Journal of Modern Physics, 2011, 3, 146-153
doi:10.4236/jmp.2011.23022 Published Online March 2011 (http://www.SciRP.org/journal/jmp)
Remarks on the Erroneous Dispersion Surfaces from a
Pair of a Hyperbolic Branch and an Elliptical Arc of the
Intersected Two Laue Spheres Based on the Usual Crude
Approximation
Tetsuo Nakajima
Saitama Institute of Technology, Advanced Science Research Laboratory, Saitama, Japan
Received February 5, 2011; revised March 7, 2011; accepted March 8, 2011
Abstract
In almost all previous works, the hyperbolic dispersion surfaces of the central proper quadrics have been
crudely derived from reduction of the degree from the bi-quadratic equation by use of some roughly inde-
finable approximate relations. Moreover, neglecting the high symmetry of the hyperbola, both the branches
have been approximated on the asymmetric surfaces composed of a pair of a branch of the hyperbola and a
vertex of the ellipse without the presentation of reasonable evidence. Based upon the same dispersion sur-
faces equation, a new original gapless dispersion surfaces could be rigorously introduced without crude
omission of even a term in the bi-quadratic equation based upon usual analogy with the extended band theory
of solid as the close approximation to the truth.
Keywords: Dynamical Theory of X-Ray Diffraction, Gapless Dispersion Surface, Gappy Dispersion Surface
1. Introduction
First of all, it could be necessarily considered that the
firm establishment of E(energy) vs. k(wave number)
curves as the dispersion relation of the electron in solids,
which have been used as the usually popular gappy dis-
persion surfaces by solid line in Figure 1 [1] in almost
all works of the dynamical theory of X-ray diffraction
(DTXD) [2-6], were carefully introduced from the solu-
tions of the secular equation based upon the experimental
and theoretical examinations in the low-energy elec-
tron-diffraction by R. M. Stern et al. [1] and have pre-
vented to foster greater understanding of DTXD. In the
band theory of the solid state physics, the energy gap at
the Brillouin zone boundary between the hyperbola and
ellipse in Figure 1 could be introduced as a perturbative
effect of the Fourier component of the periodic potential
in the crystal [7], which is the off-diagonal term in the
secular equation shown in the Section 5. It plays impor-
tant role that the band structure and its electronic struc-
ture occupied by electrons depending on their concentra-
tion could give an insulator, a metal or a semiconductor
and a semi-metal [7]. The energy band structure with the
forbidden band as the Bragg gap due to the potential bar-
rier from the band theory in Figure 1 [1] could be valid
for only the valence and conduction electrons and inci-
dent electrons in the electron diffraction [1,7,10]. On the
contrary, the off-diagonal terms of22
Cχand Cχ
g
g
KK in
Equation (2) in the Section 3 are composed of the wave
number, the polarization factor and the Fourier compo-
nent of polarizability as a response function of X-rays,
which could be rewritten by 2
χλπ
gg
RF V [3]. Here
R is the classical radius of electron, the wavelength, λ
g
F
the Fourier component of the structure factor and V
the unit cell volume. All of the factors are transparent to
X-rays contrary to the potential for electron, and there-
fore cannot construct forbidden band as the Bragg gap,
as in the Section 2 [8,9].
Secondly, the hyperbolic gappy dispersion surfaces in
DXTD have been arranged near an intersectional point at
the Lorentz point LD between the hyperbola and ellipse
in Figure 1. The hyperbola is perfectly point symmetry
about the center of the hyperbola. However, the gap in
Figure 1 is composed of a branch of the hyperbola and
an arc of the ellipse. According to each definition of the

T. NAKAJIMA
147
Figure 1. The familiar dispersion surfaces. The formation
of the Bragg gap has reasonably derived only in the elec-
tron-diffraction by Stern [1]. Superposition of the hyper-
bola over the gap between a branch of hyperbola and a
vertex of ellipse in DTXD is a misapplication by a light-
hearted interpretation.
central proper quadrics, the ellipse and hyperbola can be
formed from the loci that the sum and difference of the
two distances from the definite two points are constant,
respectively, as is written like

