Transition spontaneous

For rigid-chain crystallizable polymers, spontaneous transition into the nematic phase is accompanied by crystallization intermolecular interactions should lead to the formation of a three-dimensional ordered crystalline phase. 

Figure 2 shows the increase in the rigidity (1 - f) of macromolecules induced by the field as a function of the parameter x = e/kT + Fl/kT. As soon as the flexibility decreases to f < 0.63, a system of molecules flexible in the state of rest will undergo a spontaneous transition into a nematic oriented state upon the action of the stretching field, just as it occurs for rigid molecules at rest. 

Figure 15 describes the decrease in the flexibility f of the macromolecules during melt stretching (corresponding to an increase in /3m) with x. According to Flory s criterion, the diminution of the flexibility of molecules to the value of f < 0.63 leads to a spontaneous transition of the system into the state of parallel order. It can be seen in Fig. 15 that f = 0. is attained at x = 30 or o = 0.6 x 107 n/m2 at these stresses, the melt is organized into a nematic state. 

The only possible spontaneous transition which can occur above the saturation curve (bp > 0) is one which leads to diminution of volume when it occurs on the saturation curve the only possible transition which can occur spontaneously below... 

The emission spectmm of Co, as recorded with an ideal detector with energy-independent efficiency and constant resolution (line width), is shown in Fig. 3.6b. In addition to the expected three y-lines of Fe at 14.4, 122, and 136 keV, there is also a strong X-ray line at 6.4 keV. This is due to an after-effect of K-capture, arising from electron-hole recombination in the K-shell of the atom. The spontaneous transition of an L-electron filling up the hole in the K-shell yields Fe-X X-radiation. However, in a practical Mossbauer experiment, this and other soft X-rays rarely reach the y-detector because of the strong mass absorption in the Mossbauer sample. On the other hand, the sample itself may also emit substantial X-ray fluorescence (XRF) radiation, resulting from photo absorption of y-rays (not shown here). Another X-ray line is expected to appear in the y-spectrum due to XRF of the carrier material of the source. For rhodium metal, which is commonly used as the source matrix for Co, the corresponding line is found at 22 keV. 

The natural linewidth comes from the lifetime, r, of the upper state of a spontaneous transition, which is related to the Einstein A coefficient so that r = A l faster transitions have shorter lifetimes and vice versa, and similarly an allowed transition will have a short lifetime for the upper state whereas forbidden transitions will have a long lifetime. The lifetime consideration is very important in the laboratory where transitions have to occur on the timescale of the experiment, otherwise they are not observed. Hence in the laboratory allowed transitions are observed and in general (but not specifically) forbidden transitions are not seen. For astronomy this does not matter. So what if a forbidden transition has a lifetime of 30 million years - the Universe is 15 billion years old - if you wait long enough it will happen. The rules of spectroscopy need to be understood but in space anything goes ... 

In a celebrated paper, Einstein (1917) analyzed the nature of atomic transitions in a radiation field and pointed out that, in order to satisfy the conditions of thermal equilibrium, one has to have not only a spontaneous transition probability per unit time A2i from an excited state 2 to a lower state 1 and an absorption probability BUJV from 1 to 2 , but also a stimulated emission probability B2iJv from state 2 to 1 . The latter can be more usefully thought of as negative absorption, which becomes dominant in masers and lasers.1 Relations between the coefficients are found by considering detailed balancing in thermal equilibrium... 

Application of the F-D theorem produced [122] several significant results. Apart from the Nyquist formula these include the correct formulation of Brownian motion, electric dipole and acoustic radiation resistance, and a rationalization of spontaneous transition probabilities for an isolated excited atom. 

In this limit, therefore, the different ground states generate separate Hilbert spaces, and transitions among them are forbidden. A superselection rule [128] is said to insulate one ground state from another. A superselection rule operates between subspaces if there are neither spontaneous transitions between their state vectors and if, in addition, there are no measurable quantities with finite matrix elements between their state vectors. 

The crystals of nonequilibrium form have the value of ,04, which exceeds the minimum. Thus, under the conditions when the atoms in the crystal become mobile, the spontaneous transition to an equilibrium form becomes probable. In the system of contacting crystals, this frequently results in the decrease of A/V. 

