Correction of Q Factor Effects for Simultaneous Collection of Elemental Analysis and Relaxation Times by Nuclear Magnetic Resonance

Article abstract of DOI:10.1021/acs.analchem.9b05603

A new method for measurement of elemental analysis by nuclear magnetic resonance (NMR) of unknown samples is discussed here as a quick and robust means to measure elemental ratios without the use of internal or external calibration standards. The determination of elemental ratios was done by normalizing the signal intensities by the frequency dependent quality factor (Q) and the gyromagnetic ratios (?) for each measured nucleus. The correction for the frequency dependence was found by characterizing the output signal of the probe as a function of the quality factor (Q) and the frequency, and the correction for γwas discussed in a previous study. A Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence was used for evaluation of the relative signal intensities, which allows for derivation of elemental ratios, and was correspondingly used to simultaneously measure the T2? of samples for an added parameter for more accurate identification of unknown samples.

Full text of DOI:10.1021/acs.analchem.9b05603

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Article
Correction of Q factor effects for simultaneous collection of elemental
analysis and relaxation times by nuclear magnetic resonance
Scarlett Widgeon Paisner, Michael T. Janicke, Derrick C. Kaseman, Rachel K
Frankle, Jacob L Yoder, Marc A Alvarez, Michelle A. Espy, and Robert F Williams
Anal. Chem., Just Accepted Manuscript • DOI: 10.1021/acs.analchem.9b05603 • Publication Date (Web): 27 Apr 2020
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Page 1 of 15
Analytical Chemistry
Correction of Q factor effects for simultaneous collection of ele-
mental analysis and relaxation times by nuclear magnetic resonance
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Scarlett Widgeon Paisner1*, Michael T. Janicke2, Derrick C. Kaseman3, Rachel K.
Frankle3, Jacob L. Yoder3, Marc A. Alvarez3, Michelle A. Espy4, Robert F. Williams3
1Materials Science in Radiation and Dynamics Extremes, Los Alamos National Laboratory, Los Ala-
mos, NM USA
2Inorganic, Isotope and Actinide Chemistry, Los Alamos National Laboratory, Los Alamos, NM USA
3Bioenergy and Biome Sciences, Los Alamos National Laboratory, Los Alamos, NM USA
4Non-destructive Testing and Evaluation, Los Alamos National Laboratory, Los Alamos, NM USA
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ABSTRACT: A new method for measurement of elemental analysis by nuclear magnetic resonance (NMR)
of unknown samples has been discussed here as a quick and robust means to measure elemental ratios
without the use of internal or external calibration standards. The determination of elemental ratios was
done by normalizing the signal intensities by the frequency dependent quality factor (Q) and the gyro-
magnetic ratios (γ) for each measured nucleus. The correction for the frequency dependence was found
by characterizing the output signal of the probe as a function of the quality factor (Q) and the frequency,
and the correction for γ was discussed in a previous study. A Carr-Purcell-Meiboom-Gill (CPMG) pulse
sequence was used for evaluation of the relative signal intensities and allows for derivation of elemental
ratios, and was correspondingly used to simultaneously measure the T2* of samples for an added param-
eter for more accurate identification of unknown samples.
INTRODUCTION Quantitative analysis by nu-
clear magnetic resonance (NMR) has been used
extensively in pharmaceutical, medical, and ana-
lytical chemistry due to the non-destructive na-
ture and a low detection limit of parts per bil-
lion.1–4 In particular, NMR has been used to deter-
mine the purity of pharmaceuticals and organic
solutions and can detect relative ratios of differ-
ent stereoisomers, constitutional isomers, and en-
antiomers in solution.5–7 In order to acquire quan-
titative data, calibration standards and methods
are employed. These include external or internal
standards and the Electronic Reference To access
In vivo Concentration (ERETIC) method.8 Addition
of an internal calibration standard requires a
known quantity of a standard to be added to the
sample, which is then compared to the signal area
of the unknown substance. This requires some
knowledge unknown sample chemistry, since the
internal standard must be a substance that is inert
in the solution. The external calibration standard
requires generation of a calibration curve and is
most accurate when concentrations of the stand-
ard are close to the concentration of the unknown
sample. This method also necessitates approxi-
mate knowledge of the sample concentration and
chemistry to create an accurate calibration curve.
The ERETIC method utilizes an electrical refer-
ence signal that is implemented when collecting
the spectra of unknown samples, and this electri-
cal signal is calibrated against a standard of
known concentration. When elemental analysis of
several nuclei are desired, the process to deter-
mine the elemental ratios of a sample is time con-
suming for the three aforementioned methods
since each nucleus must be calibrated inde-
pendently, which is time consuming and ineffi-
cient. This makes NMR less desirable as a quanti-
tative tool in the aforementioned industries, and
therefore other analytical tools are more readily
applied.
