be calculated by using the equation:
1 1 hx0 ¼ k1 x2max ; nmax þ 2 2
ð9Þ
d
where xmax ¼ 2p aTF . The maximum number of bound states ðnmax Þ by using the higher anharmonic terms in the planar potential (4) is calculated from the equation
1 e þ 2n2max þ 2nmax þ 1 2 4
5e1 3 4nmax þ 6n2max þ 8nmax þ 3 þ 48 1 1 1 ¼ k1 x2max þ k2 x4max þ k3 x6max ; 2 4 6
hx0
nmax þ
ð10Þ
where Table 2 Planar atomic density ðN:dp Þn for normal lattice and for lattices disordered by various types of point defects ðN:dp Þd . Element
Ni Cu Ag V Cr Fe
Planar atomic density, N:dp (defected)
ð8Þ
where a and a0 are constants corresponding to the normal region 1=4 and the region containing point defects and are equal to cmh02k1
2
Planar atomic density ðÅ Þ Normal lattice
Octahedral
Dumb-bell
Body-centered
0.1610 0.1530 0.1198 0.1094 0.1202 0.1217
0.1711 0.1626 0.1273 0.1367 0.1503 0.1521
0.1812 0.1722 0.1348 0.1640 0.1803 0.1825
0.2013 0.1913 0.1497 – – –
ð12Þ
N:dp ðdefectedÞ N:dp ðnormalÞ
0!0
1!1
2!2
3!3
1.06
0.99977 0.99977
0.99931 0.99931
0.99931 0.99885
0.99840 0.99839
1.13
0.99914 0.99913
0.99742 0.99740
0.99570 0.99567
0.99398 0.99395
1.25
0.99690 0.99689
0.99074 0.99071
0.98461 0.98457
0.97852 0.97846
1.25
0.99690 0.99689
0.99073 0.99071
0.98460 0.98457
0.97850 0.97846
1.06
0.99977 0.99977
0.99931 0.99931
0.99885 0.99885
0.99839 0.99839
1.06
0.99977 0.99977
0.99932 0.99931
0.99886 0.99885
0.99841 0.99839
1.13
0.99913 0.99913
0.99740 0.99740
0.99567 0.98567
0.99395 0.97395
1.5
0.98980 0.98977
0.96977 0.96961
0.95013 0.94987
0.93088 0.93052
1.13
0.99913 0.99913
0.99740 0.99740
0.99567 0.99567
0.99395 0.99395
1.5
0.98980 0.98979
0.96971 0.96969
0.95003 0.94999
0.93075 0.93070
1.25
0.99689 0.99689
0.99071 0.99071
0.98457 0.98457
0.97846 0.97846
(OH)
Static 600 K Ag
0.1273
Static 600 K Ag(BC) Static 600 K Cr
0.1348
Static 600 K Cu
0.1503
Static 600 K Ni
0.1626
Static 600 K Cu
0.1711
Static 600 K Cr
0.1722
Static 600 K Ni
0.1803
Static 600 K Fe
0.1812
Static 600 K Cu
0.1825
Static 600 K
0.1913
(DB)
0.1497 (OH)
(OH)
(OH)
(DB)
(DB)
(DB)
(DB)
(BC)
2518
M.K. Abu-Assy / Nuclear Instruments and Methods in Physics Research B 267 (2009) 2515–2520
and the transition channeling probability of the particle with a state jji in the region containing a point defect to cross the second interface of this region and to be in the state jki in the normal region after passing the region containing point defect can be defined as
pj!k ¼ jhkjjij2 :
ð13Þ
Now the transition channeling probability of the particle with an initial state jii in the normal region to cross the region containing point defect and to be in the state jki in the normal region can be defined as
pi!k ¼ pi!j pj!k :
ð14Þ
In previous work [7], it has been found that in harmonic approximation, the general expression of the overlap integral hnjmi ¼ 0 if n – m while at n ¼ m it has the value
Fig. 5. Transmission coefficient for transitions 0 ! 0, 1 ! 1, 2 ! 2 and 3 ! 3 for a positron incident energy 50 MeV at the channel center of the planar direction (1 0 0) of Fe crystal containing point defect by octahedral or by dumb-bell at 600 K.
n!
hnjni ¼
2nþ1=2
nrþ1=2 X n r X Lr b ð1Þk ; ðn rÞ! k!ðr kÞ! r¼0 k¼0
ð15Þ
where
L¼
a2 a02 a2 þ a02
and b ¼
4aa0 a þ a02 2
and in anharmonic approximation up to x6 term in Eq. (4) and to the first order in e and e1 , the overlap integral is given by:
hnjni ¼ Fig. 3. Dependence of transmission probability on the ratio of planar atomic density of the plane containing point defect to that of the normal plane for cubic metals for transitions 0 ! 0, 1 ! 1, 2 ! 2, 3 ! 3 and 7 ! 7 for a positron incident energy 50 MeV in the planar direction (1 0 0).
