summaryrefslogtreecommitdiff
path: root/Master/texmf-dist/doc/xmltex/passivetex/latextei.xml
blob: af9fbb686c4d43e80e488336c319c4fc00ec883f (plain)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
912
913
914
915
916
917
918
919
920
921
922
923
924
925
926
927
928
929
930
931
932
933
934
935
936
937
938
939
940
941
942
943
944
945
946
947
948
949
950
951
952
953
954
955
956
957
958
959
960
961
962
963
964
965
966
967
968
969
970
971
972
<?xml version="1.0"?>
<!DOCTYPE TEI.2 SYSTEM "http://www.oucs.ox.ac.uk/dtds/tei-oucs.dtd" [ 
<!ENTITY aacute "&#x00E1;">
<!ENTITY ap "&#x2248;">
<!ENTITY alpha "&#x03B1;">
<!ENTITY barwed "&#x22BC;">
<!ENTITY beta "&#x03B2;">
<!ENTITY commaspace " ">
<!ENTITY gamma "&#x03B3;">
<!ENTITY delta "&#x03B4;">
<!ENTITY Delta "&#x0394;">
<!ENTITY dots "&#x2026;">
<!ENTITY epsi "&#x03B5;">
<!ENTITY gg "&#x22D9;">
<!ENTITY Gt "&#x226B;">
<!ENTITY geq "&#x2265;">
<!ENTITY infin "&#x221E;">
<!ENTITY infty "&#x221E;">
<!ENTITY int "&#x222B;">
<!ENTITY kappa "&#x03BA;">
<!ENTITY lambda "&#x03BB;">
<!ENTITY langle "&#x2329;">
<!ENTITY leq "&#x2264;">
<!ENTITY mu "&#x03BC;">
<!ENTITY nbsp "&#x00A0;">
<!ENTITY phi "&#x03C6;">
<!ENTITY pi "&#x3C0;">
<!ENTITY psi "&#x3C8;">
<!ENTITY rangle "&#x232A;">
<!ENTITY rho "&#x3C1;">
<!ENTITY sigma "&#x3C3;">
<!ENTITY Sigma "&#x03A3;">
<!ENTITY sim "&#x223C;">
<!ENTITY thinspace "&#x2009;">
<!ENTITY thickspace "&#x2005;">
<!ENTITY xi "&#x03BE;">
<!ENTITY prime "&#x2032;">
<!ENTITY isinv "&#x2208;">
<!ENTITY macr "&#x0304;">
<!ENTITY Emax "<msub><mi>E</mi><mi>max</mi></msub>">
<!ENTITY exp "E<mtext>exp</mtext>">
<!ENTITY ln "E<mtext>ln</mtext>">
<!ENTITY Rarr "&#x21D2;">
<!ENTITY rarr "&#x2192;">
<!ENTITY GEANT "GEANT">
<!ENTITY sum "&#x2211;">
]> 
<TEI.2> 
  <teiHeader> 
	 <fileDesc> 
		<titleStmt> 
		  <title>A sample article</title> 
		</titleStmt> 
		<publicationStmt> 
		  <availability><p>Converted from LaTeX by Sebastian Rahtz</p> </availability>
		</publicationStmt> 
		<sourceDesc> 
		  <p></p> 
		</sourceDesc> 
	 </fileDesc> 
	 <revisionDesc> 
		<list> 
		  <item> 
		 <date>23 Oct 1999</date> SR converted from LaTeX</item> 
		</list> 
	 </revisionDesc> 
  </teiHeader> 
  <text> 
	 <front> 
		<docTitle> 
		  <titlePart type="main">Simulation of Energy Loss Straggling</titlePart> 
		</docTitle> 
		<docAuthor>Maria Physicist</docAuthor> 
		<docDate>January 17, 1999</docDate> 
	 </front> 
	 <body> <div id="intro"> <head>Introduction</head> <p>Due to
the statistical nature of ionisation energy loss, large fluctuations
can occur in the amount of energy deposited by a particle traversing
an absorber element.  Continuous processes such as multiple scattering
and energy loss play a relevant role in the longitudinal and lateral
development of electromagnetic and hadronic showers, and in the case
of sampling calorimeters the measured resolution can be significantly
affected by such fluctuations in their active layers. The description
of ionisation fluctuations is characterised by the significance
parameter <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"><mi>&kappa;</mi></math></formula>, which is
proportional to the ratio of mean energy loss to the maximum allowed
energy transfer in a single collision with an atomic electron
 <formula
type="display"><math  xmlns="http://www.w3.org/1998/Math/MathML"><mrow> <mi>&kappa;</mi><mo>=</mo>
<mfrac>
 <mrow><mi>&xi;</mi></mrow>
 <mrow>
   &Emax;
 </mrow>
</mfrac>
</mrow>
</math></formula>
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
>&Emax;</math></formula> is the
maximum transferable energy in a single collision with an atomic electron.
<formula type="display"><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mrow>
                    &Emax;<mo>=</mo>          <mfrac><mrow><mn>2</mn><msub><mi>m</mi><mrow><mi>e</mi></mrow></msub><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><msup><mi>&gamma;</mi><mrow><mn>2</mn></mrow></msup></mrow><!--____________
--><mrow><mn>1</mn><mo>+</mo><mn>2</mn><mi>&gamma;</mi><msub><mi>m</mi><mrow><mi>e</mi></mrow></msub><mo>/</mo><msub><mi>m</mi><mrow><mi>x</mi></mrow></msub><mo>+</mo><msup><mfenced  
open='(' close=')'><msub><mi>m</mi><mrow><mi>e</mi></mrow></msub><mo>/</mo><msub><mi>m</mi><mrow><mi>x</mi></mrow></msub> </mfenced><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>,</mo>
</mrow></math></formula> where
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&gamma;</mi><mo>=</mo><mi>E</mi><mo>/</mo><msub><mi>m</mi><mrow><mi>x</mi></mrow></msub></math></formula>,
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>E</mi></math></formula> is energy and
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>m</mi><mrow><mi>x</mi></mrow></msub></math></formula> the mass of the
incident particle, <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><mo>=</mo><mn>1</mn><mo>-</mo><mn>1</mn><mo>/</mo><msup><mi>&gamma;</mi><mrow><mn>2</mn></mrow></msup></math></formula>
and <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>m</mi><mrow><mi>e</mi></mrow></msub></math></formula> is the
electron mass. <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&xi;</mi></math></formula>
comes from the Rutherford scattering crosss section and is defined as:
              <table rend="inline"><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML"><mi>&xi;</mi><mo>=</mo><mfrac><mrow><mn>2</mn><mi>&pi;</mi><msup><mi>z</mi><mrow><mn>2</mn></mrow></msup><msup><mi>e</mi><mrow><mn>4</mn></mrow></msup><msub><mi>N</mi><mrow><mi>A</mi><mi>v</mi></mrow></msub><mi>Z</mi><mi>&rho;</mi><mi>&delta;</mi><mi>x</mi></mrow><!--
     --><mrow><msub><mi>m</mi><mrow><mi>e</mi></mrow></msub><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup><mi>A</mi></mrow></mfrac>     <mo>=</mo><mn>1</mn><mn>5</mn><mn>3</mn><mo>.</mo><mn>4</mn> <mfrac><mrow><msup><mi>z</mi><mrow><mn>2</mn></mrow></msup></mrow><!--
--><mrow><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac> <mfrac><mrow><mi>Z</mi></mrow><!-- 
--><mrow><mi>A</mi></mrow></mfrac><mi>&rho;</mi><mi>&delta;</mi><mi>x</mi><mspace width='12pt'/><mi>keV </mi><mo>,</mo>                  <mtext></mtext>
</math></formula></cell></row></table>
where
</p><p><table rend="inline"><row><cell  
><formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>z</mi></math></formula></cell><cell  
>charge of the incident particle </cell>
</row><row><cell  
><formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>N</mi><mrow><mi>A</mi><mi>v</mi></mrow></msub></math></formula></cell><cell  
>Avogadro's number               </cell>
</row><row><cell  
><formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>Z</mi></math></formula></cell><cell  
>atomic number of the material</cell>
</row><row><cell  
><formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>A</mi></math></formula></cell><cell  
>atomic weight of the material </cell>
</row><row><cell  
><formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&rho;</mi></math></formula></cell><cell  
>density                               </cell>
</row><row><cell  
><formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&delta;</mi><mi>x</mi></math></formula></cell><cell  
>thickness of the material        </cell>
</row><row><cell  
>                                                                  </cell>
</row></table>
</p><p><formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&kappa;</mi></math></formula>
measures the contribution of the collisions with energy transfer close to
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
>&Emax;</math></formula>. For a given absorber,
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&kappa;</mi></math></formula> tends towards large
values if <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&delta;</mi><mi>x</mi></math></formula> is large
and/or if <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&beta;</mi></math></formula> is small.
