Calculations of the EFG tensor in DTN using the GIPAW method with CASTEP and Quantum Espresso software

One important difference between NQR and NMR is that for NQR, the transition frequency is site specific and cannot be chosen by the experimenter. In NMR studies, the excitation frequency of the nucleus is just the gyromagnetic ratio times the applied field. Thus, the difficulty in NMR is controlling the uniformity of the applied static B. However, the experimenter is free to choose an operating frequency by adjusting B. For NQR, the transition frequency is proportional to the electric field gradient (EFG) at the nuclear site, which is entirely a property of the substance and a function of temperature. While it is theoretically possible to apply a prescribed EFG in the lab, it is unfeasible as the EFG’s in crystal strcutures are often on the order of many kV/m^2. Thus the particular difficulty in NQR spectroscopy is locating the resonance in the first place, which is much like combing the desert if the apparatus runs at a fixed frequency. The home made pulsed superheterodyne spectrometer used in the trial experiments has proven to have excellent resolution. However, it has essentially no ability to perform frequency sweeps as is done in continuous wave methods. The operator must choose a relatively narrow band (< 1 MHz) to target, and design a custom bridge, tank, and receiver configuration just for small band. Band switching is non-trivial with high resolution pulsed techniques.

At one time, it was not possible to calculate the EFG for all but the simplest of structures. However, DFT calculations have become very potent and accessible in the last decade. Beginning in June of 2013, Calculations of the EFG tensor in dichloro-tetrakis-thiourea-nickel NiCl_2 [SC(NH_2)_2]_4 were made using the GIPAW method with closed source CASTEP software (by then graduatre student Dr. Tim Green at Oxford University, with advisor Dr. Jonathan = Yates). Then in October of 2013, Dr. Ari Seitsonen and I repeated the calculation at ETH in Zurich using the open source DFT-GIPAW package Quantum Espresso (QE).

The two calculations show good agreement in many respects and indicate that the 35Cl resonance frequencies in DTN are likely about an order of magnitude lower than initially believed. Because chlorine NQR transitions occurred around 30 MHz in both para-dichlorobenze and NaClO_3, DTN was initally thought to be likely to show 35Cl NQR in the 29-31 MHz band. This was tested first by Robert Baker (REU Student, 2010) and then in 2012-2013 by myself to no avail.

There is no particular reason 35Cl should have to have a resonance in this 30 MHz band. 35Cl NQR measurements have been made in many materials below 18 MHz and above 60 MHz. A high degree of symmetry in a given structure can result in very low NQR frequencies, with correspondingly very poor S/N. Though all nuclei with spin S >= 1 are guaranteed to have at least one non-zero NQR transition frequency, NQR is not always found. This is likely due to poor S/N when the EFG is small and the transition frequency is very low.

Evidence for the validity of the DFT calculations is that the prediction for NaClO3 35Cl NQR matches observation remarkably well. Additionally, since both DFT packages output the full EFG tensor, the NQR transitions for 14N were also predicted by the same calculations. There are sixteen nitrogen nuclei in each unit cell of DTN (right). DTN contains 4 independent thiourea molecules in each unit cell, each having 4 nitrogen centers.

The NQR frequencies calculated by CASTEP and QE for 14N in DTN match very well with those observed in pure thiourea from literature (David H. Smith and R. M. Cotts). The paper by Smith and Cotts quotes NQR in 14N in pure thiourea NQR at 2.6 MHz and 2.0 MHz for inequivalent sites at room temperature. In the absense of any DFT calculation, the 14N resonance in pure thourea would be a best first guess as to the NQR transition for 14N in DTN.

As the NQR frequency at a given nuclear site is directly proportional to Vzz (the electric field gradient, or stretching), crystals with a high degree of symmetry will have low NQR frequencies. Inspection of the crystal structure superficially indicates a widely symmetric nuclear distribution about the chlorine sites, as the thiourea groups are distributed symmetrically and the structure parameters show each Cl is roughly equidistant to both adjacent Ni nuclei.

