# Empirical Expression for AC Magnetization Harmonics of Magnetic Nanoparticles under High-Frequency Excitation Field for Thermometry

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Model and Methods

#### 2.1. Langevin Function

_{0}mH/k

_{B}T, H = H

_{0}sin(2πft) is the AC excitation magnetic field, f is the frequency of the applied field, k

_{B}is the Boltzmann constant, T is the absolute temperature, and m is the magnetic moment.

_{L}to be expressed as:

_{2j−1}is the amplitude of the (2j−1)-th harmonic magnetization. The Maclaurin expansion can be used to express the harmonic amplitude A

_{2j−1}as follows:

#### 2.2. Fokker–Planck Equation for Néel Relaxation

_{s}= m/V

_{c}is the saturation magnetization, and V

_{c}is the MNP volume.

_{n}(t) is the time-dependent coefficient of the n-th order spherical harmonic, and P

_{n}(cosθ) is the n-th order Legendre polynomial. Combining Equations (4) and (5), we obtain:

_{n}(t):

_{n}. The magnetization M

_{FP}in the direction of the excitation field can be calculated with the following equation:

_{FP}to be expressed as:

_{2j−1}and φ

_{2j−1}are the amplitude and phase of the (2j−1)-th harmonic magnetization, respectively. Note that an analytical expression for a

_{n}cannot be obtained, so C

_{2j−1}and φ

_{2j−1}are numerically calculated.

#### 2.3. Compensation Expression for MNP Magnetization Harmonics

_{N}

_{0}= 10 ns, 5 ns, and 1 ns). Utilizing cross-correlation principle, digital phase-sensitive detection algorithm (DPSD) can extract effectively the amplitude and phase of signal to be measured from noise [22]. We obtained the harmonic amplitudes and phases of the ensemble magnetization via DPSD.

_{2j−1}is the compensation function, φ

_{2j−1}is the phase, and A

_{2j−1}is the harmonic amplitude calculated with the Langevin function. Using G

_{2j−1}and A

_{2j−1}, the harmonic amplitude C

_{2j−1}of MNP magnetization based on the Fokker–Planck equation can be expressed as follows:

## 3. Simulation

_{0}cos(2πft), where H

_{0}was set from 1 to 15 mT with a step of 1 mT and f was set at 20 kHz.

_{0}for low H

_{0}and decreased with H

_{0}for high H

_{0}. Because higher harmonics require greater H

_{0}to reach saturation, the differences in higher harmonics keep increasing with H

_{0}in the range of H

_{0}investigated, shown in Figure 2b–d. The harmonic phase lag (−φ

_{2j−1}) decreases with increasing excitation field. The phase lag of the harmonic becomes large for higher harmonics.

_{0}. Figure 3 shows the dependence of G

_{2j−1}on H

_{0}. The symbols represent G

_{2j−1}calculated with G

_{2j−1}= C

_{2j−1}/A

_{2j−1}, and the solid lines represent polynomial curve fits given by:

_{2j−1}and harmonic phase, we reconstructed the MNP magnetization response based on Equation (10). As shown in Figure 4, the reconstructed AC M–H curve at each H

_{0}nicely fits that calculated from the Fokker–Planck equation.

## 4. Experiment and Results

_{3}O

_{4}) called SHP-20 (SHP-20, Ocean NanoTech, San Diego, CA, USA) were used as MNP samples. SHP-20 consists of iron oxide nanoparticles with a carboxylic acid group and has an iron concentration of 5 mg (Fe)/mL. The solvent of the sample is deionized H

_{2}O with 0.03% NaN

_{3}. The effective core diameter of SHP-20 was 20 nm. The MNP sample was immobilized with an epoxy resin to avoid the effect of Brownian rotational relaxation. The sample was placed in a DC excitation field with a strength of 50 mT during the immobilization process, ensuring the easy axes of MNPs aligned along the same direction.

_{0}, was set from 3 to 15 mT with a step of 2 mT at a frequency of 20 kHz. The temperature of the MNP sample was controlled at 297 K. We obtained the harmonic amplitudes (C

_{2j−1}) and phase (φ

_{2j−1}) of magnetization at each H

_{0}. As seen from Figure 5, the harmonic amplitudes increased with H

_{0}, and the phase lag (–φ

_{2j−1}) decreased with increasing H

_{0}. The higher the harmonic order is, the greater the harmonic phase is; i.e., the Néel relaxation has greater influence on the phase of higher harmonics.