22
1xa yb
.
Therefore, the hyperbola has two asymptotes that cannot
exist in the ellipse. Let it be ever so infinitesimal line
elements in each, ellipse and hyperbola are invariably
ellipse and hyperbola, respectively. Generally, approxi-
mation can round off magnitude of the quantity but can-
not change the plus and minus signs in algebra and the
symmetry in geometry. Therefore, the ellipse could not
transform into the hyperbola by approximation. By na-
ture, that’s something that cannot be, as mentioned in the
sections 3 and 4.
As a preliminary work, surpassing the above two er-
roneous points the rightly reasonable dispersion surfaces
[8,9] are carefully described based upon high analogy [3]
with the extended band theory in solid state physics as
the closest approximation to the truth [10].
2. The First New Gapless Dispersion
Surfaces in DTXD Based upon the Band
Theory of the Solid
Mainly following Kato’s scheme [2] based upon a man-
ner of the Laue method [3], we carefully examined deri-
vation of the gapless dispersion surfaces from the bi-
quadratic equation of the two wave approximation. The
vector propagation equation of an electro-magnetic wave
by the electric displacement d in a medium with a peri-
odic polarizability χ(r) has been represented by
2rot rot0Kχ dd d
.
Based upon this equation, the two wave propagation
equations from the Bloch waves of do and
g
ddefined by



 
exp exp
oog g
di i

 drk rdkr
could be derived to be


22 20
oo gg
kdKCd 

k (1a)
and


222
o0 gg
KCχdk d
g

k (1b)
in which k is the wave number in the crystal, K th e wave
number in vacuum defined by(0 12
),
where n the refraction index, C the polarization factor
and
χ
kK nK



o
,
g
χ

Fourier components of the polarizability, in
which 2
ggg from
χχ χ*
g
g
χ
χ
 by neglecting the
absorption. Here, in order to solve the two wave propa-
gation Equations (1a) and (1b), which could be repre-
sented by the simultaneous linear equations with two
unknowns, the necessary and sufficient condition, which
is satisfied for existence of solutions [6] could be repre-
sented by

22 2
og 422
ij o 222
kC
kk
Ck g
gg
Kχ
Kχ



k
Sk
k
2
k
0
22 422
oC gg
Kχ,

 kk (2)
which was designated as the bi-quadratic dispersion
surface equation called by Pinsker [4], not the secular
equation. Here, heads of o and
k
g
k(o where
kg
g
is the reciprocal lattice vector) in Figure 2 lie at the
point O and G and their initial common point1, which
chanced to be there satisfies a loci in Equation (2) in the
S
Figure 2. Diffraction in reciprocal space based upon the two
wave approximation. The points S1 and S2 are intersectional
points, through which a horizontal lines of H1 and H2 are
reference lies. The vector of OG


g is the reciprocal lattice
vector, 1o SOk


the incident wave vector and 1
g
SG

k


the
reflected wave vector.

148 T. NAKAJIMA
reciprocal space.
The diagonal terms of S11 and S22 in Equation (2) rep-
resent the two same radius circles intersected at two
points S1 and S2 in Figure 2. The roots of
X
(20k

:
positive definite) in Equation (2) can be given by

2
22 22422
oo
14.
2gg
XKχ




kk kkg C (3)
Accurately, Equation (2) could not be a form of the
secular equation but has been frequently impressed as the
secular equation in some references [2,3,6]. Just to be
sure, the intrinsic secular equation of equation (2) could
be represented from the usual definition as follows:
  
22 2
222
Det ij
g
gg
kE KCχ
KCχkE


 
k
SE k
o
ij

222 222
OO
2
gg
EkE kk kk
 
2
2224 2
O0
g
kk KCχ, kk g (4)
in which E is the eigen-value, 1for
0for
ij
ij
ij





 the
Kronecker’s delta, which represents the unit matrix. The
eigen-values of Equation (4) could be obtained to be