He points out that the variation of lifetime with glass matrix is due to at least two causes, the first being the changes in refractive index. If the wave functions of the ion remain essentially the same from host to host, the spontaneous-transition probability will increase with increasing refractive index because of the increase in density of final states. The second cause is configuration mixing of 4/ and 5d states, which must reflect the size and symmetry of the crystal field produced at the ion by the surroundings. 

Spontaneous emission of photons. This process refers to a spontaneous transition of the electron from the excited state 2 to the lower energy state 1 with emission of a photon of frequency Vi2 = ( 2 - E])lh. This process... 

By condition 3 we want to ensure that the Born-Oppenheimer approximation can be applied to the description of the simple systems, allowing definition of adiabatic potential-energy curves for the different electronic states of the systems. Since the initial-state potential curve K (f ) (dissociating to A + B) lies in the continuum of the potential curve K+(/ ) (dissociation to A + B + ), spontaneous transitions K ( )->K+(f ) + e" will generally occur. Within the Born-Oppenheimer approximation the corresponding transition rate W(R)—or energy width T( ) = hW(R) of V (R)... 

Spontaneous transitions are not the only possible transitions. Electronic transitions may be also induced by, for example, an external radiation field. According to the detailed equilibrium principle, the rate of transitions from all states of the lower level ct J into all states of the upper level aJ, caused by the absorption of photons from the radiation field, must be equal to the rates of spontaneous and induced transitions from the level a J into a J, i.e. 

The width of the spectral line equals the sum of the widths of initial and final levels. Due to the short lifetime of highly excited states with an inner vacancy, their widths, conditioned by spontaneous transitions, are very broad. The other reasons for broadening of X-ray and electronic lines (apparatus distortions, Doppler and collisional broadenings) usually lead to small corrections to natural linewidth. 

Fig. 14. Hysteresis in the transition between SSII and the oscillatory state as a function of flow rate ko with [r]0 = 0.0065 M, [ClOj] = 0.002 M, pH 1.56, and T = 25°C. Envelopes of vertical segments show upper and lower limits of pi in the oscillatory state. Numbers next to these segments indicate period of oscillation in seconds. Arrows indicate spontaneous transitions between states...

Another radiational process consists of spontaneous transitions causing a return to the initial level with rate constant r, . 

Hence, knowledge of the concentration of molecules n at level a, the rate Tj/jn of the spontaneous transition b —> c, and the upper state b relaxation rate T allows us to determine Tp from the absolute intensity measurements. 

When molecules arrive at the state with rotation quantum number J" from the state J in the process of spontaneous radiation at rate (see Fig. 3.14), a photon possessing unit spin is emitted in an arbitrary direction. Let us assume that the angular momentum carried away by the emitted photon is small, as compared with both 3 [ and Jr. This means that the angular momentum vector of each separate molecule does not change its value and does not turn in space as a result of the spontaneous transition. Consequently, the angular momenta distribution pji(0,ip) is... 

Here the first term in the righthand side determines the rate of creation of polarization moments in the spontaneous transition process, whilst the second term describes their relaxation. In writing Eq. (3.27) it is assumed that collisions do not lead to population of the state J", since it lies sufficiently high and is surrounded by non-populated rovibronic levels within the range of thermal energy kT. 

Let us consider the effect of an external magnetic field on the angular momenta distribution at a level populated in the fluorescence process see Section 3.4, Fig. 3.14. In the presence of an external magnetic field the following polarization moments are created on the lower level J" via spontaneous transitions at weak excitation, x — 0 ... 

Further, the fifth term on the righthand side of Eq. (5.23) describes reverse spontaneous transitions at the rate T j>j . In the analysis of reverse spontaneous transitions we make use of the fact that we have, as shown in [104],... 

The penultimate term in (5.92) deals with reverse spontaneous transitions. The function T jt j"(0, p 0, p ) represents the probability of the molecule with angular momentum orientation 3 (0, tp ) in the excited state arriving with orientation Ja(0, p) at the ground state. 

Reverse spontaneous transitions cannot change the orientation of the angular momentum of the molecule, hence we have Tjijn(0, p 0, p ) = Tj j"(0,p). Since the probability of spontaneous transition is invariant with respect to turn of coordinates, we may make an even stronger assertion Tj>j (0,p) = rjijn = const. According to the aforesaid we obtain a term that is responsible for reverse spontaneous transitions in the form Tj j bpQ KK Qq, which has to be added to the righthand side of Eq. (5.94). 

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