Likewise, <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&kappa;</mi></math></formula> tends
towards zero if <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&delta;</mi><mi>x</mi></math></formula> is
small and/or if <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&beta;</mi></math></formula>
approaches 1.
</p><p>The value of <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&kappa;</mi></math></formula>
distinguishes two regimes which occur in the description of ionisation fluctuations
:
</p><list type="enumerate">
<item>
<p>A
large
number
of
collisions
involving
the
loss
of
all
or
most
of
the
incident
particle
energy
during
the
traversal
of
an
absorber.
</p><p>As
the
total
energy
transfer
is
composed
of
a
multitude
of
small
energy
losses,
we
can
apply
the
central
limit
theorem
and
describe
the
fluctuations
by
a
Gaussian
distribution.
This
case
is
applicable
to
non-relativistic
particles
and
is
described
by
the
inequality
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&kappa;</mi><mo>&gt;</mo><mn>1</mn><mn>0</mn></math></formula>
(i.e.
when
the
mean
energy
loss
in
the
absorber
is
greater
than
the
maximum
energy
transfer
in
a
single
collision).
</p></item>
<item>
<p>Particles
traversing
thin
counters
and
incident
electrons
under
any
conditions.
</p><p>The
relevant
inequalities
and
distributions
are
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mn>0</mn><mo>.</mo><mn>0</mn><mn>1</mn><mo>&lt;</mo><mi>&kappa;</mi><mo>&lt;</mo><mn>1</mn><mn>0</mn></math></formula>,
Vavilov
distribution,
and
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&kappa;</mi><mo>&lt;</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>1</mn></math></formula>,
Landau
distribution.</p></item></list>
<p>An additional regime is defined by the contribution of the collisions
with low energy transfer which can be estimated with the relation
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&xi;</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub></math></formula>,
where <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub></math></formula>
is the mean ionisation potential of the atom. Landau theory assumes that
the number of these collisions is high, and consequently, it has a restriction
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&xi;</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub><mo>&Gt;</mo><mn>1</mn></math></formula>. In <code>GEANT</code> (see
URL <xptr url="http://wwwinfo.cern.ch/asdoc/geant/geantall.html"/>), the limit of Landau theory has
been set at <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&xi;</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>5</mn><mn>0</mn></math></formula>.
Below this limit special models taking into account the atomic structure of the material are
used. This is important in thin layers and gaseous materials. Figure <ptr target="fg:phys332-1"/> shows the behaviour
of <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&xi;</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub></math></formula> as
a function of the layer thickness for an electron of 100 keV and 1 GeV of kinetic
energy in Argon, Silicon and Uranium.
</p>
<p><figure  file="phys332-1" id="fg:phys332-1">
<head>The variable <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&xi;</mi><mo>/</mo><msub><mi>I</mi><mrow><mn>0</mn></mrow></msub></math></formula>
can    be    used    to    measure    the    validity    range    of    the    Landau
theory.    It    depends    on    the    type    and    energy    of    the    particle,
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>Z</mi></math></formula>,
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>A</mi></math></formula>
and the ionisation potential of the material and the layer thickness. </head>
</figure></p>
<p>In the following sections, the different theories and models for the energy loss
fluctuation are described. First, the Landau theory and its limitations are discussed,
and then, the Vavilov and Gaussian straggling functions and the methods in the thin
layers and gaseous materials are presented.
</p>
</div>
<div id="sec:phys332-1">
<head>Landau theory</head>
<p>For a particle of mass <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>m</mi><mrow><mi>x</mi></mrow></msub></math></formula> traversing
a thickness of material <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&delta;</mi><mi>x</mi></math></formula>,
the Landau probability distribution may be written in terms of the universal Landau
function <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&phi;</mi><mrow><mo>(</mo><mi>&lambda;</mi><mo>)</mo></mrow></math></formula>
as<ptr target="bib-LAND"/>:
                         <table rend="inline"><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mrow><mo>(</mo><mi>&epsi;</mi><mo>,</mo><mi>&delta;</mi><mi>x</mi><mo>)</mo></mrow>  <mo>=</mo>  <mfrac><mrow><mn>1</mn></mrow><!--
--><mrow><mi>&xi;</mi></mrow></mfrac><mi>&phi;</mi><mrow><mo>(</mo><mi>&lambda;</mi><mo>)</mo></mrow>                         <mtext></mtext>
</math></formula></cell></row></table>
where
             <table rend="inline"><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML"><mi>&phi;</mi><mrow><mo>(</mo><mi>&lambda;</mi><mo>)</mo></mrow>  <mo>=</mo>   <mfrac><mrow><mn>1</mn></mrow><!--_
--><mrow><mn>2</mn><mi>&pi;</mi><mi>i</mi></mrow></mfrac><msubsup><mo>&int;</mo>
                           <mrow><mi>c</mi><mo>+</mo><mi>i</mi><mi>&infin;</mi></mrow><mrow><mi>c</mi><mo>-</mo><mi>i</mi><mi>&infin;</mi></mrow></msubsup><mo>exp</mo><mfenced  
open='(' close=')'><mi>u</mi><mo>ln</mo><mi>u</mi><mo>+</mo><mi>&lambda;</mi><mi>u</mi></mfenced><mi>d</mi><mi>u</mi><mspace width='2cm'/><mi>c</mi><mo>&geq;</mo><mn>0</mn>             <mtext></mtext>
                 </math></formula></cell></row><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML">
               </math></formula></cell></row><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML"><mi>&lambda;</mi>  <mo>=</mo>  <mfrac><mrow><mi>&epsi;</mi><mo>-</mo><munderover accent='true'><mo>&macr;</mo><mi>&epsi;</mi><mrow></mrow></munderover></mrow><!--
 --><mrow><mi>&xi;</mi></mrow></mfrac> <mo>-</mo><mi>&gamma;</mi><mi>&prime;</mi><mo>-</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><mo>-</mo><mo>ln</mo>   <mfrac><mrow><mi>&xi;</mi></mrow><!-- ___
--><mrow>&Emax;</mrow></mfrac>                          <mtext></mtext>
                 </math></formula></cell></row><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML">
               </math></formula></cell></row><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML"><mi>&gamma;</mi><mi>&prime;</mi>  <mo>=</mo>  <mn>0</mn><mo>.</mo><mn>4</mn><mn>2</mn><mn>2</mn><mn>7</mn><mn>8</mn><mn>4</mn><mo>.</mo><mo>.</mo><mo>.</mo><mo>=</mo><mn>1</mn><mo>-</mo><mi>&gamma;</mi>                              <mtext></mtext>
                 </math></formula></cell></row><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML">
               </math></formula></cell></row><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML"><mi>&gamma;</mi>  <mo>=</mo>  <mn>0</mn><mo>.</mo><mn>5</mn><mn>7</mn><mn>7</mn><mn>2</mn><mn>1</mn><mn>5</mn><mo>.</mo><mo>.</mo><mo>.</mo><mtext>(Eulers constant)</mtext>                    <mtext></mtext>
                 </math></formula></cell></row><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML">
                </math></formula></cell></row><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML"><munderover accent='true'><mo>&macr;</mo><mi>&epsi;</mi><mrow></mrow></munderover>  <mo>=</mo>  <mtext>average energy loss</mtext>                            <mtext></mtext>
                 </math></formula></cell></row><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML">
                </math></formula></cell></row><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML"><mi>&epsi;</mi>  <mo>=</mo>  <mtext>actual energy loss</mtext>                             <mtext></mtext>
</math></formula></cell></row></table>
</p>
<div >
<head>Restrictions</head>
<p>The Landau formalism makes two restrictive assumptions :
</p><list type="enumerate">
<item>
<p>The
typical
energy
loss
is
small
compared
to
the
maximum
energy
loss
in
a
single
collision.
