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Anomalous transmission of swift electrons in thick crystals
Volume 45A, number 1
PHYSICS LETTERS
27 August 1973
ANOMALOUS TRANSMISSION OF SWIFT ELECTRONS IN THICK CRYSTALS A.Ja. BOBUDAEV, V.V. KAPLIN and S.A. VOROBIEV Nuclear Physics Research Institute of the Tomsk Polytechnical Institute, Tomsk, USSR Received 28 June 1973 A large penetration of swift electrons along the lattice crystallographic axes was observed. The center transmission peak of 2 MeV electrons along the (100) axis in 110 ~m NaCI shows a narrow dip of width at halfminimum (0.2 ±0.05). The half-width of the maximum base (0.7 ± 0.1) is in agreement with the classical mechanical ca1culati~n. In [1] was suggested the idea of stable classical trajectories in the motion of swift negative particles along the atomic rows of a ciystal lattice. This idea was confirmed experimentally in [2,31. The chan-
LL N, ~
if
neling of swift negative particles was treated in [21 on the basis of the continuum approach of atomic row attractive tory of the negatively potential charged U(r). In particle this case with themass trajecm corresponds to the one dimensional motion in the field of the effective potential Ueff(T, 1) = U(r) 2/2mr2, if the particle does not approach within +1 the critical distance rc from the row in the continuum potential treatment. This condition corresponds to the conservation of angular momentum I of the particle with regard to the row and the projection of particle trajectory in the transverse plane is a “rosette”. The bound state motion is favourable for a static lattice inside the Wigner—Seitz unit cell, where the perturbation effect of neighbouring rows is small enough. In [4] the predicted effect was measured in rather thick single crystals for beta-particles, but this did not permit all features of the anomalous distribution to be observed, Fig. la shows the dependence of the counting rate of 2 MeV electrons transmitted through a 110 pm NaC1 single crystal on the angle between the beam direction and the (100> crystallographic axis. This distribution is normalized to the transmission yield in the crystal at p = 8°.The angular divergence of the incident beam was about 0.15°.The electrons were selected from the continuous (90Sr+ 90Y) beta-ray spectrum by an analyzing magnet with a energy resolution about 3%. The scintillation counter angular resolution was less than i0~sr. The trailsmission peak in the centre of the broad dip depends
\jiJlj’
178
~
-j
-~
-i
(5)
~ (L’)
,
~
(a) ~ ‘f,,rgd
___________ ____________
I
i” i’-~(~ \j
Fig. 1.(a) The variation of electrons transmission through 110 ~jm(100) NaC1 crystal with the orientationangle. (b) The calculated variation ofparticles capture fraction with the crystal orientation angles (1) and the measured distribution (2) for the 0°to 0.6°incidence angles range. (c) The capture areas for electrons bound motion along the (100) NaC1 axis.
upon the bound motion of electrons along the (100> crystal direction. The half-width of transmission peak base (0.7 ±0.1)°is slightly larger than ~ 0.6°,calculated in the Linhard approximation = (2z 2/dE*)1/2; where E = ~puwith the 1z2e relativistic momentum. In the centre of the peak we find the pronounced minimum of electron transmission. The minimum angle of capture in a bound state [2] “mm “L(’~cfro),where rc 0.22 A is the 71
Volume 45A, number 1
PHYSICS LE~FERS
critical radius and r0 1.41 A is the effective radius of the Wigner—Seitz cell, is Wmm = 0.095°,which is in good agreement with the measured half-width at half-minimum (0.11 ±0.05)°.Fig. lb shows a calculated variation of the fraction of particles captures in a bound motion with the rotation angle of (100> NaCl single crystal relative to the incident beam axis. The angular dimensions of this curve are in good agreement with the measured distribution. The calculation was realized forarea the projectile uniform distributed over the S (fig. ic) particles on the crystal entrance side in the angular range of the crystal rotation 0°to ‘I’~.The interaction potential was taken in Thomas—Fermi form, From the condition of bound motion [2] E 1= 2+ U(r~)~ 0, ~ rc, where ‘I’ is the particle E*W incidence angle, ri is the particle incident point on the S area, rmin is the critical approach distance of the particle to the atomic row, it follows that the projectile particles with a definite incidence angle are captured in the bound motion only in the individual S area ranges, if the transverse energy E 1 of the bounded particles is constant. Fig. ic shows the I and
72
27 August 1973
II areas, corresponding to capture of negative partides with incidence angles 0.29 ~‘L and p.57 “L’
respectively. The existence of individual capture areas may cause violation of azimuthal symmetry in the angular distribution of particle transmitted through single crystals. It should be noted that the calculated minimum capture angle (fig. lb) is less than “mm’ because we take into account the fraction of tightly bounded particles (areas I’ and II’) with the angular 1mmn momentum ~ whereas where corresponds ~< to Ueff(r, 1) = 0 atrmm> r = Tc.rc, The departure in shape of the theoretical distribution from the measured one can be explained by the static lattice approach in the calculation and due to multiple scattering effects in the thick crystals. References [1] H.C.H. Nip, M.J. Hoffis and J.C. Kelly, Phys. Lett. 28A (1968) 324. [2] H.J. Kreiner et aL, Phys. Lett. 33A (1970) 135. [3] A.A. Vorobiev et al., Fmz. Tverd. Tela 14(1972) 2157. [4] A.A. Vorobiev et aL, Phys. Lett. 40A (1972) 105.
