Essential Mathematics for NMR and MRI Spectroscopists

Keith C Brown

• Publisher: Royal Society of Chemistry
• Year: 2020
• ISBN:
  9781839162961
• Edition: 1
• Topic: Physical Sciences
• Subtopic: Analytic Chemistry

Beginning with a review of the important areas of mathematics, this book then covers many of the underlying theoretical and practical aspects of NMR and MRI spectroscopy from a maths point of view. Competence in algebra and introductory calculus is needed but all other maths concepts are covered. It will bridge a gap between high level and introductory titles used in NMR or MRI spectroscopy. Uniquely, it takes a very careful and pedagogical approach to the mathematics behind NMR and MRI. It leaves out very few steps, which distinguishes it from other books in the field.

The author is an NMR laboratory manager and is sympathetic to the frustrations of trying to understand where some of the fundamental equations come from hence his desire to either explicitly derive all equations for the reader or direct them to derivations. This is an essential text aimed at graduate students who are beginning their careers in NMR or MRI spectroscopy and laboratory managers if they need an understanding of the theoretical foundations of the technique.

1 Complex Numbers

It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit.

Pierre-Simon Laplace

1.1 The History of Numbers

The first humans presumably had no need for numbers and would therefore probably not have understood the concept. Eventually, of course, numbers were invented for use in counting … taking stock of animals and items of trade. The first set of numbers was undoubtedly simply a set of integers beginning at 1 and increasing from there … we are all familiar with them as we learn them at a very early age. This is very obvious and intuitive to us; anyone that counts uses these numbers. It turned out, however, that this number system was to be altered to include zero. Why? It was originally used as a placeholder for larger numbers. Thus, in the number 1053, the zero is in the “hundreds” column and indicates that are no hundreds to be added into the number. It consists of 3 ones, 5 tens, 0 hundreds and 1 thousands. Used at first in India for practical calculations it was imported into ancient Babylonian mathematics to replace a cumbersome placeholder system of slanted wedges. Its use was only as a placeholder and as it was not used alone it was not really considered to be a number. It did, however, come to be used as a number and it is now an integral part of our number systems. So, the simple number system was altered out of necessity to produce what we now refer to as the natural numbers.

Negative numbers were introduced to represent debts, again in India and again the number system was altered to include negative numbers into what we now refer to as the integers. There were those in ancient Greece who considered negative solutions to equations to be false and “absurd” so the use of negative numbers has not been without its critics.

Next, in order to express fractional quantities another set of numbers was needed. Using fractions of integers we come up with the rational numbers. Thus:

but we still do not have a complete set of numbers. The Greek Pythagoreans considered the integers to be “perfect”. Since the rational numbers are expressible using integers, they too were considered to be perfect. However, for them, a major problem existed, ironically from the results of what we now call the Pythagorean theory. The theory says the sum of the squares of the two sides of a right-angle triangle is equal to the square of the hypotenuse and has been proven many times in many different ways. If the two sides are each one unit in length then the length of the hypotenuse must be equal to the square root of two. The square root of two cannot be expressed as a ratio of integers which made the Pythagoreans very uncomfortable since it could not be constructed from “perfect numbers”. It is now referred to as an irrational number. Another example of an irrational number is π. So yet again, the number set was added to with rational and irrational numbers. Together they make up the set of real numbers.

1.2 Why Complex Numbers?

As NMR spectroscopists we deal with nuclei that possess the property of intrinsic angular momentum or “spin” and in order to model this property we use the mathematics of classical mechanical rotations, suitably modified for quantum mechanics. It turns out that complex numbers are ideally suited to this task and, indeed, it is well-nigh impossible to do without them, as we shall see in later chapters. It is amazing that such a fundamental property of matter that was totally unknown 100 years ago can be modelled with relatively simple mathematics. Complex numbers are used in many areas of physics and engineering and the modern practitioner of these sciences must be thoroughly familiar with them. It is often puzzling though to the uninitiated (and perhaps the initiated as well) as to why one would want to use a number that is defined as the square root of a negative number. How are we to picture such a number? Where does it fit in with our “regular” numbers? Hopefully we can draw aside the veil of mystery surrounding the...