aa0 p
1=2
In;n þ n!2n
e1 þ e01
½nðn 1Þðn 2Þðn 3Þðn 4Þ 288
4ðe þ e0 Þ þ 3 e1 þ e01 ð2n 3Þ 3=2 I n6;n6 ½nðn 1Þ þ ðn 5Þ 192 n!2n6 3=2 I n4;n4 ðn 2Þðn 3Þ n!2n4
0 8ðe þ e Þð2n 1Þ þ 15 e1 þ e01 ðn2 n þ 1Þ þ 96 In2;n2 ½nðn 1Þ3=2 n!2n2
0 8ðe þ e Þð2n þ 3Þ þ 15 e1 þ e01 ðn2 þ 3n þ 3Þ 96
4ðe þ e0 Þ þ e1 þ e01 ð2n þ 5Þ 1=2 I nþ2;nþ2 ½ðn þ 1Þðn þ 2Þ 192 n!2nþ2 e1 þ e01 1=2 I nþ4;nþ4 ½ðn þ 1Þðn þ 2Þðn þ 3Þðn þ 4Þ 288 n!2nþ4 1=2 I nþ6;nþ6 ; ½ðn þ 1Þðn þ 2Þðn þ 3Þðn þ 4Þðn þ 5Þðn þ 6Þ n!2nþ6
where,
In;n ¼ n!
2p a2 þ a02
1=2 X n r¼0
nrþ1=2 X r Lr b ð1Þk ; ðn rÞ! k¼0 k!ðr kÞ!
therefore the allowed transition-channeling probability are only the n ! n transitions and then can be calculated by using the equation
pn ¼ jhnjnij2 Fig. 4. Transmission coefficient for transitions 0 ! 0, 1 ! 1, 2 ! 2 and 3 ! 3 for a positron incident energy 50 MeV at the channel center of the planar direction (1 0 0) of Cu crystal containing point defect by octahedral, dumb-bell or by body centered interstitial at 600 K.
ð16Þ
and the probability of transmission through the region containing point defect for the n ! n transition is therefore, p2n .
M.K. Abu-Assy / Nuclear Instruments and Methods in Physics Research B 267 (2009) 2515–2520
2.2. Transmission coefficient through a region containing point defects Assume that the incident particle has an energy E and move in the planar direction (1 0 0) and that the transverse potential is VðxÞ in the normal lattice region and V 0 ðxÞ is that in the region containing point defect. 2.2.1. If E > V 0 ðxÞ and E > VðxÞ Now the general solution of the wave function describing the channeled particle in the three regions can be expressed thus ð1Þ
ð1Þ
wi ðz; xÞ ¼ ðA expðikzÞ þ B expðikzÞÞ/i ðxÞ z < 0; ð2Þ wj ðz; xÞ ð3Þ wk ðz; xÞ
0
0
ð2Þ
¼ ðC expðik zÞ þ D expðik zÞÞ/j ðxÞ 0 < z < a; ¼F
ð3Þ expðikzÞ/k ðxÞ
z > a;
ð17Þ ð18Þ ð19Þ
where
h i1=2 2 k ¼ 2m0 cðE VðxÞÞ=h
h i1=2 0 2 and k ¼ 2m0 cðE V 0 ðxÞÞ=h : ð20Þ
By applying the boundary conditions at, z ¼ 0, on the wave funcð1Þ ð2Þ tions wi ðz; xÞ and wj ðz; xÞ and on their first derivatives w.r.t. z, ð2Þ ð3Þ and on the wave functions wj ðz; xÞ and wk ðz; xÞ and on their first derivatives w.r.t. z at z ¼ a we can obtain the corresponding transmission coefficient as:
T¼
" #1 2 02 FF ðk k Þ2 2 0 ¼ 1 þ sin ðk aÞ p2n : 2 02 AA 4k k
ð21Þ
Evidently, the net planar potential in the normal region has a minimum value at the midpoint between the two planes but in the region containing point defect the minimum value is shifted slightly far from the plane containing point defect. However in 0 the two cases, the wave numbers k and k as given in Eq. (20) are functions of the incident beam position x, therefore one can investigate the dependence of the transmission coefficient T on the incident beam position in the channel. 2.2.2. If E < VðxÞ and E < V 0 ðxÞ In this case we can solve the problem by substituting k in Eqs. from (17) to (20) by ik, and following the same procedure given in Section 2.2.1. Then, the corresponding transmission coefficient can be found to be 2 02
0
0
0
0
T ¼ 16k k ½ðk þ k Þ2 eðkk Þa ðk k Þ2 eðkþk Þa 2 p2n ;
2519
mission coefficient through the regions containing point defect. The calculations was performed for 50 MeV positrons in Ni, Cu, Ag, V, Cr and Fe for static lattice and for lattice at temperatures up to 600 K. The variation of transmission probability with transition state number is shown in Fig. 2 for Ag crystal containing point defects by octahedral, dumb-bell and body centered interstitial at 600 K. To indicate the dependence of transmission probability on the planar atomic density of the normal region ðN:dp Þn and that of the region containing point defect ðN:dp Þd , the planar atomic density of the normal lattice and for lattice disordered by various point defects are given in Table 2. From Table 2 it is found that the transmission probability for different bound states for all types of point defects in all metals does not depend on the planar atomic density of the plane containing point defect in the crystal lattice, but does depend on the ratio of the planar atomic density of the ðN:d Þ plane containing point defect to that of the normal one, ðN:dpp Þd . n The calculations of transmission probability for all allowed transitions has been done and that for bound states n ¼ 0; 1; 2; 3 for different planar atomic densities of one plane containing point defect has been given in Table 3 and was represented graphically in Fig. 3, for the given cubic metals for the transitions 0 ! 0, 1 ! 1, 2 ! 2, 3 ! 3 and 7 ! 7. Form the calculations given in Table 3 one can conclude that at higher temperatures the transmission probability is decreasing slightly, i.e. p2n ðat temperatureÞ 6 p2n ðstaticÞ for all metals and for various types of defects. The calculations showed that, net planar potential around the channel center is slightly higher than its value at lower temperatures. The transmission coefficient, T, for positrons penetrating through the channel center of the planar direction (1 0 0) for Cu crystal containing point defects by octahedral, dumb-bell and by body centered interstitial at 600 K was illustrated in Fig. 4 for transitions 0 ! 0, 1 ! 1, 2 ! 2 and 3 ! 3, and for Fe crystal containing point defect by octahedral and by dumb-bell configuration at 600 K was illustrated in Fig. 5. The probability distributions of the transmitted positrons, j/n ðxÞj2 T through Cu crystal containing point defect by BCI for the transition states 0 ! 0, 1 ! 1, 2 ! 2 and 3 ! 3 for a positron incident energy 50 MeV in the planar direction (1 0 0) at 600 K, where x is the distance from the disordered plane, was illustrated in Fig. 6.