The quadrupole coupling constant and NQR transitions
The quadrupole coupling constant Cq is defined as

 Cq = e*Vzz*Q/h [1]

where Vzz is the largest absolute eigenvalue, e is the electron charge, Q is the quadrupole moment and h is planck's constant.

We consider a nucleus of spin S and define

 A = eVzzQ / (4S(2S-1)) [2]

where S is spin and Q is the quadrupole moment. Then the NQR frequency or (frequencies) are given by

 f_nqr = 3|A| / h (2 |m| +1) [3]

where m is the lowest of the two levels m and m+1 over which a transition has occurred. For integral spins there are S unique transitions. For half integral spins there are S – ½ unique transitions.
NQR is observed in nuclei with I=3/2, I=5/2 and I=7/2. For I=1/2., there are no transitions. The I=5/2 and I=7/2 are indeed complicated, but for I=3/2 there is a degeneracy that causes there to be one frequency of half the quadrupole coupling constant, which makes the single transition frequency in the case of I=3/2 equal to one half the quadrupole coupling constant Cq = eVzzQ/h = e^2Qq/h.

NQR Frequencies for half-integral spins
Chlorine has I = 3/2 so there is only 1 transition. Using I=3/2 in the formula above, we obtain

 f_nqr = (1/2) eVzzQ/h = (1/2) Cq [4]

for axially symmetric field gradients. For non-axially symmetryic gradients, we define the assymetry parameter η = (Vxx-Vyy)/Vzz. η=0 in the axially symmetric case, where Vzz is the only nonzero component. When η is non-zero, the transitions for I=3/2 are

 f_nqr ~ (1/2) eVzzQ/h * (1 + 1/3 η^2)^(1/2) [5]

In general for half integral spin of I = n/2 n 3, 5, 7,... use what is given in Hahn and you should find the coefficients are 3/10, and 3/20 for I=5/2

 f_nqr_[5/2 --> 3/2] = (3/10) eVzzQ/h [6]
 f_nqr_[3/2->1/2] = (3/20) eVzzQ/h [7]

NQR frequencies for integral spins
For integral spins, we use the formula [2] again. For for I=1 and an axially symmetric field gradient (η=0) , the NQR transition frequency is

 f_nqr = (3/4) eVzzQ/h = (3/4) Cq [8]



EFG->C: https://github.com/tfgg/magres-format/blob/master/magres/constants.

CASTEP EFG DATA (DTN – 35Cl)
relaxed
Cl 1 Cq: 8.4298 (MHz) Eta: 0.1272
Cl 2 Cq: 0.1705 (MHz) Eta: 0.5898
Cl 3 Cq: 0.1866 (MHz) Eta: 0.5340
Cl 4 Cq: 8.4513 (MHz) Eta: 0.1264
unrelaxed
Cl 1 Cq: -15.7965 (MHz) Eta: 0.0089
Cl 2 Cq: -16.9778 (MHz) Eta: 0.0048
Cl 3 Cq: -16.9515 (MHz) Eta: 0.0048
Cl 4 Cq: -15.8271 (MHz) Eta: 0.0089

QE EFG DATA (DTN – 35Cl)
relaxed
 Cl 1 Cq= 9.4969 MHz Eta=-0.00000
 Cl 2 Cq= -0.9388 MHz Eta= 0.00000
 Cl 3 Cq= 9.4869 MHz Eta= 0.00000
 Cl 4 Cq= -0.9298 MHz Eta= 0.00000