However, for rapid analysis where relative ra-
tios of various elements are sufficient for identi-
fication of an unknown sample instead of absolute
concentrations, it is possible to measure the ratios
without the use of any external or internal stand-
ards. Takeda and colleagues9 have shown this pre-
viously by correcting for differences in the gyro-
magnetic ratio, γ. This was accomplished by
sweeping the magnetic field and holding the fre-
quency constant for each nucleus. The sensitivity
of each nucleus depends on γ and this causes une-
qual contributions of signal for each isotope;
therefore, differences in γ restrict and prevent di-
rect comparisons. The effect of different γ values
for different nuclei can be corrected by applying
the receptivity coefficient for constant frequency
(Dcf), and is determined to be:
|
|
(
)
퐷푐푓 = 푥푘 훾푘 퐼푘 퐼푘 + 1 (1)
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Analytical Chemistry
Page 2 of 15
where xk represents the natural abundance and Ik K describes the geometry of the receiving coil
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is the spin quantum number of nucleus k. Equation
(1) demonstrates a constant frequency measure-
ment, in which the magnetic field is varied to
measure each nucleus. By applying Dcf, Takeda and
colleagues were able to determine the relative
fraction of H, P, and Na atoms in a NaH2PO4 solu-
tion with an error of less than 4%. This is possible
since at constant frequency the efficiency of the
electrical circuit of the probe and the quality fac-
tor (Q) remain unchanged.
and is typically close to 1, η is the fill factor of the
coil, M0 is the nuclear magnetization, µ0 is the per-
meability of free space, Vc is the volume of the
coil, F describes the inherent noise of the pream-
plifier, kB is the Boltzmann constant, Tc is the tem-
perature of the coil, and Δf is the receiver band-
width. kB and µ0 are constants, and for a singular
probe configuration, K, Vc, F, Tc, and Δf are also
constant and they all can be combined into the
term, A. With carefully prepared samples in NMR
tubes with constant volume, η can also be consid-
ered as a constant and be included in A. The nu-
clear magnetization in thermal equilibrium, M0, is
defined as
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A constant frequency experiment requires a
variable field magnet which makes this approach
unfeasible for conventional, fixed field NMR.
When carrying out a fixed magnetic field experi-
ment, the resonant frequency is different for each
nucleus. This is more common compared to con-
stant frequency measurements due to the availa-
bility of high magnetic field superconducting mag-
nets. In this case, the receptivity equation is mod-
ified by incorporating a cubed dependence of γk.
This is referred to as the receptivity for constant
magnetic field (DcB) as is shown in Equation (2):9
2
(
)
푁훾ℎ 퐼 퐼 + 1 휔0
푀0 =
4휋2푘푇
푠
where N is the number of nuclei, h is Plank’s con-
stant, k is the Boltzmann constant, and Ts is the
sample temperature. Clearly M0 is proportional to
the frequency, ω0, and can be substituted in Equa-
tion (4) while the other terms are constant (as-
suming 100% natural abundance) and can be
grouped into term C.10 If the noise is constant for
a specific probe and console configuration, it can
also be incorporated into A so that frequency and
Q can be directly related to signal intensity (S). By
applying these substitutions and A to Equation
(4), the following relation is derived:
퐷푐퐵 = 푥푘 훾푘 3퐼푘 퐼푘 + 1 . (2)
|
|
(
)
The cubed factor of γk is a result of a squared
dependence from the thermal magnetization as
defined by Curie’s Law and another factor of γk
from the Larmor frequency, ω0, of each nucleus.10
Although this equation does address the issue of
different γ for the various isotopes, it does not ac-
count for the frequency dependence of the signal
from the resonant circuitry electronics. The fre-
quency dependence of Q is a measure of the effi-
ciency of an electrical circuit and is defined as a
resonator’s bandwidth with respect to the central
frequency, as is shown in Equation (3):
푆 ≈ 퐶휔10.5푄0.5. (5)
The signal intensity is then simply related to only
two easily measured parameters Q and ω0. From
this equation, the signal response of nuclei at dif-
ferent frequencies can be related.
Elemental analysis is an important parameter
for the identification of unknowns, but in some
cases, it can be misleading if only naturally abun-
dant NMR nuclei are measured. Such examples
are dichloroethane (C2H4Cl2) and acetaldehyde
(C2H4O2), which have the same C:H ratio of 2:4 but
also contain other low γ, quadrupolar nuclei that
are difficult to measure directly. Here, the accu-
racy of distinguishing between the molecules
could benefit from considering another signature
to differentiate the two, such as the transverse re-
laxation time (T2). T2 is dependent on paramag-
netic impurities in the sample, the mobility of the
molecules in solution, chemical shift anisotropy,
chemical exchange, and dipolar coupling.13 There-
fore, the T2 relaxation processes between differ-
ent molecules are likely different and the T2 time
constant can be used as an additional dimension
in the signature space. However, the measured T2
value may be shorter than the theoretical T2 value
휔0퐿
⁄
푄 =
(3)
푅
where R is an effective resistance that accounts
for the resistive losses of the circuit and dielec-
tric/resistive losses of the sample, and L is the in-
ductance of the coil. Note that in the constant fre-
quency case illustrated above, Q will be constant.
Q can be measured with a network analyzer using
the observed frequency divided by the bandwidth.