_{2j−1}. The effective core diameter of SHP-20 was set at 20 nm. Therefore, the function G

_{2j−1}= C

_{2j−1}/A

_{2j−1}associated with H

_{0}can be obtained. In Figure 6, the symbols represent G

_{2j−1}for different values of H

_{0}, and the solid lines represent polynomial curve fits using Equation (12).

_{2j−1}and harmonic phase, we reconstructed the MNP magnetization response using Equation (10). The reconstructed AC M–H curves match well with the experimental results, as shown in Figure 7.

## 5. Magnetic Nanoparticle Thermometry at High Frequency

_{1_meas}and C

_{3_meas}are the measured first and third harmonic amplitudes.

^{3}, respectively. The MNP sample had a normal core diameter of 20 nm without a core size distribution. The temperature was calculated from Equation (13) using the Levenberg–Marquardt algorithm, and then the true temperature was subtracted to give the temperature errors. As shown in Figure 8, the error increased with temperature. Although the magnetization response decreased with increasing temperature and the signal-to-noise ratio was low at high temperature, the maximum temperature error was less than 0.008 K in the temperature range 310–320 K.

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**MNP magnetization response calculated with the Langevin function and Fokker–Planck equation under different Néel relaxation times (τ

_{N}

_{0}= 10 ns, 5 ns, and 1 ns). (

**a**) The MNP magnetization response. (

**b**) M–H curves. The harmonics (

**c**) amplitude and (

**d**) phase. The excitation field has an amplitude of 1 mT and a frequency of 20 kHz. The parameters for these simulations are d

_{c}= 25 nm, T = 297 K, K = 4 kJ/m

^{3}, M

_{s}= 300 kA/m, ${\alpha}^{\prime}$ = 0.1, and $\gamma $ = 1.75 × 10

^{11}rad/s T.

**Figure 2.**The (

**a**) 1st, (

**b**) 3rd, (

**c**) 5th, and (

**d**) 7th harmonic amplitudes and phases of MNP magnetization calculated with the Langevin function and Fokker–Planck equation under different excitation field strengths. The parameters for these simulations are d

_{c}= 25 nm, T = 297 K, K = 4 kJ/m

^{3}, M

_{s}= 300 kA/m, ${\alpha}^{\prime}$ = 0.1, and γ = 1.75 × 10

^{11}rad/s T.

**Figure 3.**Dependence of the compensation function G

_{2j−1}= C

_{2j−1}/A

_{2j−1}on the excitation field strength H

_{0}.

**Figure 5.**Experimental results for the (

**a**) 1st, (

**b**) 3rd, (

**c**) 5th, and (

**d**) 7th harmonic amplitudes and phases of MNP magnetization.

**Figure 6.**Experimental results for compensation function (

**a**) G

_{1}= C

_{1}/A

_{1}, (

**b**) G

_{3}= C

_{3}/A

_{3}, (

**c**) G

_{5}= C

_{5}/A

_{5}, and (

**d**) G

_{7}= C

_{7}/A

_{7}.

**Figure 7.**(

**a**) AC M–H curves calculated from the Langevin function with a core diameter of 20 nm. (

**b**) Reconstructed MNP magnetization response based on a compensation model.

**Figure 8.**Simulation results of temperature estimation for an MNPT under Néel relaxation. The inset shows the temperature errors.

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**MDPI and ACS Style**

Du, Z.; Wang, D.; Sun, Y.; Noguchi, Y.; Bai, S.; Yoshida, T. Empirical Expression for AC Magnetization Harmonics of Magnetic Nanoparticles under High-Frequency Excitation Field for Thermometry. *Nanomaterials* **2020**, *10*, 2506.
https://doi.org/10.3390/nano10122506

**AMA Style**

Du Z, Wang D, Sun Y, Noguchi Y, Bai S, Yoshida T. Empirical Expression for AC Magnetization Harmonics of Magnetic Nanoparticles under High-Frequency Excitation Field for Thermometry. *Nanomaterials*. 2020; 10(12):2506.
https://doi.org/10.3390/nano10122506

**Chicago/Turabian Style**

Du, Zhongzhou, Dandan Wang, Yi Sun, Yuki Noguchi, Shi Bai, and Takashi Yoshida. 2020. "Empirical Expression for AC Magnetization Harmonics of Magnetic Nanoparticles under High-Frequency Excitation Field for Thermometry" *Nanomaterials* 10, no. 12: 2506.
https://doi.org/10.3390/nano10122506