# What is the Correct Approach to Solving the Deuteron Transcendental Equation?

• James_1978
In summary: I used the number you gave for the reduced mass of the proton and neutron- 1.673x10^{-27}~\rm kg- and got the two different values for b.
James_1978
Homework Statement
I believe I solved the transcendental equation but the plot does not make sense.
Relevant Equations
##k_{1} \cot{k_{1}R} = -k_{2}##
##k_{1} = \frac{\sqrt{2m(E+V_{o})}}{\hbar}##
##k_{2} = \frac{\sqrt{-2mE}}{\hbar}##
##x = -\tan{bx}##
##x = \sqrt{\frac{-(V_{o} + E)}{E}}##
Dear Forum,

I am trying to solve a problem (4.6) from the introductory nuclear physics textbook by Krane. The problem is as follows:
Solving the deuteron using the radial equations gives the transcendental function,

##k_{1} \cot{k_{1}R} = -k_{2}##

Were

##k_{1} = \frac{\sqrt{2m(E+V_{o})}}{\hbar}##

And

##k_{2} = \frac{\sqrt{-2mE}}{\hbar}##

That gives the relations between and R. Show that this equation can be written in the form,

##x = -\tan{bx}##

Where

##x = \sqrt{\frac{-(V_{o} + E)}{E}}##

Evaluate the parameter b for R = 2fm. Note that is the reduced mass. Solve the transcendental equation.

When rearranging we get ##b## as.

##b = \frac{\sqrt{-2mE}}{\hbar}*R##

For the reduced mass ##m = \frac{1.67x10^{-27}}{2} kg##
For ##\hbar = 1.054x10^{-34} J-s##
For ##E = -2.22 MeV##

We are suppose to see that when solving the transcendental equation we get ##V_{o} = 36 MeV##. However we must have something wrong because the function does not clearly show how you infer the ##V_{o} = 36 MeV##. Any help is appreciated.

How much value of b you get ? Please let me know it for checking your result.

Last edited:
Dear Anuttarasmmyak

Here is what I got for b

##b = \frac{\sqrt{-2*\frac{1.67x10^{-27}}{2}*-2.22*1.602x10^{-13}}}{1.054x10^{-34}}##

Where ##b =2.3155x10^{14} m^{-1}##

Or with R I get ##b*R = 0.4856## I get ##V_{o} = 36##. I think this is correct. Just wanted to make sure.

b has no physical dimension.

Yes. I saw that. b*R is unit-less. My mistake.

Is MeV translated to MKSA Joule ?

Yes. I think you are asking in that I multiplied MeV*1.602x10^-13 to convert MeV to Joules. Is that what you are asking?

And you say R is 2 fm.

Yes, I use 2x10-15 m.

James_1978 said:
Or with R I get
So you say b=0.4856.

Yes. That is what I got.

anuttarasammyak said:
So you say b=0.4856.
James_1978 said:
Yes. That is what I got.
I got a different value for ##b##. I calculated ##b=0.4627##.

What did you use for E and mass of proton?

I first calculated ##b## using the numbers you used and got a different answer. So I looked up the mass of a proton and neutron and found the reduced mass (##1.673\times 10^{-27}~\rm kg##) and used that to get the number above. Either way, I didn't get the value for ##b## you found. In both cases I used ##E_1 = -2.22~\rm MeV##.

In fact, I don't get the same values for the calculations you showed in post #3 for ##b## (really ##b/R##) or ##b R## (really ##b##). Moreover, your two values don't make sense if you're using ##R=2~\rm fm##.

## What is the Deuteron Transcendental Equation?

The Deuteron Transcendental Equation is a mathematical expression used in nuclear physics to describe the bound state properties of the deuteron, which is the nucleus of deuterium consisting of a proton and a neutron. The equation is derived from the Schrödinger equation and involves parameters like the binding energy and the potential well depth.

## How is the Deuteron Transcendental Equation derived?

The Deuteron Transcendental Equation is derived from the Schrödinger equation for a two-nucleon system under a potential well. By applying boundary conditions and solving the radial part of the wave function, one arrives at a transcendental equation that relates the binding energy to the parameters of the potential.

## What is the significance of the Deuteron Transcendental Equation in nuclear physics?

The Deuteron Transcendental Equation is significant because it provides insight into the binding energy and spatial structure of the deuteron. Understanding this helps physicists in studying nuclear forces, the behavior of nucleons, and the properties of other nuclear systems.

## What are the common methods for solving the Deuteron Transcendental Equation?

Common methods for solving the Deuteron Transcendental Equation include numerical techniques such as the Newton-Raphson method, graphical methods for visualizing the solutions, and approximation methods like perturbation theory. Analytical solutions are often not possible due to the transcendental nature of the equation.

## What are the parameters involved in the Deuteron Transcendental Equation?

The parameters involved in the Deuteron Transcendental Equation typically include the binding energy of the deuteron, the depth and range of the potential well, and the reduced mass of the proton-neutron system. These parameters are crucial for accurately describing the bound state of the deuteron.

Replies
30
Views
2K
Replies
2
Views
1K
Replies
10
Views
2K
Replies
29
Views
969
Replies
1
Views
876
Replies
2
Views
994
Replies
3
Views
1K
Replies
1
Views
1K
Replies
6
Views
2K
Replies
1
Views
787