# Sensitivity Enhancement of NMR Spectroscopy Receiving Chain Used in Condensed Matter Physics

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Noise Figure of NMR System Receiving Chain

#### 2.1. General Approach

#### 2.2. NMR Spectroscopy System

#### 2.3. Concept of Noise Figure

#### 2.4. Noise Model of NMR Receiving Chain

#### 2.5. Preamplifier’s Impedance Mismatch

#### 2.6. Non-Standard Temperature of the Probe

#### 2.7. Signal Averaging

#### 2.8. The Case of a Two-Stage Preamplifier

## 3. Experimental Verification of Developed Noise Model

#### 3.1. General Approach

#### 3.2. Measurements Description

^{63}Cu signal in SeCuO

_{3}(two sets of measurements) [17], while the second one was the

^{133}Cs signal in Cs

_{2}Cu

_{2}SnF

_{12}(three sets of measurements) [18]. A full description of the measurement setup is available in Table 1, while the NMR properties of the measured nuclei are presented in Table 2. We will focus on effects of different pre-amplifiers in the measurement setup, as there are a vast number of commercially available types and as it is the only active element (besides the spectrometer). While choosing the pre-amplifier, one should bear in mind that its characteristics will have a significant impact on the overall noise figure (because $G\gg 1$) and decrease the contribution of losses accumulated up to its output.

^{63}Cu and

^{65}Cu, whose abundance ratio equals 0.691:0.309. So, by recording an NMR signal of

^{63}Cu, the signal intensity will be only approximately 69% of all the copper nuclei in the sample. Hence, we need to keep the ratio of isotopes in mind when calculating what signal size we expect to see.

^{133}Cs signal has been split into seven lines. Again, as the spectral weight is preserved, the split signal amplitude will drop from the “unsplit case”. The size of quadrupolar splitting can vary from a few kHz (as for the

^{133}Cs signal) to several tens of MHz (as for the

^{63}Cu signal).

^{63}Cu measurements were the MITEQ AU-1114-SMA [19] (abbreviated as M290) and the THAMWAY N141-206AA(D) [20] (abbreviated as T77). The first pre-amplifier operated at ${T}_{0}$, while the second one was cooled to the temperature of liquid nitrogen (77 K). Both were used as single-stage pre-amplifiers. On the other hand, for the

^{133}Cs measurements, these two units were used as single-stage pre-amplifiers, but the T77 was also used as the first stage of a two-stage pre-amplifier, with a Mini-Circuits HELA -10D+ [21] (abbreviated as MC290) used as the second stage. The MC290 operated at ${T}_{0}$.

#### 3.3. Results

^{63}Cu in SeCuO

_{3}(i.e., the associated $SN{R}_{out}$), it follows that the values of NMR receiving chain $SN{R}_{in}$ were 12.17 dB and 12.14 dB for M290 and T77, respectively. The difference between the predicted values was around 0.03 dB, which is comparable to measurement uncertainty. The evaluated value of the same SNR from (1) and (2) gave 15.20 dB. Here, we took into account that the signal amplitude was reduced, due to the short ${T}_{2}$ time (to 75% of the value), broadened line-width, NQR splitting of spectral lines (to 33% of the value), number of crystallographic sites (to 50% of the value), and abundance of the

^{63}Cu isotope (69%). Out of these, the most ambiguous parameter was line broadening, because it could not be estimated with high precision. However, even if we conservatively estimate that the spectral weight was reduced to 10 % of its value, our $SN{R}_{in}$ estimation, using (1) and (2), was of an acceptable order of magnitude. To keep this estimate simple, we did not discuss the dependence of intensity on the orientation of the sample (i.e., orientation of the quadrupolar principle value with respect to external field), or NMR coil, but these effects would further reduce the signal intensity and, thus, make our result match even better.

^{133}Cs in Cs

_{2}Cu

_{3}SnF

_{12}, as shown in Figure 6a, where the M290 and T77 pre-amplifiers were used, were equal to 37.54 dB and 37.55 dB, respectively; again, showing a good consistency of the results. The evaluated value of the NMR receiving chain $SN{R}_{in}$, from (1) and (2), in this case, was 35.31 dB, which was adjusted only for the NQR splitting of spectral lines (to 25% of the value), as the line-width was only approximately 5kHz (FWHM). In this system, there was only one crystallographic site, the abundance of the

^{133}Cs isotope was 100%, and the ${T}_{2}$ time did not show any considerable effect. Therefore, the validity of the derived expression is confirmed in this case, as well.