2
22222422
oo
14
2gg
Ek KCχ,

 


kk kkg


(5)
where the first term of “”in Equation (5) only trans-
lates the origin of E, therefore ordinarily it is omitted
hereafter. It is the most important that the behavior of the
whole multinomial commonly in the right hand side in
Equations (3) and (5) should be carefully analyzed over
all k-space especially in the vicinity of the Brillouin zone
boundary. When E = 0, Equations (2) and (3) are com-
pletely equivalent to Equations (4) and (5), respectively.
Then, it is self-evident that the analytic results in both of
Equations (3) and (5) without any abbreviation of even a
term as done in the conventional crude approximation
could remain wholly valid as the close approximation to
the truth. It could be understood that the refracted beam k
in the roots in Equation (3) can be sufficiently character-
ized by AM and FM due to the three kinds of photons of
o,
2
k
k
g
k and K,which could be polarized by the periodic
electron distribution in
g
χ
in conformity with the dual
polarized photons by . C
Approximately assuming that two in-
4220
g
KCχ,
tersected circles with the same radii in Equation (3) are
shownin Figure 2. If the magnitudes of and
o
k
g
k are
close to each other, then the term of 422
4
g
KC
χ
cannot be
neglected. Thus, the amplitude of neither plane waves is
negligible. When o
g
kk, Equation (3) becomes
22
o
g
X
kKC

 and hence the ratio of to
o
d
g
d,
determined from Equation (2) is 22
g
g
KCχ:KC
χ

Hence, o11
g
d:d :. Assuming that

2
2
4g KC
χ
is
large compared with the first term under the radical sign
in Equation (3) in case of2
o
2
g
kk, X can be expanded
to be,
 
2
22
o 22 2
2χg
KC


o
1
8
2
g
gg
XKCχ  
kk
kk 



(6)

If we translate the origin of k by 2
g
and consider
the vector 2

k
g
and if we denote by x the compo-
nent of parallel to k
g

and by z the normal compo-
nent to
g

, then by using the following relations, after
more elaborate vector analysis than it looks like,
22 22
oo
2
go
k

kk 
gg
k
22
o
22
2
z








g
x
gg
kx
g
and
2
22 22
o2

 


g
kz xzxx
2
4

g
g,
a reasonable roots of X in Equation (3) from the 2nd and
3rd terms in Equation (6) can be rewritten as
2
22
4
X
g
zx
2
22
2
2g
g
KC
KC






 




xg (7)
The 4th term in the brackets in Equation (7) could be the
most important one expanded in series near the Brilluoin
zone boundary described in the above as one of main
subjects. As a result, the 4th term in Equation (7) could be
indicated by


22 222
2
gg
KC KC 





ygx
. (8)
The expressions in Equation (8) show the precisely ca-
nonical forms of the hyperbola and ellipse as

22 222
yb bxa

(9)
in which the ellipse with plus sign and hyperbola with
minus one are shown in Figures 3(a) and 3(b), together
with asymptotes of 0
yx
ba

 labeled in
1
andL2 L
hyperbola. The constants a and b in Equation (9) can be
given by Equation (8) as
2
2g aKC  g (10a)
and

T. NAKAJIMA
149
Figure 3(a). The dispersion surfaces composed of the
ellipse and hyperbola in Equation (8) with the two
asymptotes L1 and L2 defined by extended two diago-
nals of the rectangle with both sides of 2a and 2b.
3(b).This shows superposition of Figure 3(a) upon
Figure 1, which shows smooth variations of the dis-
persion surfaces in forward and backward X-ray near
the Brillouin zone boundary.
χ
g
bKC. (10b)
Therefore, 2
222g aKC  g
and 22 χ
g
bKC
from Equations (10a) and (10b) are the minor and major
axes of the ellipse, respectively and the latter of 2b also
stands for the transverse axis of the hyperbola. The gra-
dients of the two asymptotes of hyperbola defined by the
gradients of diagonals of the rectangle with both sides of
2a and 2b in Figures 3(a) and 3(b) could be expressed
by

b
a2sin
g
B
C

 
from Equations (8) and
(9) by the Bragg law, which will be discussed in the final
section. Both of the hyperbola and ellipse could stand in
a line without a gap as in Figures 3(a) and 3(b). Conse-
quently, it could be apparently proved that the expected
Bragg gap as shown in Figure 1 between hyperbola and
ellipse could not absolutely exist in Equation (8) in
DTXD and the gappy dispersion surfaces in Figure 1 can
be rigorously set to the right gapless dispersion surfaces
in Figure 3(a).
3. Examination on the Crude Approximation
to Derive the Previous Gappy Dispersion
Surfaces
According to the previous works [2-6], the dispersion
surface Equation (2) can be factorized as
22 2
222
χ
χgg
g
kKC
KC k