PHYSICS LETTERS
27 August 1973
ANOMALOUS TRANSMISSION OF SWIFT ELECTRONS IN THICK CRYSTALS A.Ja. BOBUDAEV, V.V. KAPLIN and S.A. VOROBIEV Nuclear Physics Research Institute of the Tomsk Polytechnical Institute, Tomsk, USSR Received 28 June 1973 A large penetration of swift electrons along the lattice crystallographic axes was observed. The center transmission peak of 2 MeV electrons along the (100) axis in 110 ~m NaCI shows a narrow dip of width at halfminimum (0.2 ±0.05). The half-width of the maximum base (0.7 ± 0.1) is in agreement with the classical mechanical ca1culati~n. In [1] was suggested the idea of stable classical trajectories in the motion of swift negative particles along the atomic rows of a ciystal lattice. This idea was confirmed experimentally in [2,31. The chan-
LL N, ~
if
neling of swift negative particles was treated in [21 on the basis of the continuum approach of atomic row attractive tory of the negatively potential charged U(r). In particle this case with themass trajecm corresponds to the one dimensional motion in the field of the effective potential Ueff(T, 1) = U(r) 2/2mr2, if the particle does not approach within +1 the critical distance rc from the row in the continuum potential treatment. This condition corresponds to the conservation of angular momentum I of the particle with regard to the row and the projection of particle trajectory in the transverse plane is a “rosette”. The bound state motion is favourable for a static lattice inside the Wigner—Seitz unit cell, where the perturbation effect of neighbouring rows is small enough. In [4] the predicted effect was measured in rather thick single crystals for beta-particles, but this did not permit all features of the anomalous distribution to be observed, Fig. la shows the dependence of the counting rate of 2 MeV electrons transmitted through a 110 pm NaC1 single crystal on the angle between the beam direction and the (100> crystallographic axis. This distribution is normalized to the transmission yield in the crystal at p = 8°.The angular divergence of the incident beam was about 0.15°.The electrons were selected from the continuous (90Sr+ 90Y) beta-ray spectrum by an analyzing magnet with a energy resolution about 3%. The scintillation counter angular resolution was less than i0~sr. The trailsmission peak in the centre of the broad dip depends
\jiJlj’
178
~
-j
-~
-i
(5)
~ (L’)
,
~
(a) ~ ‘f,,rgd
___________ ____________
I
i” i’-~(~ \j
Fig. 1.(a) The variation of electrons transmission through 110 ~jm(100) NaC1 crystal with the orientationangle. (b) The calculated variation ofparticles capture fraction with the crystal orientation angles (1) and the measured distribution (2) for the 0°to 0.6°incidence angles range. (c) The capture areas for electrons bound motion along the (100) NaC1 axis.
upon the bound motion of electrons along the (100> crystal direction. The half-width of transmission peak base (0.7 ±0.1)°is slightly larger than ~ 0.6°,calculated in the Linhard approximation = (2z 2/dE*)1/2; where E = ~puwith the 1z2e relativistic momentum. In the centre of the peak we find the pronounced minimum of electron transmission. The minimum angle of capture in a bound state [2] “mm “L(’~cfro),where rc 0.22 A is the 71
Volume 45A, number 1
PHYSICS LE~FERS
critical radius and r0 1.41 A is the effective radius of the Wigner—Seitz cell, is Wmm = 0.095°,which is in good agreement with the measured half-width at half-minimum (0.11 ±0.05)°.Fig. lb shows a calculated variation of the fraction of particles captures in a bound motion with the rotation angle of (100> NaCl single crystal relative to the incident beam axis. The angular dimensions of this curve are in good agreement with the measured distribution. The calculation was realized forarea the projectile uniform distributed over the S (fig. ic) particles on the crystal entrance side in the angular range of the crystal rotation 0°to ‘I’~.The interaction potential was taken in Thomas—Fermi form, From the condition of bound motion [2] E 1= 2+ U(r~)~ 0, ~ rc, where ‘I’ is the particle E*W incidence angle, ri is the particle incident point on the S area, rmin is the critical approach distance of the particle to the atomic row, it follows that the projectile particles with a definite incidence angle are captured in the bound motion only in the individual S area ranges, if the transverse energy E 1 of the bounded particles is constant. Fig. ic shows the I and
72
27 August 1973
II areas, corresponding to capture of negative partides with incidence angles 0.29 ~‘L and p.57 “L’
respectively. The existence of individual capture areas may cause violation of azimuthal symmetry in the angular distribution of particle transmitted through single crystals. It should be noted that the calculated minimum capture angle (fig. lb) is less than “mm’ because we take into account the fraction of tightly bounded particles (areas I’ and II’) with the angular 1mmn momentum ~ whereas where corresponds ~< to Ueff(r, 1) = 0 atrmm> r = Tc.rc, The departure in shape of the theoretical distribution from the measured one can be explained by the static lattice approach in the calculation and due to multiple scattering effects in the thick crystals. References [1] H.C.H. Nip, M.J. Hoffis and J.C. Kelly, Phys. Lett. 28A (1968) 324. [2] H.J. Kreiner et aL, Phys. Lett. 33A (1970) 135. [3] A.A. Vorobiev et al., Fmz. Tverd. Tela 14(1972) 2157. [4] A.A. Vorobiev et aL, Phys. Lett. 40A (1972) 105.