# What is the formla for the energy difference between the allowed nuclear spin states for #""^1 H# nuclei?

The formula is

In the presence of an applied external magnetic field,

A large magnet creates a homogeneous external magnetic field,

The energy difference

NMR spectroscopy records transitions between these spin states induced by a radio frequency electromagnetic field called the

In practice, the electrons surrounding the nucleus produce a small magnetic field

The field experienced by the proton is then

Finally, there is a proportionality constant

The final equation is

The graph below shows how the energy difference between the two spin states varies with the applied field.

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The formula for the energy difference between the allowed nuclear spin states for (^{1}H) nuclei is given by the following equation:

[ \Delta E = \gamma B ]

Where:

- ( \Delta E ) is the energy difference between the spin states.
- ( \gamma ) is the gyromagnetic ratio, which is a constant specific to the nucleus.
- ( B ) is the strength of the magnetic field.

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The formula for the energy difference between the allowed nuclear spin states for ( ^1H ) nuclei is given by the Zeeman effect. For ( ^1H ) nuclei, which have a spin quantum number ( I = \frac{1}{2} ), the energy difference between the two spin states (aligned with or against an external magnetic field) can be calculated using the following formula:

[ \Delta E = \gamma B ]

Where:

- ( \Delta E ) is the energy difference between the two spin states,
- ( \gamma ) is the gyromagnetic ratio, a fundamental constant specific to the nucleus, and
- ( B ) is the strength of the external magnetic field.

In the case of ( ^1H ) nuclei, ( \gamma ) is approximately ( 42.576 ) MHz/T, which is a characteristic value for the proton (hydrogen nucleus).

This formula is fundamental in nuclear magnetic resonance (NMR) spectroscopy, where the energy difference between nuclear spin states is manipulated and measured to obtain information about the chemical environment of ( ^1H ) nuclei in a molecule.

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