Sketch of an interatomic potential

[CAF123]

Homework Statement

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 separation 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)/U_o versus r/a, plotting the two exp terms separately.
2)Sketch the total result on the same diagram.

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/U_o 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.

 

[SammyS]

Homework Statement

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 separation 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)/U_o versus r/a, plotting the two exp terms separately.
2)Sketch the total result on the same diagram.

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/U_o 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?

 

[CAF123]

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.

 

[haruspex]

$e^{-2x+1} = 2e^{-x}$, which can't be solved analytically because of that 2.

Sure it can. Just take logs both sides.

 

[CAF123]

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..

 

[haruspex]

Does it help to note that e^2-2x = (e^1-x)²?
Does that suggest a way of writing the expression so that x only appears once?

 

[CAF123]

Does it help to note that e^2-2x = (e^1-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?

 

[CAF123]

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 modeled 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?

 

[haruspex]

I am not sure I completely understand what you mean - What expression are you talking about?

U(r) = U(ax) = U₀[e^2-2x-2e^1-x ]
e^2-2x-2e^1-x = (e^1-x)² - 2e^1-x = (e^1-x - 1)² - 1
I thought that getting it down to one occurrence of x might make it easier to study.