# Parallel and Anti-parallel Protons in NMR

I'm an Alevel Student trying to understand the concept of NMRI, I understand that when an external magnetic field is applied, the protons either line up with (parallel) or line up against (antiparallel) the external magnetic field. Sources state that protons parallel to the field have lower energy than antiparallel protons and that there are more parallel protons than antiparallel protons.

I have difficulties understanding the concept of:

1. Difference in energy between anti-parallel and parallel protons. There's an answer in quora https://www.quora.com/Why-does-a-spin-up-electron-have-slightly-more-energy-than-a-spin-down-electron explaining in terms of protons that

If the spin is already aligned with the magnetic field (up), it means that no energy is required to align it with B, which means that the spin up particle has the least potential energy.

If the spin is aligned opposite to the magnetic field (down), it means that the energy required to align it is maximum, which means that spin down particles has the most potential energy.

If the parallel protons have less potential energy, doesn't that mean that it has more kinetic energy than anti-parallel protons? Why are we only so concerned about potential energy?

1. The fact that there are more parallel than antiparallel protons due to lower energy of parallel protons. Why is this so? In this link https://www.reddit.com/r/askscience/comments/2axmfz/why_do_more_protons_align_themselves_parallel/, the only reason that I could somewhat grasp was based on part of one of the answers:

Always think about Boltzmann's population distribution when you are considering problems like this.

Using my Alevel knowledge (from Boltzmann distribution curve), I understand that in a system, majority of the particles will have lower/ intermediate energy compared to higher energy. Doesn't Boltzmann Distribution curve only apply to kinetic energy and not potential energy?

There are other reasons based on Zeeman Effect which I can't seem to understand due to my limit in knowledge at this moment, but will try to read on it. At this moment, I'm looking for simpler answers that an Alevel student could understand. Thank You.

• You are thinking of Boltzmann velocity distribution. But the distribution is more general than that. – Superfast Jellyfish Jan 31 '20 at 11:01

## 2 Answers

If the parallel protons have less potential energy, doesn't that mean that it has more kinetic energy than anti-parallel protons? Why are we only so concerned about potential energy?

This implies conservation of energy (less potential energy => more kinetic), which is true for closed systems, but here the system is not closed! We bomb this protons with external magnetic field, which pumps a lot of energy into the system. In fact, this external energy is much larger than the energy that characterize the protons movement (which is confined by their interaction with the environment). Therefore, when this magnetic field is applied, we can just omit altogether all the kinetic energy terms - they are tiny in comparison. So we are just interested in how the protons interact and respond to the external field, and here they have two options - to be in parallel to it (lower energy) or anti-parallel to it (higher energy).

Using my Alevel knowledge (from Boltzmann distribution curve), I understand that in a system, majority of the particles will have lower/ intermediate energy compared to higher energy. Doesn't Boltzmann Distribution curve only apply to kinetic energy and not potential energy?

No, the Boltzmann distribution is true for every system with some temperature. It states that the probability of the system to be in a state with some energy $$\epsilon$$ drops as $$\epsilon$$ is larger, and the "measure" against we examine this energy is the temperature. So $$\epsilon \ll k_B T$$ is favorable to $$\epsilon \gg k_b T$$. So, the lower the temperature, more and more of the protons will tend to go for the lower-energy state, which is the parallel spin configuration. The ratio between them will be determined by the temperature and the energy, as can be read from the Boltzmann distribution.

• Yeah, much better:) – StudyStudy Jan 31 '20 at 12:21

If the parallel protons have less potential energy, doesn't that mean that it has more kinetic energy than anti-parallel protons? Why are we only so concerned about potential energy

We are not dealing with classical objects here, (like soccer balls) and you are possibly thinking of conflicts with classical conservation of energy laws.

In the example you quote, we can assume for a proper description of the effect of the magnetic field, that the protons cannot move to escape it and that therefore the magnetic field that they interact with is constant. This is an ideal situation, but in real life the protons will move due to their kinetic energy.

Boltzman statistics deal with energy, not velocity, so they are an extension of Maxwell laws and they describe the distribution of energy levels.