Overtone modes

Near-infrared absorption is therefore essentially due to combination and overtone modes of higher energy fundamentals, such as C-H, N-H, and O-H stretches, which appear as lower overtones and lower order combination modes. Since the NIR absorption of polyatomic molecules thus mainly reflects vibrational contributions from very few functional groups, NIR spectroscopy is less suitable for detailed qualitative analysis than IR, which shows all (active) fundamentals and the overtones and combination modes of low-energy vibrations. On the other hand, since the vibrational intensities of near-infrared bands are considerably lower than those of corresponding infrared bands, optical layers of reasonable size (millimeters, centimeters) may be transmitted in the NIR, even in the case of liquid samples, compared to the layers of pm size which are detected in the infrared. This has important consequences for the direct quantitative study of chemical reactions, chemical equilibria, and phase equilibria via NIR spectroscopy. 

The vibrational intensities B(0 v) of fundamental and overtone modes may be related to the electric dipole moment function M(r) around the equilibrium internuclear distance of the diatomic molecule (Chackerian, 1976) via rotationless matrix elements... 

The IR range of combination and overtone modes hes above ca. 4000 cm". It is also extremely difficult to investigate this region by transmission IR spectroscopy because of the weak intensities of the respective bands which usually are only 3 to 5% of that of the fundamentals. In this context one has to keep in mind that the absolute transmission of zeolite wafers in the region of fundamental OH stretching modes is sometimes only 1% or even less (vide supra, e.g.. Sect. 5.4.1.2.3). 

The OH and OD stretching modes of free OH radicals are at 3568.0 and 2632.1 cm respectively (645), which corresponds to an isotopic shift factor of 1.3556. This value is again lower than the theoretical one and is similar to the values reported in Table 2.24 for isolated surface hydroxyls. These measured frequencies of the stretching modes are known to be affected by anharmonicity. The harmonic OH and OD frequencies (calculated on the basis of overtone modes) are at 3735.2 and 2720.9 cm, respectively. Thus, the isotopic shift factor for the harmonic frequencies is 1.3728, which is very close to the theoretical value of 1.3736. The same value for the hydroxyl anion, OH, is 1.3726. Therefore, one can conclude that the deviations of the isotopic shift of free OH radicals and ions are mainly the result of anharmonicity. Similar calculations for isolated surface OH groups (21) show that also in this case, the deviations are mainly caused by anharmonicity (isotopic shift factor of the harmonic frequencies of 1.372-1.373). 

Under normal conditions, the fundamental frequency oscillates in the quartz resonator. However, the characteristics of the external network can be utilized to promote oscillation in a higher order overtone mode. The exact overtone frequency is typically not an exact harmonic of the fundamental frequency. However, the overtone is normally close to a harmonic value. The external circuit is tuned to a frequency near the desired overtone frequency. 

Spurious modes are inharmonic modes that may also appear in the vibrational spectrum of the quartz resonator. The spurious modes usually occur in high-resistance modes, and they generally occur at frequencies that are higher than the desired oscillation frequency. The occurrence of spurious modes is increased for crystals operating in an overtone mode as shown in Fig. 3.3. 

Circuit parameter values of the crystal are largely influenced by the angle at which the quartz is cut from the raw quartz slab (precut form). One of the most popular cuts is the AT-cut, which typically is operated in a thickness shear mode between 1 and 200 MHz. Also, above 25 MHz, the AT-cut quartz crystal typically operates in an overtone mode. The AT-cut is usually made from a Y-bar, which simply indicates that the maximum dimension of the crystal is in the Y direction. Also, the actual cut is defined as a cut with angle, 9 = 35° 15 with respect to the z axis. Another popular cut is the BT-cut this is a cut made at an angle of —49° with respect to the z axis. The BT-cut crystals are usually Y-cut also, and the BT-cut crystals are more likely to be appHed in an overtone mode, although fundamental mode oscUlation is also common. 

In the radial vibration mode, besides the fundamental mode, there are some overtone modes with inharmonic frequency separation. The series resonant frequency for the n vibration mode of the piezoelectric ceramic thin disk in the radial vibration mode can be expressed as [Randeraat Setterington, 1974] ... 

For the piezoelectric ceramic disk in radial vibration mode, the ratio between the fundamental series resonant frequency and the overtones series resonant frequencies are not integers, but with a values equal to a,/ ai, in which ai is the coefficient corresponding to the fundamental mode and a is the coefficient corresponding to the n order overtone vibration mode as shown in Eq.(21). The relation between overtones mode orders and the coefficient otn and coefficient ratio ttn/tti were shown in Fig. 7, and can be fit with second order p>olynomial ... 

The coefficient a and coefficient ratio On/ai increased with the increasing of overtones mode order, but the slope of coefficient ratio On/tti curve was less than that of coefficient ctn curve. It means that the separation between vibrations modes would decreased with the increasing of overtone mode order. 

The planar effective electromechanical coupling factor kp is decreased with the increasing in overtone mode order also, as shown in Fig.8, it also can be fit with second order polynomial ... 

The frequency spectrums of different diameter-to-thickness ratio are shown in Fig.ll to Fig.l4. From the results of these figures, it found that the numbers of radial vibration overtone mode are increased with the increasing of (D/1) ratio. 

Fig. 15. Relation between (overtone mode resonant frequency/fundamental resonant frequency) ratio and (diameter/thickness) ratio.

Another example of the manifestation of the H-bond vibrations in the far infrared region is the FIR spectrum of poly(vinyl alcohol) (PVAl) [5]. The absorption band at 360 cm in this spectrum was assigned to the overtone mode 2voh.. o-... 

---