k
k
o





 
422
χ0
gg
kkkkKC
g

 kkkk
oo . (11)
In almost previous references [2-6], by use of the nu-
merical approximate relations of
O22kk K k

 and , (12) 22
g
kk K k 
the central proper bi-quadratic Equation (11) could be
further decomposed into two quadratics as follows:




2
4
g
kkK0

 kk
o (13a)





222
χ40
gg
kkKC

 kk
o. (13b)
The factorization in Equation (11) by Equation (12) is
over the limits of approach and beyond reasonable con-
ception in Equation (4).
Originally, the bi-quadratic dispersion surface Equation
(2) should be a product of the two central proper quadrics
consisting of the hyperbola and ellipse including circle as
easily understood from Figures 1 and 2. First, the central
proper quadric of the hyperbola in Equation (13a), which
could be constructed by replacing a very big product of
O
kk

and g kk

with 2 4
K
by use of Equation (12),
has been cut down without any thought for the conse-
quence. Secondly, although a product of both terms of
kk

o and g
kk

has been approximated at zero from
Equation (12), an order of magnitudes of the second term
in Equation 13(b) can be estimated to be



29to10
422222252
28to10
441010
410 1
gg
KCχKCχC
C


 



.
This is tremendously large number. The above approxi-
mate expression cannot coincide with Equations (11) and
(13b). Commonly used factors in Equations (13a) and
(13b) from Equation (12) could be only understandable
in approximation of numerical values but not in vector

150 T. NAKAJIMA
analysis of and
ko
g
k considering its direction and
orientation in the vector space. These approximations are
unmistakably self-contradictory and definitely break na-
tive goodness of the two central proper composite quad-
rics in Equation (2), which should be never allowed
without rigorous verification. This is a violation without
cause.
Reserving examination of the elimination of the de-
composed factors in Equation (13a) in the crude ap-
proximation, it could be concluded that Equation (13b)
intuitively has at least two solutions. From the condition
that the product of two variables is a constant of

222
χ4
g
KC

in Equation (13b), a simple solution
represents a rectangular hyperbola. However, this is not
practically reasonable, considering the variations of the
Bragg angle. In another solution, it could be easily un-
derstood from a well-known attribute that the product of
the two perpendiculars to the two asymptotes of the hy-
perbola from an arbitrary point on it is constant described
by




22 22
bab a, which could be easily prove from
the canonical form of Equation (13b) as
2
yx
BB
1
2cosθ2sinθ
gg
kk
KC KC 





2





,

B
0θπ2.

(14)
The fitness of Equation (14) to the hyperbola at the cen-
ture point D in Figure 1, which is composed of a hy-
perbolic branch (named as “branch 1” for
L
,k