ð22Þ
where
h i1=2 h i1=2 2 0 2 k ¼ 2m0 cðVðxÞ EÞ=h and k ¼ 2m0 cðV 0 ðxÞ EÞ= h : ð23Þ 3. Conclusions and computational results The transmission of planar channeled positron in cubic metals containing point defects has been investigated for an energetic beam of positrons in MeV region. The calculation was performed in the planar channel (1 0 0) that assumed to contain point defects by forming the configurations OH, DBC and BCI. It is well known that, the channeled positron still move in channeling regime in the normal region of crystal lattice until it encounters any type of disorder, like point defects. In this case, it may transmit through the disordered region ‘a’ containing the point defect and move in channeling regime with definite transition probability. The transmission through various types of point defects was estimated in this work by calculating the transmission probability and trans-
Fig. 6. Probability distribution of transmitted positrons, j/n ðxÞj2 T through Cu crystal containing point defect by BCI for transitions 0 ! 0, 1 ! 1, 2 ! 2 and 3 ! 3 for a positron incident energy 50 MeV in the planar direction (1 0 0) at 600 K, x is the distance from the disordered plane.
2520
M.K. Abu-Assy / Nuclear Instruments and Methods in Physics Research B 267 (2009) 2515–2520
tion and represented in Fig. 7 for Cu crystal containing point defects by octahedral, dumb-bell or by body centered interstitial at 600 K. References
Fig. 7. Transmission probability as a function of bound state number n for positron incident energy 50 MeV in the planar direction (1 0 0) of Cu crystal containing point defects by octahedral, dumb-bell or by body centered interstitial at 600 K by using anharmonic approximation.
The variation of transmission probability with bound state number was investigated by applying the anharmonic approxima-
[1] J. Lindhard, Influence of crystal lattice on motion of energetic charged particle, Mat. Fys. Medd. Dan. Vid. Selsk. 34 (14) (1965). [2] D.S. Gemmel, Channeling and related effects in the motion of charged particles through crystals, Rev. Mod. Phys. 46 (1974) 129. [3] M.K. Abu-Assy, The effect of lattice disorder on the channeling potential in fcc single crystals, Physica Scripta 65 (2002) 108. [4] A. Seppälä, R. Salonen, J. Slotte, T. Ahlgren, E. Rauhala, J. Räisänen, Lattice sites of diffused gold and platinum in epitaxial ZnSe layers, Nucl. Instr. and Meth. B 161–163 (2000) 520. [5] A. Seppälä, E. Rauhala, R. Grötzschel, Lattice location of implanted tellurium in GaN heteroepitaxial films, J. Phys. D 34 (2001) 269. [6] R.A. Johnson, A.N. Orlov (Eds.), Physics of Radiation Effects in Crystals, Elsevier Science Publishers B.V., 1968. [7] M.K. Abu-Assy, M.Y. El-Ashry, A.A. Mohamed, Dechanneling of positrons in disordered lattices – effect of anharmonic potential, Physica Scripta 72 (2005) 68. [8] B.R. Appleton, C. Erginsoy, M.W. Gibson, Channeling effects in the energy loss of 3–11 MeV protons in silicon and germanium single crystals, Phys. Rev. 161 (1967) 330. [9] M.K. Abu-Assy, E. Shalaan, A.H. Abdelaziz, Effect of lattice thermal vibrations on dechanneling of Positrons in disordered cubic metals, Arab J. Nucl. Sci. Appl. 42 (2) (in press). [10] L.I. Schiff, Quantum Mechanics, Mc Graw-Hill, New York, 1968.
