Homework 1: Solutions

1. Consider how the de Broglie’s suggestion might explain some properties of the hydrogen atom.a. Show that the assumption

p = mυ =h

λand the ‘quantization condiction’ that the length of a circular orbit be an integer multiple of the length of theelectron wavelentgh (that is: nλ = 2πr, where r is the raduis of the orbit and n an integer) imply that onlydiscrete orbits are allowed.b. Calculate the total energy (kinetic plus potential) of the electron in each orbit characterized by n.Hint: In part a find two equations descreibing the balance between the centrifugal and the Coulomb (centripetal) force. Solve for the

radius r and for the angular velocity ω. Now, in b insert these expressions into the formulae for the kinetic and potential energy.

Solution: See ECE618 Notes, Part 1.

2. Problem. Consider the tunneling problem with the potential barrier:

V (z) =

0 for z ≤ 0V > 0 for 0 < z < L0 for z ≥ L

.

Write the wavefunction as:

ψ(z) =

Aeikz + Be−ikz for z ≤ 0

Ceκz + De−κz for 0 < z < L

Feikz for z ≥ L

,

with k = (2mE)1/2/h, κ = [2m(V − E)]1/2/h. a. Write the system of four equations expressing thecontinuity of the wavefunction and its derivative at z = 0 and at z = L.b. Find the transmission coefficient, T = |F |2/|A|2. There’s no need to solve the full system. Be creative.

ECE609 Spring 2010 1

Hint: Multiply the equation expressing continuity of ψ at z = 0 by ik and add and substract it from the equation expressing

continuity of the derivatives at z = 0. Do a similar thing with the other two equations (by multiplying one by κ). Now it should be

relatively easy to solve for F in terms of A alone. This gives you T .

Solution. a. The equations expressing continuity of the wavefunction and its derivatives at z = 0 and z = Lare:

ψ(0−) = ψ(0+) → A + B = C + D , (1)

dψ(0−)

dz=

dψ(0+)

dz→ ikA− ikB = κC − κD , (2)

ψ(L−

) = ψ(L+) → Ce

κL+ De

−κL= Fe

ikL, (3)

dψ(L−)

dz=

dψ(L+)

dz→ κCeκL − κDe−κL = ikFeikL . (4)

b. Multiply Eq. (1) by ik, add and substract it from Eq. (2):

2ikA = (ik + κ)C + (ik − κ)D , (5)

2ikB = (ik − κ)C + (ik + κ)D . (6)

Multiply Eq. (3) by κ, add and substract it from Eq. (4):

2κCeκL = (ik + κ)FeikL , (7)

2κDe−κL = −(ik − κ)FeikL . (8)

From Eqns. (7) and (8) we can express C and D in terms of F . Inserting these expressions into Eq. (5) we canfinally express F in terms of A:

2ikA =eikLF

2κ[(κ2 − k2) sinh(κL) + 2iκk cosh(κL)] , (9)

ECE609 Spring 2010 2

so that the transmission coeficient is:

T =|F |2|A|2 =

4κ2k2

(κ2 − k2)2 sinh2(κL) + 4κ2k2 cosh2(κL). (10)

3. Problem. From the Schrodinger equation derive the continuity equation:

∂ρ

∂t+ ∇ · S = 0 ,

where ρ = |Ψ|2 is the ‘probability density’ and S = ih2m[Ψ∇Ψ∗ −Ψ∗∇Ψ] is the ‘probability density current’.

Solution. By the definition of the probability density ρ:

∂ρ

∂t=

∂

∂t

[Ψ

∗Ψ

]=

∂Ψ∗

∂tΨ + Ψ

∗∂Ψ

∂t.

Since ∂Ψ/∂t = 1/(ih)HΨ and ∂Ψ∗/∂t = −1/(ih)HΨ∗,

∂ρ

∂t= − 1

ih(HΨ∗)Ψ +

1

ihΨ∗HΨ =

1

ih

[h2

2mΨ∇2Ψ∗ − V ΨΨ∗ − h2

2mΨ∗∇2Ψ + V Ψ∗Ψ

].

But the second and fourth terms inside the square bracket cancel, so that:

∂ρ

∂t=

−ih

2m

[Ψ∇2

Ψ∗ − Ψ

∗∇2Ψ

]. (11)

On the other hand, using the definition of the probability-flux S:

∇ · S =ih

2m∇ · [

Ψ∇Ψ∗ − Ψ

∗∇Ψ]

=ih

2m

[∇Ψ · ∇Ψ

∗+ Ψ∇2

Ψ∗ − ∇Ψ

∗ · ∇Ψ − Ψ∗∇2

Ψ]

=ih

2m

[Ψ∇2Ψ∗ − Ψ∗∇2Ψ

]. (12)

ECE609 Spring 2010 3

From Eqns. (11) and (12) we have ∂ρ/∂t = −∇ · S, which is the desired result.

4. Problem. The Wentzel-Kramers-Brillouin (WKB) approximation to solve the Schrodinger equation consists inwriting the solution of the time-independent problem:

− h2

2m

d2ψ(x)

dx2+ V (x)ψ(x) = Eψ(x) ,

as

ψ(x) ≈ 1

k1/2exp

{i

∫ x

k(x′) dx

′}

,

where k(x) = {2m[E − V (x)]}1/2/h. This is a good approximation if the potential V (x) varies slowly(that is, it does not change much compared to the electron energy E when x varies over several de Brogliewavelengths). If E − V (x) < 0, the WKB wavefunction becomes

ψ(x) ≈ 1

k1/2exp

{−

∫ x

κ(x′) dx′}

,

where now κ(x) = {2m[V (x) − E]}1/2/h.

Let’s now ignore the factor k−1/2 (which simply ensures continuity of probability current). Consider now theprevious tunneling problem (problem 2) and identify the WKB approximation to the transmission coefficient as:

TWKB = |ψ(L)|2 = exp

{−2

∫ L

0κ(x′) dx′

}. (13)

Compare TWKB with the ‘exact’ transmission coefficient T of the previous tunneling problem (problem 2) inthe limit in which κL >> 1.Solution. From Eq. (10) above, for κ >> L we have sinh(κL) ≈ cosh(κL) ≈ eκL/2, so

T ≈ 16κ2k2e−2κL

(κ2 + k2)2. (14)

ECE609 Spring 2010 4

On the other hand, from the WKB expression, Eq. (13), we have trivially (since inside the barrier (0 < x ≤ L)

κ(x) = [2m(E − V )]1/2/h is a constant independent of x):

TWKB ∼ e−2κL . (15)

Except for prefactors which we have ignored in the WKB expression, Eq. (15) agrees with the exponentialbehavior of Eq. (14).