^{63}Cu in SeCuO

_{3}, the receiving chain with SOA generated a $SN{R}_{out}$ 2.16 dB greater than that of M290. This corresponds to a 65% enhancement in linear scale with SOA (i.e., to reach equal $SN{R}_{out}$, SOA would, in our configuration, measure 1.65 times faster). This is a very significant decrease in measurement time. Furthermore, for

^{133}Cs measurements in Cs

_{2}Cu

_{3}SnF

_{12}, this difference rose to 2.76 dB in logarithmic, or 89% in linear scale. Here, the measurement time was reduced by almost two times, which is a drastic decrease. These two predictions prove that pre-amplifier properties, along with those of the spectrometer, are one of the most important aspects of a NMR system.

## 4. Improvement Suggestions

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Spectrometer Noise Figure Determination

**Figure A2.**RF generator’s ${N}_{SSB}-f$ chart and linear approximation of its curve (red line). The figure corresponds to Rohde & Schwarz SMC100A generator (with permission from Rohde & Schwarz).

## References

- Kleckner, I.R.; Foster, M.P. An introduction to NMR-based approaches for measuring protein dynamics. Biochim. Biophys. Acta (BBA) Proteins Proteom.
**2011**, 1814, 942–968. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Richards, M.G.; Andrews, A.R.; Lusher, C.P.; Schratter, J. Cryogenic GaAs FET amplifiers and their use in NMR detection. Rev. Sci. Instrum.
**1986**, 57, 404–409. [Google Scholar] [CrossRef] - Abragam, A. The Principles of Nuclear Magnetism; Clarendon Press: Oxford, UK, 1989. [Google Scholar]
- Motchenbacher, C.D.; Connelly, J.A. Low-Noise Electronic System Design; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 1993. [Google Scholar]
- Pozar, D.M. Microwave Engineering; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 2012. [Google Scholar]
- Hoult, D.; Richards, R. The signal-to-noise ratio of the nuclear magnetic resonance experiment. J. Magn. Reson. (1969)
**1976**, 24, 71–85. [Google Scholar] [CrossRef] - Fukushima, E.; Roeder, S.B.W. Experimental Pulse NMR: A Nuts and Bolts Approach; Addition-Wesley Publishing Company: Reading, MA, USA, 1981. [Google Scholar]
- Rahman, A.; Choudhary, M.; Wahab, A. Solving Problems with NMR Spectroscopy, 2nd ed.; Elsevier Academic Press: San Diego, CA, USA, 2016. [Google Scholar]
- Pelc, D.; Grafe, H.J.; Gu, G.D.; Požek, M. Cu nuclear magnetic resonance study of charge and spin stripe order in La
_{1.875}Ba_{0.125}CuO_{4}. Phys. Rev. B**2017**, 95, 054508. [Google Scholar] [CrossRef] - Zheng, X.; Wang, Z.G.; Huang, Y.C. Implementation of a two-stage digital AGC for spectrum analyzer. In Proceedings of the 2011 International Conference on Applied Superconductivity and Electromagnetic Devices, Sydney, NSW, Australia, 14–16 December 2011; pp. 37–40. [Google Scholar] [CrossRef]
- Levitt, M.H. Spin Dynamics: Basics of Nuclear Magnetic Resonance; John Wiley & Sons, Ltd.: Chichester, UK, 2008. [Google Scholar]
- Hiebel, M. Fundamentals of Vector Network Analysis; Rohde & Schwarz: Munich, Germany, 2014. [Google Scholar]
- Keysight Technologies. Fundamentals of RF and Microwave Noise Figure Measurements. Available online: http://literature.cdn.keysight.com/litweb/pdf/5952-8255E.pdf (accessed on 6 June 2019).
- Davenport, W.B., Jr.; Root, W.L. An Introduction to the Theory of Random Signals and Noise; McGraw-Hill Book Company, Inc.: New York, NY, USA, 1958. [Google Scholar]
- Traficante, D.D. Time averaging. Does the noise really average toward zero? Concepts Magn. Reson.
**1991**, 3, 83–87. [Google Scholar] [CrossRef] - Robinson, F.N.H. Noise and Fluctuations in Electronic Devices and Circuits; Clarendon Press: Oxford, UK, 1974. [Google Scholar]
- Cvitanić, T.; Šurija, V.; Prša, K.; Zaharko, O.; Kupčić, I.; Babkevich, P.; Frontzek, M.; Požek, M.; Berger, H.; Magrez, A.; et al. Singlet state formation and its impact on the magnetic structure in the tetramer system SeCuO
_{3}. Phys. Rev. B**2018**, 98, 054409. [Google Scholar] [CrossRef] - Cvitanić, T.; Lukas, M.; Grbić, M.S. Two-axis goniometer for single-crystal nuclear magnetic resonance measurements. Rev. Sci. Instrum.
**2019**, 90, 043903. [Google Scholar] [CrossRef] [PubMed] - MITEQ AU-1114. Available online: https://nardamiteq.com/docs/1114-1606276-L0910.PDF (accessed on 6 June 2019).
- THAMWAY N141-206AA. Available online: http://www.thamway.co.jp/english/product07-08_e.html (accessed on 6 June 2019).
- Mini-Circuits HELA -10D+. Available online: https://www.minicircuits.com/WebStore/dashboard.html?model=HELA-10%2B (accessed on 6 June 2019).
- Kolar, P.; Hrabar, S.; Grbić, M.S. Towards optimal noise properties of NMR antenna-receiver chain. In Proceedings of the 2017 11th European Conference on Antennas and Propagation (EUCAP), Paris, France, 19–24 March 2017; pp. 1054–1056. [Google Scholar] [CrossRef]
- Mizuno, T.; Takegoshi, K. Development of a cryogenic duplexer for solid-state nuclear magnetic resonance. Rev. Sci. Instrum.
**2009**, 80, 124702. [Google Scholar] [CrossRef] [PubMed] - Bialkowski, M.E.; Ibrahim, S.Z.; Abbosh, A.M. Wideband performance of 3 dB microstrip-slot coupler using different substrates. Microw. Opt. Technol. Lett.
**2011**, 53, 1618–1624. [Google Scholar] [CrossRef] - Moskau, D. Application of real time digital filters in NMR spectroscopy. Concepts Magn. Reson.
**2002**, 15, 164–176. [Google Scholar] [CrossRef] - Giovannetti, G.; Hartwig, V.; Viti, V.; Gaeta, G.; Francesconi, R.; Landini, L.; Benassi, A. Application of undersampling technique for the design of an NMR signals digital receiver. Concepts Magn. Reson. Part B Magn. Reson. Eng.
**2006**, 29B, 107–114. [Google Scholar] [CrossRef] - NMR Spectroscopy Rx Chain Noise Figure Calculator—GitHub. Available online: https://github.com/5ARK/NMR_F_Calc (accessed on 6 June 2019).