For a single probe configuration, Q is then de-
pendent on the dielectric and resistive losses of
the sample, the coil resistance, and the frequency,
where the former can be related to the polarity of
the sample. Q is related to the signal:noise ratio
(SNR) (ψ) as described by Equation (4):10–12
0.5
휇0푄휔0푉
푐
훹 = 퐾휂푀0 (
⁄
)
. (4)
4퐹푘퐵푇 Δ푓
푐
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Analytical Chemistry
and isopropyl methyl phosphonate (IMP)
caused by molecular diffusion in an inhomogene-
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ous magnetic field. Equation (6) shows the rela-
(Synquest Laboratories) solution was used neat
(without dilution). Trifluoro acetic acid (TFA)
(Sigma-Aldrich) was prepared in deuterated wa-
ter and deuterated acetonitrile (Cambridge Iso-
topes) at concentrations ranging from 1M to 5M
TFA to determine the effect of solvent polarity on
our NMR measurements. In order to observe the
relationship of Q and frequency, samples of water,
acetonitrile (Sigma-Aldrich), toluene (Sigma-Al-
drich), and 5M NaCl (Sigma-Aldrich), were also
prepared. Precisely 1 mL of the above-mentioned
solutions were placed in 5 mm NMR tubes for
analysis.
tionship of the measured T2 value (referred to as
*
T2 ), which is a sum of the inverse of the inherent
T2 and the additional T2 contributions that arise
from molecular diffusion in an inhomogeneous
field (T2i).
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=
+
푇
2푖
(6)
∗
푇
푇
2
2
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The Carr-Purcell-Meiboom-Gill (CPMG) pulse
*
sequence is routinely used to determine T2 , and
consists of a 90o pulse followed by a train of 180o
pulses. Between each 180o pulse, a spin echo
forms. The spin echo intensity decreases with
each successive 180o pulse, due to T2* processes.
The intensity decrease follows an exponential de-
cay with time constant T2*.14,15 Although, quantita-
tive NMR measurements are typically carried out
using a single-pulse experiment, herein CPMG is
also used to collect quantitative information.
Quantitation has been previously demonstrated
on amino acids and proteins by using the maxi-
mum amplitude of the first CPMG echo.16–18
1
All H, 19F, and 31P NMR measurements were
collected using a Tecmag Redstone LF2 NMR con-
sole with a Cryogenic Ltd. 54 mm bore variable
field (0 –7 T) cryogen free magnet. The field has a
homogeneity of <20 ppm over a 10 mm diameter
spherical volume. The broadband channel of a
5mm Bruker 400 MHZ standard bore probe was
used to collect all measurements so that the same
coil and electronic circuit were used for each nu-
cleus. For measurements collected at a magnetic
field of 1 T for 1H, 19F, and 31P nuclei, the observed
frequencies were 42.58, 40.05, and 17.24 MHz, re-
spectively, and measurements collected at 3.5T
were observed at frequencies of 149.0, 140.2, and
60.3 MHz, respectively. Constant frequency ex-
periments were collected using a frequency of
42.6 MHz corresponding to magnetic fields of 1,
1.06 and 2.47 T for 1H, 19F, and 31P nuclei, respec-
tively. The Carr-Purcell-Meiboom-Gill (CPMG)
pulse sequence was employed to acquire both T2
relaxation and quantitative data, where all acqui-
sition parameters were held constant with excep-
tion of the 90o and 180o pulse lengths and the re-
cycle delay times. The time between successive
180o pulses was set at 345 µs for all nuclei. Nuta-
In this study, CPMG was used to determine el-
emental ratios for 1H, 19F, and 31P. We have inves-
tigated the impact of varying solvent dielectric
constants, different magnetic fields (1 to 3.5 T),
and molecular structure on the error of H:F:P ele-
mental ratios. At a field of 3.5 T, these errors are
as large as 32%. We demonstrate that these errors
may be corrected by measuring Q and normalizing
the signal by both Q and Dcf. After applying these
corrections, elemental ratios with errors of less
than 6% are achieved.
EXPERIMENTAL SECTION Bis-(2,2,2 trifluoro-
ethyl)methylphosphonate (BTFEMP) was synthe-
sized by slowly adding 2,2,2-trifluoroethanol
(sigma Aldrich) to methylphosphonic dichloride
1
tion curves were collected for H, 19F, and 31P nu-
(Sigma-Aldrich)
in
tetrahydrofuran
(THF)
clei to determine the 90o and 180o pulse lengths.
The CPMG echo trains used for quantitation were
collected using 8192 echoes and 64 acquisition
points for an acquisition time of 0.32 ms for each
echo. The decay of the signal intensity of the echo
train was fit using an exponential curve shown by
equation (7),
(Sigma-Aldrich) at -12 oC. Triethyl amine (Sigma-
Aldrich) was then added in excess and the solution
stirred overnight. THF was removed by evapora-
tion and the resulting solution was evaluated by
thin layer chromatography (TLC). Purification of
BTFEMP was accomplished by column chromatog-
raphy on silica with a mobile phase of 90:10 hex-
−푥
ane/ethyl acetate. 31P, 13C, and H NMR spectra
1
y = 푦0 + 퐴0푒푥푝(
), (7)
⁄
푡
were collected and show a product purity of >98%
for each nuclei. This BTFEMP was used without
further purification. A 1M triphenyl phosphate
(TPP) (Sigma-Aldrich) solution was prepared us-
ing deuterated dimethyl sulfoxide (DMSO) (Cam-
where the pre-exponential factor, A0, was used for
quantitation by applying the receptivity factor, Dcf
or DcB, to determine the elemental ratios. The ef-
fect of polarity/ionicity of the sample on Q was
measured as a function of frequency at magnetic
fields of 0.47, 1.17, 1.88, 2.58, and 3.29 T which
correspond to frequencies of approximately 20,
bridge
Isoptopes).