5. Problem. Calculate the matrix element between two wavefunctions of the form

ψ(k, r) =1

V 1/2eik·r

and ψ(k′, r) =1

V 1/2eik′·r

,

and the pertubation potentials of the form:a. H ∝ eiq·rb. H ∝ δ(r)c. H ∝ |r|−2

d. H ∝ e−|r|/r0Polar coordinates are useful in c and d.Solution. In each case we must evaluate the integral∫

V

1

V 1/2e−ik′·r H(r)

1

V 1/2eik·r dr =

1

V

∫V

ei(k−k′)·r H(r) dr .

So:a. For H(r) = eiq·r:

1

V

∫Vei(k−k′+q)·r dr = δ(k − k′ + q) ,

thanks to the definition of the δ ‘function’.

ECE609 Spring 2010 5

b. For H(r) = δ(r):1

V

∫Vei(k−k′)·r

δ(r) dr =1

V.

c. For H(r) = |r|−2, going to polar coordinates and assuming a spherical volume 4πR3/3 = V :

1

V

∫V

ei(k−k′)·r

|r|2 dr =1

V

∫ 2π

0dφ

∫ π

0dθ sin θ

∫ R

0dr r

2 ei∆kr cos θ

r2=

= −2π

V

∫ R

0dr

∫ π

0d(cos θ) e

i∆kr cos θ,

where θ is the polar angle and we have assumed (without loss of generality) that the (polar) z-axis is along thedirection of k − k′ and ∆k = |k − k′|. With a change of integration variable µ = cos θ, this can be writtenas:

2π

V

∫ R

0dr

ei∆krµ

i∆krµ

∣∣∣∣∣µ=1

µ=−1

=2π

V

2

∆k

∫ R

0dr

sin(∆kr)

r→ 4π

V ∆k

π

2

in the limit R → ∞, since∫ ∞0 sin(x)/x = π/2 (see any text of tables of integrals: The integral can be

calculated rigorously by contour integrals in the complex plane).

d. For H(r) = e−|r|/r0, going again to polar coordinates:

1

V

∫V

ei(k−k′)·r e−|r|/r0 dr =1

V

∫ 2π

0dφ

∫ π

0dθ sin θ

∫ R

0dr r2ei∆kr cos θ e−r/r0 =

= −2π

V

∫ R

0dr r2 e−r/r0

∫ π

0d(cos θ) ei∆kr cos θ .

ECE609 Spring 2010 6

As before, let’s set µ = cos θ:

2π

V

∫ R

0dr r2 e−r/r0 ei∆krµ

i∆krµ

∣∣∣∣∣µ=1

µ=−1

=2π

V

2

∆k

∫ R

0dr r sin(∆kr) e−r/r0 .

The last integral can be solved by replacing sin(∆kr) with ei∆kr and retaining only the imaginary part of theresult. The integral is of the form: ∫ R

0dr r e

αr,

where α = −1/r0 + i∆k. Integrating by parts:

∫ R

0dr r eαr =

r

αeαr

∣∣∣∣R

0−

∫ R

0dr

1

αeαr =

r

αeαr

∣∣∣∣R

0+

eαr

α2

∣∣∣∣R

0=

1

α2,

in the limit R → ∞. Therefore:

2π

V

2

∆k

∫ R

0dr r sin(∆kr) e−r/r0 =

4π

V ∆kIm

[∫ R

0dr r er(i∆k−1/r0)

]=

4π

V

2

r0(1/r2

0 + ∆k2)2

.

6. Problem. Show in detail the equivalence between the two formulations of Bloch theorem:

ψ(k, r + Rl) = eik·Rl ψ(k, r) , (16)

and:

ψ(k, r) = eik·r uk(r) , (17)

where uk(r) is periodic:

uk(r + Rl) = uk(r) .

ECE609 Spring 2010 7

Solution. From Eq. (17):

ψ(k, r + Rl) = eik·(r+Rl) uk(r + Rl) . (18)

By the periodicity of u, uk(r + Rl) = uk(r), and from the previous equation:

ψ(k, r + Rl) = eik·Rle

ik·ruk(r) = e

ik·Rl ψ(k, r) , (19)

which is Eq. (16).

7. Problem. Find the reciprocal lattice vectors of the fcc lattice. As fundamental translation vectors use

a =a

2(x + y) , b =

a

2(y + z) , c =

a

2(z + x) .

b. Find the volume of the primitive cell and the density of valence electrons in cm−3 for Si (a = 0.543 nm).c. Find the volume of the BZ.d. Using a computer program, list the first 50-to-100 G-vectors in order of increasing magnitude.Solution. a. Using the definition of reciprocal lattice vectors A, B, and C (see notes, page 36):

A =2π

a(x + y − z) , B =

2π

a(−x + y + z) , C =

2π

a(x − y + z) .

b. The volume of the primitive cell will be:

|a · b × c| =a3

4.

Since we have 2 Si atoms in the primitive cell, each with 4 valence electrons, the electron density will be32a−3 ≈ 2 × 1023 cm−3.

ECE609 Spring 2010 8

c. The volume of the BZ will be:

|A · B × C| = 4

(2π

a

)3

≈ 6.2 × 1023

cm−3

d. The G-vectors are of the form:G = l1A + l2B + l3C ,

where A, B, and C are the fundamental translation vectors in reciprocal space given above and l1, l2, and l3are integers. Therefore the first G-vectors are (in units of 2π/a):the 8 vectors:

(1, 1, 1) (−1, 1, 1) (1,−1, 1) (1, 1,−1)

(−1,−1, 1) (1,−1,−1) (−1, 1,−1) (−1,−1,−1) of length√

3 ;

the 6 vectors:

(2, 0, 0) (−2, 0, 0) (0, 2, 0) (0,−2, 0) (0, 0, 2) (0, 0,−2) of length√

4 ;

the 12 vectors:

(2, 2, 0) (−2, 2, 0) (2,−2, 0) (−2,−2, 0)(2, 0, 2) (−2, 0, 2) (2, 0,−2) (−2, 0,−2)

(0, 2, 2) (0,−2, 2) (0, 2,−2) (0,−2,−2) of length√

8 ;

the 24 vectors:

(3, 1, 1) (3,−1, 1) (3, 1,−1) (3,−1,−1)(−3, 1, 1) (−3,−1, 1) (−3, 1,−1) (−3,−1,−1)(1, 3, 1) (−1, 3, 1) (1, 3,−1) (−1, 3,−1, )(1, 3, 1) (−1, 3, 1) (1, 3,−1) (−1, 3,−1, )(1, 1, 3) (−1, 1, 3) (1,−1, 3) (−1,−1, 3)

(1, 1,−3) (−1, 1,−3) (1,−1,−3) (−1,−1,−3) of length√

11 ;

ECE609 Spring 2010 9

the 8 vectors:

(2, 2, 2) (−2, 2, 2) (2,−2, 2) (2, 2,−2)

(−2,−2, 2) (2,−2,−2) (−2, 2,−2) (−2,−2,−2) of length√

12 ;

...etc.