This
restriction
is
removed
in
the
Vavilov
theory
(see
section
<ptr target="vavref"/>).
</p></item>
<item>
<p>The
typical
energy
loss
in
the
absorber
should
be
large
compared
to
the
binding
energy
of
the
most
tightly
bound
electron.
For
gaseous
detectors,
typical
energy
losses
are
a
few
keV
which
is
comparable
to
the
binding
energies
of
the
inner
electrons.
In
such
cases
a
more
sophisticated
approach
which
accounts
for
atomic
energy
levels<ptr target="bib-TALM"/>
is
necessary
to
accurately
simulate
data
distributions.
In
<code>GEANT</code>,
a
parameterised
model
by
L.
Urb&aacute;n
is
used
(see
section
<ptr target="urban"/>).</p></item></list>
<p>In addition, the average value of the Landau distribution is infinite.
Summing the Landau fluctuation obtained to the average energy from the
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>d</mi><mi>E</mi><mo>/</mo><mi>d</mi><mi>x</mi></math></formula>
tables, we obtain a value which is larger than the one coming from the table. The
probability to sample a large value is small, so it takes a large number of steps
(extractions) for the average fluctuation to be significantly larger than zero. This
introduces a dependence of the energy loss on the step size which can affect
calculations.
</p><p>A solution to this has been to introduce a limit on the value of the
variable sampled by the Landau distribution in order to keep the average
fluctuation to 0. The value obtained from the <code>GLANDO</code> routine is:
<formula type="display"><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mrow>
                  <mi>&delta;</mi><mi>d</mi><mi>E</mi><mo>/</mo><mi>d</mi><mi>x</mi><mo>=</mo><mi>&epsi;</mi><mo>-</mo><munderover accent='true'><mo>&macr;</mo><mi>&epsi;</mi><mrow></mrow></munderover><mo>=</mo><mi>&xi;</mi><mrow><mo>(</mo><mi>&lambda;</mi><mo>-</mo><mi>&gamma;</mi><mi>&prime;</mi><mo>+</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><mo>+</mo><mo>ln</mo>   <mfrac><mrow><mi>&xi;</mi></mrow><!-- ___
--><mrow>&Emax;</mrow></mfrac> <mo>)</mo></mrow>
</mrow></math></formula>
In order for this to have average 0, we must impose that:
<formula type="display"><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mrow>
                        <munderover accent='true'><mo>&macr;</mo><mi>&lambda;</mi><mrow></mrow></munderover><mo>=</mo><mo>-</mo><mi>&gamma;</mi><mi>&prime;</mi><mo>-</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><mo>-</mo><mo>ln</mo>   <mfrac><mrow><mi>&xi;</mi></mrow><!-- ___
--><mrow>&Emax;</mrow></mfrac>
</mrow></math></formula>
</p><p>This is realised introducing a <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>&lambda;</mi><mrow><mtext>max</mtext></mrow></msub><mrow><mo>(</mo><munderover accent='true'><mo>&macr;</mo><mi>&lambda;</mi><mrow></mrow></munderover><mo>)</mo></mrow></math></formula>
such that if only values of <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&lambda;</mi><mo>&leq;</mo><msub><mi>&lambda;</mi><mrow><mtext>max</mtext></mrow></msub></math></formula>
are accepted, the average value of the distribution is
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><munderover accent='true'><mo>&macr;</mo><mi>&lambda;</mi><mrow></mrow></munderover></math></formula>.
</p><p>A parametric fit to the universal Landau distribution has been performed, with following result:
<formula type="display"><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mrow>
    <msub><mi>&lambda;</mi><mrow><mtext>max</mtext></mrow></msub><mo>=</mo><mn>0</mn><mo>.</mo><mn>6</mn><mn>0</mn><mn>7</mn><mn>1</mn><mn>5</mn><mo>+</mo><mn>1</mn><mo>.</mo><mn>1</mn><mn>9</mn><mn>3</mn><mn>4</mn><munderover accent='true'><mo>&macr;</mo><mi>&lambda;</mi><mrow></mrow></munderover><mo>+</mo><mrow><mo>(</mo><mn>0</mn><mo>.</mo><mn>6</mn><mn>7</mn><mn>7</mn><mn>9</mn><mn>4</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>5</mn><mn>2</mn><mn>3</mn><mn>8</mn><mn>2</mn><munderover accent='true'><mi>&lambda;</mi><mrow></mrow><mo>&macr;</mo></munderover><mo>)</mo></mrow><mo>exp</mo><mrow><mo>(</mo><mn>0</mn><mo>.</mo><mn>9</mn><mn>4</mn><mn>7</mn><mn>5</mn><mn>3</mn><mo>+</mo><mn>0</mn><mo>.</mo><mn>7</mn><mn>4</mn><mn>4</mn><mn>4</mn><mn>2</mn><munderover accent='true'><mi>&lambda;</mi><mrow></mrow><mo>&macr;</mo></munderover><mo>)</mo></mrow>
</mrow></math></formula> only values
smaller than <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>&lambda;</mi><mrow><mtext>max</mtext></mrow></msub></math></formula>
are accepted, otherwise the distribution is resampled.