A
5M
diisopropyl
methylphosphonate (DIMP) (Alfa Aesar) was pre-
pared in deuterated acetonitrile (Sigma-Aldrich)
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Analytical Chemistry
Page 4 of 15
*
50, 80, 110, and 140 MHz, respectively. Q was
measured using a Copper Mountain TR1300/1 vec-
tor network analyzer and was calculated by divid-
ing the resonant frequency by the bandwidth (-6
dB).19 Additionally, Q was also measured for fre-
quencies ranging from 17 to 140 MHz for BTFEMP
and TPP to generate a trend line for the relation-
ship of 푀푧 ≈ 퐴휔1.5푄0.5, frequency, and signal inten-
sity. Nutation curves for 1H and 31P were collected
for each sample at all frequencies, and T1 relaxa-
tion times were measured using the conventional
inversion recovery experiment so that a 90o pulse
was applied for excitation and 5*T1 was used for
the recycle delay to acquire quantitatively compa-
rable results. These measurements were carried
out for frequencies ranging from 17 to 140 MHz
for both 1H and 31P nuclei.
separating into the T2 for each particular envi-
*
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ronment. The T2 values have been measured and
are reported in Table 1. For 1H, 19F, and 31P meas-
*
urements, the data shows that T2 decreases with
increasing frequency. This is explained by the var-
iable magnetic field strength that is applied for
measurements at different frequencies. It has
been noted previously that the field homogeneity
of B0 decreases as B0 increases, and a reduction in
*
9
field homogeneity has a direct influence on the T2
relaxation rate.20 Since higher frequencies were
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achieved by increasing the strength of B0, it can
*
be expected that T2 values decrease with fre-
quency.
RESULTS AND DISCUSSION The CPMG experi-
ment yields an echo train where the amplitude of
the echo exponentially decays with time constant
*
T2 . The decay of the signal intensity is the time
that it takes for the spins in the transverse plane
to lose coherence, designated as the T2 relaxation
time, and can be a result of molecular diffusion,
chemical shift anisotropy, and dipolar coupling.
Figure 1 shows the decay of the 1H, 19F, and 31P sig-
nals for BTFEMP at 1 T along with an exponential
fit for each curve. The signal intensity for 31P is
considerably smaller than the signal intensity for
1H and 19F due to the lower concentration and gy-
romagnetic ratio of the phosphorous. Conse-
quently, the 31P decay curve was multiplied by a
factor of 25 to accentuate the exponential nature
of the curve. The pre-exponential factor, A0, in
Equation (7) is a measure of the signal intensity
and depends on the concentration of a particular
isotope in solution. The T2* relaxation time meas-
ured in this study is representative of all environ-
ments in the sample and does not account for sep-
FIGURE 1. Exponential decay of the magnitude signal
intensity of neat BTFEMP collected using CPMG for
1H (blue), 19F (red) and 31P (green). The decay of the
signal in the echo train is simulated using an expo-
nential curve (black lines). The fits provide T2* relax-
ation times, as well as the pre-exponential factor
that is used for quantitative measurements. Note:
the 31P intensity was multiplied by 25 for clarity.
Fitting the decay curves results in values for
A0 for each isotope, which is used to determine the
elemental ratios. The values of A0 are inherently
tied to the concentration as well as the receptivity
(γ) of the nucleus, but the receptivity effects may
be corrected by using the receptivity for constant
field, DcB, as shown in Equation (2). Consequently,
the H:F:P elemental ratios were calculated using
A0 from a CPMG echo train at 1 and 3.5 T and were
corrected using DcB, as shown in Table 1. The the-
oretical H:F:P ratio for BTFEMP is 7:6:1, and the
calculated elemental ratios have been normalized
to the theoretical number of hydrogen atoms. For
both fields, the H:F ratios have low errors of less
than 3% and are due mostly to statistical instru-
mental variance. All spectra were acquired in trip-
licates, and variance between subsequent spectra
showed errors of roughly 1 %. Due to the similar-
ity in γ for 1H and 19F (42.6 and 40.1 MHz/T,
1
arate H molecular environments. This is due to
(1) low magnetic fields that do not allow for sep-
aration of chemical shift information, (2) mag-
netic field inhomogeneity that causes signals to be
broad, and (3) the time domain analysis of the
data. By fitting the decay of the signal, both pa-
rameters can be simultaneously determined, and
such an analysis can be expedited by using a
Matlab or Python code to generate and tabulate
this information within seconds. All exponential
fits were calculated with a single decay constant,
which appeared to be independent of the number
1
of H, 31P, or 19F nuclei in the molecule, which is
likely an artifact of the lack of resolution at the
magnetic field strengths employed. In this case,
*
the T2 of all environments of isotope k are com-
bined together to give a single value rather than
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Analytical Chemistry
TABLE 1. Elemental ratios determined using A0 from the signal decay of the echo train in CPMG experiments
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2
3
4
5
6
7
8
1
for BTFEMP. Gyromagnetic ratios, γ, for H, 19F, and 31P are 42.58, 40.05, and 17.24 MHz/T. The theoretical
H:F:P ratios for BTFEMP is 7:6:1 and experimental elemental ratios have been normalized relative to the the-
oretical hydrogen value. Note that Ao values have been normalized by DcB and DcF.