On the next pages I list the FORTRAN program used to generate G-vectors up to a magnitude of√

59 and theoutput list.

ECE609 Spring 2010 10

Gvectors_FORTRAN* Define A,B,C A(1) = 1 A(2) = 1 A(3) = -1 B(1) = -1 B(2) = 1 B(3) = 1 C(1) = 1 C(2) = -1 C(3) = 1* Define cutoff magnitude Gmax = 59.0* Define max magnitude of G(l1,l2,l3) n = 31* Generate G-vectors ig = 1 do l1 = 1, 2*n+1 do l2 = 1, 2*n+1 do l3 = 1, 2*n+1 g(1,ig) = (l1-n)*A(1) + (l2-n)*B(1) + (l3-n)*C(1) g(2,ig) = (l1-n)*A(2) + (l2-n)*B(2) + (l3-n)l3*C(2) g(3,ig) = (l1-n)*A(3) + (l2-n)*B(3) + (l3-n)*C(3) g(4,ig) = g(1,ig)**2 + g(2,ig)**2 + g(3,ig)**2 if(g(4,ig).le.gmax**2) then ig = ig + 1 else go to 1 endif enddo enddo enddo 1 continue ng = ig* Reorder them by increasing magnitude do ig = 1, ng do jg = 1, ng if(g(4,jg).ge.g(4,ig)) go to 20 do ir = 1, 4 q = g(ir,ig) g(ir,ig) = g(ir,jg) g(ir,jg) = q enddo enddo enddo 20 continue

Page 1

Gvectors_list G-vector number: 1 G = .0 .0 .0 |G|^2 = .0

2 G = 1.0 1.0 -1.0 |G|^2 = 3.0 3 G = -1.0 -1.0 -1.0 |G|^2 = 3.0 4 G = 1.0 -1.0 1.0 |G|^2 = 3.0 5 G = 1.0 -1.0 -1.0 |G|^2 = 3.0 6 G = -1.0 1.0 -1.0 |G|^2 = 3.0 7 G = 1.0 1.0 1.0 |G|^2 = 3.0 8 G = -1.0 -1.0 1.0 |G|^2 = 3.0 9 G = -1.0 1.0 1.0 |G|^2 = 3.0

10 G = .0 .0 -2.0 |G|^2 = 4.0 11 G = .0 2.0 .0 |G|^2 = 4.0 12 G = .0 -2.0 .0 |G|^2 = 4.0 13 G = -2.0 .0 .0 |G|^2 = 4.0 14 G = .0 .0 2.0 |G|^2 = 4.0 15 G = 2.0 .0 .0 |G|^2 = 4.0

16 G = .0 -2.0 -2.0 |G|^2 = 8.0 17 G = 2.0 .0 -2.0 |G|^2 = 8.0 18 G = .0 2.0 2.0 |G|^2 = 8.0 19 G = -2.0 .0 2.0 |G|^2 = 8.0 20 G = .0 -2.0 2.0 |G|^2 = 8.0 21 G = 2.0 -2.0 .0 |G|^2 = 8.0 22 G = .0 2.0 -2.0 |G|^2 = 8.0 23 G = 2.0 .0 2.0 |G|^2 = 8.0 24 G = -2.0 .0 -2.0 |G|^2 = 8.0 25 G = -2.0 2.0 .0 |G|^2 = 8.0 26 G = 2.0 2.0 .0 |G|^2 = 8.0 27 G = -2.0 -2.0 .0 |G|^2 = 8.0

28 G = 1.0 3.0 1.0 |G|^2 = 11.0 29 G = -1.0 -3.0 1.0 |G|^2 = 11.0 30 G = -1.0 -1.0 -3.0 |G|^2 = 11.0 31 G = 1.0 1.0 -3.0 |G|^2 = 11.0 32 G = 1.0 3.0 -1.0 |G|^2 = 11.0 33 G = 1.0 -1.0 -3.0 |G|^2 = 11.0 34 G = 1.0 -1.0 3.0 |G|^2 = 11.0 35 G = 1.0 3.0 -1.0 |G|^2 = 11.0 36 G = -1.0 -3.0 -1.0 |G|^2 = 11.0 37 G = 1.0 -3.0 1.0 |G|^2 = 11.0 38 G = 1.0 1.0 3.0 |G|^2 = 11.0 39 G = -1.0 1.0 -3.0 |G|^2 = 11.0 40 G = -1.0 3.0 1.0 |G|^2 = 11.0 41 G = -1.0 -1.0 3.0 |G|^2 = 11.0 42 G = -1.0 3.0 -1.0 |G|^2 = 11.0 43 G = -1.0 1.0 3.0 |G|^2 = 11.0 44 G = 3.0 1.0 1.0 |G|^2 = 11.0 45 G = -3.0 1.0 -1.0 |G|^2 = 11.0 46 G = 3.0 1.0 -1.0 |G|^2 = 11.0 47 G = -3.0 -1.0 1.0 |G|^2 = 11.0 48 G = -3.0 -1.0 -1.0 |G|^2 = 11.0 49 G = 3.0 -1.0 -1.0 |G|^2 = 11.0 50 G = -3.0 1.0 1.0 |G|^2 = 11.0 51 G = 3.0 -1.0 1.0 |G|^2 = 11.0

52 G = 2.0 -2.0 -2.0 |G|^2 = 12.0 53 G = -2.0 -2.0 2.0 |G|^2 = 12.0 54 G = -2.0 -2.0 -2.0 |G|^2 = 12.0 55 G = 2.0 2.0 -2.0 |G|^2 = 12.0 56 G = 2.0 2.0 2.0 |G|^2 = 12.0

Page 1

Gvectors_list 57 G = -2.0 2.0 -2.0 |G|^2 = 12.0 58 G = 2.0 -2.0 2.0 |G|^2 = 12.0 59 G = -2.0 2.0 2.0 |G|^2 = 12.0

60 G = .0 -4.0 .0 |G|^2 = 16.0 61 G = .0 .0 -4.0 |G|^2 = 16.0 62 G = 4.0 .0 .0 |G|^2 = 16.0 63 G = .0 .0 4.0 |G|^2 = 16.0 64 G = .0 4.0 .0 |G|^2 = 16.0 65 G = -4.0 .0 .0 |G|^2 = 16.0

