## Sketch of an interatomic potential

1. The problem statement, all variables and given/known data
Consider the interatomic potential between 2 atoms: $$U(r) = U_o \left[exp\left(\frac{-2(r-a)}{a}\right) - 2exp \left(\frac{-(r-a)}{a} \right) \right],$$ where r is the seperation between the two atoms and ##U_o## and a are constants.
1)Evaluate the expressions in the square brackets for a number of values of r/a and thereby make a plot of U(r)/Uo versus r/a, plotting the two exp terms separately.
2)Sketch the total result on the same diagram.

3. The attempt at a solution

I have attached two graphs of the attractive and repulsive component of this potential. Conveniently, the potential can be expressed in the form y =y(x), with y = U/Uo and x =r/a. I don't know what a is so I am struggling to see if my graphs make any physical sense. As r/a decreases, however, I note that the repulsive term increases with limit of e2 in the repulsive case and -2e in the attractive case. To obtain these graphs, I simply plotted the two exp terms against r/a separetely, for random values of r/a. Can someone check these graphs?
Also, I am a bit unsure of how to combine them into one graph.
Many thanks.
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 Quote by CAF123 1. The problem statement, all variables and given/known data Consider the interatomic potential between 2 atoms: $$U(r) = U_o \left[exp\left(\frac{-2(r-a)}{a}\right) - 2exp \left(\frac{-(r-a)}{a} \right) \right],$$ where r is the seperation between the two atoms and ##U_o## and a are constants. 1)Evaluate the expressions in the square brackets for a number of values of r/a and thereby make a plot of U(r)/Uo versus r/a, plotting the two exp terms separately. 2)Sketch the total result on the same diagram. 3. The attempt at a solution I have attached two graphs of the attractive and repulsive component of this potential. Conveniently, the potential can be expressed in the form y =y(x), with y = U/Uo and x =r/a. I don't know what a is so I am struggling to see if my graphs make any physical sense. As r/a decreases, however, I note that the repulsive term increases with limit of e2 in the repulsive case and -2e in the attractive case. To obtain these graphs, I simply plotted the two exp terms against r/a separetely, for random values of r/a. Can someone check these graphs? Also, I am a bit unsure of how to combine them into one graph. Many thanks.
Letting x = r/a is indeed the way to handle the parameter, a .

Both terms are basic exponential graphs. They don't flatten out for small x = r/a, unlike what you show .

$\displaystyle e^{-2r/a+2}=e^2\,e^{-2r/a}\ \$ and $\displaystyle \ \ 2e^{-r/a+1}=(2e)\,e^{-r/a}\ .\$

What is each term at r=a, i.e., when x = 1 ?

At what value of x, are the two terms equal in magnitude?
 At r/a = x = 1, the attractive term is -2 and the repulsion term 1. In solving for when they are equal, I get to ##e^{-2x+1} = 2e^{-x}##, which can't be solved analytically because of that 2. ( I could use Newton Raphson although I feel that is not the purpose of the problem) What values of r/a do you suggest? Is numbers like 1,2,3.. okay? EDIT: What I have now is two exp looking graphs. In the attractive case, I have a curve coming from ##-\infty## and tending to 0. Similarly, in the repulsive case, I have a graph starting at ##+\infty## and again decaying to zero. This looks better - thanks. How to combine them? I suppose if we are concerned only with +ve x (i.e their separation cannot be negative since this would imply one of the other particles passes the other, I draw from x ##\geq##0 i.e from y = ##-2e^2## in attractive case and y = ##e## in repulsive case, and then both tending to 0.

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## Sketch of an interatomic potential

 Quote by CAF123 ##e^{-2x+1} = 2e^{-x}##, which can't be solved analytically because of that 2.
Sure it can. Just take logs both sides.

 Quote by haruspex Sure it can. Just take logs both sides.
Hmm.. not sure how I missed that first time!
But does it help? The term I am dealing with is -2exp.. not 2exp..
 Recognitions: Homework Help Science Advisor Does it help to note that e2-2x = (e1-x)2? Does that suggest a way of writing the expression so that x only appears once?

 Quote by haruspex Does it help to note that e2-2x = (e1-x)2? Does that suggest a way of writing the expression so that x only appears once?
I am not sure I completely understand what you mean - What expression are you talking about?
 Ok, I got both graphs now. One quick question about something later in the problem. It asks to derive the force between these two atoms. Now I think the interatomic force(s) is/are conservative so I can apply the straightforward -derivative of potential. One thing that I was trying to understand was why the attractive compt is modelled by the negative term in the potential. The way I tried to describe it is: The attractive component tends to lower U. Since we deal with a conservative force, T + U = const, and so T must increase. This makes sense, as the particles get closer, there is more of a attraction. Now at some point, which is the minimum of the potential, the particles gets repelled, corresponding to an increase in U and so a decrease in T, which also makes sense since if the particle is so far away, there is barely any attraction. In between these two extremes, T and U vary so as to preserve the constant. Is this a reasonable explanation?

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