Ground state energy for exciton in Si

[adamaero]

Homework Statement

Estimate the ground state energy (eV) for an exciton in Si.
ε_Si = 12
ε = 1.0359×10⁻¹⁰

Effective masses​

m_e* = 0.26m_e
m_h* = 0.36m_e
effective mass = 0.15m_e

Values of h​

6.626×10⁻³⁴ J⋅s
4.136×10⁻¹⁵ eV⋅s

Values of ħ Units​

1.055×10⁻³⁴ J⋅s
6.582×10⁻¹⁶ eV⋅s

Homework Equations

E₁ = me4/(8*h²ε₀²)
E = m^*e4/(8*h²ε²)

The Attempt at a Solution

0.15(9.11e-31)e^4/[8(6.626e-34)^2(1.0359e-10)^2] =
0.15(9.11e-31)54.59815/[8(6.626e-34)^2(1.0359e-10)^2]
[link to wolfram for this calculation]
= 1.98×10⁵⁶ J
This seems way too big!
It's even bigger in eV: 1.24×10⁷⁵

I would really appreciate it if someone could help point out where I went wrong. It seems like most want to beat around the bush...or maybe everyone that's helped me here is just in a rush. That epsilon I'm not 100% sure about either.

 

[kuruman]

0.15(9.11e-31)e^4/[8(6.626e-34)^2(1.0359e-10)^2]
What do you understand the term highlighted in red to be?
Wolfram understands to be (Exp[1.0])⁴. The same understanding applies to all the e's in your expression.

 

[adamaero]

0.15(9.11e-31)e^4/[8(6.626e-34)^2(1.0359e-10)^2]
What do you understand the term highlighted in red to be?
Wolfram understands to be (Exp[1.0])⁴. The same understanding applies to all the e's in your expression.

Yes. I don't see any problem with that. The link to Wolfram clearly shows that without the caret operator (^) is needed for an exponential. 10e10 = 100 billion.

 

[kuruman]

E1 = me4/(8*h2ε0²)

What does "e4" mean in your posted equation and how do you implement it?

 

[adamaero]

That's a typo for e⁴.

 

[kuruman]

And does "e" stand for?

 

[adamaero]

I believe it's the constant: 2.71828

 

[kuruman]

Which constant? What is its value?

 

[adamaero]

Okay, do you actually know the answer to this minor tangent?! If so, why are you beating around the bush?

I understand probing for thinking and such is the main idea of Physics Forums, but this is a mere tangent about an equation.

 

[adamaero]

1.6×10⁻¹⁹
the charge of an electron?
That's what someone else said from a different site.

 

[kuruman]

I believe it's the constant: 2.71828

I don't think so. It's the charge of the electron.

 

[adamaero]

0.15(9.11e-31)1.6e-19/[8(6.626e-34)^2(1.0359e-10)^2] =
5.8×10³⁵ J
or 36×10⁴⁵ eV

I don't know what the ballpark number should be (so I have no idea if this is near correct).

 

[kuruman]

I understand probing for thinking and such is the main idea of Physics Forums, but this is a mere tangent about an equation.

Maybe so, but it looks like you did not understand what the symbols in the equation stood for and therefore you did not understand the equation and couldn't use it correctly. I wouldn't call that a "mere tangent".

 

[kuruman]

0.15(9.11e-31)1.6e-19/[8(6.626e-34)^2(1.0359e-10)^2] =
5.8×10³⁵ J
or 36×10⁴⁵ eV

Where is it that you raise the charge of the electron to the fourth power in this expression?

 

[adamaero]

Ya, I got it. Ate up about 8 hours though.
14.8mJ
It is a mere equation, but I don't have time to explain out why. It should not take an hour to point out that the equation is wrong, and what it actually should be! I mean, a guy from another site is the one who actually helped.

 

[kuruman]

You are welcome.

 

[adamaero]

I mean, a guy from another site is the one who actually helped.