66 G = -1.0 -3.0 -3.0 |G|^2 = 19.0 67 G = 1.0 -3.0 -3.0 |G|^2 = 19.0 68 G = 3.0 -1.0 -3.0 |G|^2 = 19.0 69 G = 1.0 3.0 -3.0 |G|^2 = 19.0 70 G = -1.0 3.0 -3.0 |G|^2 = 19.0 71 G = 1.0 3.0 3.0 |G|^2 = 19.0 72 G = -1.0 -3.0 3.0 |G|^2 = 19.0 73 G = 1.0 -3.0 3.0 |G|^2 = 19.0 74 G = -1.0 3.0 3.0 |G|^2 = 19.0 75 G = -3.0 1.0 3.0 |G|^2 = 19.0 76 G = 3.0 -3.0 1.0 |G|^2 = 19.0 77 G = 3.0 3.0 1.0 |G|^2 = 19.0 78 G = 3.0 -3.0 -1.0 |G|^2 = 19.0 79 G = -3.0 -3.0 -1.0 |G|^2 = 19.0 80 G = 3.0 -1.0 3.0 |G|^2 = 19.0 81 G = 3.0 3.0 -1.0 |G|^2 = 19.0 82 G = -3.0 3.0 -1.0 |G|^2 = 19.0 83 G = -3.0 1.0 -3.0 |G|^2 = 19.0 84 G = 3.0 1.0 -3.0 |G|^2 = 19.0 85 G = -3.0 -1.0 -3.0 |G|^2 = 19.0 86 G = -3.0 3.0 1.0 |G|^2 = 19.0 87 G = 3.0 1.0 3.0 |G|^2 = 19.0 88 G = -3.0 -1.0 3.0 |G|^2 = 19.0 89 G = -3.0 -3.0 1.0 |G|^2 = 19.0

90 G = .0 4.0 2.0 |G|^2 = 20.0 91 G = .0 2.0 4.0 |G|^2 = 20.0 92 G = .0 -4.0 -2.0 |G|^2 = 20.0 93 G = 2.0 .0 -4.0 |G|^2 = 20.0 94 G = -2.0 .0 4.0 |G|^2 = 20.0 95 G = .0 -2.0 -4.0 |G|^2 = 20.0 96 G = -2.0 -4.0 .0 |G|^2 = 20.0 97 G = 4.0 -2.0 .0 |G|^2 = 20.0 98 G = .0 -4.0 2.0 |G|^2 = 20.0 99 G = .0 -2.0 4.0 |G|^2 = 20.0 100 G = .0 4.0 -2.0 |G|^2 = 20.0 101 G = 2.0 -4.0 .0 |G|^2 = 20.0 102 G = 2.0 .0 4.0 |G|^2 = 20.0 103 G = -4.0 2.0 .0 |G|^2 = 20.0 104 G = 2.0 4.0 .0 |G|^2 = 20.0 105 G = -4.0 .0 -2.0 |G|^2 = 20.0 106 G = 4.0 2.0 .0 |G|^2 = 20.0 107 G = -2.0 4.0 .0 |G|^2 = 20.0 108 G = 4.0 .0 -2.0 |G|^2 = 20.0 109 G = 4.0 .0 2.0 |G|^2 = 20.0 110 G = .0 2.0 -4.0 |G|^2 = 20.0 111 G = -4.0 .0 2.0 |G|^2 = 20.0 112 G = -4.0 -2.0 .0 |G|^2 = 20.0 113 G = -2.0 .0 -4.0 |G|^2 = 20.0

114 G = 2.0 2.0 4.0 |G|^2 = 24.0 115 G = -2.0 -2.0 -4.0 |G|^2 = 24.0

Page 2

Gvectors_list 116 G = 2.0 -2.0 -4.0 |G|^2 = 24.0 117 G = -2.0 2.0 4.0 |G|^2 = 24.0 118 G = 4.0 2.0 -2.0 |G|^2 = 24.0 119 G = 2.0 -4.0 -2.0 |G|^2 = 24.0 120 G = -2.0 -4.0 -2.0 |G|^2 = 24.0 121 G = -2.0 4.0 -2.0 |G|^2 = 24.0 122 G = -2.0 2.0 -4.0 |G|^2 = 24.0 123 G = 4.0 -2.0 2.0 |G|^2 = 24.0 124 G = 4.0 -2.0 -2.0 |G|^2 = 24.0 125 G = 2.0 4.0 -2.0 |G|^2 = 24.0 126 G = 2.0 4.0 2.0 |G|^2 = 24.0 127 G = -2.0 -4.0 2.0 |G|^2 = 24.0 128 G = -4.0 2.0 -2.0 |G|^2 = 24.0 129 G = -2.0 4.0 2.0 |G|^2 = 24.0 130 G = -4.0 -2.0 2.0 |G|^2 = 24.0 131 G = -2.0 -2.0 4.0 |G|^2 = 24.0 132 G = 2.0 -2.0 4.0 |G|^2 = 24.0 133 G = 2.0 -4.0 2.0 |G|^2 = 24.0 134 G = 2.0 2.0 -4.0 |G|^2 = 24.0 135 G = -4.0 -2.0 -2.0 |G|^2 = 24.0 136 G = -4.0 2.0 2.0 |G|^2 = 24.0 137 G = 4.0 2.0 2.0 |G|^2 = 24.0

138 G = 1.0 1.0 5.0 |G|^2 = 27.0 139 G = -1.0 1.0 -5.0 |G|^2 = 27.0 140 G = 1.0 5.0 -1.0 |G|^2 = 27.0 141 G = -1.0 5.0 -1.0 |G|^2 = 27.0 142 G = -1.0 -5.0 -1.0 |G|^2 = 27.0 143 G = 1.0 5.0 1.0 |G|^2 = 27.0 144 G = -1.0 -5.0 1.0 |G|^2 = 27.0 145 G = 1.0 -1.0 5.0 |G|^2 = 27.0 146 G = -1.0 -1.0 5.0 |G|^2 = 27.0 147 G = 1.0 -1.0 -5.0 |G|^2 = 27.0 148 G = 1.0 -5.0 1.0 |G|^2 = 27.0 149 G = -1.0 -1.0 -5.0 |G|^2 = 27.0 150 G = -1.0 1.0 5.0 |G|^2 = 27.0 151 G = 1.0 1.0 -5.0 |G|^2 = 27.0 152 G = 3.0 -3.0 3.0 |G|^2 = 27.0 153 G = 1.0 -5.0 -1.0 |G|^2 = 27.0 154 G = 3.0 -3.0 -3.0 |G|^2 = 27.0 155 G = -5.0 1.0 -1.0 |G|^2 = 27.0 156 G = -3.0 -3.0 3.0 |G|^2 = 27.0 157 G = 5.0 1.0 -1.0 |G|^2 = 27.0 158 G = -5.0 -1.0 1.0 |G|^2 = 27.0 159 G = 5.0 1.0 1.0 |G|^2 = 27.0 160 G = -3.0 -3.0 -3.0 |G|^2 = 27.0 161 G = -3.0 3.0 -3.0 |G|^2 = 27.0 162 G = -5.0 1.0 1.0 |G|^2 = 27.0 163 G = -1.0 5.0 1.0 |G|^2 = 27.0 164 G = 5.0 -1.0 1.0 |G|^2 = 27.0 165 G = 3.0 3.0 3.0 |G|^2 = 27.0 166 G = 5.0 -1.0 -1.0 |G|^2 = 27.0 167 G = -5.0 -1.0 -1.0 |G|^2 = 27.0 168 G = 3.0 3.0 -3.0 |G|^2 = 27.0 169 G = -3.0 3.0 3.0 |G|^2 = 27.0

170 G = .0 4.0 4.0 |G|^2 = 32.0 171 G = .0 -4.0 -4.0 |G|^2 = 32.0 172 G = 4.0 .0 -4.0 |G|^2 = 32.0 173 G = -4.0 .0 4.0 |G|^2 = 32.0 174 G = 4.0 4.0 .0 |G|^2 = 32.0 175 G = .0 -4.0 4.0 |G|^2 = 32.0 176 G = 4.0 -4.0 .0 |G|^2 = 32.0

Page 3

Gvectors_list 177 G = -4.0 4.0 .0 |G|^2 = 32.0 178 G = -4.0 -4.0 .0 |G|^2 = 32.0 179 G = .0 4.0 -4.0 |G|^2 = 32.0 180 G = 4.0 .0 4.0 |G|^2 = 32.0 181 G = -4.0 .0 -4.0 |G|^2 = 32.0

182 G = -1.0 5.0 -3.0 |G|^2 = 35.0 183 G = 1.0 -5.0 3.0 |G|^2 = 35.0 184 G = 1.0 3.0 -5.0 |G|^2 = 35.0 185 G = -1.0 -3.0 5.0 |G|^2 = 35.0 186 G = 1.0 -5.0 -3.0 |G|^2 = 35.0 187 G = 1.0 -3.0 -5.0 |G|^2 = 35.0 188 G = 1.0 5.0 -3.0 |G|^2 = 35.0 189 G = -1.0 -3.0 -5.0 |G|^2 = 35.0 190 G = -1.0 -5.0 -3.0 |G|^2 = 35.0 191 G = 1.0 5.0 3.0 |G|^2 = 35.0 192 G = 1.0 -3.0 5.0 |G|^2 = 35.0 193 G = -1.0 3.0 -5.0 |G|^2 = 35.0 194 G = -1.0 -5.0 3.0 |G|^2 = 35.0 195 G = 1.0 3.0 5.0 |G|^2 = 35.0 196 G = 5.0 1.0 -3.0 |G|^2 = 35.0 197 G = -1.0 3.0 5.0 |G|^2 = 35.0 198 G = -5.0 -1.0 3.0 |G|^2 = 35.0 199 G = -1.0 5.0 3.0 |G|^2 = 35.0

200 G = -3.0 -1.0 -5.0 |G|^2 = 35.0 201 G = 3.0 -5.0 1.0 |G|^2 = 35.0 202 G = 5.0 1.0 3.0 |G|^2 = 35.0 203 G = -3.0 5.0 1.0 |G|^2 = 35.0 204 G = 5.0 -3.0 -1.0 |G|^2 = 35.0 205 G = -5.0 -1.0 -3.0 |G|^2 = 35.0 206 G = 5.0 3.0 1.0 |G|^2 = 35.0 207 G = -5.0 -3.0 1.0 |G|^2 = 35.0 208 G = 3.0 -1.0 5.0 |G|^2 = 35.0 209 G = 5.0 3.0 -1.0 |G|^2 = 35.0 210 G = 3.0 5.0 -1.0 |G|^2 = 35.0 211 G = 3.0 5.0 1.0 |G|^2 = 35.0 212 G = 3.0 1.0 -5.0 |G|^2 = 35.0 213 G = -3.0 1.0 -5.0 |G|^2 = 35.0 214 G = -3.0 5.0 -1.0 |G|^2 = 35.0 215 G = 3.0 1.0 5.0 |G|^2 = 35.0 216 G = -5.0 1.0 -3.0 |G|^2 = 35.0 217 G = 5.0 -1.0 3.0 |G|^2 = 35.0 218 G = -5.0 3.0 -1.0 |G|^2 = 35.0 219 G = 5.0 -1.0 -3.0 |G|^2 = 35.0 220 G = -5.0 3.0 1.0 |G|^2 = 35.0 221 G = 5.0 -3.0 1.0 |G|^2 = 35.0 222 G = 3.0 -1.0 -5.0 |G|^2 = 35.0 223 G = -3.0 -5.0 1.0 |G|^2 = 35.0 224 G = 3.0 -5.0 -1.0 |G|^2 = 35.0 225 G = -5.0 -3.0 -1.0 |G|^2 = 35.0 226 G = -3.0 -1.0 5.0 |G|^2 = 35.0 227 G = -3.0 -5.0 -1.0 |G|^2 = 35.0 228 G = -5.0 1.0 3.0 |G|^2 = 35.0 229 G = -3.0 1.0 5.0 |G|^2 = 35.0

230 G = 2.0 4.0 -4.0 |G|^2 = 36.0 231 G = -2.0 -4.0 4.0 |G|^2 = 36.0 232 G = 4.0 -2.0 -4.0 |G|^2 = 36.0 233 G = 2.0 -4.0 4.0 |G|^2 = 36.0 234 G = 4.0 2.0 4.0 |G|^2 = 36.0 235 G = -6.0 .0 .0 |G|^2 = 36.0 236 G = 4.0 -4.0 2.0 |G|^2 = 36.0

Page 4

Gvectors_list 237 G = 4.0 4.0 -2.0 |G|^2 = 36.0 238 G = .0 .0 -6.0 |G|^2 = 36.0 239 G = 2.0 4.0 4.0 |G|^2 = 36.0 240 G = .0 6.0 .0 |G|^2 = 36.0 241 G = 4.0 -2.0 4.0 |G|^2 = 36.0 242 G = -4.0 -4.0 -2.0 |G|^2 = 36.0 243 G = -2.0 4.0 -4.0 |G|^2 = 36.0 244 G = 6.0 .0 .0 |G|^2 = 36.0 245 G = 4.0 4.0 2.0 |G|^2 = 36.0 246 G = .0 .0 6.0 |G|^2 = 36.0 247 G = 4.0 -4.0 -2.0 |G|^2 = 36.0 248 G = .0 -6.0 .0 |G|^2 = 36.0 249 G = -4.0 4.0 -2.0 |G|^2 = 36.0 250 G = -4.0 4.0 2.0 |G|^2 = 36.0 251 G = 4.0 2.0 -4.0 |G|^2 = 36.0 252 G = -2.0 4.0 4.0 |G|^2 = 36.0 253 G = -4.0 -2.0 4.0 |G|^2 = 36.0 254 G = -2.0 -4.0 -4.0 |G|^2 = 36.0 255 G = -4.0 2.0 -4.0 |G|^2 = 36.0 256 G = 2.0 -4.0 -4.0 |G|^2 = 36.0 257 G = -4.0 -2.0 -4.0 |G|^2 = 36.0 258 G = -4.0 -4.0 2.0 |G|^2 = 36.0 259 G = -4.0 2.0 4.0 |G|^2 = 36.0

