Effective spectroscopic factor of the doublet ground state: A useful tool for comparison with outcome of EPR experiments

Journal Pre-proof Effective spectroscopic factor of the doublet ground state: a useful tool for comparison with outcome of EPR experiments

R. Pełka, P. Konieczny, D. Czernia PII:

S0921-4526(19)30839-7

DOI:

https://doi.org/10.1016/j.physb.2019.411960

Reference:

PHYSB 411960

To appear in:

Physica B: Physics of Condensed Matter

Received Date:

29 October 2019

Accepted Date:

19 December 2019

Please cite this article as: R. Pełka, P. Konieczny, D. Czernia, Effective spectroscopic factor of the doublet ground state: a useful tool for comparison with outcome of EPR experiments, Physica B: Physics of Condensed Matter (2019), https://doi.org/10.1016/j.physb.2019.411960

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Journal Pre-proof Effective spectroscopic factor of the doublet ground state: a useful tool for comparison with outcome of EPR experiments R. Pełka1,*, P. Konieczny1, D. Czernia1 1

Institute of Nuclear Physics Polish Academy of Sciences, Radzikowskiego 152, 31-342 Kraków, Poland *

Corresponding author. E-mail address: [email protected]. Abstract

A mutual relation between the effective spectroscopic factors of the doublet ground state implied by the magnetization at saturation and the low-temperature susceptibility is established. Useful tools for the comparison and consistency check of the outcome of the EPR experiment and the results of the magnetic measurements at low-temperature are developed. Numerical values of the effective spectroscopic factor implied by the magnetization at saturation are provided for instant reference. The presented findings may be of valuable assistance in rationalizing the physical implications of the powder sample EPR experiment. Keywords: ground state doublet, EPR, spectroscopic tensor, magnetization, magnetic susceptibility. 1. Introduction Contemporary research more and more frequently involves complementary experiments. For magnetic systems at low temperature the SQUID measurements of the magnetization and susceptibility [1,2] together with the electronic paramagnetic resonance (EPR) studies of the constituent magnetic ions [3-5] has become the usual standard. An important subclass of such systems is constituted by those involving magnetic ions with the doublet ground state. The major outcome of the EPR experiment for the doublet ground state is the threesome of the principal values of the spectroscopic tensor gˆ  [ g xx , g yy , g zz ] . At the same time, the amplitudes of the isothermal magnetization or susceptibility detected in the magnetometric experiments depend crucially, albeit not always explicitly, on the components of the spectroscopic tensor. Therefore, the magnetometric results may play an auxiliary role in the interpretation of the EPR signal. Several comments are here in order. Firstly, a compound can comprise more than one paramagnetic species and it is possible that not all of them give rise to EPR signals (e.g. the non-Kramers ions). Such EPR-silent species can at the same time contribute to bulk magnetization preventing from the prediction of total magnetization or susceptibility of a compound on the basis of the EPR experiment. Secondly, if exchange coupling between neighboring doublets is stronger than hyperfine couplings (such an exchange coupling can still be too small to be resolved in a SQUID measurement) and the principal axes of the gˆ tensor have different orientation for different crystal sites, one will measure an exchange-averaged gˆ tensor. This is not the genuine gˆ tensor and sometimes it is possible to obtain the true one by isomorphic diamagnetic dilution, replacing for instance the Cu(II) ions by the Zn(II) ions. Although the issue is not addressed here, if one computes magnetization or susceptibility from the exchange-averaged spectroscopic tensor, one may or

Journal Pre-proof may not find deviations beyond experimental uncertainty of the SQUID measurement. Finally, the paramagnetic species in a compound can be an impurity or - vice versa - the sample can contain diamagnetic impurities. While the EPR spectrum can be in such a case soundly physically interpreted, magnetization and susceptibility cannot be correctly predicted from it. The findings reported in this contribution refer therefore to systems with isolated EPR-active paramagnetic species not contaminated with any paramagnetic or diamagnetic impurities. The magnetic signals for powder samples may be expressed in terms of the effective spectroscopic factor geff which is a function of gxx, gyy, and gzz. It is thus desirable to have some convenient means to estimate geff on the basis of the EPR outcome for the two major magnetic characteristics, i.e. the magnetization at saturation and the low-temperature susceptibility. In this way it can readily be compared and checked for consistency with the values of geff implied directly by these magnetic quantities pushing the interpretation of the EPR signal into the physically relevant direction. It is the intention of the present contribution to provide simple tools to estimate the effective value of the spectroscopic factor geff for an arbitrary configuration of the components gxx, gyy, and gzz implied by the EPR experiment. 2. Methods Numerical calculations involved in the present work were all performed with the in-built procedures of the Mathematica8.0 environment. 3. Theory Let us consider the ground state doublet characterized by the diagonal spectroscopic tensor gˆ  diag[ g xx , g yy , g zz ] . Without the loss of generality let us assume that the diagonal components are ordered such that g xx  g yy  g zz . This can be always achieved by the appropriate rotation of the reference frame. The Hamiltonian corresponding to the interaction  of the doublet with the external magnetic field H  [ H x , H y , H z ] has the following form

