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Define and solve formula syntax problem

Karl S
Posted 3 years ago

Hello,

I some problems with the Mathematica syntax for defining and solving equations

I have following equations:

electrondensity[x_, Z_, Avog_] := (7*Avog*Z)/(118) * (1/x)^3

electrondensitycrit[\[Lambda]_] := 1.1 10^21/\[Lambda]^2

and try to solve them:

NSolve[electrondensity[x_, 12, Avog] == electrondensitycrit[10.6], x_]

This however works only if the underscores "_" are used for the variable x If "x" instead of "x_", the solution is {}

Why is it? and why the following error message in case of "x_" appears? (but still solves the equation):

Solve::ratnz: Solve was unable to solve the system with inexact coefficients. The answer was obtained by solving a corresponding exact system and numericizing the result.

2nd question:

I want to define a formula:

enter image description here

Do I need to define the formula in this way? (indicating the variables?)

Kramer[Z_, \[Lambda]_, T_, x_] := 
 108*10^-4 *Z * 
  Log[4*T/100 *\[Lambda]*2 \[Pi]/(0.577*6.626*10^-34 *3*10^8)]/((T/
        100)^(3/2)*\[Lambda]^2 ) *((electrondensity[
       x])/(electrondensitycrit[\[Lambda]]))^2

The error appears:

"Tag Times in (5.20617*10^-6\ [Lambda]^2\ Log[6.57374*10^25\ \ [Lambda]]\ Z)[Z,[Lambda],T,x_] is Protected"
POSTED BY: Karl S
7 Replies
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I am getting no errors:

Kramer[Z_, \[Lambda]_, T_, x_] := 
  108*10^-4*Z*
   Log[Konst[\[Lambda]]]/((T/100)^(3/
         2)*\[Lambda]^2)*((electrondensity[x, Z, 
        Avog])/(electrondensitycrit[\[Lambda]]))^2;
Integrate[Kramer[Z, \[Lambda], T, x], x]
POSTED BY: Gianluca Gorni
Karl S
Karl S
Posted 3 years ago 
Kramer[Z_, \[Lambda]_, T_, x_] := 
      108*10^-4*Z*
       Log[Konst[\[Lambda]]]/((T/100)^(3/
             2)*\[Lambda]^2)*((electrondensity[x, Z, 
            Avog])/(electrondensitycrit[\[Lambda]]))^2;
    Plot[Integrate[Kramer[12, 10.6, 30, x], x], {x, 0, 40}]

I get

SetDelayed::write: Tag Times in (5.20617*10^-6 [Lambda]^2 Log[6.57374*10^25 [Lambda]] Z)[Z,[Lambda],T,x_] is Protected.

and the plot is empty
POSTED BY: Karl S

You must give a numerical value Avog and decide what function is Konst. Also, you must wrap Integrate in Evaluate, to avoid the trap of the order of evaluation:

electrondensity[x_, Z_, Avog_] := (7*Avog*Z)/(118)*(1/x)^3;
electrondensitycrit[\[Lambda]_] := 1.1 10^21/\[Lambda]^2;
Kramer[Z_, \[Lambda]_, T_, x_] := 
  108*10^-4*Z*
   Log[Konst[\[Lambda]]]/((T/100)^(3/
         2)*\[Lambda]^2)*((electrondensity[x, Z, 
        Avog])/(electrondensitycrit[\[Lambda]]))^2;
Block[{Konst = Identity, 
  Avog = Quantity[1, "AvogadroConstant"] // QuantityMagnitude}, 
 Plot[Evaluate[Integrate[Kramer[12, 10.6, 30, x], x]], {x, 0, 40}]]
POSTED BY: Gianluca Gorni
Rohit Namjoshi
Rohit Namjoshi
Posted 3 years ago

Probably because of preexisting conflicting definitions. Quit the kernel and start over.

Integrate[Kramer[12, 10.6, 30, x], x]
(* -((7.42291*10^-42 Avog^2 (50.0099 + Log[T]))/x^5) *)

Avog and T do not have numeric values so a Plot cannot be generated.
POSTED BY: Rohit Namjoshi

Also, beware of the expression 4*T/100 *\[Lambda]*2 \[Pi].

Mathematica reads it as 4*(T/100) *\[Lambda]*2 \[Pi], which may not be what you have in mind.
POSTED BY: Gianluca Gorni

This works well enough for me:

electrondensity[x_, Z_, Avog_] := (7*Avog*Z)/(118)*(1/x)^3
electrondensitycrit[\[Lambda]_] := 1.1 10^21/\[Lambda]^2;
Solve[electrondensity[x, 12, Avog] == 
  electrondensitycrit[10.6], x, Reals]

It is somewhat tricky, because Mathematica may interpret the cubic root in the complex sense, while you may expect real values only. As for indicating the variables, it depends on what you want to do with the formula. Do you need only to display it, or for actual calculation?
POSTED BY: Gianluca Gorni
Karl S
Karl S
Posted 3 years ago

Ok this works for me also now.

electrondensity[x_, Z_, Avog_] := (7*Avog*Z)/(118) * (1/x)^3
electrondensitycrit[\[Lambda]_] := 1.1 10^21/\[Lambda]^2
Konst[\[Lambda]_] := 
 4*T/(100 *\[Lambda]*2 \[Pi])/(0.577*6.626*10^-34 *3*10^8)

I want to integrate the following formula for x:

Kramer[Z_, \[Lambda]_, T_, x_] := 
 108*10^-4 *Z * 
  Log[Konst[\[Lambda]]]/((T/100)^(3/
        2)*\[Lambda]^2 ) *((electrondensity[
       x])/(electrondensitycrit[\[Lambda]]))^2

However, with defining this equation, the error:

SetDelayed::write: Tag Times in (5.20617*10^-6 [Lambda]^2 Log[6.57374*10^25 [Lambda]] Z)[Z,[Lambda],T,x_] is Protected.

occurs with the output: $Failed
POSTED BY: Karl S
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