Contents lists available at ScienceDirect
Nuclear Instruments and Methods in Physics Research B journal homepage: www.elsevier.com/locate/nimb
Effect of temperature on transmission of planar channeled positrons in cubic metals containing point defects M.K. Abu-Assy * Physics Department, Faculty of Science, Suez-Canal University, Ismailia 41522, Egypt
a r t i c l e
i n f o
Article history: Received 2 March 2009 Received in revised form 23 May 2009 Available online 18 June 2009 PACS: 61.85.+p Keywords: Planar channeling in crystals Disorder lattice Point defects Cubic metals
a b s t r a c t Characterization the point defects in cubic metals can be attained by investigation the transmission of planar channeled positively charged particles thorough the regions containing point defects in the metal. A quantum-mechanical calculation for the transmission coefficient and the probability distribution of the transmitted positrons after passing the region containing point defect was performed in the planar channel (1 0 0). The effect of temperature was taken into consideration in the calculations which was made at metal temperatures up to 600 K by using the Debye approximation of lattice thermal vibrations. The calculations showed that the net planar potential at the channel center is slightly higher at higher temperatures and the transmission probability is decreasing with increasing the temperature. The incident energy of the channeled positron was taken in MeV region. The effect of higher anharmonic terms on the planar channeling potential was considered where new bound states was found. Ó 2009 Elsevier B.V. All rights reserved.
1. Introduction If a positively charged particle penetrates through a metal in the direction of the major axis or planes, its trajectory will be governed by a corresponding axial or planar potential of the lattice. Lindhard [1] introduced a continuum model approximation for the potential by averaging the potential over a direction parallel to an axis or a plane. The stability of the particle motion in the open channels (axial or planar) of the crystal lattice to move in channeling regime, depends mainly on the incidence angle of the particle beam with respect to the channel direction. This angle must be less than or equal to a definite angle, called critical angle for channeling, and its value depends on the particle incidence energy and on the string or planar atomic density in the case of axial or planar channeling respectively [2]. I.e. on the channeling potential, and for the same incident particle energy and crystal lattice it depends on the channel direction. Therefore, under these conditions for planar channeling, the particle can propagate through the normal lattice between two parallel planes under the influence of a net planar potential transverse to the direction of the particle propagation in the channel. However, if there exist any disorder in the lattice, as that caused by the presence of any point defects such as self interstitials or vacancies, this will alter the planar potential of the channel [3].
* Tel.: +20 12 4177447; fax: +20 64 3226307. E-mail address: [email protected] 0168-583X/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.nimb.2009.05.065
Ion channeling measurements have been successfully used as a tool for determining the lattice location of dopant and impurity atoms in crystals [2] and to study crystalline defects in compound semiconductor materials [4,5], which strongly influence on the physical properties. In cubic metals there may occur three different types of point defects, namely octahedral (OH), dumb-bell configuration (DBC) and body-centered interstitial (BCI) [6]. This work concentrated on such point defects by using the energetic positrons as planar channeled particles in the metal. In the case of planar channeled positrons, the transverse motion can be described by a one-dimensional Schrödinger wave equation and by solving it one can obtain transverse eigenstates and eigenvalues by using a planar potential function in harmonic oscillator approximation or a planar potential function including harmonic and higher anharmonic terms [7]. Now the planar channeled positrons in the normal region of the lattice have discrete energy states dependent on the planar potential in this region, but in the region containing point defect (disordered region) the corresponding energy states of the channeled positrons will be altered according to the type of the defect. The transmission of the channeled positrons from the normal region into the disordered one depends on a transition-channeling probability for a channeled particle in the normal region to penetrate and still move in the channeling regime into the disordered region in the lattice. The wave function of the channeled positron in the normal and disordered regions respectively was used in the calculation of the transition probability for each case of lattice defect and hence one can calculate the trans-
2516
M.K. Abu-Assy / Nuclear Instruments and Methods in Physics Research B 267 (2009) 2515–2520