Hˆ   B ( g xx H x Sˆ x  g yy H y Sˆ y  g zz H z Sˆ z ) ,

(1)

where μB is the Bohr magneton and Sˆi  1/2ˆ i (i  x, y, z ) are the spin operator components with ˆ i denoting the well-known Pauli matrices. The above Hamiltonian is a 2×2 matrix which can be readily diagonalized. The two eigenvalues

    B g xx2 H x2  g yy2 H y2  g zz2 H z2 / 2

are

used

to

calculate

the

partition

function

Z  e   e  , where β=1/kBT. The differentiation of the partition function with respect to the field components yields the components of the isothermal magnetization Mi 

N A B 2

g ii2 H i

1  2 tanh   B g xx2 H x2  g yy H y2  g zz2 H z2  . 2  g H g H g H 2 xx

2 x

2 yy

2 y

2 zz

2 z

(2)

 The measureable quantity for a given applied magnetic field H is the projection of the magnetization pseudovector onto the field vector

Journal Pre-proof   (M / N A B )  H  , m |H |

(3)

where m is dimensionless. The saturation value of the measured magnetization ms  lim m  | H |

becomes the function of the couple of spherical angles ( ,  ) parametrizing the orientation of   the applied field: H | H | [sin  cos  , sin  sin  , cos  ] . It can be expressed in terms of the orientation dependent spectroscopic factor, ms ( ,  )  1 / 2 g ( ,  )  1 / 2 g zz  ( ,  ) , where

 ( ,  )   xy2 ( )  [1   xy2 ( )] cos 2  ,

(4)

where

 x2   y2

 xy ( ) 

2



 y2   x2 2

cos(2 ) ,

(5)

where  i  g ii / g zz (i  x, y ) . The powder sample measurement yields the magnetization saturation value which corresponds to the directional average of the spectroscopic factor

m 

1 4

2



 d  sin d  ( ,  ) . 0

(6)

0

The integration with respect to the polar angle in Eq. (6) may be performed explicitly yielding 1 m ( x ,  y )  4

 1  1   2 ( )    xy2 ( ) xy   0 d 1  1   2 ( ) ln   xy ( )   .  xy   

2

(7)

The integration with respect to the azimuthal angle must be performed numerically. The particular value of the mean spectroscopic factor  m is implied by the magnetization powder measurement in high field. Another measure of the effective spectroscopic factor derives from the consideration of the zero-field magnetic susceptibility pertinent to the studied system. Its definition reads

  M H  .   lim  | H |  0 | H |2

(8)

Straightforward calculation using Eq. (2) gives immediately

 ( ,  ) 

N A  B2 2 2 g zz  ( ,  ) . 4k BT

(9)

Directional averaging of this quantity can be performed explicitly and yields the powder sample susceptibility

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

N A  B2 2 2 g zz   , 4k BT

(10)

where

 

 x2   y2  1

(11)

3

The above value of the mean spectroscopic factor is implied by the powder sample measurement of the zero-field susceptibility. For the sake of argument developed in the following section let us additionally consider the third effective measure of the spectroscopic factor, defined as the arithmetic mean of the diagonal values, which in terms of the reduced quantities reads

a 

x   y 1 3

.

(12)

4. Results and discussion Numerical calculations performed within the Mathematica8.0 environment revealed mutual relations between the three different measures of the effective spectroscopic factor given respectively in Eqs. (7), (11), and (12). Figure 1 shows the values of   ,  m , and  a as a function of the ratios  x and  y in the area 0   x   y  1 corresponding to the assumed condition 0  g xx  g yy  g zz .

Fig. 1: The three measures of the effective Fig. 2: Relative difference percentage between spectroscopic factor: ρχ (blue), ρm (red), ρm and ρχ (blue), and between ρm and ρa (green). and ρa (green).

Journal Pre-proof It is apparent that for all values of the ratios  x and  y the following inequality holds

a  m    .

(13)

Hence the value of the effective spectroscopic factor based on the magnetization high-field powder measurement ρm is bounded by that based on the susceptibility powder measurement ρχ from above and by the arithmetic mean of the principal values ρa from below. This immediately allows one to compare the measured magnetization value at saturation with the outcome of the EPR measurement, i.e. g xx  g yy  g zz 6

1  ms  2

2 g xx2  g yy  g zz2

3

.

(14)

The level of accuracy of this comparison is quantified by the relative difference (RD) between the values of ρm and ρχ, and between those of ρm and ρa: RD  100 (   ( a )   m ) /  m [%] . The RD values for all relevant values of the ratios  x and  y are shown in Fig. 2. It can be seen that the discrepancies between ρm and ρχ or ρa increase with decreasing values of ρx and ρy attaining the extrema of c. 15% and c. -35%, respectively, in the limit of ρx=ρy=0. A better estimation might be achieved by approximating the ρm values with a linear function of ρx and ρy, which boils down to constructing a plane sharing the three vertices (  x ,  y ,  m )  (0,0,0.5), (0,1,  m (0,1)), (1,1,1) of the  m (  x ,  y ) surface bounded by the pertinent variable range 0   x   y  1 . It is easy to find that the corresponding linear approximation reads

 la  0.5  [1   m (0,1)] x  [  m (0,1)  0.5] y ,

(15)

where  m (0,1)  0.785 (rounded to the nearest thousandth). Its plot is shown in Fig. 3 together with the  m (  x ,  y ) surface.