</p>
</div>
</div>
<div id="vavref">
<head>Vavilov theory</head>
<p>Vavilov<ptr target="bib-VAVI"/> derived a more accurate straggling distribution by introducing the kinematic
limit on the maximum transferable energy in a single collision, rather than using
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
>&Emax;<mo>=</mo><mi>&infin;</mi></math></formula>. Now
we can write<ptr target="bib-SCH1"/>:
                       <table rend="inline"><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi> <mfenced  
open='(' close=')'><mi>&epsi;</mi><mo>,</mo><mi>&delta;</mi><mi>s</mi></mfenced>  <mo>=</mo>  <mfrac><mrow><mn>1</mn></mrow><!--
--><mrow><mi>&xi;</mi></mrow></mfrac><msub><mi>&phi;</mi><mrow><mi>v</mi></mrow></msub> <mfenced  
open='(' close=')'><msub><mi>&lambda;</mi><mrow><mi>v</mi></mrow></msub><mo>,</mo><mi>&kappa;</mi><mo>,</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup></mfenced>                       <mtext></mtext>
</math></formula></cell></row></table>
where
         <table rend="inline"><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&phi;</mi><mrow><mi>v</mi></mrow></msub> <mfenced  
open='(' close=')'><msub><mi>&lambda;</mi><mrow><mi>v</mi></mrow></msub><mo>,</mo><mi>&kappa;</mi><mo>,</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup></mfenced>  <mo>=</mo>   <mfrac><mrow><mn>1</mn></mrow><!--_
--><mrow><mn>2</mn><mi>&pi;</mi><mi>i</mi></mrow></mfrac><msubsup><mo>&int;</mo>
                            <mrow><mi>c</mi><mo>+</mo><mi>i</mi><mi>&infin;</mi></mrow><mrow><mi>c</mi><mo>-</mo><mi>i</mi><mi>&infin;</mi></mrow></msubsup><mi>&phi;</mi><mfenced  
open='(' close=')'><mi>s</mi></mfenced><msup><mi>e</mi><mrow><mi>&lambda;</mi><mi>s</mi></mrow></msup><mi>d</mi><mi>s</mi><mspace width='2cm'/><mi>c</mi><mo>&geq;</mo><mn>0</mn>                  <mtext></mtext>
                 </math></formula></cell></row><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML">
               </math></formula></cell></row><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML"><mi>&phi;</mi><mfenced  
open='(' close=')'><mi>s</mi></mfenced>  <mo>=</mo>  <mo>exp</mo><mfenced  
open='[' close=']'><mi>&kappa;</mi><mrow><mo>(</mo><mn>1</mn><mo>+</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><mi>&gamma;</mi><mo>)</mo></mrow></mfenced><mo>exp</mo><mfenced  
open='[' close=']'><mi>&psi;</mi> <mfenced  
open='(' close=')'><mi>s</mi></mfenced></mfenced><mo>,</mo>                         <mtext></mtext>
                 </math></formula></cell></row><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML">
               </math></formula></cell></row><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML"><mi>&psi;</mi> <mfenced  
open='(' close=')'><mi>s</mi></mfenced>  <mo>=</mo>  <mi>s</mi><mo>ln</mo><mi>&kappa;</mi><mo>+</mo><mrow><mo>(</mo><mi>s</mi><mo>+</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><mi>&kappa;</mi><mo>)</mo></mrow><mfenced  
open='[' close=']'><mo>ln</mo><mrow><mo>(</mo><mi>s</mi><mo>/</mo><mi>&kappa;</mi><mo>)</mo></mrow><mo>+</mo><msub><mi>E</mi><mrow>
<mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>s</mi><mo>/</mo><mi>&kappa;</mi><mo>)</mo></mrow></mfenced><mo>-</mo><mi>&kappa;</mi><msup><mi>e</mi><mrow><mo>-</mo><mi>s</mi><mo>/</mo><mi>&kappa;</mi></mrow></msup><mo>,</mo>         <mtext></mtext>
</math></formula></cell></row></table>
and
           <table rend="inline"><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow>  <mo>=</mo><msubsup>  <mo>&int;</mo>
                       <mrow><mi>&infin;</mi></mrow><mrow><mi>z</mi></mrow></msubsup><msup><mi>t</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><msup><mi>e</mi><mrow><mo>-</mo><mi>t</mi></mrow></msup><mi>d</mi><mi>t</mi><mspace width='1cm'/><mtext>(the exponential integral)</mtext>           <mtext></mtext>
                </math></formula></cell></row><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML">
              </math></formula></cell></row><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&lambda;</mi><mrow><mi>v</mi></mrow></msub>  <mo>=</mo>  <mi>&kappa;</mi><mfenced  
open='[' close=']'><mfrac><mrow><mi>&epsi;</mi><mo>-</mo><munderover accent='true'><mo>&macr;</mo><mi>&epsi;</mi><mrow></mrow></munderover></mrow><!--
  --><mrow><mi>&xi;</mi></mrow></mfrac>  <mo>-</mo><mi>&gamma;</mi><mi>&prime;</mi><mo>-</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup></mfenced>                                 <mtext></mtext>
</math></formula></cell></row></table>
</p><p>The Vavilov parameters are simply related to the Landau parameter by
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>&lambda;</mi><mrow><mi>L</mi></mrow></msub><mo>=</mo><msub><mi>&lambda;</mi><mrow><mi>v</mi></mrow></msub><mo>/</mo><mi>&kappa;</mi><mo>-</mo><mo>ln</mo><mi>&kappa;</mi></math></formula>. It can be shown that
as <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&kappa;</mi><mo>&rarr;</mo><mn>0</mn></math></formula>, the distribution of
the variable <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>&lambda;</mi><mrow><mi>L</mi></mrow></msub></math></formula> approaches
that of Landau. For <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&kappa;</mi><mo>&leq;</mo><mn>0</mn><mo>.</mo><mn>0</mn><mn>1</mn></math></formula>
the two distributions are already practically identical. Contrary to what many textbooks
report, the Vavilov distribution <emph>does not</emph> approximate the Landau distribution for small
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&kappa;</mi></math></formula>, but rather the
distribution of <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>&lambda;</mi><mrow><mi>L</mi></mrow></msub></math></formula>
defined above tends to the distribution of the true
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&lambda;</mi></math></formula> from
the Landau density function. Thus the routine <code>GVAVIV</code> samples the variable
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>&lambda;</mi><mrow><mi>L</mi></mrow></msub></math></formula> rather
than <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>&lambda;</mi><mrow><mi>v</mi></mrow></msub></math></formula>.
For <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&kappa;</mi><mo>&geq;</mo><mn>1</mn><mn>0</mn></math></formula>
the Vavilov distribution tends to a Gaussian distribution (see next section).
</p>
</div>
<div >
<head>Gaussian Theory</head>
<p>Various conflicting forms have been proposed for Gaussian straggling functions, but most
of these appear to have little theoretical or experimental basis. However, it has been shown<ptr target="bib-SELT"/>
that for <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&kappa;</mi><mo>&geq;</mo><mn>1</mn><mn>0</mn></math></formula>
the Vavilov distribution can be replaced by a Gaussian of the form:
            <table rend="inline"><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML"><mi>f</mi><mrow><mo>(</mo><mi>&epsi;</mi><mo>,</mo><mi>&delta;</mi><mi>s</mi><mo>)</mo></mrow><mo>&ap;</mo>         <mfrac><mrow><mn>1</mn></mrow><!--________
--><mrow><mi>&xi;</mi><msqrt><!--<mi>&radical;</mi>
   ______________--><mfrac><mrow><mn>2</mn><mi>&pi;</mi></mrow><!--
 --><mrow><mi>&kappa;</mi></mrow></mfrac>  <mfenced  
open='(' close=')'><mn>1</mn><mo>-</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>2</mn></mfenced></msqrt></mrow></mfrac><mo>exp</mo><mfenced  
open='[' close=']'><mfrac><mrow><msup><mrow><mo>(</mo><mi>&epsi;</mi><mo>-</mo><munderover accent='true'><mo>&macr;</mo><mi>&epsi;</mi><mrow></mrow></munderover><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow><!--
    --><mrow><mn>2</mn></mrow></mfrac>           <mfrac><mrow><mi>&kappa;</mi></mrow><!-- _______
--><mrow><msup><mi>&xi;</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mn>1</mn><mo>-</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>2</mn><mo>)</mo></mrow></mrow></mfrac></mfenced>                <mtext></mtext>
</math></formula></cell></row></table>
thus implying
                <table rend="inline"><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML"><mi>mean</mi>  <mo>=</mo>  <munderover accent='true'><mo>&macr;</mo><mi>&epsi;</mi><mrow></mrow></munderover>                                       <mtext></mtext>
                    </math></formula></cell></row><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML">
                  </math></formula></cell></row><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup>  <mo>=</mo>  <mfrac><mrow><msup><mi>&xi;</mi><mrow><mn>2</mn></mrow></msup></mrow><!--
 --><mrow><mi>&kappa;</mi></mrow></mfrac> <mrow><mo>(</mo><mn>1</mn><mo>-</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>2</mn><mo>)</mo></mrow><mo>=</mo><mi>&xi;</mi><msub><mi>E</mi><mrow><mi>
max</mi></mrow></msub><mrow><mo>(</mo><mn>1</mn><mo>-</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><mn>2</mn><mo>)</mo></mrow>                <mtext></mtext>
</math></formula></cell></row></table>
</p>
</div>
<div id="urban">
<head>Urb&aacute;n model</head>
<p>The method for computing restricted energy losses with
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&delta;</mi></math></formula>-ray
production above given threshold energy in <code>GEANT</code> is a Monte Carlo method that
can be used for thin layers. It is fast and it can be used for any thickness of a
medium. Approaching the limit of the validity of Landau's theory, the loss
distribution approaches smoothly the Landau form as shown in Figure <ptr target="fg:phys332-2"/>.
