Diffraction theory of phonon scattered electrons Page: 2 of 2
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A 2-D lattice vibration model: For phonon excitations, since q, = KrW2E = 0. the 'IDS stie.ik> a:e :::.i;n!\
eeneralcd hv the phonon modes with woe sectors parallel to the diIfraclivin plane. These acouMk mode-., lei whuii
io tends to zero when q approaches zero, are mainly generated by the atomic vibrations within the plane perpendietji.tr
it' the incident beam direction B = [hkl] near a main crystal zone axis. This is actually a 2-D lankc \ih; .11,0:1 model.
The main contributions to the I DS streaks are from the acoustic branches; optical branches contribute only a Jotu-'C
background. Titus the TDS streaks can be predicted by examining the terms in the first {.. ) bracket in Fq. o and is
StlxshI BKJ.I) - 2, 1/ t0j( TI, (Kb
.1 j=1
u here co.(l) is the dispersion surface of the acoustic branches, and is determined by the 2-D atomic interactions of the
nearest neighbors falling in the same (hklt plane. The TDS streaks are defined by the tx-tj lines satisfy ing i - < t j - 0.
For a monatomic h.c.c. crystal oriented in [001J, if only the interactions with the atoms located at raw iOdi and
±a<>( 010) arc considered, one has
((Op;)1 2 = (4/M) [(F+G) sin-(qX N an/2) + (F-G) sin2(q_v,xuo/2)], (11)
'.‘.here ao is lire lattice constant, and F and G are the atomic force constants. In the central force approximation i F = G I.
so dial
STDS-(l/lsin(W2)!)+ l/lsin(isao/2)l). (12)
Thus sharp TDS streaks should appear along tx = 0 ([100]) and iy = 0 ([010]) directions (sec Fig. 1). The <11()>
streaks would appear in the pattern if die phonon modes created by the vibrational coupling of the atoms located at
(000) and a<y2{ 111) were strongly excited. The absence of the <110> streaks in Fig. 1 therefore supports the validity
of die simplified 2-D model discussed above. In practice, the (F-G) term in Eq. (11) determines die streak sharpness
and width. Therefore, the observed finite width of TDS streaks is mainly the result of non-ccntral interaction forces.
This 2-D model can systematically interpret the directions and sharpness of the TDS streaks observed in die diffraction
patterns near the [001 ]. [Ill] and [110] zone axes of Mo (b.c.c.), Au and A! (f.c.c.), and Si (diamond cubic). If B is
far from main zone axes, die contributions of optical branches may become important.
In practice, TDS streaks arc located on die lines satisfying r»ri - 0, i.c., along the t = B x ri = [(kLj-IKp), (IH;-hLp),
(hKj-kHj)] direction in reciprocal space, where rj = (H]ai,Kja2,Lia3) is the relative position of the first nearest
neighbors in die 2-D plane, and a;s arc crystal lattice constants.3 4 This is a general rule for predicting TDS streaks
without any numerical calculations, and is analogue to the g»b = 0 rule for determining dislocation Burgers \ cctors in
diffraction contrast imaging. The TDS streaks predicted by this rule fit most die observations of N^Al, AhSc, NiAl
and Fe3Al intermetalies.
The TDS TEM image: The HREM image formed by the TDS electrons in a TEM can also be derived from Eq. (6),
IT(b)= {XG2(b)IY(b,0,zn)l2}® IFoB(b) I2, (13)
n
where the generation function of TDS is
G2(b,zn) = (aoAz)2{£Xai2 f Vv,(r-R(h)-r(!)) | :); (14)
where a;2 = | d(0 p(o>) A]2(co) is the atomic mean square vibration amplitude; p(w) is die phonon density of states;
and Fob is die inverse Fourier transform of die objective lens contrast transfer function. This formulation is equivalent
to incoherent imaging theory. The image resolution may be higher dian that formed purely by elastically scattered
electrons, but the "inclined incidence effect" (i.c., a small off-axis correction of momentum transfer q to incident wave
vector K in die clastic wave calculation) of phonon scattering may distort the image. The phase coupling of vibrating
atoms docs not affect the calculations for images but docs affect diffraction patterns. Thus die image simulations can
be carried out based on die Einstein model if the correct vibration amplitude for each atom is used.
It can be proved from Eqs. (3) and (6) that the semi-classical approach utilizing a "frozen" lattice model for TDS is
equivalent to the result of inelastic quantum scattering dicory (Eq. (8)) if kgT «Jia> and the "inclined incidence effect”
is negligible. The former condition is satisfied if die temperature is not much higher than room temperature.^
1. M. J. Whelan, J. Appl. Phys. (1965) 36, 2103.
2. H. Yoshioka, J. Phys. Soc. Japan. (1957) 12, 618.
3. Z. L. Wang and J. Bentley, in Proc. of Xllth Intern.
Cong, for EM (Seattle), San Francisco Press (1990)
Vol. 2, p532; Z. L. Wang, Acta Cryst. A (1991).
4. This research was sponsored by die Division of
Materials Sciences, U.S. Department of Energy,
under contract DE-AC05-840R21400 with Marlin
Marietta Energy Systems, Inc. FIG. I,-Diffraction pattern of Mo [001],
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Wang, Z. L. Diffraction theory of phonon scattered electrons, article, January 1, 1991; United States. (https://digital.library.unt.edu/ark:/67531/metadc1092237/m1/2/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.