Fig. 3: Linear approximation ρla (cyan) of the Fig.

4:

Relative

difference

percentage

Journal Pre-proof effective spectroscopic factor ρm implied by between ρm and ρla. The discrepancies do not the magnetization measurement at saturation exceed 11%. (red). Due to the convexity of the  m (  x ,  y ) surface the linear approximation provides the upper limit for the accurate ρm values. Figure 4 shows the relative difference (RD) between ρm and ρla. It is apparent that the discrepancies do not exceed 11%. This implies the following constraint for the measured magnetization at saturation

ms  [1   m (0,1)]g xx  [  m (0,1)  0.5]g yy  0.5 g zz ,

(16)

and may be readily used for a comparison with the EPR outcome for the spectroscopic factors. Finally, Table 1 collects the values of ρm determined numerically for a grid of the ratios ρx and ρy for instant reference. Table 1: The values of ρm determined numerically for a decimal grid of the ratios ρx and ρy. The last digit was rounded to the nearest thousandth. ρy 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 ρx 0 0.5 0.508 0.525 0.548 0.575 0.606 0.638 0.673 0.709 0.747 0.785 0.1 0.515 0.531 0.554 0.581 0.610 0.643 0.677 0.713 0.750 0.789 0.2 0.547 0.568 0.594 0.623 0.655 0.688 0.724 0.761 0.799 0.3 0.588 0.613 0.641 0.672 0.705 0.740 0.776 0.814 0.4 0.637 0.664 0.694 0.726 0.760 0.796 0.832 0.5 0.690 0.719 0.750 0.784 0.818 0.855 0.6 0.747 0.778 0.810 0.844 0.880 0.7 0.807 0.839 0.872 0.907 0.8 0.870 0.902 0.936 0.9 0.934 0.967 1 1 The above considerations serve as a good opportunity to discuss the precision of magnetic SQUID measurements. The accuracy (sensitivity limit) of the Quantum Design MPMS®3 SQUID magnetometer for magnetization measurement amounts to ΔM=6·10-7 emu for fields above 2.5 kOe. To see what limitations this level of precision sets for the determination of the anisotropy of the spectroscopic gˆ tensor let us consider the situation where its principal values deviate slightly from the isotropic value g0=2.0, i.e. gxx=g0(1-δx), gyy=g0(1-δy), gzz=g0(1+δx+δy). Let us next compare the difference, Mdiff=Mexact-M0, between the exact powder average of the molar magnetization Mexact and the isotropic estimate M 0  1 / 2 N A  B g 0 , where

M exact 

 1  x 1  y  1 , N A  B g 0 (1   x   y )  m  , 1    1     2 x y x y  

(17)

with the SQUID instrument precision ΔM. Let us note that Eq. (14) implies that Mdiff ≥ 0. Figure 5 shows Mdiff as a function of δx and δy (red) together with the reference level ΔM (blue). It is apparent that in terms of the quantities δx and δy the magnetization measurement is capable of resolving the anisotropy on the order of 4·10-5.

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Fig. 5: Mdiff as a function of δx and δy (red) together with the reference level ΔM (blue). 5. Conclusions A mutual relation between the effective spectroscopic factors implied by the magnetization at saturation (ρm) and the low-temperature susceptibility (ρχ) has been established (see Eq. (13)). Useful tools for the comparison and consistency check of the outcome of the EPR experiment and the results of the magnetic measurements at low-temperature have been developed in the form of the inequalities given in Eqs. (14) and (16). Finally, the numerical values of the effective spectroscopic factor implied by the magnetization at saturation have been provided in Table 1 for instant reference. The above results may come in useful while trying to rationalize the physical implications of the powder sample EPR experiment.

References [1] O. Kahn, Molecular Magnetism, VCH Publishers, Inc., New York, 1993. [2] C. Benelli, D. Gatteschi, Introduction to Molecular Magnetism, Wiley-VCH Verlag GmbH & Co., Weinheim, Germany, 2015. [3] S. A. Al'tshuler, B. M. Kozyrev, Electron Paramagnetic Resonance, Academic Press, Amsterdam, 1964. [4] A. Abragam, B. Bleaney, Electron Paramagnetic Resonance of Transition Ions, Clarendon Press, Oxford, 1970. [5] A. Bencini, D. Gatteschi, Electron Paramagnetic Resonance of Exchange Coupled Systems, Springer Verlag GmbH, Heidelberg, 1990.

Journal Pre-proof Robert Pełka: Conceptualization, Methodology, Software, Formal Analysis, Visualization, Writing - Original Draft, Writing - Review&Editing. Piotr Konieczny: Validation, Writing Review&Editing, Visualization. Dominik Czernia: Validation, Resources, Writing Review&Editing.

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Declaration of interests

× The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

☐ The authors declare the following financial interests/personal relationships which may be considered as potential competing interests: