NSolve and Solve run forever or give errors with big numbers

Questions
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1

Hello,

I didn’t use Mathematica for years and I am VERY rusty regarding the best way to solve this problem :

2*vectk.vectG = G^2,

k = 2*pi/lambda

I have the G vector and I want to find all vector k that can fit into this previous equation. Vect k should use some “primitive” vectors b1, b2 and b3.

Clear["Global`*"];
la1 =  6.2696*10^-10;
la2 =  40.3246*10^-10;
la3 =  9.8488*10^-10;
alpha = 90;
beta = 99.464;
gamma = 90;
a1 = {la1, 0, 0};
a2 = {0, la2, 0};
a3 = {la3*Cos[beta Degree], 0, la3*Sin[beta Degree]};
omega = 2456.07*(10^-10)^3;
lambda = 1.5*10^-10;
b1 = 2*Pi*(Cross[a2, a3]/omega);
b2 = 2*Pi*(Cross[a3, a1]/omega);
b3 = 2*Pi*(Cross[a1, a2]/omega);
G[h_, k_, l_] := h*b1 + k*b2 + l*b3;
G = G[1, 7, 0]
h = 1;
k = 7;
l = 0;
Solve[2*Dot[k1*b1 + k2*b2 + k3*b3, h*b1 + k*b2 + l*b3] == 
   Norm[h*b1 + k*b2 + l*b3]^2 && 
  Norm[{k1*b1 + k2*b2 + k3*b3}] <= 2*Pi*1.01/lambda, {k1, k2, 
  k3}, Integers, WorkingPrecision -> 100]

When I execute it processes for ever.

I don’t know if Solve function can solve such a problem or if there is a problem with my syntax. If Solve cannot find such a solution, would you have some recommandations ? It is for a school project…

Thanks

NSolve bug.nb (6.74 KB)

2

There are many problems with your code. The object {k1*b1, k2*b2, k3*b3} is a matrix. What do you mean by its Norm? The maximum singular value? Your first equation seems of the form vector==scalar. How can you hope to find integer solutions to an equation with inexact coefficients?

3

You say your vector g is given, { g1, g2, g3 } say.

Now chose your vector k = { k1, k2, k3 } and form vectors ks1 and ks2

g = {g1, g2, g3};
k = {k1, k2, k3};
ks1 = k - g.k/g.g g
ks2 = Cross[g, k]

Then for arbitrary u and v

k and (g.k/g.g g + u ks1 + v ks2) have the same sclar product, which means your problems has infinitly many solutions:

g.k
g.(g.k/g.g g + u ks1 + v ks2) // FullSimplify
4

Thank you for your answer. I was looking for the Norm[vector]. I corrected the code.

Also, I thought

(k1*b1 + k2*b2 + k3*b3).( h*b1 + k*b2 + l*b3)

was a vector. I corrected it too with

Dot[k1*b1 + k2*b2 + k3*b3, h*b1 + k*b2 + l*b3]
5

I don’t really understand the reasoning, but from my visualization of the problem, there should be a limited quantity of solutions. We only want the nodes in “the plan normal to vect.G passing throw (1/2)*vect.G” touched by vect.k who has a fixed length starting from the origin…

I used a completely different path. I made a list of the nodes of the lattice using Table, then extracted those included in the plan (an almost flat cylinder) with Select, then extracted again those at distance k from the origin.

I am curious to understand why my first approach didn’t work. But this is not vital either : /

6

What do you mean by “reasoning”? You asked for vectors having the same scalar-product g.k = G^2 / 2, and I gave you a method to construct such vectors. And there are infinitly many. Due to u and v for me it is a two-dimensional manifold. And of course you can normalize them and give them any lengths you want.

But perhaps I don’t understand your problem correctly. What do you mean by “nodes”? What do you want to do?