o
k
0
gkk) and an elliptic arc near a vertex on the major
axis side (similarly “branch 2” for ) has
not yet been correctly examined from a geometrical
viewpoint to decide whether to support Equation (13b)
on the principle of being fair and just.
, 0
g
kkkk
o
4. Geometrical Examination on the
Conventional Gappy Dispersion Surfaces
In DTXD
It is thoughtless beyond mathematical knowledge to as-
similate both substantially different extremities from a
different nature of symmetry between hyperbola and
ellipse, by simple numerical approximation in Figure 1.
In Figure 2, the oblique lines of T1 and T2 represent tan-
gential lines of two circles at the intersection point S1. It
is very important to note that the intersected curves
above and below H1 show quietly an obvious asymmetry,
which could be paraphrased as follows: the vertically
opposite angles at the intersectional point S1 of the two
composite arcs O1S1G1 and O2S1G2 defined by intersec-
tional angle included by both of the tangential lines T1
and T2 at S1 are identical. The angle included by two tan-
gential lines at the points O1 and G1 on both arcs S1O1and
S1G1 can increase with increasing distances to both
points O1 and G1 from the point S1 and become mono-
tonically larger than the vertically opposite angles in the
vicinity of S1. But a variation of the corresponding angles
included by two tangential lines at the points O2 and G2
on both arcs S
1O2 and S 1G2 can become entirely vice
versa. As easily imagined from Figure 2, both of the
whole closed curves of oval S1G2G3S2O3O2S1 and co-
coon-s hap ed S
1O1O4S2G4G1S1 constricted in the middle
could be redrawn as hyperbolic and elliptic dispersion
surfaces with the Bragg gap as in Figure 1 for low-
energy electron-diffraction by Stern [1], not for X-ray
diffraction. This is a rigorous proof of asymmetry of arcs
O1S1G1 and O2S1G2, from which the symmetric branches
of the hyperbola can never be constructed. Therefore, the
gappy dispersion surfaces in Figure 1 cannot be applied
to TDXD as the hyperbola. In the previous works, a pair
of asymmetric arcs of O1S1G1 and O2S1G2 has been in-
sensitively replaced by the hyperbola of Equation (13b).
This is the second violation without cause. For a long
term, the previous works [2-6] have been just an attempt
to put together the indefinable hybrid dispersion surfaces
composed of a pair of the hyperbolic branch and elliptic
arc to lay basis for today’s TDXD. However, by nature
that is something that cannot be. Consequently, ap-
proximation can round off magnitude of the quantity but
cannot substantially change the plus and minus sign and
geometrical symmetry. Therefore, the ellipse could not
transform into the hyperbola by reasonably scientific
approximation, whose use is totally outrageous and
should not be fundamentally permitted.
5. Origin of the Energy Gap in the Band
Theory of Solid and Brief
Characterization of the Off-Diagonal
Term in the Dispersion Surface Equation
(2)
There are very close resemblance in physical treatment
between the energy gap in conduction band [1,7,10] and
anomalous transmission of X-rays [2-6] but the definite
necessary results reveal that the off-diagonal terms of the
Fourier component of potential in electrons create a for-
bidden energy gap [1,7,10] and those in photons in Equ-
ation (2) closely relate with the different ratio between
the absorption and transmission of photons [2-6].
According to Kittel [7], two different standing waves
of electron in solids could be derived from the two trav-
eling waves of exp


ix/a and exp

ixa as

ψexp exp2cos
ix ix
x
a
aa

 
 
 
  (15a)

T. NAKAJIMA
151
and


i
ψexp exp
2sin
ix x
aa
ixa.


 

 
 




(15b)
The two standing waves


  and


ψ pile up
electrons at different regions, and therefore the two
waves have different values of the potential energy. This
is the origin of the energy gap. The probability density ρ
of a particle is equal to *2 ψψ ψ
k
i
. For a pure traveling
wave exp(x), ρ is equal to exp(x)exp(- ikx)=1 so that
the charge density is constant in Figure 4(b).
k
i
From the standing waves


ψ in Equation (15a),
the probability density could be expressed by
 
22
ρψ cos
x
a. 
The function piles up negative charge on the positive
ions at the periodic lattice points of x = na (n = 0, 1, 2,
), where the potential energy is the lowest. For the
standing wave



ψ in Equation (15b) the probability
density is given by
 
22
ρψ sinπ
x
a, 
which concentrate electrons away from the ion cores.
Consequently, the wave function


ψ

piles up elec-
tronic charge on the cores of the positive ions, thereby
lowering the potential energy in comparison with the
average potential energy seen by a traveling wave in
Figure 4(b). The wave function

ψ piles up charge
in the region between the ions, thereby raising the poten-
tial energy in comparison with that seen by a traveling
wave in Figure 4(b). When the expectation values of the
potential energy could be calculated over these three
Figure 4(a) Variation of electrostatic potential energy of
conduction electron in the field of ion cores of a linear lat-
tice. 4(b) Distribution of probability density ρin the lattice
for
 
22
ψsin π

x
a (solid line),
 
22
ψcos
x
a
charge distributions, it is in the nature of things that po-
tenti


ρ

trave
(dotted line) and for a traveling wave (one point broken line)
[7].
al energy is lower than that of the traveling
wave, whereas the potential energy of

ρ is higher
than that of theling wave. If energy difference be-
tween


ρ

and


ρ

is equal to
g
E, than energy
gap of width becomes
en
g
E Assuming that the potential
energylectron in the crystal at the point x could be
expressed by
in e


cos 2Ux Uxa ,
the energy difference between two staing wave states
is
nd
 


122
dExUx  
0
g




22
2 dcos2cossin
x
Uxaxa xaU


.