Isotope
1H
Ao
Elemental Ratio
7.00
Error (%)
-
T2* (s)
0.758
1.100
4.009
0.932
1.311
3.535
0.758
1.105
3.889
Field/Frequency
2.50 × 105
1.73 × 105
2.80 × 103
1.14 × 105
8.29 × 104
1.43 × 103
2.50 × 105
2.00 × 105
1.38 × 104
Constant Field:
B = 1T
19F
31P
1H
5.84
2.74
18.20
-
9
1.18
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7.00
Constant Field:
B = 3.5T
19F
31P
1H
6.10
1.65
32.22
-
1.32
7.00
Constant Frequency:
19F
31P
5.93
1.03
4.77
ω = 42.6 MHz
0.95
respectively), the signal response is assumed to be
similar and produces lower errors in the H:F ra-
tios, as shown in Table 1. The H:P ratio, on the
other hand, is higher than expected and has an er-
ror of 18.2 and 32.2% for the ratios collected at 1
and 3.5T, respectively, even after taking the con-
stant field receptivity, DcB, into consideration. In
this case, DcB, which only corrects for differences
in γ, is not sufficient to account for the different
frequencies used in a constant field experiment.
The error is exaggerated at higher magnetic field
(32.2% at 3.5 T compared to 18.2% at 1 T) and is
attributed to the response of the probe at the dif-
ferent frequencies of the two nuclei, where Δω is
88.7 and 25.4 MHz, respectively. According to
Equation (3), the difference in frequency between
1H and 31P has an effect on Q, and the dependence
of Q on frequency is shown in Figure 2 for
BTFEMP, as well as other solutions, which will be
discussed in detail below. This plot clearly illus-
trates that Q increases with frequency for most
solutions, therefore the probe efficiency and out-
put signal increases with frequency as well.
H:F:P ratios for constant frequency measure-
ments are 7:5.77:1.22, which yield errors of 1.03
and 4.77 % for fluorine and phosphorous. While
the error for the H:F ratio is not affected, it was
significantly reduced for the H:P ratio and can be
considered as experimental error. Because the
frequency was held constant, the Q factor was the
same for all three nuclei, and therefore any error
in these experiments is likely due to statistical in-
strumental variance, data processing, or the infe-
rior signal:noise ratio (S/N) of the 31P measure-
ments.
To determine how to correct Q, 1H and 31P sin-
gle-pulse NMR spectra where collected for neat
BTFEMP and 1M TPP in deuterated DMSO at fre-
quencies ranging from 17 to 140 MHz. The relation
in Equation (5) was used to describe the depend-
ence of frequency and Q (ω1.5Q0.5) with signal,
therefore Q was measured for all frequencies, and
this is plotted in Figure 3. The results of the
ω1.5Q0.5 versus signal data are shown in a double
logarithmic plot in Figure 2, and the data has been
fit using a linear regression. Figure 3 shows data
from both 1H and 31P measurements for several so-
lutions, and the average of the slopes of the linear
fit for all data sets is 1.025 ± 0.006. The low var-
iance in the slope values signifies that for differ-
ent isotopes and samples with varying Q, the rate
of the increasing signal intensity with frequency
remains unchanged. It further suggests that the
signal intensity can be corrected for differences in
frequency between nuclei for samples of various
compositions and polarities. Finally, this Figure
proves that the relation given in Equation 5 is
valid over the range of frequencies investigated
here. This is exemplified with 1H and 31P data that
In order to eliminate the differences in probe
efficiency, H:F:P ratios were also evaluated using
the constant frequency method, and results are
shown in Table 1. These results confirm the rela-
tionship of greater frequency difference with
greater relative error. The frequency was held
constant at 42.6 MHz and the magnetic field was
1
swept from 1.0 to 2.5 T to collect spectra for H,
19F, and 31P. Using this method, Q does not have to
be taken into account, and only γ is responsible
for any differences in receptivity for these meas-
urements. Thus, signal may be corrected using Dcf
to determine the elemental ratios. The observed
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Page 6 of 15
was collected for BTFEMP at a constant magnetic efficiency. The Q correction has been applied to
1
2
3
4
5
6
7
8
field of 1.17 T with frequencies of 50.0 and 20.26
MHz, respectively. First, the 31P signal intensity,
acquired at 20.26 MHz, was corrected so that it is
analogous to the expected 31P signal intensity of
BTFEMP at 50 MHz. The slope of the magnetiza-
tion (Mz) versus ω1.5Q0.5 fit, which is defined as
푚 = 푦2 − 푦1 푥2 − 푥1 , is used to relate the fre-
quency and Q difference to the acquired and ex-
pected signal intensities, as shown in Equation
(8):
DIMP and TPP as well and both show errors of less
than 6%, as is tabulated in Table 2. While deter-
mination of the probe efficiency was carried out
using single pulse experiments, the two-step cor-
rection for Q can also be applied to CPMG results,
where the A0 is used in place of the signal inten-
sity. It should be noted here that this relationship
is valid for only a given system, which includes
the console and probe, but if either of these com-
ponents is changed, then the slope of the effi-
ciency correction factor must be reevaluated.