mission coefficient and also the probability distribution of the transmitted positrons after leaving the defected region. In this work, a quantum mechanical calculation was presented for the transmission coefficient and the probability distribution of the transmitted positrons through the disordered planar channel (1 0 0) at metal temperatures up to 600 K. Using the Debye approximation for lattice thermal vibrations, the continuum potential experienced by the particles is obtained by averaging over the thermal displacements of the atoms [2]. A planar potential function based on the Moliêre model modified by the effect of lattice thermal vibrations as given by Appleton et al. [8] was used. The effect of anharmonic terms on the planar potential function was taken into consideration by expanding the planar potential function in a series form around x ¼ 0, where x is measured from the midpoint between the two planes. The series includes x2 ; x4 ; x6 ; . . . . . .. terms, so we can consider the higher anharmonic terms like x4 ; x6 ; . . . in the potential function. Here, we considered the higher terms up to x6 term. 2. Quantum mechanical calculation of transmission coefficient In order to execute the calculations of transmission probability and transmission coefficient through the disordered region in cubic metals we assume that the incident positron moves in the xz-plane in the direction of z-axis (the planar direction (1 0 0)) as shown in Fig. 1 for crystals containing a point defect by self interstitial where there are one plane contains point defect in the disordered region. The positron comes from the left side in the normal lattice region (region I, where z < 0) and penetrates through the region contains point defect in the lattice (region II where 0 < z < a), where a, is the disordered region containing the point defect, then transmits into the normal lattice region (region III, where z > a). In this situation, the positron motion is influenced by a transverse planar potential in the direction of the x-axis. By using the appropriate planar potential function in each region we can solve the wave equation of the channeled positron in the three regions respectively to obtain an eigenfunctions describe the spatial distribution of the channeled positron in the transverse x-direction namely, ð1Þ ð2Þ ð3Þ /i ðxÞ; /j ðxÞ and /k ðxÞ together with an eigenvalues of the channeled positrons. We note that the normal lattice regions I and III have the same transverse planar potential function, denoted here by VðxÞ. In this work, an expression for the planar potential at large distance from one plane was used which based on the Moliêre ionatom potential due to one plane modified by the effect of lattice thermal vibration and given by [8]:
VðxÞ ¼
3 X
V i ðxÞ exp u21 =2a2i ;
ð1Þ
i¼1
where
V i ðxÞ ¼ 2pðN:dpÞz1 z2 e2 aTF
ai bi
ebi x=aTF ;
ð2Þ
where N is the bulk density, dp is the interplanar distance, z1 and z2 are the charge numbers of the channeled positron and the crystal 1=2 2=3 2=3 is the Thoatoms respectively, and aTF ¼ 0:8853a0 = z1 þ z2 mas–Fermi screening radius and a0 ¼ 0:529 Å is the Bohr radius and fai g ¼ f0:1; 0:55; 0:35g and fbi g ¼ f6:0; 1:2; 0:3g. Also, u1 is the thermal vibration amplitude and may be estimated from the Debye approximation, indicating that VðxÞ increasing slightly with increasing temperature at large x (at x dp =2). In this work, the transmission of positrons channeled through (1 0 0) direction and around the midpoint between the two planes was considered i.e. at large distance from the plane. Therefore, we used in the calculations the planar potential function as given in (1). Hence, the net planar potential function due to two planes is given as:
VðxÞ ¼ 2pðN:dpÞz1 z2 e2 aTF
3 X ai bi d2p x=aTF bi d2p þx=aTF u1 =2a2i e e þe : b i¼1 i ð3Þ
In previous work [9], the anharmonic effects has been considered in the calculations, by expanding VðxÞ as given in (3) around x ¼ 0, where x is measured from the midpoint between the two planes. We get
1 1 1 1 VðxÞ ¼ V 0 þ k1 x2 þ k2 x4 þ k3 x6 þ k4 x8 þ ; 2 4 6 8
ð4Þ
where 3 X ai bi dp =2aTF u1 =2a2i e e ; bi i¼1 3 X ai bi 2 bi dp =2aTF u1 =2a2i ¼ 2pðNdp Þz1 z2 e2 aTF e e ; bi aTF i¼1 3 X 1 ai bi 4 bi dp =2aTF u1 =2a2i ¼ pðNdp Þz1 z2 e2 aTF e e ; 3 bi aTF i¼1 3 X 1 ai bi 6 bi dp =2aTF u1 =2a2i ¼ pðNdp Þz1 z2 e2 aTF e e ; 60 bi aTF i¼1 3 X 1 ai bi 2n bi dp =2aTF u1 =2a2i ¼ e e ; 2pðNdp Þz1 z2 e2 aTF ð2n 1Þ! bi aTF i¼1
V 0 ¼ 4pðNdp Þz1 z2 e2 aTF k1 k2 k3 kn
where k1 is the force constant of the harmonic oscillator and k2 ; k3 ; . . . represents the force constants of the higher anharmonic oscillators. The eigenfunctions of the channeled positron in harmonic approximation in the three regions are given respectively as [10]: ð1Þ
/i ðxÞ ¼ ð2Þ /j ðxÞ
a
pffiffiffiffi i p2 i!
!1=2 Hi ðaxÞ expða2 x2 =2Þ;
ð5Þ
Hj ða0 xÞ expða02 x2 =2Þ;
ð6Þ
!1=2
a0 ¼ pffiffiffiffi j p2 j!
Table 1 Maximum number of bound states, nmax , for positrons with incident energy 50 MeV move in (1 0 0) direction in normal static lattice and for normal lattice at 600 K of the given cubic metals.
Fig. 1. The transmission of positrons in the planar direction (1 0 0) passing through the region of one plane containing point defect.
Cubic metal
Ni
Cu
Ag
V
Cr
Fe
Static lattice 600 K
11 12
12 12
15 16
7 7
7 7
7 7
2517
M.K. Abu-Assy / Nuclear Instruments and Methods in Physics Research B 267 (2009) 2515–2520
e ¼ 3hk2 =4cm0 x0 k1 and e1 ¼ k3 h 2 =k21 cm0 :
ð11Þ
The calculated values of nmax are given in Table 1 for the normal lattice of the given cubic metals. It was found that the maximum number of bound states nmax , of the channeled positron is increasing as the planar atomic density ðN:dp Þ is increasing and also is slightly increasing with increasing the temperature as shown in Table 1. In addition, we found that the higher anharmonic terms give a new bound states [7]. 2.1. Transition channeling probability between two states in two regions
Fig. 2. Transmission probability as a function of bound state number n for a positron incident energy 50 MeV in the planar direction (1 0 0) of Ag crystal containing point defect by octahedral, dumb-bell or by body centered interstitial at 600 K.
!1=2
ð3Þ
/k ðxÞ ¼
a pffiffiffiffi k p2 k!