It is found that the gap is equal to the Fourier compent
of the crystal potential, which is the off-diagonal term in
on
the matrix


ij
S in the secular Equation (4) and can
cause an insulator, a metal, a semimetal etc in the above.
In DTXD, by use of og

kk
g
, the amplitudes of the
Bloch waves in the section 2 [5] could be given by





exp exp
oog g
ii

 drdkrdkr



 
exp exp
og o
ii

 dd
g
rk. r
From this, the intensity of the wave-field can bgiven by e



222
o12cosId dRRC

 
g
r, (16)
in which the amplitude ratio is represented by
go gogg
RddCK CK

 

 (17)
where





2
ooo
α21
o
Kk
χ
kk
and




2
21
ggg
αKkχkk
g
from off-diagonal term in the matrix


ij
S
gr
. For simplic-
wave field inten-

ity, instead of pile up of electrons, the
sity is modulated by the factor

cos in Equation
(16), which has maxima atn
g
r. It corresponds to an
atomic plane and minima at 
212gr n with in-
tegral n. Therefore, by givifferent reading r in-
stead of x in Figure 4(b), the mima of the
standing wave occur at or halfway between the atomic
planes. Including a role of the polarization factor C, it is
important to stress that all of the off-diagonal terms in
Equation (2) are transparent to X-rays as aforementioned
and never construct the forbidden energy gaps in Figure
1 by splitting of the energy bands. The existence of the
off-diagonal terms in Equation (2) could give energy
ng a di
axima and min
dispersive X-ray diffraction depending on the ratios from

152 T. NAKAJIMA
transmission to absorption due to the photoelectric effect.
For example, in the 200 reflection in NaCl, the structure
factor
g
F is positive. Hence,
g
χ
is negative and the
wave-field in branch 2 is absorbed more strongly than
that in branch 1shown in Figure 5. The effect is well
known as the Borrmann effect [5]
6. Results and Discussion
.
sion
be c
In DTXD, the asymmetric disper
solid line in Figure 1 that could
surfaces shown by
omposed of hyper-
olic ch and elliptic arcs have been aggressively b bran
camouflaged with the gappy dispersion surfaces devised
from the crude approximation beyond the fundamental
algebra and geometry in Equation (13b) mentioned
mainly the Sections 3 and 4 in which they have been
constructed out of the quadratics crumbled from the
bi-quadratic equation based upon the crude approximated
relation in Equation (12). It means that these surfaces
have been lacking fundamental consistency based upon
the physical necessity. Therefore, the branch 2 has not
the intrinsic asymptotes, because the root is from a part
of the vertex of the ellipse. Hence, the previously wide-
spread analyses in DTXD by use of Equation (13b)
should be reexamined based upon the new gapless dis-
persion surfaces in Figure 3(a) in Equation (7).
From Equation (14), another canonical form of the
hyperbola in Equation (13b) could be represented as fol-
lows,


2
22 22
2cos θsinθ10 θπ2.kk ,
KC





yB
x B


(1
racteristic that the in of the factor in the
large
value aentioned in the se3 and the disp
te
B
verse
quation (18) is equal to very
ction
g