</p>
<p><figure file="phys332-2" id="fg:phys332-2">
<head>Energy loss distribution for a 3 GeV electron in Argon as given by
standard GEANT. The width of the layers is given in centimeters.</head>
</figure></p>
<p>It is assumed that the atoms have only two energy levels with binding energy
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub></math></formula> and
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub></math></formula>.
The particle--atom interaction will then be an excitation with energy loss
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub></math></formula> or
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub></math></formula>, or
an ionisation with an energy loss distributed according to a function
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mo>&sim;</mo><mn>1</mn><mo>/</mo><msup><mi>E</mi><mrow><mn>2</mn></mrow></msup></math></formula>:
<formula type="equation"><math  xmlns="http://www.w3.org/1998/Math/MathML">
                        <mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mo>=</mo><mfrac><mrow><mrow><mo>(</mo>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow><mi>I</mi></mrow><!--
    --><mrow>&Emax;</mrow></mfrac>
<mfrac><mrow><mn>1</mn></mrow><!-- _
--><mrow><msup><mi>E</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac>
</math></formula></p><p>The
macroscopic cross-section for excitations (<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"
><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn></math></formula>)
is <formula id="eq:sigex" type="equation"><math  xmlns="http://www.w3.org/1998/Math/MathML">
                   <msub><mi>&Sigma;</mi><mrow><mi>i</mi></mrow></msub><mo>=</mo><mi>C</mi> <mfrac><mrow><msub><mi>f</mi><mrow><mi>i</mi></mrow></msub></mrow><!-- 
--><mrow><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub></mrow></mfrac> <mfrac><mrow><mo>ln</mo><mrow><mo>(</mo><mn>2</mn><mi>m</mi><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><msup><mi>&gamma;</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mo>-</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup></mrow><!-- 
 --><mrow><mo>ln</mo><mrow><mo>(</mo><mn>2</mn><mi>m</mi><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup><msup><mi>&gamma;</mi><mrow><mn>2</mn></mrow></msup><mo>/</mo><mi>I</mi><mo>)</mo></mrow><mo>-</mo><msup><mi>&beta;</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac> <mrow><mo>(</mo><mn>1</mn><mo>-</mo><mi>r</mi><mo>)</mo></mrow>               </math></formula>and
the macroscopic cross-section for ionisation is
<formula id="eq:sigion" type="equation"><math  xmlns="http://www.w3.org/1998/Math/MathML">
                    <msub><mi>&Sigma;</mi><mrow><mn>3</mn></mrow></msub><mo>=</mo><mi>C</mi>            <mfrac><mrow>&Emax;</mrow><!-- ________________
--><mrow><mi>I</mi><mrow><mo>(</mo>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow><mo>ln</mo><mrow><mo>(</mo><mfrac><mrow>&Emax;<mo>+</mo><mi>I</mi></mrow><!-- 
      --><mrow><mi>I</mi></mrow></mfrac>      <mo>)</mo></mrow></mrow></mfrac><mi>r</mi>                 </math></formula>
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML">&Emax;</math></formula>
is the <code>GEANT</code> cut for <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&delta;</mi></math></formula>-production,
or the maximum energy transfer minus mean ionisation energy, if it is smaller than
this cut-off value. The following notation is used:
</p><p><table rend="inline"><row><cell  
><formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>r</mi><mo>,</mo><mi>C</mi></math></formula></cell><cell  
>parameters of the model</cell>
</row><row><cell  
><formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub></math></formula></cell><cell  
>atomic energy levels      </cell>
</row><row><cell  
><formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>I</mi></math></formula></cell><cell  
>mean ionisation energy  </cell>
</row><row><cell  
><formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>f</mi><mrow><mi>i</mi></mrow></msub></math></formula></cell><cell  
>oscillator strengths       </cell>
</row></table>
</p><p>The model has the parameters <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>f</mi><mrow><mi>i</mi></mrow></msub></math></formula>,
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub></math></formula>,
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>C</mi></math></formula> and
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>r</mi><mrow><mo>(</mo><mn>0</mn><mo>&leq;</mo><mi>r</mi><mo>&leq;</mo><mn>1</mn><mo>)</mo></mrow></math></formula>. The oscillator
strengths <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>f</mi><mrow><mi>i</mi></mrow></msub></math></formula> and the
atomic level energies <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub></math></formula>
should satisfy the constraints
                              <table rend="inline"><row><cell><formula type="subeqn" id="eq:fisum"><math  xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>f</mi><mrow><mn>2</mn></mrow></msub>  <mo>=</mo>  <mn>1</mn>                     
                                   </math></formula></cell></row><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML">
                      </math></formula></cell></row><row><cell><formula type="subeqn" id="eq:flnsum"><math  xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>f</mi><mrow><mn>1</mn></mrow></msub><mo>ln</mo><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>f</mi><mrow><mn>2</mn></mrow></msub><mo>ln</mo><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub>  <mo>=</mo>  <mo>ln</mo><mi>I</mi>                   
</math></formula></cell></row></table>
The parameter <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>C</mi></math></formula>
can be defined with the help of the mean energy loss
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>d</mi><mi>E</mi><mo>/</mo><mi>d</mi><mi>x</mi></math></formula> in the following way: The
numbers of collisions (<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>n</mi><mrow><mi>i</mi></mrow></msub></math></formula>,
i = 1,2 for the excitation and 3 for the ionisation) follow the Poisson distribution with a mean
number <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mi>i</mi></mrow></msub><mo>>;</mo></mrow></math></formula>. In a step
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&Delta;</mi><mi>x</mi></math></formula> the mean number
of collisions is <formula type="display"><math  xmlns="http://www.w3.org/1998/Math/MathML">
                                            <mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mi>i</mi></mrow></msub><mo>>;</mo></mrow><mo>=</mo><msub><mi>&Sigma;</mi><mrow><mi>i</mi></mrow></msub><mi>&Delta;</mi><mi>x</mi>                                        
</math></formula>The
mean energy loss <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>d</mi><mi>E</mi><mo>/</mo><mi>d</mi><mi>x</mi></math></formula>
in a step is the sum of the excitation and ionisation contributions
<formula type="display"><math  xmlns="http://www.w3.org/1998/Math/MathML">
            <mfrac><mrow><mi>d</mi><mi>E</mi></mrow><!--
--><mrow><mi>d</mi><mi>x</mi></mrow></mfrac> <mi>&Delta;</mi><mi>x</mi><mo>=</mo><mfenced  
open='[' close=']'><msub><mi>&Sigma;</mi><mrow><mn>1</mn></mrow></msub><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>&Sigma;</mi><mrow><mn>2</mn></mrow></msub><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mi>&Sigma;</mi><mrow><mn>3</mn></mrow></msub><msubsup><mo>&int;</mo>
    <mrow><mi>I</mi></mrow><mrow>&Emax;<mo>+</mo><mi>I</mi></mrow></msubsup><mi>E</mi><mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mi>d</mi><mi>E</mi></mfenced><mi>&Delta;</mi><mi>x</mi>        