Hk ðaxÞ expða2 x2 =2Þ
ð7Þ
For the channeled particle in region I to transmit into region II and still move in channeling regime D E is governed by the value of ð1Þ ð2Þ the overlap integral /i ðxÞj/j ðxÞ and for the channeled particle in region II to transmit into region III and still D move in channeling E ð2Þ ð3Þ regime is governed by the overlap integral /j ðxÞj/k ðxÞ . In this respect the transition channeling probability of the particle with an initial state jii to cross the first interface and to be in the state jji in a region containing point defect can be defined as
pi!j ¼ jhjjiij2
Table 3 Transmission probabilities for bound states n ¼ 0; 1; 2; 3 of static lattice and for lattices at 600 K for different planar atomic densities of the disordered plane ðN:dp Þd .
and the eigenvalues are given by:
1 En ¼ V 0 þ n þ hx0 ; 2
Ag
by substituting k1 by its numerical value in the normal region and 1=2 . in the region containing point defects respectively, x0 ¼ ckm10 The maximum number of bound states ðnmax Þ in harmonic approximation can be calculated by using the equation:
1 1 hx0 ¼ k1 x2max ; nmax þ 2 2
ð9Þ
d
where xmax ¼ 2p aTF . The maximum number of bound states ðnmax Þ by using the higher anharmonic terms in the planar potential (4) is calculated from the equation
1 e þ 2n2max þ 2nmax þ 1 2 4
5e1 3 4nmax þ 6n2max þ 8nmax þ 3 þ 48 1 1 1 ¼ k1 x2max þ k2 x4max þ k3 x6max ; 2 4 6
hx0
nmax þ
ð10Þ
where Table 2 Planar atomic density ðN:dp Þn for normal lattice and for lattices disordered by various types of point defects ðN:dp Þd . Element
Ni Cu Ag V Cr Fe
Planar atomic density, N:dp (defected)
ð8Þ
where a and a0 are constants corresponding to the normal region 1=4 and the region containing point defects and are equal to cmh02k1
2
Planar atomic density ðÅ Þ Normal lattice
Octahedral
Dumb-bell
Body-centered
0.1610 0.1530 0.1198 0.1094 0.1202 0.1217
0.1711 0.1626 0.1273 0.1367 0.1503 0.1521
0.1812 0.1722 0.1348 0.1640 0.1803 0.1825
0.2013 0.1913 0.1497 – – –
ð12Þ
N:dp ðdefectedÞ N:dp ðnormalÞ
0!0
1!1
2!2
3!3
1.06
0.99977 0.99977
0.99931 0.99931
0.99931 0.99885
0.99840 0.99839
1.13
0.99914 0.99913
0.99742 0.99740
0.99570 0.99567
0.99398 0.99395
1.25
0.99690 0.99689
0.99074 0.99071
0.98461 0.98457
0.97852 0.97846
1.25
0.99690 0.99689
0.99073 0.99071
0.98460 0.98457
0.97850 0.97846
1.06
0.99977 0.99977
0.99931 0.99931
0.99885 0.99885
0.99839 0.99839
1.06
0.99977 0.99977
0.99932 0.99931
0.99886 0.99885
0.99841 0.99839
1.13
0.99913 0.99913
0.99740 0.99740
0.99567 0.98567
0.99395 0.97395
1.5
0.98980 0.98977
0.96977 0.96961
0.95013 0.94987
0.93088 0.93052
1.13
0.99913 0.99913
0.99740 0.99740
0.99567 0.99567
0.99395 0.99395
1.5
0.98980 0.98979
0.96971 0.96969
0.95003 0.94999
0.93075 0.93070
1.25
0.99689 0.99689
0.99071 0.99071
0.98457 0.98457
0.97846 0.97846
(OH)
Static 600 K Ag
0.1273
Static 600 K Ag(BC) Static 600 K Cr
0.1348
Static 600 K Cu
0.1503
Static 600 K Ni
0.1626
Static 600 K Cu
0.1711
Static 600 K Cr
0.1722
Static 600 K Ni
0.1803
Static 600 K Fe
0.1812
Static 600 K Cu
0.1825
Static 600 K
0.1913
(DB)
0.1497 (OH)
(OH)
(OH)
(DB)
(DB)
(DB)
(DB)
(BC)
2518
M.K. Abu-Assy / Nuclear Instruments and Methods in Physics Research B 267 (2009) 2515–2520
and the transition channeling probability of the particle with a state jji in the region containing a point defect to cross the second interface of this region and to be in the state jki in the normal region after passing the region containing point defect can be defined as
pj!k ¼ jhkjjij2 :
ð13Þ
Now the transition channeling probability of the particle with an initial state jii in the normal region to cross the region containing point defect and to be in the state jki in the normal region can be defined as
pi!k ¼ pi!j pj!k :
ð14Þ
In previous work [7], it has been found that in harmonic approximation, the general expression of the overlap integral hnjmi ¼ 0 if n – m while at n ¼ m it has the value
Fig. 5. Transmission coefficient for transitions 0 ! 0, 1 ! 1, 2 ! 2 and 3 ! 3 for a positron incident energy 50 MeV at the channel center of the planar direction (1 0 0) of Fe crystal containing point defect by octahedral or by dumb-bell at 600 K.
n!
hnjni ¼
2nþ1=2
nrþ1=2 X n r X Lr b ð1Þk ; ðn rÞ! k!ðr kÞ! r¼0 k¼0
ð15Þ
where
L¼
a2 a02 a2 þ a02
and b ¼
4aa0 a þ a02 2
and in anharmonic approximation up to x6 term in Eq. (4) and to the first order in e and e1 , the overlap integral is given by:
hnjni ¼ Fig. 3. Dependence of transmission probability on the ratio of planar atomic density of the plane containing point defect to that of the normal plane for cubic metals for transitions 0 ! 0, 1 ! 1, 2 ! 2, 3 ! 3 and 7 ! 7 for a positron incident energy 50 MeV in the planar direction (1 0 0).
