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7 Light in Biology and Medicine
momentum of system, and l is the angular momentum quantum number
(l = 0, 1, 2, . . .). For a molecule with a line axis of symmetry, such as
carbon dioxide, generating spin around the symmetry axis will only be seen
in exposure to the far ultraviolet, which has sufficient energy to distort the
molecule to bulge radially. Otherwise, there would be no way to detect the
spin. Rotation of the molecule perpendicular to the axis of symmetry is far
easier, and occurs by exposure to infrared light.
In exciting the molecule to rotate, the photon, which always carries one
unit of angular momentum, 1 ¯
h, changes l for the molecule to l ± 1. Thus,
the resonant absorption frequencies will be given by f = (l + 1)h/(4π 2 I )
and will show a linear increase in their separation in the absorption
spectrum. Taking, for example, carbon monoxide, we have an I of about
2 × 10 −40 g/cm 2 giving a light frequency (for l = 0 to 1 transition)
of f = 8 × 10 12 Hz and a wavelength of 1.2 × 10 −3 cm. Rotational
“lines” or bands in the absorption spectrum of small molecules typically
lie in the far infrared. So, if one sees a set of IR transmission dips whose
separation increases linearly with absorption frequency, one should suspect
IR rotational excitations are the cause for this part of the IR absorption
spectra.
(b) Molecular vibrational excitations: Small and large molecules can be
excited into vibrational modes by light. Since the natural vibrational
frequencies depend very sensitively on the strength of the various bonds
which are stretching during vibration, the absorption spectra of a substance
is an important tool in identifying the molecular structure of the material.
To estimate the frequency of light required to excite vibrational modes,
we can use the quantum mechanical result that the energy stored in a
vibrational system with a small amplitude of vibration is given by E =
(1/2 + n)hf o , where n is a quantum number (n = 0, 1, 2, . . .) and f o
is the natural vibrational frequency of the system. Note that the energies
are evenly spaced, so the absorption spectrum will show evenly spaced
absorption dips.
If the bond length is changing during the vibration, vibrational frequencies are given by (1/(2π)
∂ 2 V /∂x 2 )
x 0
/(2μ), where μ is the reduced
mass of the two objects forming the bond and V (x) is the potential energy
stored in the bond as a function of bond separation. The second derivative
is evaluated at the average bond separation, x 0 , when the system is in its
lowest vibrational quantum state. (Quantum fluctuations do not allow a
fixed separation.)
An isolated water molecule has three vibrational modes: a symmetric
stretching mode with wave-number (k = f/c, in units of cm −1 ) 3657;
an antisymmetric deformation mode with wave-number 1595; and an
antisymmetric stretching mode with wave-number 3756.
7 Light in Biology and Medicine
momentum of system, and l is the angular momentum quantum number
(l = 0, 1, 2, . . .). For a molecule with a line axis of symmetry, such as
carbon dioxide, generating spin around the symmetry axis will only be seen
in exposure to the far ultraviolet, which has sufficient energy to distort the
molecule to bulge radially. Otherwise, there would be no way to detect the
spin. Rotation of the molecule perpendicular to the axis of symmetry is far
easier, and occurs by exposure to infrared light.
In exciting the molecule to rotate, the photon, which always carries one
unit of angular momentum, 1 ¯
h, changes l for the molecule to l ± 1. Thus,
the resonant absorption frequencies will be given by f = (l + 1)h/(4π 2 I )
and will show a linear increase in their separation in the absorption
spectrum. Taking, for example, carbon monoxide, we have an I of about
2 × 10 −40 g/cm 2 giving a light frequency (for l = 0 to 1 transition)
of f = 8 × 10 12 Hz and a wavelength of 1.2 × 10 −3 cm. Rotational
“lines” or bands in the absorption spectrum of small molecules typically
lie in the far infrared. So, if one sees a set of IR transmission dips whose
separation increases linearly with absorption frequency, one should suspect
IR rotational excitations are the cause for this part of the IR absorption
spectra.
(b) Molecular vibrational excitations: Small and large molecules can be
excited into vibrational modes by light. Since the natural vibrational
frequencies depend very sensitively on the strength of the various bonds
which are stretching during vibration, the absorption spectra of a substance
is an important tool in identifying the molecular structure of the material.
To estimate the frequency of light required to excite vibrational modes,
we can use the quantum mechanical result that the energy stored in a
vibrational system with a small amplitude of vibration is given by E =
(1/2 + n)hf o , where n is a quantum number (n = 0, 1, 2, . . .) and f o
is the natural vibrational frequency of the system. Note that the energies
are evenly spaced, so the absorption spectrum will show evenly spaced
absorption dips.
If the bond length is changing during the vibration, vibrational frequencies are given by (1/(2π)
∂ 2 V /∂x 2 )
x 0
/(2μ), where μ is the reduced
mass of the two objects forming the bond and V (x) is the potential energy
stored in the bond as a function of bond separation. The second derivative
is evaluated at the average bond separation, x 0 , when the system is in its
lowest vibrational quantum state. (Quantum fluctuations do not allow a
fixed separation.)
An isolated water molecule has three vibrational modes: a symmetric
stretching mode with wave-number (k = f/c, in units of cm −1 ) 3657;
an antisymmetric deformation mode with wave-number 1595; and an
antisymmetric stretching mode with wave-number 3756.