Category Archives: Michael

References and Mathematica Code

References are here.

1)  Jacobsen, Neil E. NMR Spectroscopy Explained: Simplified Theory, Applications and Examples for Organic Chemistry and Structural Biology. Hoboken, NJ: Wiley-Interscience, 2007. Print.


Mathematica Code and the PowerPoint for other figures.  To download the Mathematica notebooks, right click the links and Save As.:



The Finished Structure of 3,3-dimethyl-2-butanol

Now that we’ve analyzed both the ^{1}H-NMR and the ^{13}C-NMR spectra, we can see how the results come together and give the structure of 3,3-dimethyl-2-butanol.  Below we bring back the two figures that show the labeled hydrogens and carbons of the molecule:

If we did not already know the identity of the molecule, we could have determined it using only the four spectra previously shown–the ^{1}H-NMR, the ^{13}C-NMR, the DEPT-90, and the DEPT-135–by following the guidelines laid out in the previous posts.  Peaks on the four spectra work together to give  bits of information about  pieces of the molecule, and those pieces are put back together at the end of the analysis.

To summarize, we began by taking advantage of the magnetic properties of ^{1}H and ^{13}C nuclei to learn about the environment of each nucleus in the molecule.  Radio waves were pulsed at the sample to record small changes in the resonant Larmor frequency of the nuclei.  The change happened because electrons around each nucleus became a current when influenced by the NMR’s external field, and generated their own fields in the opposite direction according to Lenz’s Law.  This new, effective magnetic field resulted in changes in the Larmor frequency of each nucleus.  It is these changes that are displayed on the spectra as chemical shifts.  The chemical shift is measured in ppm of the original Larmor frequency of the nucleus.

Using the Larmor frequency equation below, and the shifted frequencies, we calculated the shielding factor \sigma for each nucleus.  The shielding factor is another measure of the relative electron density around each nucleus.

(1)   \begin{equation*} \nu_{0}=\frac{\gamma B_{0}(1-\sigma)}{2\pi} \end{equation*}

Combining the information from the spectra gives the structure of 3,3-dimethyl-2-butanol, which is shown again below.

It is the ability of the NMR spectrometer to detect such small changes in resonant frequency, and convert them into data we can use to piece together molecular structures, that makes NMR spectroscopy so powerful and important in the research world.

Interpreting a C-13 NMR spectrum

In Vassar’s 300 MHz NMR, the Larmor frequency of an unaltered hydrogen nucleus with shielding factor \sigma=0 is 300 MHz.  ^{13}C nuclei are different than ^{1}H nuclei, and have a different Larmor frequency in the spectrometer’s 7.046 Tmagnetic field.  According to Jacobsen in “NMR Spectroscopy Explained,” the Larmor frequency of unaltered ^{13}C nuclei with \sigma=0 is 75.43 MHz in this magnetic field.  The magnetogyric ratio \gamma, is 672.650*10^{5}\frac{rad}{Ts}  We bring back the equation for Larmor Frequency:

(1)   \begin{equation*} \nu_{0}=\frac{\gamma B_{0}(1-\sigma)}{2\pi} \end{equation*}

You might ask why ^{13}C nuclei are the focus of carbon NMR instead of the much more common and ordinary ^{12}C nuclei.  It comes down to the composition of the nucleus.  Only nuclei with nonzero spin are magnetically active.   Spin is a type of angular momentum intrinsic to subatomic particles.  Particles with nonzero spin can have magnetic moments that can be influenced by a magnetic field.  Any nucleus with an even number of protons and an even number of neutrons will not be magnetically active because its spin is 0 and it has no magnetic moment.  Carbon-12 has 6 protons and 6 neutrons, so it can’t be studied with NMR.  Carbon-13 has 6 protons and 7 neutrons, so we can study it with NMR.  The structure and ^{13}C-NMR spectrum of 3,3-dimethyl-2-butanol is shown below.

At first glance, the spectrum looks almost the same as the H-NMR spectrum.  However, there is little indication of how many carbons a single peak is referring to.  Each peak corresponds to a type of carbon in the molecule.  The triplet peak at 77 ppm represents the deuterated chloroform, CDCl_{3}, used to dissolve the sample.  Excluding that triplet, the molecule has four different kinds of carbon.  To help us find what carbons are represented here, we turn to the other two spectra of interest.  They are called the DEPT-90, and the DEPT 135.  DEPT stands for Distortionless Enhancement by Polarization Transfer.  The technique hits the sample with five successive radio pulses designed to excite either hydrogen or ^{13}C nuclei.  The sequence of pulses is:

  • A hydrogen pulse at 90° from the x-axis
  • A hydrogen pulse at 180° from the x-axis  plus a carbon pulse at 90° from the x-axis.
  • A carbon pulse at 180° from the y-axis plus a hydrogen pulse at either 45°, 90°, or 135° from the y-axis.