aa0 p
1=2
In;n þ n!2n
e1 þ e01
½nðn 1Þðn 2Þðn 3Þðn 4Þ 288
4ðe þ e0 Þ þ 3 e1 þ e01 ð2n 3Þ 3=2 I n6;n6 ½nðn 1Þ þ ðn 5Þ 192 n!2n6 3=2 I n4;n4 ðn 2Þðn 3Þ n!2n4
0 8ðe þ e Þð2n 1Þ þ 15 e1 þ e01 ðn2 n þ 1Þ þ 96 In2;n2 ½nðn 1Þ3=2 n!2n2
0 8ðe þ e Þð2n þ 3Þ þ 15 e1 þ e01 ðn2 þ 3n þ 3Þ 96
4ðe þ e0 Þ þ e1 þ e01 ð2n þ 5Þ 1=2 I nþ2;nþ2 ½ðn þ 1Þðn þ 2Þ 192 n!2nþ2 e1 þ e01 1=2 I nþ4;nþ4 ½ðn þ 1Þðn þ 2Þðn þ 3Þðn þ 4Þ 288 n!2nþ4 1=2 I nþ6;nþ6 ; ½ðn þ 1Þðn þ 2Þðn þ 3Þðn þ 4Þðn þ 5Þðn þ 6Þ n!2nþ6
where,
In;n ¼ n!
2p a2 þ a02
1=2 X n r¼0
nrþ1=2 X r Lr b ð1Þk ; ðn rÞ! k¼0 k!ðr kÞ!
therefore the allowed transition-channeling probability are only the n ! n transitions and then can be calculated by using the equation
pn ¼ jhnjnij2 Fig. 4. Transmission coefficient for transitions 0 ! 0, 1 ! 1, 2 ! 2 and 3 ! 3 for a positron incident energy 50 MeV at the channel center of the planar direction (1 0 0) of Cu crystal containing point defect by octahedral, dumb-bell or by body centered interstitial at 600 K.
ð16Þ
and the probability of transmission through the region containing point defect for the n ! n transition is therefore, p2n .
M.K. Abu-Assy / Nuclear Instruments and Methods in Physics Research B 267 (2009) 2515–2520
2.2. Transmission coefficient through a region containing point defects Assume that the incident particle has an energy E and move in the planar direction (1 0 0) and that the transverse potential is VðxÞ in the normal lattice region and V 0 ðxÞ is that in the region containing point defect. 2.2.1. If E > V 0 ðxÞ and E > VðxÞ Now the general solution of the wave function describing the channeled particle in the three regions can be expressed thus ð1Þ
ð1Þ
wi ðz; xÞ ¼ ðA expðikzÞ þ B expðikzÞÞ/i ðxÞ z < 0; ð2Þ wj ðz; xÞ ð3Þ wk ðz; xÞ
0
0
ð2Þ
¼ ðC expðik zÞ þ D expðik zÞÞ/j ðxÞ 0 < z < a; ¼F
ð3Þ expðikzÞ/k ðxÞ
z > a;
ð17Þ ð18Þ ð19Þ
where
h i1=2 2 k ¼ 2m0 cðE VðxÞÞ=h
h i1=2 0 2 and k ¼ 2m0 cðE V 0 ðxÞÞ=h : ð20Þ
By applying the boundary conditions at, z ¼ 0, on the wave funcð1Þ ð2Þ tions wi ðz; xÞ and wj ðz; xÞ and on their first derivatives w.r.t. z, ð2Þ ð3Þ and on the wave functions wj ðz; xÞ and wk ðz; xÞ and on their first derivatives w.r.t. z at z ¼ a we can obtain the corresponding transmission coefficient as:
T¼
" #1 2 02 FF ðk k Þ2 2 0 ¼ 1 þ sin ðk aÞ p2n : 2 02 AA 4k k
ð21Þ
Evidently, the net planar potential in the normal region has a minimum value at the midpoint between the two planes but in the region containing point defect the minimum value is shifted slightly far from the plane containing point defect. However in 0 the two cases, the wave numbers k and k as given in Eq. (20) are functions of the incident beam position x, therefore one can investigate the dependence of the transmission coefficient T on the incident beam position in the channel. 2.2.2. If E < VðxÞ and E < V 0 ðxÞ In this case we can solve the problem by substituting k in Eqs. from (17) to (20) by ik, and following the same procedure given in Section 2.2.1. Then, the corresponding transmission coefficient can be found to be 2 02
0
0
0
0
T ¼ 16k k ½ðk þ k Þ2 eðkk Þa ðk k Þ2 eðkþk Þa 2 p2n ;
2519
mission coefficient through the regions containing point defect. The calculations was performed for 50 MeV positrons in Ni, Cu, Ag, V, Cr and Fe for static lattice and for lattice at temperatures up to 600 K. The variation of transmission probability with transition state number is shown in Fig. 2 for Ag crystal containing point defects by octahedral, dumb-bell and body centered interstitial at 600 K. To indicate the dependence of transmission probability on the planar atomic density of the normal region ðN:dp Þn and that of the region containing point defect ðN:dp Þd , the planar atomic density of the normal lattice and for lattice disordered by various point defects are given in Table 2. From Table 2 it is found that the transmission probability for different bound states for all types of point defects in all metals does not depend on the planar atomic density of the plane containing point defect in the crystal lattice, but does depend on the ratio of the planar atomic density of the ðN:d Þ plane containing point defect to that of the normal one, ðN:dpp Þd . n The calculations of transmission probability for all allowed transitions has been done and that for bound states n ¼ 0; 1; 2; 3 for different planar atomic densities of one plane containing point defect has been given in Table 3 and was represented graphically in Fig. 3, for the given cubic metals for the transitions 0 ! 