CASTEP EFG DATA (DTN – 14N)
relaxed
N 1 Cq: -3.7067 (MHz) Eta: 0.3400
N 2 Cq: -3.3560 (MHz) Eta: 0.4226
N 3 Cq: -3.6918 (MHz) Eta: 0.3463
N 4 Cq: -3.3621 (MHz) Eta: 0.4256
N 5 Cq: -3.6917 (MHz) Eta: 0.3468
N 6 Cq: -3.3533 (MHz) Eta: 0.4231
N 7 Cq: -3.7068 (MHz) Eta: 0.3410
N 8 Cq: -3.3535 (MHz) Eta: 0.4228
N 9 Cq: -3.6920 (MHz) Eta: 0.3466
N 10 Cq: -3.3611 (MHz) Eta: 0.4251
N 11 Cq: -3.7067 (MHz) Eta: 0.3409
N 12 Cq: -3.3524 (MHz) Eta: 0.4230
N 13 Cq: -3.7073 (MHz) Eta: 0.3399
N 14 Cq: -3.3563 (MHz) Eta: 0.4225
N 15 Cq: -3.6927 (MHz) Eta: 0.3469

unrelaxed
N 1 Cq: 3.0165 (MHz) Eta: 0.8408
N 2 Cq: -2.6850 (MHz) Eta: 0.9198
N 3 Cq: 3.0154 (MHz) Eta: 0.8423
N 4 Cq: -2.6885 (MHz) Eta: 0.9216
N 5 Cq: 3.0164 (MHz) Eta: 0.8419
N 6 Cq: -2.6841 (MHz) Eta: 0.9213
N 7 Cq: 3.0179 (MHz) Eta: 0.8414
N 8 Cq: -2.6854 (MHz) Eta: 0.9186
N 9 Cq: 3.0188 (MHz) Eta: 0.8421
N 10 Cq: -2.6833 (MHz) Eta: 0.9207
N 11 Cq: 3.0214 (MHz) Eta: 0.8411
N 12 Cq: -2.6802 (MHz) Eta: 0.9177
N 13 Cq: 3.0199 (MHz) Eta: 0.8406
N 14 Cq: -2.6798 (MHz) Eta: 0.9189
N 15 Cq: 3.0198 (MHz) Eta: 0.8417
N 16 Cq: -2.6788 (MHz) Eta: 0.9203

QE EFG DATA (DTN – 14N)
 relaxed
 N 45 Cq= -3.5069 MHz eta= 0.36826
 N 46 Cq= -3.5084 MHz eta= 0.36870
 N 47 Cq= -3.5084 MHz eta= 0.36870
 N 48 Cq= -3.1864 MHz eta= 0.38355
 N 49 Cq= -3.1864 MHz eta= 0.38355
 N 50 Cq= -3.1880 MHz eta= 0.38408
 N 51 Cq= -3.1864 MHz eta= 0.38355
 N 52 Cq= -3.1864 MHz eta= 0.38355
 N 53 Cq= -3.1880 MHz eta= 0.38408
 N 54 Cq= -3.5069 MHz eta= 0.36826
 N 55 Cq= -3.5084 MHz eta= 0.36870
 N 56 Cq= -3.5084 MHz eta= 0.36870
 N 57 Cq= -3.5069 MHz eta= 0.36826
 N 58 Cq= -3.1880 MHz eta= 0.38408
 N 59 Cq= -3.1880 MHz eta= 0.38408
 N 60 Cq= -3.5069 MHz eta= 0.36826
Q= 2.04 1e-30 m^2
 


For NaClO3, CASTEP arrived at an average NQR frequency (over 4 sites) of 28.9 MHz. For chlorine in DTN, CASTEP calculated NQR at 8.5 MHz (unrelaxed) and 4.5 MHz (relaxed). For 14N in DTN, the calculation yields 2.9 MHz and 1.6 MHz for inequivalent sites in the relaxed EFG calculation. The difference between the relaxed and unrelaxed calculation lies in the structural data preparation. In the unrelaxed case, structure data from direct measurements in literature is fed directly into CASTEP and used. In the relaxed case, the structure parameters itself are adjusted using GIPAW, and then the EFG is calculated.

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