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The actual solution of (14.63) may be found graphically, as shown in Fig 14.13. The slope of the right side of (14.63) at z = 1 is ('Y - 1)( e 4fJ < - 1) c= (1 + e 2fJ <)2 Thus if c < 1 the only solution is z = 1. If c > 1, there are three solutions z = 1_ zo, ljzo, of which z = 1 is discarded by comparison with the solution in the Bragg-Williams approximation. The solution llzo does not lead to anything ne\\ since it merely means interchanging up spin with down spin. It too will be ignored. Let us define the critical temperature Tc by the equation ('Y - 1) e 4 </kT, - - - - - : : - =1 (14.68 I (1 + e 2 </kTc )2 which leads to the explicit expression

f(z)

log [ I 'Y - 2)] 'Y (

(14.69 1

Graphical solution of (14.63).

log['Y!('Y- 2)]

z= 1

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known as the spiders ). The ascending trendline is quite long, beginning in late 1994 and being violated in the autumn of 2000. After prices cracked below the trendline, the decline was swift and terrible. This illustrates an important principle with trendlines: the length of the trendline is a strong indicator of how meaningful the trend change will be once that line has been crossed. A trendline spanning 10 days is probably meaningless, whereas a trendline spanning ten years is terribly important. You certainly don t have to look just at the Internet bubble to nd examples of trendline violations and their aftermath. Krispy Kreme (Figure 7.6), the makers of incredibly delicious (particularly when fresh) doughnuts was the darling of 2000 and 2001. It held onto that distinction since virtually every other stock was getting clobbered. Perhaps people were gobbling down doughnuts to distract themselves from their sorrows of this major bear market. In any event, as the Atkins diet craze started gathering momentum, Krispy starting to weaken. Late in 2003, KKD broke below its trendline. Had you been long this stock, you would have already suffered a 20 percent loss from the stock s peak. But the value of trendlines isn t that they guarantee you will make as much pro t as possible. Instead, they try to give you as much time as pos-

(14.70)

(14.71)

5 4 3

In this case we have spontaneous magnetization. Of the thermodynamic functions we only consider the specific heat, which can be shown to be

(14.72) and which, in contradistinction the Bragg-Williams result, does not vanish for T> Tc : 2y(2 e 2</kT (14.73) (T> TJ (kT)2 (1 + e 2</kT)2 A more detailed calculation yields the graph shown in Fig. 14.14 for the specific heat. For comparison the Bragg-Williams result is also shown.

The one-dimensional Ising model is a chain of N spins, each spin interacting only with its two nearest neighbors and with an external magnetic field. The energy for

the configuration specified by {Sl'

EJ = -(

S2' ... , SN}

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