</math></formula>From
this, using the equations (<ptr target="eq:sigex"/>), (<ptr target="eq:sigion"/>), (<ptr target="eq:fisum"/>) and (<ptr target="eq:flnsum"/>), one can define the parameter
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>C</mi></math></formula>
<formula type="display"><math  xmlns="http://www.w3.org/1998/Math/MathML">
                              <mi>C</mi><mo>=</mo><mfrac><mrow><mi>d</mi><mi>E</mi></mrow><!--
--><mrow><mi>d</mi><mi>x</mi></mrow></mfrac>                           
</math></formula>
</p><p>The following values have been chosen in <code>GEANT</code> for the other parameters:
<formula type="display"><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mrow>
                     <mtable equalrows='false' equalcolumns='false'><mtr><mtd><msub><mi>f</mi><mrow><mn>2</mn></mrow></msub><mo>=</mo><mfenced  
open='{' ><mtable equalrows='false' equalcolumns='false'><mtr><mtd><mn>0</mn>    </mtd><mtd><mi>if</mi><mi>Z</mi><mo>&leq;</mo><mn>2</mn></mtd>
</mtr><mtr><mtd><mn>2</mn><mo>/</mo><mi>Z</mi></mtd><mtd><mi>if</mi><mi>Z</mi><mo>&gt;</mo><mn>2</mn></mtd>
</mtr><mtr><mtd>     </mtd></mtr></mtable>         </mfenced></mtd><mtd><mo>&Rarr;</mo></mtd><mtd><msub><mi>f</mi><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn><mo>-</mo><msub><mi>f</mi><mrow><mn>2</mn></mrow></msub>     </mtd>
                     </mtr><mtr><mtd><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub><mo>=</mo><mn>1</mn><mn>0</mn><msup><mi>Z</mi><mrow><mn>2</mn></mrow></msup><mi>eV </mi>    </mtd><mtd><mo>&Rarr;</mo></mtd><mtd><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub><mo>=</mo><msup><mfenced  
open='(' close=')'>  <mfrac><mrow><mi>I</mi></mrow><!--___
--><mrow><msubsup><mi>E</mi><mrow><mn>2</mn></mrow><mrow><msub><mi>f</mi><mrow><mn>2</mn></mrow></msub></mrow></msubsup></mrow></mfrac>  </mfenced><mrow> <mfrac><mrow><mn>1</mn></mrow><!-- _
--><mrow><msub><mi>f</mi><mrow><mn>1</mn></mrow></msub></mrow></mfrac> </mrow></msup></mtd>
                     </mtr><mtr><mtd><mi>r</mi><mo>=</mo><mn>0</mn><mo>.</mo><mn>4</mn>            </mtd><mtd>  </mtd><mtd>          </mtd>
                     </mtr><mtr><mtd>             </mtd></mtr></mtable>
</mrow></math></formula> With these values
the atomic level <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub></math></formula>
corresponds approximately the K-shell energy of the atoms and
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>Z</mi><msub><mi>f</mi><mrow><mn>2</mn></mrow></msub></math></formula> the number of
K-shell electrons. <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>r</mi></math></formula>
is the only variable which can be tuned freely. It determines the relative contribution
of ionisation and excitation to the energy loss.
</p><p>The energy loss is computed with the assumption that the step length (or the relative
energy loss) is small, and---in consequence---the cross-section can be considered
constant along the path length. The energy loss due to the excitation is
<formula type="display"><math  xmlns="http://www.w3.org/1998/Math/MathML">
                         <mi>&Delta;</mi><msub><mi>E</mi><mrow><mi>e</mi></mrow></msub><mo>=</mo><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><msub><mi>E</mi><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub><msub><mi>E</mi><mrow><mn>2</mn></mrow></msub>                      
</math></formula>where
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub></math></formula> and
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub></math></formula>
are sampled from Poisson distribution as discussed above. The
loss due to the ionisation can be generated from the distribution
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow></math></formula> by
the inverse transformation method:
                       <table rend="inline"><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mi>F</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow>  <mo>=</mo><msubsup>  <mo>&int;</mo>
                                     <mrow><mi>I</mi></mrow><mrow><mi>E</mi></mrow></msubsup><mi>g</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mi>d</mi><mi>x</mi>                     <mtext></mtext>
                              </math></formula></cell></row><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML">
                     </math></formula></cell></row><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML"><mi>E</mi><mo>=</mo><msup><mi>F</mi><mrow><mo>-</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow>  <mo>=</mo>       <mfrac><mrow><mi>I</mi></mrow><!--____
--><mrow><mn>1</mn><mo>-</mo><mi>u</mi>  <mfrac><mrow>&Emax;</mrow><!-- ___
--><mrow>&Emax;<mo>+</mo><mi>I</mi></mrow></mfrac> </mrow></mfrac>                
                              </math></formula></cell></row><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML">
                              </math></formula></cell></row><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML">                                
</math></formula></cell></row></table>
where <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>u</mi></math></formula> is a uniform random
number between <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>F</mi><mrow><mo>(</mo><mi>I</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></math></formula> and
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>F</mi><mrow><mo>(</mo>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></math></formula>. The contribution from the
ionisations will be <formula type="display"><math  xmlns="http://www.w3.org/1998/Math/MathML">
                                 <mi>&Delta;</mi><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub><mo>=</mo><msubsup><mo>&sum;</mo>
    <mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></mrow></msubsup>          <mfrac><mrow><mi>I</mi></mrow><!--________
--><mrow><mn>1</mn><mo>-</mo><msub><mi>u</mi><mrow><mi>j</mi></mrow></msub>   <mfrac><mrow>&Emax;</mrow><!-- ___
--><mrow>&Emax;<mo>+</mo><mi>I</mi></mrow></mfrac> </mrow></mfrac>                                      
</math></formula>where
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></formula> is the
number of ionisation (sampled from Poisson distribution). The energy loss in a step will
then be <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&Delta;</mi><mi>E</mi><mo>=</mo><mi>&Delta;</mi><msub><mi>E</mi><mrow><mi>e</mi></mrow></msub><mo>+</mo><mi>&Delta;</mi><msub><mi>E</mi><mrow><mi>i</mi></mrow></msub></math></formula>.
</p>
<div >
<head>Fast simulation for <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>&geq;</mo><mn>1</mn><mn>6</mn></math></formula></head>
<p>If the number of ionisation <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></formula>
is bigger than 16, a faster sampling method can be used. The possible energy loss
interval is divided in two parts: one in which the number of collisions is large and the
sampling can be done from a Gaussian distribution and the other in which
the energy loss is sampled for each collision. Let us call the former interval
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mrow><mo>[</mo><mi>I</mi><mo>,</mo><mi>&alpha;</mi><mi>I</mi><mo>]</mo></mrow></math></formula> the interval A,
and the latter <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mrow><mo>[</mo><mi>&alpha;</mi><mi>I</mi><mo>,</mo>&Emax;<mo>]</mo></mrow></math></formula> the
interval B. <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&alpha;</mi></math></formula> lies
between 1 and <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
>&Emax;<mo>/</mo><mi>I</mi></math></formula>.
