150 T. NAKAJIMA

analysis of and

ko

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

gkk) 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

a

aa

(15a)

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