260 G = -2.0 -6.0 .0 |G|^2 = 40.0 261 G = 2.0 6.0 .0 |G|^2 = 40.0 262 G = .0 -2.0 6.0 |G|^2 = 40.0 263 G = .0 -2.0 -6.0 |G|^2 = 40.0 264 G = .0 2.0 -6.0 |G|^2 = 40.0 265 G = -2.0 .0 -6.0 |G|^2 = 40.0 266 G = .0 2.0 6.0 |G|^2 = 40.0 267 G = 2.0 .0 6.0 |G|^2 = 40.0 268 G = -2.0 6.0 .0 |G|^2 = 40.0 269 G = -6.0 .0 -2.0 |G|^2 = 40.0 270 G = 2.0 -6.0 .0 |G|^2 = 40.0 271 G = 2.0 .0 -6.0 |G|^2 = 40.0 272 G = .0 6.0 2.0 |G|^2 = 40.0 273 G = 6.0 2.0 .0 |G|^2 = 40.0 274 G = -6.0 2.0 .0 |G|^2 = 40.0 275 G = 6.0 .0 -2.0 |G|^2 = 40.0 276 G = .0 6.0 -2.0 |G|^2 = 40.0 277 G = -6.0 .0 2.0 |G|^2 = 40.0 278 G = -6.0 -2.0 .0 |G|^2 = 40.0 279 G = .0 -6.0 -2.0 |G|^2 = 40.0 280 G = .0 -6.0 2.0 |G|^2 = 40.0 281 G = -2.0 .0 6.0 |G|^2 = 40.0 282 G = 6.0 -2.0 .0 |G|^2 = 40.0 283 G = 6.0 .0 2.0 |G|^2 = 40.0

284 G = 3.0 -3.0 5.0 |G|^2 = 43.0 285 G = 3.0 3.0 5.0 |G|^2 = 43.0 286 G = 3.0 -3.0 -5.0 |G|^2 = 43.0 287 G = 3.0 3.0 -5.0 |G|^2 = 43.0 288 G = -3.0 -3.0 5.0 |G|^2 = 43.0 289 G = -3.0 3.0 -5.0 |G|^2 = 43.0 290 G = -3.0 -3.0 -5.0 |G|^2 = 43.0 291 G = -3.0 3.0 5.0 |G|^2 = 43.0 292 G = -5.0 3.0 -3.0 |G|^2 = 43.0 293 G = -3.0 5.0 3.0 |G|^2 = 43.0 294 G = -5.0 -3.0 3.0 |G|^2 = 43.0 295 G = 3.0 -5.0 3.0 |G|^2 = 43.0 296 G = -3.0 5.0 -3.0 |G|^2 = 43.0 297 G = -3.0 -5.0 -3.0 |G|^2 = 43.0

Page 5

Gvectors_list 298 G = 5.0 -3.0 3.0 |G|^2 = 43.0 299 G = 5.0 3.0 3.0 |G|^2 = 43.0 300 G = 3.0 5.0 3.0 |G|^2 = 43.0 301 G = 3.0 5.0 -3.0 |G|^2 = 43.0 302 G = -3.0 -5.0 3.0 |G|^2 = 43.0 303 G = 5.0 3.0 -3.0 |G|^2 = 43.0 304 G = 3.0 -5.0 -3.0 |G|^2 = 43.0 305 G = -5.0 -3.0 -3.0 |G|^2 = 43.0 306 G = -5.0 3.0 3.0 |G|^2 = 43.0 307 G = 5.0 -3.0 -3.0 |G|^2 = 43.0

308 G = 2.0 6.0 -2.0 |G|^2 = 44.0 309 G = 2.0 -6.0 -2.0 |G|^2 = 44.0 310 G = -2.0 -2.0 -6.0 |G|^2 = 44.0 311 G = 2.0 -6.0 2.0 |G|^2 = 44.0 312 G = -2.0 -6.0 -2.0 |G|^2 = 44.0 313 G = 2.0 6.0 2.0 |G|^2 = 44.0 314 G = -2.0 -2.0 6.0 |G|^2 = 44.0 315 G = -2.0 -6.0 2.0 |G|^2 = 44.0 316 G = 2.0 -2.0 6.0 |G|^2 = 44.0 317 G = 2.0 2.0 6.0 |G|^2 = 44.0 318 G = -2.0 2.0 -6.0 |G|^2 = 44.0 319 G = 2.0 2.0 -6.0 |G|^2 = 44.0 320 G = -2.0 6.0 -2.0 |G|^2 = 44.0 321 G = -2.0 6.0 2.0 |G|^2 = 44.0 322 G = -2.0 2.0 6.0 |G|^2 = 44.0 323 G = 2.0 -2.0 -6.0 |G|^2 = 44.0 324 G = -6.0 -2.0 -2.0 |G|^2 = 44.0 325 G = -6.0 2.0 2.0 |G|^2 = 44.0 326 G = 6.0 2.0 2.0 |G|^2 = 44.0 327 G = 6.0 -2.0 -2.0 |G|^2 = 44.0 328 G = -6.0 2.0 -2.0 |G|^2 = 44.0 329 G = 6.0 2.0 -2.0 |G|^2 = 44.0 330 G = 6.0 -2.0 2.0 |G|^2 = 44.0 331 G = -6.0 -2.0 2.0 |G|^2 = 44.0

332 G = 4.0 4.0 -4.0 |G|^2 = 48.0 333 G = 4.0 -4.0 -4.0 |G|^2 = 48.0 334 G = -4.0 4.0 -4.0 |G|^2 = 48.0 335 G = -4.0 -4.0 4.0 |G|^2 = 48.0 336 G = 4.0 4.0 4.0 |G|^2 = 48.0 337 G = 4.0 -4.0 4.0 |G|^2 = 48.0 338 G = -4.0 -4.0 -4.0 |G|^2 = 48.0 339 G = -4.0 4.0 4.0 |G|^2 = 48.0

340 G = -1.0 5.0 -5.0 |G|^2 = 51.0 341 G = 1.0 -5.0 5.0 |G|^2 = 51.0 342 G = -1.0 -5.0 5.0 |G|^2 = 51.0 343 G = 1.0 5.0 5.0 |G|^2 = 51.0 344 G = 1.0 5.0 -5.0 |G|^2 = 51.0 345 G = -1.0 -5.0 -5.0 |G|^2 = 51.0 346 G = 5.0 1.0 -5.0 |G|^2 = 51.0 347 G = -1.0 -1.0 -7.0 |G|^2 = 51.0 348 G = 1.0 -1.0 7.0 |G|^2 = 51.0 349 G = 7.0 1.0 1.0 |G|^2 = 51.0 350 G = 1.0 7.0 -1.0 |G|^2 = 51.0 351 G = -1.0 7.0 -1.0 |G|^2 = 51.0 352 G = -5.0 1.0 -5.0 |G|^2 = 51.0 353 G = -1.0 -7.0 -1.0 |G|^2 = 51.0 354 G = 1.0 1.0 7.0 |G|^2 = 51.0 355 G = -5.0 -5.0 1.0 |G|^2 = 51.0 356 G = 1.0 -1.0 -7.0 |G|^2 = 51.0 357 G = 1.0 7.0 1.0 |G|^2 = 51.0

Page 6

Gvectors_list 358 G = -1.0 -7.0 1.0 |G|^2 = 51.0 359 G = -1.0 5.0 5.0 |G|^2 = 51.0 360 G = 1.0 -5.0 -5.0 |G|^2 = 51.0 361 G = -7.0 -1.0 -1.0 |G|^2 = 51.0 362 G = 7.0 -1.0 1.0 |G|^2 = 51.0 363 G = -7.0 -1.0 1.0 |G|^2 = 51.0 364 G = -1.0 7.0 1.0 |G|^2 = 51.0 365 G = -5.0 -5.0 -1.0 |G|^2 = 51.0 366 G = 1.0 -7.0 1.0 |G|^2 = 51.0 367 G = 7.0 -1.0 -1.0 |G|^2 = 51.0 368 G = -7.0 1.0 1.0 |G|^2 = 51.0 369 G = 5.0 -1.0 5.0 |G|^2 = 51.0 370 G = -7.0 1.0 -1.0 |G|^2 = 51.0 371 G = 5.0 5.0 1.0 |G|^2 = 51.0 372 G = 1.0 -7.0 -1.0 |G|^2 = 51.0 373 G = 5.0 5.0 -1.0 |G|^2 = 51.0 374 G = -1.0 -1.0 7.0 |G|^2 = 51.0 375 G = 7.0 1.0 -1.0 |G|^2 = 51.0 376 G = -5.0 5.0 -1.0 |G|^2 = 51.0 377 G = -5.0 -1.0 -5.0 |G|^2 = 51.0 378 G = 5.0 -5.0 1.0 |G|^2 = 51.0 379 G = -1.0 1.0 7.0 |G|^2 = 51.0 380 G = 1.0 1.0 -7.0 |G|^2 = 51.0 381 G = 5.0 1.0 5.0 |G|^2 = 51.0 382 G = -1.0 1.0 -7.0 |G|^2 = 51.0 383 G = -5.0 -1.0 5.0 |G|^2 = 51.0 384 G = 5.0 -5.0 -1.0 |G|^2 = 51.0 385 G = 5.0 -1.0 -5.0 |G|^2 = 51.0 386 G = -5.0 5.0 1.0 |G|^2 = 51.0 387 G = -5.0 1.0 5.0 |G|^2 = 51.0

388 G = 4.0 6.0 .0 |G|^2 = 52.0 389 G = 6.0 4.0 .0 |G|^2 = 52.0 390 G = -4.0 -6.0 .0 |G|^2 = 52.0 391 G = -6.0 -4.0 .0 |G|^2 = 52.0 392 G = -4.0 .0 -6.0 |G|^2 = 52.0 393 G = -4.0 6.0 .0 |G|^2 = 52.0 394 G = .0 -4.0 6.0 |G|^2 = 52.0 395 G = .0 -4.0 -6.0 |G|^2 = 52.0 396 G = 4.0 .0 6.0 |G|^2 = 52.0 397 G = .0 4.0 6.0 |G|^2 = 52.0 398 G = 4.0 -6.0 .0 |G|^2 = 52.0 399 G = .0 4.0 -6.0 |G|^2 = 52.0 400 G = -6.0 4.0 .0 |G|^2 = 52.0 401 G = 4.0 .0 -6.0 |G|^2 = 52.0 402 G = 6.0 .0 -4.0 |G|^2 = 52.0 403 G = 6.0 .0 4.0 |G|^2 = 52.0 404 G = .0 6.0 -4.0 |G|^2 = 52.0 405 G = -6.0 .0 -4.0 |G|^2 = 52.0 406 G = .0 -6.0 4.0 |G|^2 = 52.0 407 G = 6.0 -4.0 .0 |G|^2 = 52.0 408 G = .0 -6.0 -4.0 |G|^2 = 52.0 409 G = -6.0 .0 4.0 |G|^2 = 52.0 410 G = -4.0 .0 6.0 |G|^2 = 52.0 411 G = .0 6.0 4.0 |G|^2 = 52.0

412 G = 2.0 -4.0 -6.0 |G|^2 = 56.0 413 G = -2.0 4.0 6.0 |G|^2 = 56.0 414 G = 2.0 6.0 4.0 |G|^2 = 56.0 415 G = -2.0 -4.0 -6.0 |G|^2 = 56.0 416 G = 2.0 4.0 6.0 |G|^2 = 56.0 417 G = 4.0 -2.0 -6.0 |G|^2 = 56.0 418 G = -2.0 -6.0 -4.0 |G|^2 = 56.0

Page 7

Gvectors_list 419 G = -4.0 2.0 6.0 |G|^2 = 56.0 420 G = -6.0 -2.0 -4.0 |G|^2 = 56.0 421 G = 6.0 2.0 4.0 |G|^2 = 56.0 422 G = -2.0 -4.0 6.0 |G|^2 = 56.0 423 G = -4.0 -2.0 -6.0 |G|^2 = 56.0 424 G = -4.0 6.0 -2.0 |G|^2 = 56.0 425 G = -4.0 -6.0 2.0 |G|^2 = 56.0 426 G = 6.0 4.0 2.0 |G|^2 = 56.0 427 G = 4.0 6.0 -2.0 |G|^2 = 56.0 428 G = -2.0 -6.0 4.0 |G|^2 = 56.0 429 G = 4.0 2.0 -6.0 |G|^2 = 56.0 430 G = 6.0 4.0 -2.0 |G|^2 = 56.0 431 G = 4.0 6.0 2.0 |G|^2 = 56.0 432 G = 2.0 -6.0 4.0 |G|^2 = 56.0 433 G = -2.0 6.0 -4.0 |G|^2 = 56.0 434 G = 4.0 2.0 6.0 |G|^2 = 56.0 435 G = -4.0 2.0 -6.0 |G|^2 = 56.0 436 G = -2.0 4.0 -6.0 |G|^2 = 56.0 437 G = -4.0 -6.0 -2.0 |G|^2 = 56.0 438 G = 6.0 2.0 -4.0 |G|^2 = 56.0 439 G = 4.0 -2.0 6.0 |G|^2 = 56.0 440 G = 2.0 6.0 -4.0 |G|^2 = 56.0 441 G = 4.0 -6.0 2.0 |G|^2 = 56.0 442 G = -4.0 6.0 2.0 |G|^2 = 56.0 443 G = -6.0 -2.0 4.0 |G|^2 = 56.0 444 G = -4.0 -2.0 6.0 |G|^2 = 56.0 445 G = 2.0 -4.0 6.0 |G|^2 = 56.0 446 G = 2.0 4.0 -6.0 |G|^2 = 56.0 447 G = -6.0 -4.0 2.0 |G|^2 = 56.0 448 G = -6.0 -4.0 -2.0 |G|^2 = 56.0 449 G = 4.0 -6.0 -2.0 |G|^2 = 56.0 450 G = 6.0 -2.0 4.0 |G|^2 = 56.0 451 G = 2.0 -6.0 -4.0 |G|^2 = 56.0 452 G = -2.0 6.0 4.0 |G|^2 = 56.0 453 G = -6.0 4.0 2.0 |G|^2 = 56.0 454 G = 6.0 -4.0 2.0 |G|^2 = 56.0 455 G = 6.0 -4.0 -2.0 |G|^2 = 56.0 456 G = -6.0 2.0 -4.0 |G|^2 = 56.0 457 G = -6.0 4.0 -2.0 |G|^2 = 56.0 458 G = -6.0 2.0 4.0 |G|^2 = 56.0 459 G = 6.0 -2.0 -4.0 |G|^2 = 56.0