The angle of the last hydrogen pulse is the number in the DEPT label.  DEPT-90 ends with a hydrogen pulse at 90° from the y-axis.  The pulse sequence is modeled below for DEPT-90 as an example.  Red waves are hydrogen pulses, green waves are carbon pulses:

DEPT-90 Pulse 1

DEPT-90 Pulse 2

DEPT-90 Pulse 3

These combinations of pulses result in transfers of polarization between the hydrogen and carbon nuclei.  Different combinations pick out carbons with one, two, or three hydrogens attached to them.  The DEPT-90 and DEPT-135 spectra are shown below:

DEPT-90 spectrum

DEPT-135 Spectrum

The chemical shifts of the peaks on these two spectra are the same as they are on the ^{13}C-NMR spectrum.  The sizes of the peaks on the two spectra are different, and this size difference is the key to understanding them.  You may have noticed the peak at 35 ppm on the ^{13}C-NMR spectrum doesn’t appear on either of the DEPT spectra.  This is because DEPT spectra only show carbons with hydrogens attached to them.  Therefore the 35 ppm peak represents the carbon in 3,3-dimethyl-2-butanol with no hydrogens on it.

The three peaks on the DEPT spectra are distinguished by their relative directions and sizes on each spectrum.

  • Peaks that point up on the DEPT-135, point up on the DEPT-90, and are larger on the DEPT-90, represent carbons with one hydrogen attached.
  • Peaks that point down on the DEPT-135 represent carbons with two hydrogens attached.
  • Peaks that point up on the DEPT-135, point up on the DEPT-90, and are smaller on the DEPT-90 represent carbons with three hydrogens attached.
  • Carbons without hydrogens don’t appear on either the DEPT-135 or DEPT-90

These are not the most general rules for interpreting DEPT-90 and DEPT-135 spectra.  Normally, carbons with two and three hydrogens don’t appear on the DEPT-90 spectra.  However, in this experiment the radio pulse used to collect the data was too long.  This causes the DEPT-90 spectrum to contain a small fraction of the DEPT-135 spectrum’s information.

Armed with this new information from the DEPT spectra, we can determine which carbons are represented by which peaks on the ^{13}C-NMR spectrum, and calculate their shielding factors.  A table of chemical shift and shielding factor values for each ^{13}C nucleus in 3,3-dimethyl-2-butanol is below.


The same trend that applies to hydrogen nuclei applies to ^{13}C nuclei.  The more negative the shielding constant, the lower the electron density around the nucleus, and the higher the effective magnetic field felt by the nucleus.  ^{13}C nuclei also experience higher chemical shifts compared to hydrogen nuclei because they are bonded to more atoms and have more opportunity to be deshielded when those atoms take electron density.

The figures below shows the structure of 3,3-dimethyl-2-butanol with the carbons labeled with their corresponding chemical shifts.  This is the final step in reconstructing the molecule from the NMR data.




Interpreting a H-NMR Spectrum

The NMR Spectrometer at Vassar college is graded at 300 MHz.  This means the Larmor frequency of a single, unaltered hydrogen nucleus and its electron with \sigma=0, is 300 MHz.  The magnetogyric ratio \gamma of a hydrogen nucleus is 267.513*10^{6}\frac{rad}{T•s} .  A quick calculation using:

(1)   \begin{equation*} \nu_{0}=\frac{\gamma B_{0}(1-\sigma)}{2\pi} \end{equation*}

shows the magnetic field strength of the NMR is roughly 7.046 T.

Using this information and the ^{1}H-NMR spectrum, we can calculate shielding factors for each type of hydrogen nucleus in 3,3-dimethyl-2-butanol, and determine what each spectrum peak actually means.

Let’s start with the peak farthest to the right on the spectrum.  It is a single peak, called a singlet, that represents nine hydrogens, and is centered at approximately 0.9 ppm on the x-axis.  The ppm scale measures how much the Larmor frequency of hydrogen is changed by the effective magnetic field.  The Larmor frequency of the hydrogens represented by the first peak was increased by 0.9 ppm of the initial 300 MHz frequency.  A calculation of these hydrogens’ shielding factor follows using equation (1):

300 MHz+(\frac{0.9 ppm}{10^{6}}*300 MHz)=\frac{267.513*10^{6}\frac{rad}{T•s}*7.046T*(1-\sigma)}{2\pi}

300000270 Hz=299990611 Hz(1-\sigma)


Shielding factors tend to be small for most hydrogen nuclei.  A table of chemical shift and shielding factor values for each hydrogen nucleus in 3,3-dimethyl-2-butanol is below.

A more negative shielding factor corresponds with a lower electron density around the hydrogen nucleus and with a larger effective magnetic field influencing the nucleus.  The presence of very electronegative atoms, like oxygen, near the hydrogen causes increased chemical shifts like the 3.5 ppm shift in the table.

One peak in the spectrum above is split into two peaks centered around the chemical shift  1.1 ppm.  This splitting occurs because there is another magnetically active hydrogen nucleus nearby in the molecule.  The rule for split peaks is: the number of nearby hydrogens is given by n-1, where n is the number of peaks.  The definition of “nearby” is usually 1 carbon atom over in the molecule from the one the original hydrogen is attached to.

The figure below shows the structure of 3,3-dimethyl-2-butanol with the hydrogens labeled with their corresponding chemical shifts.  This will be the first step in reconstructing the molecule from the NMR data.


Preliminary Results

The magnetic field inside an NMR spectrometer is generated by a superconducting solenoid.  The solenoid is cooled by a three-layer cooling system.  The outer layer is one of liquid nitrogen at 77K.  The middle layer is an evacuated cavity that prevents heat conduction by air.  The innermost layer contains the solenoid submerged in 4K liquid helium that is kept below atmospheric pressure.  Initially, the magnetic field in an NMR will not be homogenous because of interferences resulting from the magnetization of the sample, and environmental interferences like iron or other metals used in construction.

Initial NMR Magnetic Field

Sample NMR Starting Magnetic Field

Note the somewhat ordered field that already exists in the Z-direction.  This is what the initial field looks like most of the time:  the inhomogeneities are a complex function in three dimensions.  In order to make the magnetic field homogenous, it is modified using the shim system. The shim system is a series of uncooled, current carrying coils that generate fields in all three directions.  Each coil attempts to create a field with a function that precisely cancels the imperfections.  Numerous simple 3-D functions like X, Z, XY^{2}, Z^{2}, Z^{3}, Z^{4}, XZ, YZ, and X^{2}-Y^{2} are available to add to the magnetic field created by shimming.  Each set of coils creates a magnetic field gradient in its direction.  Below is a model of a correction field that uses the function Z^{2}:

Z^{2} Correction Field

All but the most experienced and exacting users of NMR use functions other than Z and Z^{2} in the correction field because the original field is usually close to homogenous.  The optimal field for doing NMR spectroscopy is shown below-it is perfectly homogenous and only in the Z-direction.

Optimal NMR Magnetic Field

Once the magnetic field is homogenized, the analysis of a sample can begin.  Each nucleus in the atom to be tested has its own resonant frequency somewhere in the radio section of the electromagnetic spectrum.  NMR takes advantage of this frequency’s dependence on the strength on an external magnetic field to gather information about each single nucleus, all at one time.  When the sample is placed in the now-homogenized magnetic field, the electrons around each nucleus become a current, and generate their own magnetic field in the opposite direction.  This effect is shown below, where the field of the NMR is blue, and the field created by the sample is red.

Magnetic field generated by a sample

The smaller, red magnetic field decreases the influence of the NMR’s magnetic field on each nucleus and causes a very small shift in their resonant frequencies.  These shifts, which are dependent on the electron density around each nucleus, are calculated using the equation:

(1)   \begin{equation*} \delta=10^{6}(\sigma_{0}-\sigma) \end{equation*}

The peaks that appear on NMR spectra are created when radio waves that match the now-altered resonant frequencies are absorbed and reradiated by the nuclei.  The four NMR spectra collected for 3,3-dimethyl-2-butanol are shown below as a sample of the gathered data.


^{1}H-NMR Spectrum of 3,3-dimethyl-2-butanol

Each peak in a ^{1}H-NMR spectrum represents a certain number of hydrogen nuclei and describes their neighborhood in the molecule.

^{13} C-NMR spectrum of 3,3-dimethyl-2-butanol

Each peak in a ^{13} C-NMR spectrum represents a specific “type” of carbon in the molecule.  One peak can represent more than one carbon nucleus if they have the same electron densities.

The two spectra below are meant to be read together.

Top:  DEPT-135 Spectrum; Bottom:  DEPT-90 Spectrum

The peaks on these spectra describe the number of hydrogen atoms bound to each carbon atom.  The frequencies of the peaks are the same as they are on the ^{13}C-NMR.

