# Phase transitions in a superconductor

• Silviu
In summary: Ah, I see your point. I had not looked carefully at the given info. I Have to think about it more then. Would it be long to show the solution the steps they follow to get their expression for...
Silviu

## Homework Statement

The phase transition to a superconducting state moves to lower temperatures as the applied magnetic field, H, increases. The magnetic moment, M, for a system of volume V is given by: $$M=-\frac{HV}{4\pi}$$
for ##H<H_C(T)## (superconducting) and $$M=0$$ ##H>H_C(T)## (normal). Changes in the internal energy, U, of such a system can be represented by the expression: $$dU=TdS+HdM$$ a) Show that for this system the constant field and constant magnetization heat capacities are equal: ##C_H=C_M## b) The phase transition between superconducting and normal phases takes place along a path ##H_C(T)##. Find an expression for the slope of the critical field, ##dH_C(T)/dT##, in terms of ##H_C(T)##, ##V## and the entropies of the two phases at the transitions.

## The Attempt at a Solution

a) So they give a solution, with a pretty long derivation and I think I am missing something. The way I was planning to do it was like this: $$C_H=\frac{dU}{dT}|_H=\frac{dU}{dT}|_{-4\pi M/V}$$ Assuming the volume is constant, which I assume is the case by "a system of volume V" this is equivalent to $$C_H=\frac{dU}{dT}|_M=C_M$$ I assume something is wrong, otherwise they wouldn't have a much longer derivation, but I am not sure what is wrong. b) I was trying to use the Gibbs free energy, making it equal for normal and superconducting at the boundary between the 2. So I have $$G=U+pV-TS$$ and going to the differential form I get $$dG=dU+pdV+Vdp-TdS-SdT = TdS+HdM-TdS-SdT=HdM-SdT$$ where I assumed that the pressure and volume are constant. However in their solution they get $$dG=-SdT-MdH$$ I assume that $$HdM=MdH$$ by the formula that connects them, but I am not sure how do they get a minus sign there. Can someone help me here?

Silviu said:

## Homework Statement

The phase transition to a superconducting state moves to lower temperatures as the applied magnetic field, H, increases. The magnetic moment, M, for a system of volume V is given by: $$M=-\frac{HV}{4\pi}$$
for ##H<H_C(T)## (superconducting) and $$M=0$$ ##H>H_C(T)## (normal). Changes in the internal energy, U, of such a system can be represented by the expression: $$dU=TdS+HdM$$ a) Show that for this system the constant field and constant magnetization heat capacities are equal: ##C_H=C_M## b) The phase transition between superconducting and normal phases takes place along a path ##H_C(T)##. Find an expression for the slope of the critical field, ##dH_C(T)/dT##, in terms of ##H_C(T)##, ##V## and the entropies of the two phases at the transitions.

## The Attempt at a Solution

a) So they give a solution, with a pretty long derivation and I think I am missing something. The way I was planning to do it was like this: $$C_H=\frac{dU}{dT}|_H=\frac{dU}{dT}|_{-4\pi M/V}$$ Assuming the volume is constant, which I assume is the case by "a system of volume V" this is equivalent to $$C_H=\frac{dU}{dT}|_M=C_M$$ I assume something is wrong, otherwise they wouldn't have a much longer derivation, but I am not sure what is wrong. b) I was trying to use the Gibbs free energy, making it equal for normal and superconducting at the boundary between the 2. So I have $$G=U+pV-TS$$ and going to the differential form I get $$dG=dU+pdV+Vdp-TdS-SdT = TdS+HdM-TdS-SdT=HdM-SdT$$ where I assumed that the pressure and volume are constant. However in their solution they get $$dG=-SdT-MdH$$ I assume that $$HdM=MdH$$ by the formula that connects them, but I am not sure how do they get a minus sign there. Can someone help me here?
How did you get $$HdM=MdH?$$ Using an integration by parts and assuming vanishing surface term, we get $$HdM=-MdH$$

nrqed said:
How did you get $$HdM=MdH?$$ Using an integration by parts and assuming vanishing surface term, we get $$HdM=-MdH$$
Oh, I just assumed $$HdM=(\frac{-4\pi M}{V})d(-\frac{HV}{4 \pi})=MdH$$ assuming constant ##V##. Is this wrong? (I mean I know it is wrong, but I am not sure why)

Silviu said:
Oh, I just assumed $$HdM=(\frac{-4\pi M}{V})d(-\frac{HV}{4 \pi})=MdH$$ assuming constant ##V##. Is this wrong? (I mean I know it is wrong, but I am not sure why)
Ah, I see your point. I had not looked carefully at the given info. I Have to think about it more then. Would it be long to show the solution the steps they follow to get their expression for dG?

nrqed said:
Ah, I see your point. I had not looked carefully at the given info. I Have to think about it more then. Would it be long to show the solution the steps they follow to get their expression for dG?
The statement is here (last problem) and the solution is here

nrqed said:
Ah, I see your point. I had not looked carefully at the given info. I Have to think about it more then. Would it be long to show the solution the steps they follow to get their expression for dG?
Any idea? At least for part a)?

## 1. What is a superconductor?

A superconductor is a material that has the ability to conduct electricity with zero resistance when cooled below a certain critical temperature. This allows for the flow of current without any energy loss, making them highly efficient for certain applications.

## 2. What are phase transitions in a superconductor?

Phase transitions in a superconductor refer to the change in the material's properties at different temperatures. As a superconductor is cooled below its critical temperature, it undergoes a phase transition from a normal state to a superconducting state, where it exhibits zero resistance and other unique characteristics.

## 3. How does the critical temperature affect the superconducting properties?

The critical temperature, also known as the transition temperature, is the temperature at which a superconductor undergoes a phase transition. It is a crucial factor in determining the superconducting properties of a material. The higher the critical temperature, the easier it is to achieve superconductivity and the more stable the superconducting state is.

## 4. What are the types of phase transitions in superconductors?

There are two main types of phase transitions in superconductors - the first-order phase transition and the second-order phase transition. The first-order phase transition involves a sudden change in the material's properties, while the second-order phase transition is a gradual change. The type of transition depends on the material's properties and the conditions under which it is cooled.

## 5. How do external factors affect phase transitions in superconductors?

External factors such as magnetic fields, pressure, and impurities can affect the phase transitions in superconductors. These factors can alter the material's critical temperature and other properties, which can impact its superconducting behavior. Understanding and controlling these external factors is crucial for developing practical superconducting materials.

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