**Figure 1.**NMR spectroscopy system: (

**a**) Basic schematic diagram and (

**b**) schematic of the probe used in condensed matter physics (co-axial cable (orange) connected to the coil at the bottom by two variable capacitors (blue)). Thin stainless steel tubes parallel to the cable keep the structure stable, while transverse plates block external radiation from reaching the coil space.

**Figure 4.**Losses of coaxial cables per unit length in the case of standard coaxial cable (red) and high-quality coaxial cable produced by Fujikura company (blue).

**Figure 6.**NMR spectra measured to check the validity of NMR receiving chain’s noise figure calculation: (

**a**)

^{133}Cs in Cs

_{2}Cu

_{3}SnF

_{12}(the splitting of the spectra due to quadrupolar splitting into 7 lines is visible), (

**b**) central line of

^{63}Cu in SeCuO

_{3}(quadrupolar satellites are too far apart to be excited by a single excitation pulse).

**Figure 7.**Calculated $\mathit{SNR}$ deterioration of the weakest signal that has ${\mathit{SNR}}_{\mathit{in}}=-14.7$ dB along NMR receiving chain, for various conditions: existing chain with SOA (red) or M290 (black). The gray area depicts an estimated range of NMR signal’s $\mathit{SNR}$ deterioration by placing low-quality (l.q.) elements in the chain.

Compound | SeCuO_{3} | Cs_{2}Cu_{3}SnF_{12} | |
---|---|---|---|

Parameter | Symbol | Values | |

Measurement frequency (MHz) | - | 147.20 | 33.50 |

Coil and sample temperature (K) | ${T}_{coil}$ | 20 | 30 |

DC magnetic field (T) | ${B}_{0}$ | 11.90 | 6 |

Input cable loss (dB) | ${L}_{1}$ | 0.28 | 0.10 |

Duplexer loss (dB) | ${L}_{2}$ | 0.27 | 0.43 |

M290 gain (dB) | ${G}_{3}$ | 36.13 | 36.52 |

M290 noise figure (dB) | ${F}_{3}$ | 1.11 | 1.14 |

M290 reflection coefficient (dB) | ${S}_{{11}_{3}}$ | −16.43 | −13.50 |

T77 gain (dB) | ${G}_{3}$, ${G}_{3a}$ | 28.54 | 27.89 |

T77 noise figure (dB) | ${F}_{3}$, ${F}_{3a}$ | 0.32 | 1.07 |

T77 reflection coefficient (dB) | ${S}_{{11}_{3}}$, ${S}_{{11}_{3a}}$ | −7.57 | −8.05 |

