Daniel K. Sodickson^{1,2}

The principle of reciprocity, as it applies to magnetic resonance, is both remarkably powerful and regrettably easy to misconstrue. As evidence of this fact, each of us with an interest in the operation of radiofrequency coils need only recall the time we have spent trying to understand, in our guts, the difference between transmit and receive sensitivity patterns. As a respectful supplement to Dr. Hoult’s seminal explications, we here provide a highly streamlined derivation, aimed at bolstering intuition, and offer a simple but fundamental mnemonic to keep your pluses and your minuses straight.

Introduction

Since Hoult’s seminal paper characterizing the signal-to-noise ratio of the nuclear magnetic resonance experiment appeared in 1976,**Figure 1** illustrates the geometry of our simple derivations to follow, in
which an external conductive loop receives signal from a precessing spin
embedded in a body. **Figure
2** begins with an expression for the magnetic moment of the spin. In Equation 1, the real quantity **M**_{spin}(**r**,t) is expressed as the real part of a time-invariant complex
moment multiplied by a harmonic time dependence with frequency ω (the Larmor frequency). This single convention
for complex quantities will apply throughout.
Positive precession of the spin magnetic moment is clearly associated
with the complex combination of unit vectors **x**–i**y**.

Equation 2 begins with Hoult’s expression for the measured MR
signal voltage^{1}. The
dot product of the loop’s magnetic field with the spin’s magnetic moment
immediately identifies the signal sensitivity with B_{1}^{(-)}.

Equation 3
simply decomposes the loop’s **B**_{1}
field, originally expressed in Cartesian coordinates, into combinations involving
**x**–i**y** and **x**+i**y**
(leaving the details to be verified by the reader). The field coefficient co-rotating with the
spin (i.e. sharing its **x**–i**y** directionality, and therefore
stationary in its rotating frame) is obviously B_{1}^{(+)}!

In other words, the receive sensitivity is characterized by B_{1}^{(-)}
because it results from a *dot product
with* the spin magnetic moment,
whereas the transmit sensitivity goes as B_{1}^{(+)} because
the transmit field *precesses with* the
spin magnetic moment.

What is the origin of the dot product in the MR signal expression in
Equation 2? **Figure 3** contains a simple derivation of this expression, beginning
from a definition of the signal voltage as an electromotive induced in the loop
by the time-varying electric field of the precessing dipole. Reciprocity
may then be applied in a traditional manner to exchange the E field and current
density of the loop with that of the spin.
Treatment of the spin as an equivalent infinitesimal current loop,
followed by an application of Stokes’ Theorem and the Maxwell-Faraday Equation,
yields Hoult’s original signal expression.

In short, the dot product derives directly from the inductive character
of signal detection. When we measure an induced voltage or current in an RF
coil, we are really measuring **E**∙**dl**, or **J**∙**E** , or **B**∙**da**. In signal reception, we are measuring the work done by the field of the precessing spin, whereas what matters
for excitation is the field itself.

This perspective on reciprocity is offered as a guide for students
of MR, young and old, who wish to keep track of why signal reception should be associated
with B_{1}^{(-)} and transmission with B_{1}^{(+)}. The answer, in these terms, is simple: reception involves a dot
product, and transmission does not.

This distinction, of course, may also be understood in more fundamental
terms, with reference to creation and annihilation operators involving
exchanges of energy from the coil to the spin and vice versa, or by analogy to
the flip-flip term in the dipole-dipole coupling Hamiltonian, which takes a
similar form. Not surprisingly, the ever-rigorous
Dr. Hoult has explored such explanations as well.^{4} Let us
close with one final question for the stout of heart: were a practical noninductive
mechanism of routine MR signal detection to be devised^{5}, what might
its pattern of sensitivity be?

1. Hoult DI, Richards RE. The signal-to-noise ratio of the nuclear magnetic resonance experiment. J Magn Reson 1976;24:71-85.

2. Hoult DI. The Principle of Reciprocity in Signal Strength Calculations - A Mathematical Guide. Concepts Magn Reson 2000;12:173-187.

3. Collins CM, Yang QX, Wang JH, Zhang X, Liu H, Michaeli S, Zhu X-H et al. Different excitation and reception distributions with a single-loop transmit-receive surface coil near a head-sized spherical phantom at 300 MHz. Magn Reson Med 2002; 47(5): 1026-1028.

4. Hoult DI, Bhakar B. NMR signal reception: Virtual photons and coherent spontaneous emission. Concepts in Magnetic Resonance A 1997;9(5):277-297.

5. Hennig, JH. My dream high-field MR scanner. 2016 i2i workshop (www.cai2r.net/i2i).

Figure 3: Derivation of Hoult’s
original expression for measured MR signal voltage.^{1} Note the origin of the dot product, which is
blamed, herein, for the change of apparent precession sense between
transmission and reception.