More information on exactly how to interpret each spectrum, and translate all of their information into a molecular structure, is forthcoming.



Project Proposal

Project Proposal

Nuclear magnetic resonance is a technique for determining the structure of organic molecules in solution.  It takes advantage of the magnetic properties of the nucleus to sense the proximity of double bonds, electronegative atoms like oxygen, and other magnetic nuclei in the molecular structure.  The technique can identify these bond types and functional groups in a molecule.  The same principles this project will study apply in MRI (Magnetic Resonance Imaging) machines that create detailed images of a medical patient without using dangerous, ionizing radiation.  An NMR spectrometer is a scaled-down MRI designed to look at small samples of a substance dissolved in solution [1].

The magnetic field generated by an NMR spectrometer, and the effective field experienced by a nucleus will be modeled.  These field’s effects on atomic nuclei will be studied and modeled.  The shim system, which is the process by which the magnetic field is homogenized, will also be explained.

The Larmor frequency is the resonant frequency of a given nucleus and is only dependent on the strength of the external magnetic field, and the magnet strength of the nucleus.  It is this frequency that allows us to select what elements to analyze in a given test.  The following equation is used to calculate a Larmor frequency [1]:

(1)   \begin{equation*} \nu_{0}=\frac{\gamma B_{eff}}{2\pi} \end{equation*}

where \nu_{0} is the Larmor frequency, \gamma is the magnetogyric ratio, or nuclear magnet strength, and B_{eff} is the effective magnetic field strength.  The effective magnetic field strength is given by the following equation [1]:

(2)   \begin{equation*} B_{eff}=B_{0}(1-\sigma) \end{equation*}

where B_{0} is the external magnetic field strength and \sigmais the shielding factor.  When the atom is polarized and stretched by a magnetic field, a small magnetic field is generated in the opposite direction, in accordance with Lenz’s Law, that also affects the nucleus.  The shielding factor is a direct measurement of electron density around a nucleus, and is important because the electron density determines the effect of the external magnetic field on the nucleus:  more electron density causes less of an effect, and vice-versa.  The resonant frequency can then be rewritten to show the dependence of frequency on external magnetic field strength:

(3)   \begin{equation*} \nu_{0}=\frac{\gamma B_{0}(1-\sigma)}{2\pi} \end{equation*}

Most of the information from NMR comes from the chemical shift, which is the change in a nucleus’ resonant frequency as the external magnetic field’s influence on the nucleus increases or decreases.  To standardize the results of NMR experiments, we express the chemical shift in parts-per-million so it is the same regardless of how powerful the instrument’s magnet is.  The formula for chemical shift is [1]:

(4)   \begin{equation*} \delta=10^{6}(\sigma_{0}-\sigma) \end{equation*}

where \delta is the chemical shift, and \sigma_{0} is the reference shielding factor used to define the zero-point of the chemical shift scale.  It is most often the shielding factor for a compound called tetramethylsilane, or TMS, because of the peak’s consistent and reliable strength and shape.


There are four experiments in particular that are performed using NMR.  1H-NMR focuses on hydrogen nuclei, while 13C-NMR, DEPT-90, and DEPT-135 focus on 13C nuclei.  They are so widely used because most organic molecules are made primarily of hydrogen and carbon atoms, so much information can be gleaned these four spectra.  To that end, these four tests will be performed on                   3,3-dimethyl-2-butanol



Figure 1:  3,3-dimethyl-2-butanol  

Mathematica, Wolfram Alpha, Excel and the NMR software Topspin will be used in the study and to create figures.  The book “NMR Spectroscopy Explained” by Neil E. Jacobsen will be used as the primary theoretical tool.  Karen Wovkulich, the Chemistry Instrumentation Manager at Vassar College, will also be consulted.


1)                   Jacobsen, Neil E. NMR Spectroscopy Explained: Simplified Theory, Applications and Examples for Organic Chemistry and Structural Biology. Hoboken, NJ: Wiley-Interscience, 2007. Print.


Nuclear Magnetic Resonance Spectroscopy

The magnetic field and radio frequency pulses generated in an NMR spectrometer will be modeled, and their effects on atomic nuclei will be studied.  Mathematica, Excel, Wolfram Alpha, and the NMR software Topspin are the computational tools most likely to be used.  The book “NMR Spectroscopy Explained” by Neil E. Jacobsen will be used as the primary theoretical tool.  Visual pieces will include a model of the magnetic field inside an NMR, a model of a radio frequency pulse, and one or more spectra of an as-yet-undetermined molecule that will be used to explain the physics behind the NMR process.