(
)⁄(
)
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푙푛(푀푧,50.0 푀퐻푧) = 푒{푙푛(푀푧,20.3 푀퐻푧) +
푚 푙푛 휔1.5푄0.5
− 푙푛 휔1.5푄0.5
}. (8)
The above correction is described for the case
where a variable field magnet is used. However,
since a majority of NMR labs house static mag-
nets, the correction should also be considered for
this configuration. In order to test the viability of
[
(
)
(
)
]
20.3 푀퐻푧
50.0 푀퐻푧
Since Q was measured previously as a function
of frequency, the ln(ω1.5Q0.5) for both frequency
points can be applied to Equation (8) along with
the acquired signal intensity. This leaves only the
expected signal intensity at 50 MHz, which has
been corrected for ω and Q.
1
this method under static field conditions, H and
31P NMR spectra were collected at a field of 1.17 T
(ωH = 50 MHz and ωP = 20.26 MHz) using a CPMG
pulse sequence. Ao was measured, then divided by
Dcf. Takeda has shown that Dcf should be used for
measurements at constant frequency only since it
corrects for γ. However, it is used here for con-
stant field experiments because the signal first
needs to be corrected for γ before it can be cor-
rected for ω and Q. After applying Dcf, a double
logarithmic plot similar to Figure 3 can be gener-
ated. This yields a plot where the data points can
be fit with a line of slope 1.096. In comparison,
the slope of the line fit with the variable field data
was 1.005, proving that this correction can be car-
ried out under constant field conditions as well.
TABLE 2. Elemental ratios of BTFEMP, DIMP, and
TPP using Q efficiency correction factor, with ele-
mental H:P ratios with error of less than or equal to
~6%. All data was processed using time domain sig-
nal intensities for single pulse experiments, unless
otherwise stated.
FIGURE 2. Dependence of Q on frequency, at frequen-
cies ranging from 20 to 140 MHz for 5M NaCl (purple
pentagon), water (orange inverted triangle), ace-
tonitrile (red square), toluene (black triangle),
BTFMP (blue circle), and TPP (green star) samples.
The Q increases exponentially for toluene, acetoni-
trile, and water, but Q for 5M NaCl decreases at
higher frequencies.
Calcu-
lated
H:P
Theoreti-
cal H:P
Sample
BTFEMP
(CPMG)
BTFEMP
DIMP
TPP
Error (%)
7:1
7:1
17:1
15:1
7:0.96
7:0.94
17:0.94
15:1.02
3.6
1
Now, the comparison of the H and 31P signal
5.88
6.05
2.52
can be considered as a pseudo-constant frequency
experiment, and Dcf can be used. Here, the differ-
1
ences in γ for H and 31P can be accounted for so
that the signal intensities can be compared. Ap-
plying the correction for ω and Q and the recep-
tivity for γ to the 31P data for BTFEMP gives a H:P
ratio of 7:0.95 (theoretical ratio is 7:1) with an
error of 5.88%. Previously, when only the DcB was
applied to the data collected under constant field
conditions at 1 T, the error was 18.20 %. This is
significantly reduced by correcting for the probe
1H single-pulse NMR spectra were also col-
lected and analyzed for toluene, acetonitrile, wa-
ter, and 5M NaCl to determine the effect of polar-
ity on the Q dependence of signal intensity and the
results are plotted in Figures 2 and 3. The relative
polarities for toluene, acetonitrile, water, and
DMSO are 0.1, 0.5, 1.0, and 0.421 and this set of
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Analytical Chemistry
solvents delineates how the slope is effected by sponse can be directly related to the effective re-
1
polarity. The average slope for the linear fit of the
1H and 31P data for all six solutions are 1.025 ±
0.006, as shown in Figure 3, demonstrating that
polarity has no effect on the relationship between
the signal intensity and ω1.5Q0.5, and therefore this
relationship stays constant for other nuclei and
solutions of differing polarity. This relationship
also holds up for the ionic solution of 5M NaCl,
even though ionic solutions tend to be problematic
in NMR due to the lossiness of the sample.21–23 The
effect of polar and ionic samples has been re-
ported previously, and is a result of dielectric and
resistive losses, caused by the interaction of RF
electric fields with the sample, which lowers Q
and the SNR.22 By examining the dependence of Q
on frequency, the NaCl solution shows a much dif-
ferent behavior than other solvents, as shown in
Figure 2. This is a consequence of the resistive
losses typical of ionic solutions. Toluene, acetoni-
trile, and water all show a similar exponential
growth of Q with frequency, while 5M NaCl in-
crease to a maximum Q at 80 MHz and then de-
creases with higher frequencies. Although 5M
NaCl has a much different behavior of Q with re-
spect to frequency, the slope of the linear fit for
ω1.5Q0.5 is the same as for the other samples (Fig-
ure 3). As Q decreases at higher frequencies for
5M NaCl, the signal also decreases by a propor-
tional amount, thus keeping the trend constant,
and characterizing the relationship between fre-
quency, Q, and signal.
sistance of the samples. For the analysis of the
TFA standards in different solvents, the only term
that differs is Rs, and the higher effective re-
sistance of water over acetonitrile is expected due
to polarity differences. The effect of the resistive
losses is seen in the relative intensity of the two
calibration curves, where the signal intensity of
the deuterated water curve is lower than that of
the acetonitrile for similar fluorine concentra-
tions. The slope of the curve is also lower for the
former because the sample becomes more lossy as
the concentration of the TFA increases, which has
previously been seen in other polar or ionic solu-
tions.22 According to Equation (3), the effective re-
sistance of the sample is inversely proportional to
Q, so highly polar or ionic samples should exhibit
a lower Q value.