MC290 gain (dB) | ${G}_{3b}$ | - | 10.75 |

MC290 noise figure (dB) | ${F}_{3b}$ | - | 4.22 |

MC290 reflection coefficient (dB) | ${S}_{{11}_{3b}}$ | - | −27.58 |

Output cable loss (dB) | ${L}_{4}$ | 0.46 | 0.62 |

Spectrometer noise figure (dB) | ${F}_{5}$ | 33.50 | 38.40 |

Number of measurements | ${n}_{meas}$ | 200 | 400 |

Input impedance M290 ($\Omega $) | ${Z}_{in}$ | 51.20 | 72.80 |

Output impedance M290 ($\Omega $) | ${Z}_{out}$ | 54.50 | 45.60 |

Input impedance T77 ($\Omega $) | ${Z}_{in}$ | 45.77 | 34.70 |

Output impedance T77 ($\Omega $) | ${Z}_{out}$ | 101.46 | 38.20 |

Input impedance MC290 ($\Omega $) | ${Z}_{in}$ | 50 | 50 |

Output impedance MC290 ($\Omega $) | ${Z}_{out}$ | 50 | 50 |

Compound | Nucleus | $\mathit{\gamma}$ (MHz/T) | Spin | Abundance (%) | Quadrupole Splitting |
---|---|---|---|---|---|

SeCuO_{3} | ^{63}Cu | 11.285 | 3/2 | 69.1 | 48.05 MHz |

Cs_{2}Cu_{3}SnF_{12} | ^{133}Cs | 5.5844 | 7/2 | 100 | 9.54 kHz |

**Table 3.**Results of experimental verification of the derived expression for NMR spectroscopy receiving chain.

Compound | SeCuO_{3} | Cs_{2}Cu_{3}SnF_{12} |
---|---|---|

Frequency (MHz) | 147.20 | 33.50 |

M290 | ||

Measured $SN{R}_{out}$ (dB) | 30.11 | 56.00 |

Determined ${F}_{NM{R}_{Rx}}$ (dB) | −17.95 | −18.46 |

Calculated $SN{R}_{in}$ via (14) and (3) (dB) | 12.17 | 37.54 |

T77 | ||

Measured $SN{R}_{out}$ (dB) | 24.97 | 48.32 |

Determined ${F}_{NM{R}_{Rx}}$ (dB) | −12.83 | −10.77 |

Calculated $SN{R}_{in}$ via (14) and (3) (dB) | 12.14 | 37.55 |

T77 with MC290 | ||

Measured $SN{R}_{out}$ | - | 55.65 |

Determined ${F}_{NM{R}_{Rx}}$ (dB) | - | −19.41 |

Calculated $SN{R}_{in}$ via (15) and (3) (dB) | - | 36.24 |

Calculated${\mathit{SNR}}_{\mathit{in}}$via (1) (dB) | 15.20 | 35.31 |

Compound | SeCuO_{3} | Cs_{2}Cu_{3}SnF_{12} |
---|---|---|

Frequency (MHz) | 147.20 | 33.50 |

$SN{R}_{in}$ (dB) | 12.17 | 37.54 |

$SN{R}_{out}$ using M290 (dB) | 30.11 | 56.00 |

Predicted $SN{R}_{out}$ using SOA (dB) | 32.28 | 58.77 |

$SN{R}_{out}$ enhancement (dB) | 2.16 | 2.76 |

$SN{R}_{out}$ enhancement (%) | 65 | 89 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kolar, P.; Grbić, M.S.; Hrabar, S.
Sensitivity Enhancement of NMR Spectroscopy Receiving Chain Used in Condensed Matter Physics. *Sensors* **2019**, *19*, 3064.
https://doi.org/10.3390/s19143064

**AMA Style**

Kolar P, Grbić MS, Hrabar S.
Sensitivity Enhancement of NMR Spectroscopy Receiving Chain Used in Condensed Matter Physics. *Sensors*. 2019; 19(14):3064.
https://doi.org/10.3390/s19143064

**Chicago/Turabian Style**

Kolar, Petar, Mihael S. Grbić, and Silvio Hrabar.
2019. "Sensitivity Enhancement of NMR Spectroscopy Receiving Chain Used in Condensed Matter Physics" *Sensors* 19, no. 14: 3064.
https://doi.org/10.3390/s19143064