460 G = 1.0 -3.0 -7.0 |G|^2 = 59.0 461 G = 3.0 -1.0 -7.0 |G|^2 = 59.0 462 G = -3.0 1.0 7.0 |G|^2 = 59.0 463 G = -1.0 3.0 7.0 |G|^2 = 59.0 464 G = -3.0 -5.0 -5.0 |G|^2 = 59.0 465 G = -3.0 5.0 5.0 |G|^2 = 59.0 466 G = 7.0 3.0 -1.0 |G|^2 = 59.0 467 G = 3.0 -5.0 -5.0 |G|^2 = 59.0 468 G = -7.0 -3.0 1.0 |G|^2 = 59.0 469 G = -7.0 -3.0 -1.0 |G|^2 = 59.0 470 G = 5.0 -3.0 -5.0 |G|^2 = 59.0 471 G = 1.0 -7.0 -3.0 |G|^2 = 59.0 472 G = 7.0 3.0 1.0 |G|^2 = 59.0 473 G = 3.0 5.0 5.0 |G|^2 = 59.0 474 G = -1.0 7.0 3.0 |G|^2 = 59.0 475 G = -5.0 3.0 5.0 |G|^2 = 59.0 476 G = 7.0 -3.0 -1.0 |G|^2 = 59.0 477 G = -1.0 7.0 -3.0 |G|^2 = 59.0 478 G = -7.0 1.0 -3.0 |G|^2 = 59.0 479 G = 1.0 3.0 7.0 |G|^2 = 59.0 480 G = -3.0 -1.0 -7.0 |G|^2 = 59.0

Page 8

Gvectors_list 481 G = 1.0 7.0 -3.0 |G|^2 = 59.0 482 G = -1.0 -3.0 -7.0 |G|^2 = 59.0 483 G = -5.0 -5.0 3.0 |G|^2 = 59.0 484 G = -1.0 3.0 -7.0 |G|^2 = 59.0 485 G = 7.0 -3.0 1.0 |G|^2 = 59.0 486 G = -3.0 -5.0 5.0 |G|^2 = 59.0 487 G = 3.0 7.0 -1.0 |G|^2 = 59.0 488 G = 5.0 5.0 3.0 |G|^2 = 59.0 489 G = -3.0 5.0 -5.0 |G|^2 = 59.0 490 G = 3.0 7.0 1.0 |G|^2 = 59.0 491 G = -1.0 -7.0 -3.0 |G|^2 = 59.0 492 G = -5.0 3.0 -5.0 |G|^2 = 59.0 493 G = 7.0 -1.0 3.0 |G|^2 = 59.0 494 G = -3.0 7.0 -1.0 |G|^2 = 59.0 495 G = 5.0 3.0 -5.0 |G|^2 = 59.0 496 G = 3.0 -5.0 5.0 |G|^2 = 59.0 497 G = -1.0 -3.0 7.0 |G|^2 = 59.0 498 G = 3.0 1.0 -7.0 |G|^2 = 59.0 499 G = -5.0 -3.0 -5.0 |G|^2 = 59.0 500 G = 5.0 3.0 5.0 |G|^2 = 59.0 501 G = 3.0 5.0 -5.0 |G|^2 = 59.0 502 G = 5.0 -5.0 3.0 |G|^2 = 59.0 503 G = -7.0 3.0 1.0 |G|^2 = 59.0 504 G = 7.0 1.0 -3.0 |G|^2 = 59.0 505 G = 1.0 7.0 3.0 |G|^2 = 59.0 506 G = -3.0 -7.0 1.0 |G|^2 = 59.0 507 G = -3.0 1.0 -7.0 |G|^2 = 59.0 508 G = -7.0 3.0 -1.0 |G|^2 = 59.0 509 G = 1.0 -3.0 7.0 |G|^2 = 59.0 510 G = -3.0 -7.0 -1.0 |G|^2 = 59.0 511 G = 3.0 -1.0 7.0 |G|^2 = 59.0 512 G = 3.0 -7.0 -1.0 |G|^2 = 59.0 513 G = -1.0 -7.0 3.0 |G|^2 = 59.0 514 G = -5.0 5.0 3.0 |G|^2 = 59.0 515 G = -5.0 -5.0 -3.0 |G|^2 = 59.0 516 G = 5.0 -3.0 5.0 |G|^2 = 59.0 517 G = 7.0 -1.0 -3.0 |G|^2 = 59.0 518 G = -5.0 -3.0 5.0 |G|^2 = 59.0 519 G = 3.0 -7.0 1.0 |G|^2 = 59.0 520 G = -7.0 -1.0 3.0 |G|^2 = 59.0 521 G = 7.0 1.0 3.0 |G|^2 = 59.0 522 G = 3.0 1.0 7.0 |G|^2 = 59.0 523 G = -7.0 -1.0 -3.0 |G|^2 = 59.0 524 G = -7.0 1.0 3.0 |G|^2 = 59.0 525 G = -3.0 -1.0 7.0 |G|^2 = 59.0 526 G = 5.0 5.0 -3.0 |G|^2 = 59.0 527 G = 5.0 -5.0 -3.0 |G|^2 = 59.0 528 G = -5.0 5.0 -3.0 |G|^2 = 59.0 529 G = 1.0 -7.0 3.0 |G|^2 = 59.0 530 G = -3.0 7.0 1.0 |G|^2 = 59.0 531 G = 1.0 3.0 -7.0 |G|^2 = 59.0

Page 9