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The effect of polarity and ionicity is an issue
when dealing with calibrations using external
standards to build a calibration curve due to the
aforementioned resistive losses. 19F calibration
curves, constructed using TFA in acetonitrile and
deuterated water, are plotted in Figure 4. The re-
sulting slopes of the two calibration curves are
drastically different and stemm from the use of
the two different solvents, acetonitrile and deu-
terated water, which have polarities of 0.46 and
1.11, respectively. The deuterated water, a higher
polarity solvent, leads to more losses due to the
higher effective resistance of the sample. The re-
lationship of the SNR to the resistance is:
FIGURE 3. Double logarithmic plot of the signal
as a function of ω1.5Q0.5 for various solvents. Data
1
points for H (open symbols) and 31P (solid sym-
bols) are shown for TPP (green star), BTFEMP
(blue circle), 5M NaCl (purple pentagon), water
(orange inverted triangle), acetonitrile (red
square), and toluene (black upright triangle). The
regression line of best fit for each data set is also
plotted, and all lines have a slope of 1.025 ±
0.006.
This suggests that the use of external calibra-
tion standards is not a viable method to determine
elemental ratios of unknown samples where the
polarity is not previously identified. For rapid
analyses, the more robust process is to apply the
efficiency correction that not only considers dif-
ference in γ but also corrects for the effects of Q.
This approach can be widely used for ionic, or-
ganic polar and non-polar samples to identify the
elemental ratios of various samples and can be es-
pecially useful in biological specimens, since wa-
ter and salt contents in such samples can vary
widely. While this method may be used for meas-
urement of elemental ratios, it should be noted
1
푆푁푅 ∝
, (9)
)
√푅 (푇 +푇 )+푅 (푇 +푇
푝푎 푝푎
푠
푠
푐
푐
where Rs and Rc are the resistance values of the
sample and coil, respectively, and Ts, Tpa, and Tc
are the temperatures of the sample, preamplifier,
and coil, respectively.11 For the system used in this
study, it was found that noise is constant and
therefore Equation (9) can be directly related to
signal intensity and the differences in signal re-
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here that this does not identify the absolute con-
Page 8 of 15
and a nutation curve must be generated at
1
2
3
4
5
6
7
8
centration of a particular nucleus as would be de-
termined by using an internal or external calibra-
tion standard. However, the detection of ele-
mental ratios for individual samples or substances
does have advantages especially in the case of
chemical warfare agents or concentrated pharma-
ceuticals which tend to have a specific number of
P and/or F atoms for a certain number of H atoms
and therefore elemental ratios are sufficient in
these instances. By implementing CPMG to deter-
each frequency.
2. The Q at each frequency should needs to be
recorded.
3. A plot similar to that shown in Figure 3 is
generated to determine the signal response
over a range of frequencies by carrying out
a linear regression. This only needs to be
carried out on one nucleus since the re-
sponse is constant regardless of nucleus or
polarity of the sample.
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*
mine the elemental ratios, T2 values can be sim-
ultaneously measured to add another dimension
to the signature space for identification of
unknown samples.
4. The linear regression is used to generate the
line of best fit. Takeda’s correction (Dcf) is
then used to divide the signal intensity. The
slope of the best fit line will be applied to
Equation (8) to determine elemental ratios
for other samples. There is no need to re-
characterize the probe for future use.
Constant magnetic field case:
1. A single sample should also be used in the
case of a constant field system. Here, since
the field cannot be swept, different frequen-
cies should be probed by acquiring spectra
on each different nuclei. For example, 13C
and 29Si could be measured on a tetramethyl
silane or 13C and 31P on a methyl phosphate.
These spectra need to be collected using a
π/2 pulse and a recycle delay of 5*T1. It
should be noted that this only works for a
single channel, such as the broadband chan-
nel. The H/F channel would need to be char-
acterized independently by collecting 1H and
19F spectra.
FIGURE 4. Calibration curves collected using 19F
NMR of trifluoro-acetic acid in either acetonitrile
(red) or D2O (blue). The calibration curves show sig-
nificantly different slopes and the intensity/isotope
changes drastically at higher concentrations, which
is due to differences in the polarity of the two sol-
vents. All 19F NMR spectra were collected at a mag-
netic field of 1 T and a frequency of 40.05 MHz.
2. Q is recorded at each frequency.
3. The signal intensity, A0, is then measured
for each spectrum and normalized to make
the nuclear ratios 1:1. For example, the 13C
signal intensity in Si(CH3)4 needs to be di-
vided by 4. The normalized signal is then di-
vided by Dcf to correct for γ.
This method can be applied to NMR spectrom-
eter systems that utilize either static or variable
magnetic fields. In both cases, the circuit needs to
be characterized by generating a ln(ω1.5Q0.5) ver-
sus ln(signal) plot, as shown in Figure 3. The fol-
lowing procedure describes the steps that are re-
quired for either static or variable field systems
to enable rapid detection of elemental ratios with-
out using internal or external standards.
4. A double logarithmic plot, similar to Figure
3, is constructed using the signal divided by
for γ for each nucleus and then plotting
against ω1.5Q0.5.