A collision with a loss in the interval A happens with the probability
<formula type="display" id="eq:phys332-5"><math  xmlns="http://www.w3.org/1998/Math/MathML">
                <mi>P</mi><mrow><mo>(</mo><mi>&alpha;</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mo>&int;</mo>
   <mrow><mi>I</mi></mrow><mrow><mi>&alpha;</mi><mi>I</mi></mrow></msubsup><mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mi>d</mi><mi>E</mi><mo>=</mo><mfrac><mrow><mrow><mo>(</mo>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow><mrow><mo>(</mo><mi>&alpha;</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow></mrow><!--
      --><mrow>&Emax;<mi>&alpha;</mi></mrow></mfrac>                
</math></formula>The
mean energy loss and the standard deviation for this type of collision are
<formula type="display"><math  xmlns="http://www.w3.org/1998/Math/MathML">
                <mrow><mo>&lt;</mo><mi>&Delta;</mi><mi>E</mi><mrow><mo>(</mo><mi>&alpha;</mi><mo>)</mo></mrow><mo>>;</mo></mrow><mo>=</mo>   <mfrac><mrow><mn>1</mn></mrow><!--___
--><mrow><mi>P</mi><mrow><mo>(</mo><mi>&alpha;</mi><mo>)</mo></mrow></mrow></mfrac><msubsup><mo>&int;</mo>
          <mrow><mi>I</mi></mrow><mrow><mi>&alpha;</mi><mi>I</mi></mrow></msubsup><mi>E</mi><mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mi>d</mi><mi>E</mi><mo>=</mo><mfrac><mrow><mi>I</mi><mi>&alpha;</mi><mo>ln</mo><mi>&alpha;</mi></mrow><!--
 --><mrow><mi>&alpha;</mi><mo>-</mo><mn>1</mn></mrow></mfrac>            
</math></formula>and <formula type="display"><math  xmlns="http://www.w3.org/1998/Math/MathML"><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>&alpha;</mi><mo>)</mo></mrow><mo>=</mo>   <mfrac><mrow><mn>1</mn></mrow><!--___
--><mrow><mi>P</mi><mrow><mo>(</mo><mi>&alpha;</mi><mo>)</mo></mrow></mrow></mfrac><msubsup><mo>&int;</mo>
          <mrow><mi>I</mi></mrow><mrow><mi>&alpha;</mi><mi>I</mi></mrow></msubsup><msup><mi>E</mi><mrow><mn>2</mn></mrow></msup><mi>g</mi><mrow><mo>(</mo><mi>E</mi><mo>)</mo></mrow><mi>d</mi><mi>E</mi><mo>=</mo><msup><mi>I</mi><mrow><mn>2</mn></mrow></msup><mi>&alpha;</mi><mfenced  
open='(' close=')'><mn>1</mn><mo>-</mo> <mfrac><mrow><mi>&alpha;</mi><msup><mo>ln</mo><mrow><mn>2</mn></mrow></msup><mi>&alpha;</mi></mrow><!--_
--><mrow><msup><mrow><mo>(</mo><mi>&alpha;</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac></mfenced>         
</math></formula>If the
collision number is high, we assume that the number of the type A collisions can be
calculated from a Gaussian distribution with the following mean value and standard
deviation:
                     <table rend="inline"><row><cell><formula type="subeqn" id="eq:phys332-1"><math  xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow>  <mo>=</mo>  <msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mi>P</mi><mrow><mo>(</mo><mi>&alpha;</mi><mo>)</mo></mrow>                         
                         </math></formula></cell></row><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML">
                     </math></formula></cell></row><row><cell><formula type="subeqn" id="eq:phys332-2"><math  xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>&sigma;</mi><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msubsup>  <mo>=</mo>  <msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mi>P</mi><mrow><mo>(</mo><mi>&alpha;</mi><mo>)</mo></mrow><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mi>P</mi><mrow><mo>(</mo><mi>&alpha;</mi><mo>)</mo></mrow><mo>)</mo></mrow>                 
</math></formula></cell></row></table>
It is further assumed that the energy loss in these collisions has a Gaussian
distribution with
                       <table rend="inline"><row><cell><formula type="subeqn" id="eq:phys332-3"><math  xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>&lt;</mo><mi>&Delta;</mi><msub><mi>E</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow>  <mo>=</mo>  <msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mrow><mo>&lt;</mo><mi>&Delta;</mi><mi>E</mi><mrow><mo>(</mo><mi>&alpha;</mi><mo>)</mo></mrow><mo>>;</mo></mrow>                  
                            </math></formula></cell></row><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML">
                       </math></formula></cell></row><row><cell><formula type="subeqn" id="eq:phys332-4"><math  xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mi>&sigma;</mi><mrow><mi>E</mi><mo>,</mo><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msubsup>  <mo>=</mo>  <msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><msup><mi>&sigma;</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo><mi>&alpha;</mi><mo>)</mo></mrow>                     
</math></formula></cell></row></table>
The energy loss of these collision can then be sampled from the Gaussian
distribution.
</p><p>The collisions where the energy loss is in the interval B are sampled directly from
<formula type="display"><math  xmlns="http://www.w3.org/1998/Math/MathML">
                 <mi>&Delta;</mi><msub><mi>E</mi><mrow><mi>B</mi></mrow></msub><mo>=</mo><msubsup><mo>&sum;</mo>
    <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>-</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub></mrow></msubsup>             <mfrac><mrow><mi>&alpha;</mi><mi>I</mi></mrow><!--_________
--><mrow><mn>1</mn><mo>-</mo><msub><mi>u</mi><mrow><mi>i</mi></mrow></msub> <mfrac><mrow>&Emax;<mo>+</mo><mi>I</mi><mo>-</mo><mi>&alpha;</mi><mi>I</mi></mrow><!-- 
   --><mrow>&Emax;<mo>+</mo><mi>I</mi></mrow></mfrac>    </mrow></mfrac>             
</math></formula>The
total energy loss is the sum of these two types of collisions:
<formula type="display"><math  xmlns="http://www.w3.org/1998/Math/MathML">
                          <mi>&Delta;</mi><mi>E</mi><mo>=</mo><mi>&Delta;</mi><msub><mi>E</mi><mrow><mi>A</mi></mrow></msub><mo>+</mo><mi>&Delta;</mi><msub><mi>E</mi><mrow><mi>B</mi></mrow></msub>                     
</math></formula></p>
<p>The approximation of equations (<ptr target="eq:phys332-1"/>), (<ptr target="eq:phys332-2"/>), (<ptr target="eq:phys332-3"/>) and (<ptr target="eq:phys332-4"/>) can be used under the following
conditions:
                           <table rend="inline"><row><cell><formula type="subeqn" id="eq:phys332-6"><math  xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow><mo>-</mo><mi>c</mi><msub><mi>&sigma;</mi><mrow><mi>A</mi></mrow></msub>  <mo>&geq;</mo>  <mn>0</mn>                    
                                   </math></formula></cell></row><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML">
                           </math></formula></cell></row><row><cell><formula type="subeqn" id="eq:phys332-7"><math  xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow><mo>+</mo><mi>c</mi><msub><mi>&sigma;</mi><mrow><mi>A</mi></mrow></msub>  <mo>&leq;</mo>  <msub><mi>n</mi><mrow><mn>3</mn></mrow></msub>                   
                                   </math></formula></cell></row><row><cell><formula type="subeqn"><math  xmlns="http://www.w3.org/1998/Math/MathML">
                       </math></formula></cell></row><row><cell><formula type="subeqn" id="eq:phys332-8"><math  xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>&lt;</mo><mi>&Delta;</mi><msub><mi>E</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow><mo>-</mo><mi>c</mi><msub><mi>&sigma;</mi><mrow><mi>E</mi><mo>,</mo><mi>A</mi></mrow></msub>  <mo>&geq;</mo>  <mn>0</mn>                    
</math></formula></cell></row></table>
where <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>c</mi><mo>&geq;</mo><mn>4</mn></math></formula>. From
the equations (<ptr target="eq:phys332-5"/>), (<ptr target="eq:phys332-1"/>) and (<ptr target="eq:phys332-3"/>) and from the conditions (<ptr target="eq:phys332-6"/>) and (<ptr target="eq:phys332-7"/>) the following limits can be
<formula type="display"><math  xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>&alpha;</mi><mrow><mi>min</mi></mrow></msub><mo>=</mo><mfrac><mrow><mrow><mo>(</mo><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>+</mo><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mo>(</mo>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow></mrow><!--
--><mrow><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mrow><mo>(</mo>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow><mo>+</mo><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup><mi>I</mi></mrow></mfrac> <mo>&leq;</mo><mi>&alpha;</mi><mo>&leq;</mo><msub><mi>&alpha;</mi><mrow><mtext>max</mtext></mrow></msub><mo>=</mo><mfrac><mrow><mrow><mo>(</mo><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>+</mo><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup><mo>)</mo></mrow><mrow><mo>(</mo>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow></mrow><!--
--><mrow><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup><mrow><mo>(</mo>&Emax;<mo>+</mo><mi>I</mi><mo>)</mo></mrow><mo>+</mo><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mi>I</mi></mrow></mfrac>         
</math></formula>This
conditions gives a lower limit to number of the ionisations
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></formula> for which the fast
<formula  type="display"><math  xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>&geq;</mo><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup>                                                  </math></formula>As
in the conditions (<ptr target="eq:phys332-6"/>), (<ptr target="eq:phys332-7"/>) and (<ptr target="eq:phys332-8"/>) the value of
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>c</mi></math></formula> is as minimum
4, one gets <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>&geq;</mo><mn>1</mn><mn>6</mn></math></formula>.