It is cha
first parentheses E
s m
8)
ersive
rms in the second factors consisting of two terns in
parentheses are remarkably modulated by the sine
squared B
θ plus cosine squared B θ in Figure 6. In the
region of B
0π4θ , the gradients of the hyperbola
are steep and the intersectional angles of the asymptotes
are the ace angles. And the majoaxis of the reference
ellipse curve of 2 2
yB B
cos sin1θθk is parallel to the
ky-axis. When
ut r
2
Bπ4θ, it becomes the rectangular hy-
perbola and the rrcle. Moreover, in eference curve is a ci
B
π4π2θ, ntersectional angles become the
obtuse angles. he offered ellipse for reference is
those i
That of t
Whereas, from
parallel to the kx-axis.
Equation (9), the canonical form of the
hyperbola and ellipse in Equation (8) could be repre-
sented as
Figure 5. Intensity of the two wave-fields at the exact Bragg
condition in the 200 refection. Note that branch 1 and 2
wavesimum intensity and a m at th
ig
have a minaximum onee
atomic positions, which are analogous to the electrons in
Figure 4(b).
Fure 6. The main part of the dispersion surfaces of
22 22
cossin 1kk

θθ
yBxB
in Equation (18), together with the
pses of 22 22 cossin 1kk elli


B
θθ
yBx as a reference. The Bragg
angles from (1) 10 to 8 (80) in every ten degree should be
slope hyperbola to the gentry slope one
in order.
assigned to the steep

22
y
kk

2102.
2
x
gg
,g
k
KC KCχg



 





(19)
The major axis of ellipse in Equation (19) can b
stantly parallel to the ky-axis because it is constant
ger than the minor axis like

e con-
ly lar-
2
2χ
g
g
KCgK C


since
K
g

in the order of magnitude in the region of
02
g
k

, which could corresp onds to the region of the
Bragg angles of 02
B



 in Figure 2. It is consid-
ered thatppearance of the scattering vector a
g
only in
the parameter a in Equation (9) could be appropriate.
Further the gradieymptotes of the hyperbola nt of the as
in Equation (19) can be represented as follows,
-
6
-
4
-
2
2 4
6
x
-
6
-
4
-
2
2
4
6
y
(1) (2) (3) (4)
(5)
(6)
) (7
(8)
BRANCH 1BRANCH 2

T. NAKAJIMA
153

2
b
asin
B
g
C


 .
If the Bragg angle
B

is larger than 05.by a r
estimation of the a expression, tintersectional
angles of the asymptotes could be the acute angle and
reasonable.
rom
Dr. R. Negishi and the late
ful stimulating discussions.
Prof. Dr. H. Kobaya
. References
] R. M. Stern, J. J. Perry and D. S. Boudreaux, Reviews of
8
ough
Morden Physics, Vol. 41, 1969, pp. 275-295.
[1
doi:10.1103/RevModPhys.41.275
[2] N. Kato, Kaisetsu and Sanran (in
bovehe
Conclusively, it is very important to pote that the
afore-mentioned misapplication of Equitation (13b)
should be reasonably set to rights and validity of the new
proposed gapless dispersion surfaces in Equation (19)
an
Japanese), “Diffraction
y Diffraction,”
rys-
opography,” Perga-
tsu (in Japanese), “X-Ray Dif-
6th Edi-
ajima, ECM26, Darmstadt, 2010.
Physics,” Chapter
and Scattering,” Chapter 9, Toyko, 1978.
[3] A. Authier, “Dynamical Theory of X-Ra
Chapter 4, Oxford university press, New York, 2001.
[4] Z. G. Pinsker, “Dynamical Scattering of X-Rays in C
tals,” Springer-Verlag, Berlin, 1978.
[5] B. K. Tanner, “X-Ray Diffraction T
d should be strictly examined by thorough investiga-
tion.
7. Acknowledgments
The author would like cordially thanks to Prof. Dr. T.
ukamachi, Associate Prof.
mon Press, Oxford, 1976.
[6] S. Miyake, X sen no Kaise
fraction,” 3rd Edition, Chapter 5, Tokyo, 1988.
[7] C. Kittel, “Introduction to Solid State Physics,”
tion, Chapter 7, John Wiley & Sons, Inc., New York,
1986.
[8] T. Nak
F
M. Yoshizawa in SIT for use
e is also indebted to Emeritus Hkawa [9] T. Nakajima, XTOP, Warwick, 2010.
of KEK for useful suggestion and drawings of Figures 2,
3 and 6 by Mathematica.
[10] R. Kubo and T. Nagamiya, “Solid State
4, McGraw-Hill Book Co., Inc., Hong Kong.