Variable magnetic field case:
5. Step 4 from the variable magnetic field pro-
cedure is followed to determine the ele-
mental ratios.
1. Using a sample, the magnetic field should be
swept so that spectra can be collected for a
given nucleus at different frequencies. The
spectra must be collected using a π/2 pulse
and a recycle delay of 5*T1, therefore the T1
CONCLUSION The ability to measure ele-
mental ratios without the use of internal or exter-
nal standards has been discussed here as a method
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Analytical Chemistry
for rapid detection of unknown solutions, espe- (2) Harris, R. K. Applications of Solid-State
1
2
3
4
5
6
7
8
cially at low magnetic fields (1 and 3.5 T). Because
the polarity of the solvent has an effect on the sig-
nal output due to dielectric losses, the implemen-
tation of calibration curves using external calibra-
tion standards is not a robust method for rapid de-
tection. NMR signal intensities were collected us-
ing CPMG time domain analysis then corrected for
(1) the Q of the probe at different frequencies and
(2) the γ of each nucleus. The application of such
a correction yielded elemental ratios with a 6% or
less error for various material solutions. The re-
lationship of ω1.5Q0.5 and the signal intensity is
constant for a variety of samples that have polar-
ities ranging from 0.1 to 1.0 and even holds con-
stant for ionic solutions. This allows the 2-step
correction factor to be applied broadly for many
different types of samples and eliminates the need
for internal or external calibration standards if el-
emental ratios are satisfactory for the analysis of
the unknown sample. The correction also works
on static magnetic field systems, as described
NMR to Pharmaceutical Polymorphism and Re-
lated Matters *. J. Pharm. Pharmacol. 2007, 59,
225–239. https://doi.org/10.1211/jpp.59.2.0009.
(3) Rosen, B. R.; Belliveau, J. W.; Vevea, J. M.;
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Agents. Magn. Reson. Med. 1990, 14, 249–265.
(4) Stanley, P. D. Organofluorines, 1st ed.;
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(5) Flores-parra, A.; Gutierrez-avella, D. M.;
Contreras, R.; Khuong-Huu, F. 13C and 1H NMR In-
vestigations of Quinic Acid Derivativesꢀ : Complete
Spectral Assignment and Elucidation of Preferred
Conformations. Magn. Reson. Chem. 1989, 27,
544–555.
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Based Metabolomics at Work in Phytochemistry.
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(7) Diakos, C. I.; Messerle, B. A.; Murdoch, P.
S.; Parkinson, J. A.; Sadler, P. J.; Fenton, R. R.;
Hambley, T. W. Identification by NMR Spectros-
copy of the Two Stereoisomers of the Platinum
Complex [ PtCl 2 ( S -Ahaz )] ( S -Ahaz ) 3 ( S ) -
Aminohexahydroazepine ) Bound to a DNA 14-Mer
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*
above. The T2 is simultaneously measured when
the CPMG pulse sequence is employed, and there-
fore this analytical method allows an additional
signature to be simultaneously measured to iden-
tify unknown or potentially dangerous chemicals
rapidly and accurately.
(8) Akoka, S.; Barantin, L.; Trierweiler, M.
Concentration Measurement by Proton NMR Us-
ing the ERETIC Method. Anal. Chem. 1999, 71 (95),
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AUTHOR INFORMATION
Corresponding Author
(9) Takeda, K.; Ichijo, N.; Noda, Y.; Takegoshi,
K. Elemental Analysis by NMR. J. Magn. Reson.
* Email: scarlettw@lanl.gov
2012,
224,
48–52.
ACKNOWLEDGMENT
https://doi.org/10.1016/j.jmr.2012.09.004.
(10) Abragam, A. Principles of Nuclear Mag-
netism; Oxford University Press: Oxford, 1961.
(11) Hoult, D. I.; Richards, R. E. The Signal-to-
Noise Ratio of the Nuclear Magnetic Resonance
Experiment. J. Magn. Reson. 1976, 24, 71–85.
(12) Hill, H. D. .; Richards, R. E. Limits of Meas-
urement in Magnetic Resonance. J. Phys. E. 1968,
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The
authors
gratefully
acknowledge
R.
Michalczyk, M.W. Malone, P. Magnelind, and A.V.
Urbaitis for intellectual support and informative
conversations in support for this study. This pro-
ject was supported by the Department of Home-
land Security (DHS) Science & Technology Direc-
torate (S&T) through Interagency Agreement
HSHQPM-16-X-00213 with Los Alamos National
Laboratories. This work is released for publica-
tion in accordance with Los Alamos National La-
boratory LA-UR-19-26783 by Triad National Secu-
rity, LLC (Los Alamos, NM, USA) operator of the
Los Alamos National Laboratory under Contract
No. 89233218CNA000001 with the U.S. Depart-
ment of Energy.
(13) Levitt, M. H. Spin Dynamics Basics of Nu-
clear Magnetic Resonance; John Wiley & Sons, Ltd:
West Sussex, 2008; pp 543–595.
(14) Meiboom, S.; Gill, D. Modified Spin-Echo
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ACS Paragon Plus Environment

Analytical Chemistry
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289x202mm (300 x 300 DPI)
ACS Paragon Plus Environment

Page 15 of 15
Analytical Chemistry
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272x208mm (300 x 300 DPI)
ACS Paragon Plus Environment