In order to speed the simulation, the maximum value is used for
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&alpha;</mi></math></formula>.
</p><p>The number of collisions with energy loss in the interval B (the number of interactions
which has to be simulated directly) increases slowly with the total number of collisions
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></formula>.
The maximum number of these collisions can be estimated as
<formula type="equation"><math  xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mrow><mi>B</mi><mo>,</mo><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>=</mo><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>-</mo><msub><mi>n</mi><mrow><mi>A</mi><mo>,</mo><mi>m</mi><mi>i</mi><mi>n</mi></mrow></msub><mo>&ap;</mo><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mrow><mo>(</mo><mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow><mo>-</mo><msub><mi>&sigma;</mi><mrow><mi>A</mi></mrow></msub><mo>)</mo></mrow>              
</math></formula>From the previous
expressions for <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mi>A</mi></mrow></msub><mo>>;</mo></mrow></math></formula> and
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>&sigma;</mi><mrow><mi>A</mi></mrow></msub></math></formula> one can derive the
<formula type="equation"><math  xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>n</mi><mrow><mi>B</mi></mrow></msub><mo>&leq;</mo><msub><mi>n</mi><mrow><mi>B</mi><mo>,</mo><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub><mo>=</mo> <mfrac><mrow><mn>2</mn><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup></mrow><!--_
--><mrow><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>+</mo><msup><mi>c</mi><mrow><mn>2</mn></mrow></msup></mrow></mfrac>                                       
</math></formula>The following
values are obtained with <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>c</mi><mo>=</mo><mn>4</mn></math></formula>:
</p><p><table rend="inline"><row><cell  
><formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></formula></cell><cell  
><formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>n</mi><mrow><mi>B</mi><mo>,</mo><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></math></formula></cell><cell  
></cell><cell  
><formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub></math></formula></cell><cell  
><formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>n</mi><mrow><mi>B</mi><mo>,</mo><mi>m</mi><mi>a</mi><mi>x</mi></mrow></msub></math></formula></cell>
</row><row><cell  
>16                                                                                                     </cell><cell  
>16                                                                                                     </cell><cell  
></cell><cell  
>                                                               200</cell><cell  
>                                                             29.63</cell>
</row><row><cell  
>20                                                                                                     </cell><cell  
>17.78                                                                                                 </cell><cell  
></cell><cell  
>                                                               500</cell><cell  
>                                                             31.01</cell>
</row><row><cell  
>50                                                                                                     </cell><cell  
>24.24                                                                                                 </cell><cell  
></cell><cell  
>                                                              1000</cell><cell  
>                                                             31.50</cell>
</row><row><cell  
>100                                                                                                   </cell><cell  
>27.59                                                                                                 </cell><cell  
></cell><cell  
><formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&infin;</mi></math></formula></cell><cell  
>                                                             32.00</cell>
</row></table>
</p>
</div>
<div >
<head>Special sampling for lower part of the spectrum</head>
<p>If the step length is very small (<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mo>&leq;</mo><mn>5</mn></math></formula>
mm in gases, <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mo>&leq;</mo></math></formula>
2-3 <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>&mu;</mi></math></formula>m in solids)
the model gives 0 energy loss for some events. To avoid this, the probability of 0 energy loss is
<formula type="display"><math  xmlns="http://www.w3.org/1998/Math/MathML"><mi>P</mi><mrow><mo>(</mo><mi>&Delta;</mi><mi>E</mi><mo>=</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><msup><mi>e</mi><mrow><mo>-</mo><mrow><mo>(</mo><mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mn>1</mn></mrow></msub><mo>></mo></mrow><mo>+</mo><mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mn>2</mn></mrow></msub><mo>></mo></mrow><mo>+</mo><mrow><mo>&lt;</mo><msub><mi>n</mi><mrow><mn>3</mn></mrow></msub><mo>></mo></mrow><mo>)</mo></mrow></mrow></msup>                                
</math></formula>If the
probability is bigger than 0.01 a special sampling is done, taking into account the fact that in
these cases the projectile interacts only with the outer electrons of the atom. An energy level
<formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><msub><mi>E</mi><mrow><mn>0</mn></mrow></msub><mo>=</mo><mn>1</mn><mn>0</mn></math></formula> eV is chosen
to correspond to the outer electrons. The mean number of collisions can be calculated from
<formula type="display"><math  xmlns="http://www.w3.org/1998/Math/MathML"><mrow><mo>&lt;</mo><mi>n</mi><mo>></mo></mrow><mo>=</mo> <mfrac><mrow><mn>1</mn></mrow><mrow><msub><mi>E</mi><mrow><mn>0</mn></mrow></msub></mrow></mfrac>
</math></formula>The number
of collisions <formula><math  xmlns="http://www.w3.org/1998/Math/MathML"  
><mi>n</mi></math></formula>
is sampled from Poisson distribution. In the case of the thin layers, all the
collisions are considered as ionisations and the energy loss is computed as
<formula type="display"><math  xmlns="http://www.w3.org/1998/Math/MathML"><mi>&Delta;</mi><mi>E</mi><mo>=</mo><msubsup><mo>&sum;</mo>
    <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup>
<mfrac><mrow><msub><mi>E</mi><mrow><mn>0</mn></mrow></msub></mrow><mrow><mn>1</mn><mo>-</mo>
<mfrac><mrow>&Emax;</mrow>
<mrow>&Emax;<mo>+</mo><msub><mi>E</mi><mrow><mn>0</mn></mrow></msub></mrow></mfrac>
<msub><mi>u</mi><mrow><mi>i</mi></mrow></msub></mrow></mfrac>
</math></formula> 
</p> </div> </div> <div
type="star"> <head>References</head> <list type="bibliography"> <item
id="bib-LAND"> <p>L.Landau.  On the Energy Loss of Fast Particles by
Ionisation.  Originally published in <emph>J.  Phys.</emph>, 8:201,
1944.  Reprinted in D.ter Haar, Editor, <emph>L.D.Landau, Collected
papers</emph>, page 417.  Pergamon Press, Oxford, 1965.
</p></item> <item id="bib-SCH1"> <p>B.Schorr.  Programs for
the Landau and the Vavilov distributions and the corresponding random
numbers.  <emph>Comp.  Phys.  Comm.</emph>, 7:216, 1974.
</p></item> <item id="bib-SELT"> <p>S.M.Seltzer and
M.J.Berger.  Energy loss straggling of protons and mesons.  In
<emph>Studies in Penetration of Charged Particles in Matter</emph>,
Nuclear Science Series 39, Nat.  Academy of Sciences, Washington DC,
1964.  </p></item> <item id="bib-TALM"> <p>R.Talman.  On the
statistics of particle identification using ionization.  <emph>Nucl.
Inst.  Meth.</emph>, 159:189, 1979.  </p></item> <item
id="bib-VAVI"> <p>P.V.Vavilov.  Ionisation losses of high energy
heavy particles.  <emph>Soviet Physics JETP</emph>, 5:749,
1957.</p></item></list>  </div>
 </body> </text> </TEI.2>