0, 1 ! 1, 2 ! 2, 3 ! 3 and 7 ! 7. Form the calculations given in Table 3 one can conclude that at higher temperatures the transmission probability is decreasing slightly, i.e. p2n ðat temperatureÞ 6 p2n ðstaticÞ for all metals and for various types of defects. The calculations showed that, net planar potential around the channel center is slightly higher than its value at lower temperatures. The transmission coefficient, T, for positrons penetrating through the channel center of the planar direction (1 0 0) for Cu crystal containing point defects by octahedral, dumb-bell and by body centered interstitial at 600 K was illustrated in Fig. 4 for transitions 0 ! 0, 1 ! 1, 2 ! 2 and 3 ! 3, and for Fe crystal containing point defect by octahedral and by dumb-bell configuration at 600 K was illustrated in Fig. 5. The probability distributions of the transmitted positrons, j/n ðxÞj2 T through Cu crystal containing point defect by BCI for the transition states 0 ! 0, 1 ! 1, 2 ! 2 and 3 ! 3 for a positron incident energy 50 MeV in the planar direction (1 0 0) at 600 K, where x is the distance from the disordered plane, was illustrated in Fig. 6.
ð22Þ
where
h i1=2 h i1=2 2 0 2 k ¼ 2m0 cðVðxÞ EÞ=h and k ¼ 2m0 cðV 0 ðxÞ EÞ= h : ð23Þ 3. Conclusions and computational results The transmission of planar channeled positron in cubic metals containing point defects has been investigated for an energetic beam of positrons in MeV region. The calculation was performed in the planar channel (1 0 0) that assumed to contain point defects by forming the configurations OH, DBC and BCI. It is well known that, the channeled positron still move in channeling regime in the normal region of crystal lattice until it encounters any type of disorder, like point defects. In this case, it may transmit through the disordered region ‘a’ containing the point defect and move in channeling regime with definite transition probability. The transmission through various types of point defects was estimated in this work by calculating the transmission probability and trans-
Fig. 6. Probability distribution of transmitted positrons, j/n ðxÞj2 T through Cu crystal containing point defect by BCI for transitions 0 ! 0, 1 ! 1, 2 ! 2 and 3 ! 3 for a positron incident energy 50 MeV in the planar direction (1 0 0) at 600 K, x is the distance from the disordered plane.
2520
M.K. Abu-Assy / Nuclear Instruments and Methods in Physics Research B 267 (2009) 2515–2520
tion and represented in Fig. 7 for Cu crystal containing point defects by octahedral, dumb-bell or by body centered interstitial at 600 K. References
Fig. 7. Transmission probability as a function of bound state number n for positron incident energy 50 MeV in the planar direction (1 0 0) of Cu crystal containing point defects by octahedral, dumb-bell or by body centered interstitial at 600 K by using anharmonic approximation.
The variation of transmission probability with bound state number was investigated by applying the anharmonic approxima-
[1] J. Lindhard, Influence of crystal lattice on motion of energetic charged particle, Mat. Fys. Medd. Dan. Vid. Selsk. 34 (14) (1965). [2] D.S. Gemmel, Channeling and related effects in the motion of charged particles through crystals, Rev. Mod. Phys. 46 (1974) 129. [3] M.K. Abu-Assy, The effect of lattice disorder on the channeling potential in fcc single crystals, Physica Scripta 65 (2002) 108. [4] A. Seppälä, R. Salonen, J. Slotte, T. Ahlgren, E. Rauhala, J. Räisänen, Lattice sites of diffused gold and platinum in epitaxial ZnSe layers, Nucl. Instr. and Meth. B 161–163 (2000) 520. [5] A. Seppälä, E. Rauhala, R. Grötzschel, Lattice location of implanted tellurium in GaN heteroepitaxial films, J. Phys. D 34 (2001) 269. [6] R.A. Johnson, A.N. Orlov (Eds.), Physics of Radiation Effects in Crystals, Elsevier Science Publishers B.V., 1968. [7] M.K. Abu-Assy, M.Y. El-Ashry, A.A. Mohamed, Dechanneling of positrons in disordered lattices – effect of anharmonic potential, Physica Scripta 72 (2005) 68. [8] B.R. Appleton, C. Erginsoy, M.W. Gibson, Channeling effects in the energy loss of 3–11 MeV protons in silicon and germanium single crystals, Phys. Rev. 161 (1967) 330. [9] M.K. Abu-Assy, E. Shalaan, A.H. Abdelaziz, Effect of lattice thermal vibrations on dechanneling of Positrons in disordered cubic metals, Arab J. Nucl. Sci. Appl. 42 (2) (in press). [10] L.I. Schiff, Quantum Mechanics, Mc Graw-Hill, New York, 1968.